Quantum Quench Dynamics and Entanglement

TIANCI ZHOU

DISSERTATION

Submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 2018

Urbana, Illinois

Doctoral Committee:

Assistant Professor Thomas Faulkner, Chair Professor Michael Stone, Director of Research Professor Peter Abbamonte Assistant Professor Lucas Wagner Abstract

Quantum quench is a non-equilibrium process where the Hamiltonian is suddenly changed during the quantum evolution. The change can be made by spatially local perturbations (local quench) or globally switching to a completely diﬀerent Hamiltonian (global quench). This thesis investigates the post-quench non- equilibrium dynamics with an emphasis on the time dependence of the quantum entanglement. We inspect the scaling of entanglement entropy (EE) to learn how correlation and entanglement built up in a quench.

We begin with two local quench examples. In Chap.2, we apply a local operator to the groundstate of the quantum Lifshitz model and monitor the change of the EE. We ﬁnd that the entanglement grows according to the dynamical exponent z = 2 and then saturates to the scaling dimension of the perturbing operator – a value representing its strength. In Chap.3, we study the evolution after connecting two diﬀerent one-dimensional critical chains at their ends. The Loschmidt echo which measures the similarity between the evolved state and the initial one decays with a power law, whose exponent is the scaling dimension of the defect (junction). Among other conclusions, we see that the local quench dynamics contain universal information of the (critical) theory.

In the global quench scenario, the change of the Hamiltonian affects all parts of the system. In this thesis, we focus on the global chaotic quench driven by generic non-integrable Hamiltonians. In Chap.4, we propose to use the operator entanglement entropy of the unitary operator as a probe. Its fast linear entanglement production is sharply contrasted to the slow logarithmic spreading of the many-body localized system. The entanglement saturation suggests that the evolution operator in the long time can be modeled by a random unitary matrix. In Chap.5, we construct a random tensor network which consists of random unitary matrices connected locally to model chaotic evolution with local interactions. We find that the entanglement dynamics is mapped to the statistical mechanics of interacting random walks. This appealing emergent picture allows us to understand the universal linear growth as well as the fluctuations of entanglement in a chaotic quench.

ii To my parents.

iii Acknowledgments

I am deeply grateful to my advisor Prof. Michael Stone. Mike’s Physics 508 at 9am on Monday was my very

ﬁrst class in UIUC. His instructions have persisted since then. It is fortunate to have his patient guidance in my starter projects and the often knowledgeable and constructive comments to all my later scientiﬁc works.

Mike’s research notes demonstrate how a combination of masterful mathematical skills and physical insights make solving physics problem an aesthetically appealing work, to which I wish to have my own contribution.

I thank Mike wholeheartedly for the freedom to choose research topics, sincere career advices, and all the jokes and academia anecdotes in our delightful conversations.

Among others, I have my gratitude to all the people who taught me physics, including Prof. Thomas

Faulkner, Prof. Eduardo Fradkin, Prof. Shinsei Ryu, Prof. Nigel Goldenfeld, Prof. Stuart Shapiro and many others. Their oﬃce doors and minds are always open for my questions and confusions. It is my pleasure to collaborate with some of them, and other physicists such as Dr. Xiao Chen, Dr. David Luitz, Dr. Mao

Lin, Prof. David Huse, Prof. Cenke Xu and Dr. Adam Nahum, who inspired and educated me through numerous board discussions, emails and online video conversations on the problems of our common interest.

I would like to acknowledge many friends at UIUC for their accompany and support. Special thanks must go to our study group members Xiongjie Yu, Bo Han and Ye Zhuang, with whom I consult and casually chat from time to time. I also appreciate the research, career and life experiences shared by many (then-) fellow graduate students Ching-Kai Chiu, Po-yao Chang, Xiao Chen, Vatsal Dwivedi, Xueda Wen, Di Zhou,

Chang-tse Hsieh, Pok-an Chen, Matthew Lapa (four-year oﬃce mate) and other postdocs. I thank Po-yao

Chang, Yueqing Chang and Luke Yeo for generously lending their clean and exhaustive course notes for me to review and catch up. There are so many memorable moments with these folks, and other friends like

Kridsanaphong Limtragool, Da Ku, Lixiang Li, Can Zhang, Chi Xue, Yizhi Fang, Wenchao Xu, Zihe Gao,

Yang Bai, Hu Jin, Mao-Chuang Yeh, Peiwen Tsai, Han-Yi Chou, Mao Lin, Tianhe Li, Huacheng Cai, Di

Luo and many others not mentioned here. The dinners and barbecues we had, the Frisbee games we played, the road trips we enjoyed eased my mind to be less turbulent and stressful.

Finally, I own everything to my dear parents. Their unconditional support for my career, regardless of

iv its popularity, their own economic and health status, is extremely valuable. This thesis is dedicated to them.

For part of the thesis work, I was supported by the National Science Foundation Grant NSF DMR

1306011.

v Table of Contents

List of Abbreviations ...... viii

Chapter 1 Introduction to Quantum Quench ...... 1 1.1 Introduction to Quantum Quench...... 1 1.2 Entanglement in Quantum Quench...... 5

Chapter 2 Entanglement Entropy of Local Operators in Quantum Lifshitz Theory . . . 14 2.1 Introduction...... 14 2.2 Introduction to Quantum Lifshitz Model and Its Dynamics...... 15 2.3 Excess Entanglement Entropy...... 18 2.4 Results and Discussion...... 25 2.5 Summary...... 31

Chapter 3 Bipartite Fidelity and Loschmidt Echo of the Bosonic Conformal Interface . 33 3.1 Introduction...... 33 3.2 Bosonic Conformal Interface...... 35 3.3 Bipartite Fidelity and Loschmidt Echo...... 39 3.4 Discussion...... 50 3.5 Conclusion...... 53

Chapter 4 Operator Entanglement Entropy of the Time Evolution Operator ...... 54 4.1 Introduction...... 54 4.2 Operator Entanglement Entropy (opEE) of the Time Evolution Operator...... 56 4.3 General Behavior of opEE...... 58 4.4 Models...... 62 4.5 Saturation Value...... 64 4.6 Growth...... 70 4.7 Conclusion...... 74

Chapter 5 Entanglement of Quantum Quench by Random Unitary Circuits ...... 76 5.1 Introduction...... 76 5.2 Overview of Results...... 79 5.3 Mapping to a ‘Lattice Magnet’...... 87 5.4 Entanglement Production Rates...... 93 5.5 The Entanglement Line Tension n ...... 96 5.6 Fluctuations and the Replica LimitE ...... 99 5.7 Numerical Checks Using the Operator Entanglement...... 110 5.8 Saturation at Late Time and Page’s Formula...... 111 5.9 Dynamics without Noise...... 114 5.10 Outlook...... 115

vi Appendix A Appendices for the Lifshitz Quench Problem ...... 117 A.1 Boundary Reproducing Kernel...... 117 A.2 Equal Space Green Function for a Simply Connected Region...... 118 A.3 Alternative Calculation for the Equal Space Green Function on the Half Plane...... 120 A.4 Schwinger Parameter Calculation...... 123 A.5 Distributional Boundary Integral...... 124

Appendix B Appendices for the Conformal Interface Problem ...... 126 B.1 General Boundary State Amplitude...... 126 B.2 Alternative Approach to DN λ Amplitude...... 127 B.3 Corrections to the Free Energy→...... 129 B.4 Winding Modes of Compact Bosons...... 131 B.5 Conformal Interface in Free Bosonic Lattice...... 134 B.6 A Determinant Identity for the Boundary State Amplitude...... 136 B.7 Numerical Computation of Bipartite Fidelity and Loschmidt Echo...... 137

Appendix C Appendices for the Operator Entanglement Entropy ...... 143 C.1 Channel-State Duality...... 143 C.2 Average opEE of Random Unitary Operator...... 144 C.3 Lin Table Algorithm for Sz = 0 Sector...... 148

Appendix D Appendices for the Random Tensor Network Problem ...... 150 1 D.1 Combinatorial Calculation of Z3(t) at Order q2 ...... 150 D.2 Slope-dependent Line Tension (v)...... 153 E3 D.3 Line Tension 3(v) Close to Lightcone...... 154 D.4 The WeingartenE Function...... 157 D.5 Exact Weights with 1 Incoming Domain Wall...... 158 D.6 Exact Weights for N≤= 3...... 160 D.7 Perturbative Calculation of the Triangle Weights...... 161 D.8 Continuum Interaction Constants...... 164

References...... 165

vii List of Abbreviations

EE Entanglement Entropy opEE operator Entanglement Entropy CFT Conformal Field Theory

RK Rokhsar-Kivelson MPS Matrix Product State MPO Matrix Product Operator AdS Anti-de Sitter

MBL Many Body Localization KPZ Kardar-Parisi-Zhang

viii Chapter 1

Introduction to Quantum Quench

1.1 Introduction to Quantum Quench

Quantum quench is an idealized protocol to investigate non-equilibrium dynamics in an isolated quantum

system. The procedures typically involve two steps: one ﬁrst prepares an initial state ψ which can be the | i groundstate of some Hamiltonian H, and then evolves it with a diﬀerent Hamiltonian H0. The state ψ | i is in general not the eigenstate of the new Hamiltonian H0, and the observables such as the expectation values of local operators, correlation functions[1] will therefore be time dependent. The system regains its

equilibrium when these observables approach new stationary values macroscopically.

In the past few decades, this rather theoretical setup has become experimentally accessible thanks to the

development of the ultra-cold atom systems (see review in [2]). Because of the extremely weak coupling to

the environment, the cold-atom systems can be regarded as isolated for the time scale we are interested in[3].

The quench requires a dynamical change of the parameters of the Hamiltonian in a short period of time.

This is made possible by the Feshbach resonance. The strong anisotropy of the optical lattice[4] enables us

to perform experiments on eﬀectively one dimensional system, where often exact solutions[1,5] are available

to be compared with.

There are also many interesting theoretical problems of post quench non-equilibrium behaviors in ex-

tended systems. We here brieﬂy introduce some of them following the line of entanglement spreading that

is relevant to the later chapters of the thesis.

One way to motivate this is the question of how isolated quantum system thermalizes under unitary

evolution. The long time-evolved state is thermalized if the expectation value of the (spatially) local operators

are given by the various statistical ensemble (microcanonical, canonical, etc.) averages. This is possible if we

only look at a subsystem and treat the remaining part as a reservoir. This suggests us to pay attention to the

dynamics of the subsystem’s reduced density matrix, whose evolution is not unitary. Perhaps the simplest

basis independent measure is the eigenvalues (entanglement spectrum) of the reduced density matrix of the

subsystem, which crudely speaking quantiﬁes the subsystem’s quantum entanglement with the reservoir (see

1 introduction in Sec. 1.2.1).

A quantum quench on an initially unentangled state can be used to study thermalization. This process is accompanied by the spreading of quantum entanglement. The scaling of the dynamics of entanglement and the relevant time scales therefore provide another angle than the relaxation of the local correlators. For example, systems with interaction induced localization – the many-body localized systems[6,7] – have a logarithmic increase of entanglement when quenched from unentangled state, showing its failure to thermalize. In contrast, systems that are able to thermalize have a much faster sub-linear power law entanglement growth[8,6]. We give our account about this scaling in Chap.4 and how a linear growth is obtained in a chaotic evolution in Chap.5.

One can also quench the system by a local perturbation (generally called a local quench, see Sec. 1.1.1), for instance acting on a local operator. Then the entanglement dynamics are diagnostic about the nature of this excitation. For example, the time elapsed before the entanglement changes indicates how fast the excitation travels. In 2d conformal ﬁeld theory (CFT), it is also possible to extract the quantum dimension from the long time value of the entanglement[9]. We consider a similar problem in Chap.2. The local change can also be made by connecting two initially disconnected systems. The quench thus probes the properties of the junction. Chap.3 deals with the problem of gluing two diﬀerent critical chains by inspecting the post quench dynamics of the Loschmidt echo (overlap of the evolved wavefunction with the initial state).

1.1.1 Quench Protocol

In this subsection, we discuss three general types of quench protocols. They were introduced in the study of the quench processes in conformal field theory. Since the scaling behaviors and the associated quasi-particle picture largely influence our understanding of the quench, we here use it as a general classification to guide the topics in this thesis.

1. Global quench. The global quench protocol evolves the ground state of H with a globally diﬀerent

Hamiltonian H0. Consequently, the initial state has an extensive amount of extra energy compared to

the ground state of H0 and the system changes dramatically afterwards.

2. Local quench. In contrast, the local quench protocol only weakly perturbs the initial system. One

example of this (usually called “local quench” or “cut and join” protocol in the literature) is to prepare

two identical 1d ground states of H and then join them together and evolve with the Hamiltonian of

the same form but in the doubled system. The only diﬀerence lies at the connecting points where

the dynamics that used to be set by boundary conditions is now determined by a bulk term in the

Hamiltonian of the doubled system.

2 3. Local operator quench. This is a local quench similar to the “cut and join”. As the name suggests, we

let a local operator act on the initial state and then evolve with the same Hamiltonian. Equivalently,

one sets the new Hamiltonian H0 as the sum of H and a delta function pulse of local operators at the moment just before the quench.

1.1.2 Overview of the Results

In this subsection, we give an overview of the main results in this thesis. It consists of four interrelated projects, covering all the protocols in the Sec. 1.1.1. We use different probes – primary entanglement entropy – to detect the fascinating physics occurred after the quench. Entanglement entropy (EE) is an information theoretical measure about quantum entanglement of the pure state. We defer its introduction to Sec. 1.2. At this point, we only need to know that it quantifies the entanglement between two parts of the wavefunction. The dynamics of quantum quench generate correlation between different parts, and hence increase the entanglement. This thesis will therefore focus on the scaling of EE.

In the case of the local quench, we care about how fast the local perturbation spreads out. In integrable systems, the quasi-particles are responsible for the information propagation (see Sec. 1.2.2), and the EE scaling is compatible with this picture. We investigate two examples based on the following works:

• Tianci Zhou. Entanglement entropy of local operators in quantum Lifshitz theory. Journal of Statistical Mechanics: Theory and Experiment, 2016(9):093106, 2016. DOI: 10.1088/1742-5468/2016/09/093106

In Chap.2 we present the result of a local operator quench in quantum Lifshitz model. We ﬁnd

that the excess EE, i.e. the growing part of the EE, increases as t2, which is consistent with the

system’s dynamical exponent z = 2. It thus suggests a quasi-particle interpretation for the post

quench evolution.

• Tianci Zhou and Mao Lin. Bipartite ﬁdelity and Loschmidt echo of the bosonic conformal inter-

Chap.3 presents another example of local quench, which is intimately related to the “cut-and-join”

protocol in Sec. 1.1.1. We consider joining two diﬀerent one dimensional critical systems at the end

of each chain. However unlike the “cut-and-join” protocol, the connection point will neither be com-

pletely transparent nor reﬂective, but interpolating between the two limiting cases. The quantity we

calculate is the Loschmidt echo and bipartite ﬁdelity, which bears a power law decaying exponent

3 that characterizes the interface (the connecting bond). The defect’s scaling dimension suggests the

existence of some exotic quasi-particles that mediating the two sides of the chain.

The remaining two chapters are devoted to the global quench by quantum chaotic evolution. They are based on the following articles:

• Tianci Zhou and David J. Luitz. Operator entanglement entropy of the time evolution operator in

Chap.4 proposes a new perspective of the quantum quench problem. Previous studies focus on

the quench on initially unentangled state. However, what we really want to extract in many quench

problems are the properties of the Hamiltonian governing the dynamics. So we propose instead to study

the operator entanglement entropy (opEE) of the unitary evolution operator. We obtain scalings of

the opEE growth and saturation in the localized, Floquet and chaotic quenches by either numerical

means or random matrix technique. The results along with the saturation values demonstrate how the

presence or absence of integrability and conserved quantities constrain the entanglement propagation.

• Tianci Zhou and Adam Nahum. Emergent statistical mechanics of entanglement in random unitary circuits. arXiv:1804.09737, April 2018

Chap.5 is a detailed study of EE growth in a random tensor network, which is expected to capture the

universal scaling behaviors of all chaotic quenches with local interactions. Evolution by a single random

unitary matrix can not tell us how the entanglement is built up1 since thermalization is completed

in a single step of applying this operator with system size. Because of the unitary invariance, it can

not distinguish local and non-local operators. The random unitary tensor network resolves both issues

by breaking the evolution matrix into a network of small independent random matrices with causal

structure, and so is a minimal model of quantum chaotic quench with local interactions.

We ﬁnd that the random tensor network problem maps to the statistical mechanics of interacting

random walks, whose free energy is the time dependent EE. We therefore understand the leading order

entanglement behaviors by analyzing the energy and entropy of the domain walls. The interaction

1 induced ﬂuctuation leads to the subleading t 3 Kardar-Parisi-Zhang (KPZ) scaling of the EE, conﬁrming

its conjectured universal presence in the global quench by chaotic Hamiltonians[10, 11].

1This can be avoided by considering the time evolution of a random Hamiltonian, but the locality of the interactions is still missing.

4 1.2 Entanglement in Quantum Quench

1.2.1 Introduction to Entanglement Entropy

In this subsection, we brieﬂy review the concept of entanglement entropy and its general behaviors in many-

body systems.

The Hilbert space of a many-body system is the tensor product of the single particle Hilbert spaces (sub-

ject to the statistics of the constituent particles). So the wavefunction can accommodate entanglement that

does not exist in a direct sum of classical phase spaces. Entanglement entropy is a measure of entanglement

for pure quantum state.

Formally, suppose ψ is a pure state. We divide the Hilbert space into two disjoint parts = . | i H HA ⊗ HB The whole wave function can be decomposed as

ψ = ψ i j , (1.1) | i ij| iA| iB ij X where i and j are orthonormal basis in subsystem A and B respectively. We can do a Schmidt | iA iB decomposition about the coeﬃcient matrix

ψij = Uia λaVaj (1.2) p such that2

ψ = λ a b , (1.3) | i a| iA| iB a X p where λ 0 are the eigenvalues. The entanglement entropy is the von Neumann entropy of these eigenvalues a ≥ (entanglement spectrum)

S = λ ln λ . (1.4) A − a a Xλa

Alternatively, we can deﬁne the reduced density matrix ρA of a subsystem A by tracing out the degrees of freedom in subsystem B

ρ = Tr ( ψ ψ ) = b ρ b . (1.5) A B | ih | Bh | | iB b X| i From Eq. (1.3), we see that in the a basis, the reduced density matrix is diagonal | i

ρ = λ a a (1.6) A a| iAAh | a X 2The Schmidt eigenvalue of a matrix is non-negative, so it is always possible to take a square root.

5 with λa as its diagonal entries. A basis invariant way to deﬁne entanglement entropy is thus

S = Tr(ρ ln ρ ). (1.7) A − A A

Elementary Example

Let us do an elementary example to understand what it means operationally to take the partial trace.

Take a two-site state

ψ = cos θ + sin θ . (1.8) | i |↑iA|↓iB |↓iA|↑iB

The linear superposition in this state is what distinguishes quantum mechanics from classical mechanics: there are entanglement between the two sites. Following the prescriptions given above, the density matrix is

2 ρ = cos θ A BA B + cos θ sin θ A BA B |↑i |↓i h↑| h↓| |↑i |↓i h↓| h↑| (1.9) + cos θ sin θ + sin2 θ . |↓iA|↑iBAh↑|Bh↓| |↓iA|↑iBAh↓|Bh↑|

We take the partial trace where the inner product is understood to be taken only within the B subspace, for example

With this, we have

ρ = cos2 θ + sin2 θ . (1.11) A |↑iAAh↑| |↓iAAh↓|

Hence the EE is

S = ln cos2 θ ln sin2 θ. (1.12) A − −

The maximal value is taken at θ = π . ± 4 R´enyi Entanglement Entropy and Replica Trick

The nth R´enyi entanglement entropy is deﬁned as

1 1 S = Trρn = λn, (1.13) n −n 1 A −n 1 i i − − X where the integer n is the R´enyi index. Because of the constraint TrρA = 1 (from the normalization of the wavefunction), we see λ = 1, λ [0, 1]. So if we generalize the R´enyi entropy to have a complex index i i i ∈ α P

1 α 1 α Sα = Trρ = λ , α C, (1.14) −α 1 A −α 1 i ∈ i − − X

6 the series in Eq. (1.14) is absolutely convergent for Re(α) > 1, and therefore analytic in this region. We can analytically continue to α 1+ to obtain the von Neumann entropy[12] →

SA = lim Sα. (1.15) α 1+ →

The replica trick does the analytic continuation only with the knowledge of the R´enyi entropy on integer value n. The analytic continuation is then not necessarily unique, unless one can show that the conditions of Carlson’s theorem[13] is satisﬁed. In practice, to avoid this subtlety we should always check the replica trick result on physical ground or against other means.

Entanglement in Many-Body Systems

The behaviors of entanglement become more interesting in many-body systems with interactions. Its scaling behaviors with respect to the system size often reveal universal information of the system.

In general, if the state only has local correlation, we expect the degrees of freedom are only entangled in the vicinity of the entanglement cut. Hence the EE should scale with the area of the spatial cut[14, 15, 16].

This has been rigorously proved by Hastings[15] in one dimension as a theorem: for a gapped Hamiltonian with local interactions of ﬁnite strength, the EE of the groundstate is bounded by a number independent of the system size. Since in one dimensional system, a segment as a subsystem only has two end points as its boundary, the area law implies constant EE.

The area law in one dimensional gapped system seeds the success of the numerical method – the density matrix renormalization group(DMRG) and more generally the matrix product state (MPS) representation of quantum state. The area law guarantees that only a ﬁxed number of Schmidt eigenvalues are important, and the basic idea is therefore to systematically throw away unimportant ones as a good approximation. We will introduce MPS’s graphical representation in Sec. 1.2.3.

Violation of the area law occurs when the correlation length is comparable to the system size. One notable example is the one dimensional critical system, where the EE scales logarithmically with the subsystem size

LA[17] c S ln L . (1.16) A ∼ 3 A

The coeﬃcient c is the central charge of the CFT. Hence EE shows that the critical system are entangled over a large range of the system, rather than the vicinity of the cut; it also extracts the universal data

(central charge in this case) of the critical system.

There is an appealing geometric way to understand the structure of entanglement in CFT. It was noted in [18] that the EE of region A in d + 1 dimensional CFT can be interpreted as the area of the static (Ryu-

7 Takayanagi) minimal surface in the d + 2 dimensional anti-de Sitter space that has the same boundary as A

(see Fig. 1.1).

Figure 1.1: Geometric point of view of entanglement. (Left) The boundary circle represents a (strongly interacting) conformal ﬁeld theory, the interior is the anti-de Sitter spacetime. The entanglement entropy (EE) of the red region on the boundary theory is equal to the length of the geodesics in the bulk geometry. (Right) The dots and lines constitute a tensor network state. The entanglement entropy of boundary states in red region is bounded by the length of the minimal cut through the bulk structure of the tensor network.

Following this proposal, there are various attempts[19, 20] to construct tensor network that are able to reproduce the Ryu-Takayanagi minimal surface picture. One of the motivations of these constructions is to understand more about the non-perturbative eﬀects on the quantum gravity side. On the other hand, the geometric relation between tensor network and entanglement is interesting on its own right. One of the proposals[20] use random tensor network and map the entanglement problem to the statistical mechanics of the Ising model. This partly motivates us to study a similar unitary random tensor network for the chaotic evolution in Chap.5.

These are static properties of EE in the groundstate. In the next subsection, we will turn to the main topic of this thesis: the dynamical properties of the EE in the quench process.

1.2.2 Behaviors of Entanglement Entropy in a Quantum Quench

Entanglement will grow in the quench protocols considered in this thesis, because of the extra energy injected to the system. So the questions are how the entanglement grows (scaling) and how it saturates. This thesis will focus on these two aspects of the entanglement in quench3.

Our current understanding of the entanglement in a quench process is largely based on the heuristic pictures from several exactly solvable examples. Among them, the quasi-particle picture is perhaps simplest and wildly applicable interpretation, in which the extra energy compared to the ground state of H0 is assumed to be carried by coherent quasi-particle pairs. The subsequent time evolution separates the individual quasi- particles and EE is gained when one of them in the pairs crosses the entanglement cut. In short, the

3Chap.3 discusses the Loschmidt echo and bipartite ﬁdelity, not the entanglement. But the spirit of using the scaling behaviors of quench to probe the defect is in line with the analysis of entanglement here.

8 generation of excess EE is ascribed to the proliferation and propagation of those quasi-particle pairs. So we will begin by reviewing this framework in the context of the 1+1 dimensional CFT.

The global quench starts with an initial state with ﬁnite correlation length. It is not the groundstate of, but is subsequently quenched by the critical Hamiltonian H0. The EE will growth linearly and ultimately saturates to a value that is proportional to the size of the subsystem[21].

∆S ∆S

lA t t l excitation

Figure 1.2: Quasi-particle picture for global quench Figure 1.3: Quasi-particle picture for local primary protocol. This is a space time diagram where at ﬁeld excitation in 1+1d CFT. Figure at the bottom time t, region A and B are labelled by red and shows A and B to be two semi-inﬁnite systems. blue lines. The coherent pair of quasi-particles are The local excitation is located at a distance l to generated uniformly on each point and radiated in the entanglement cut. The excess EE remains zero the direction of the light cone. The green region before quasi-particle’s arrival and bursts into log of encloses sites where part of the quasi-particle pair the quantum dimension afterwards. is in region A at time t. The length of green region grows linearly and saturates to value lA after t = lA 2 .

This can be understood as follows. The excess energy compared with the true ground state of the

Hamiltonian H0 is distributed across the system. For a translational invariant Hamiltonian, same types of quasi-particles are radiated on each point of the system. The number of entangled pairs between region

A and B is proportional to the area of the green region in Fig. 1.2, which grows linearly and saturates to

lA the maximal value after t > 2 (lA is the length of subsystem A). This gives rise to the linear growth and extensive saturation value[22, 23].

There is an equilibrium value after the time of saturation. We can deﬁne seq to be the equilibrium entropy density, then the quantity 1 dS vE = (1.17) seq dt

(in 1d) is the normalized rate of change of EE and is called the entanglement velocity. Since the carrier of entanglement is the quasi-particle here, the entanglement velocity is the speed of light in the global quench.

The local quench example is the “cut-and-join” protocol, where both chains are critical, and the quench

9 is performed by connecting them. Here the extra energy in “cut-and-join” protocol is only distributed in the vicinity of the joint point. If we choose the region A to be a single interval that has distance l to the

c 4 5 joint point, then the EE will keep the ground state value of 3 ln lA + constant until the time lA when the quasi-particle traveling at the speed of light arrives the entanglement cut[21, 23]. It will grow logarithmically afterwards. This picture naturally gives rise to the horizon eﬀect.

Finally one can also apply a local operator (e.g. vertex operator) to the groundstate of critical Hamilto- nian and then evolve. In this case the quasi-particle is created exactly at the point of the operator insertion.

Again if the excitation point has a distance to the entanglement cut, causality constraint will force the entanglement to be unchanged until the arrival of quasi-particles, see Fig. 1.3. Here the EE will not be extensive since the local excitation only add a very small amount of single particle energy to the ground state. Its strength can be quantiﬁed by the quantum dimension, which represents the degrees of freedom of the quasi-particles. We see that the saturation value of excess EE is indeed proportional to this strength[9].

The quasi-particle picture becomes less useful when we study quantum chaotic evolution in Chap.4 and

Chap.5, in which cases the quasi-particle does not exist. Due to the rapid growth (in fact at almost maximal rate) of entanglement and evidences from holography calculations[24, 25], there is a so-called tsunami picture[24] for the chaotic global quench, which is also applicable in higher dimensions[26]. It basically ascribes the growth of entanglement to the ﬁctitious wave emerged from the entanglement cuts, whose wave front propagates with entanglement velocity vE. Hence the entanglement growth is only constrained by this speed and the area of the wave front dS s v Area. (1.18) dt ≤ eq E

There are also proposals [26] that this bound along with the causality are the only constraints of EE growth in the chaotic evolution described by relativistic quantum ﬁeld theory.

Our studies of quantum chaotic evolution in Chap.4 and Chap.5 are restricted to one dimension, so will not really challenge the tsunami picture. What we do is to conﬁrm this linear growth and understand how the entanglement velocity arises in the emergent domain wall statistical mechanical problem (Chap.5).

We believe that this complementary domain wall (or minimal cut, minimal membrane) picture and the associated coarse grained line tension are needed in place of the quasi-particle picture for the EE in the chaotic quantum quench.

4c is the central charge, which represents the degrees of freedom of CFT. 5Speed of light is set to 1 to avoid confusion with the central charge c.

10 1.2.3 Introduction to Random Tensor Network

In this subsection, we introduce the diagrammatic representation of the tensor network. It will become useful

when we refer to the matrix product states and matrix product operators in Chap.4, and more importantly

the random tensor network in Chap.5.

The basic operations of tensor is shown in Fig. 1.4. A rank-n tensor Tµ µ , ,µ is represented as an 1 2 ··· n object with n external legs. This way of representing tensor gets rid of the explicit symbols for indices and makes the tensor product easy: simply drawing two objects and that is their tensor product. Tensors

can contract with each other to form new tensors, for example Tµ1µ3 = Uµ1µ2 Vµ2µ3 is a rank-2 tensor by contracting one of the indices in U and one in V . Contraction is performed by connecting two legs of the tensors in the diagram.

µ3 µ3

µ4 µ2 µ4 µ1 µ3

µ5 T µ1 µ5 T U V

µ6 µ8 µ6 µ8

µ7 µ7

Figure 1.4: Diagrammatic representation of tensor and its contraction. (Left) A tensor of rank-8. (Middle)

A rank-6 tensor constructed from contracting a rank-8 tensor. (Right) The example Tµ1µ3 = Uµ1µ2 Vµ2µ3 in the text.

A tensor network is a collection of tensors, part of whose legs are contracted within the collection. It is

a very useful way to represent quantum states and operators.

As an example, we review the diagrammatic representation of the matrix product state (MPS). For

1 clarity we take the Hilbert space to be a one dimensional spin- 2 chain. A matrix product state with open boundary condition is

σ1 σ2 σi σL ψ = Mµ1 Mµ1µ2 Mµi 1µi MµL σ1, σ2, , σL , (1.19) | i ··· − ··· | ··· i

where the σs are physical spin indices and µs are the matrix auxiliary indices. This state in the spin basis

has L free indices: it is a rank-L tensor with L external legs. All the matrix indices are contracted, we thus

have the diagram in Fig. 1.5, where the M matrices at diﬀerent sites are represented by circles.

The matrix product state has a natural decomposition if we cut the system along the bond between σi

and σi+1,

σ σ σ σ ψ = A 1··· i B i+1··· L σ , , σ σ σ , (1.20) | i µi µi | 1 ··· ii| i+1 ··· Li

11 σ1 σ2 σ3 σ4 σ5 σ6

ψ = M M M M M M | i

Figure 1.5: A 6-site matrix product state. The external legs are physical spin indices, the internal legs are contracted matrix indices. The bond dimension in the middle cut gives an upper bound of the entanglement.

where

σ1 σi σ1 σ2 σi σi+1 σL σi+1 σL Aµi ··· = Mµ1 Mµ1µ2 Mµi 1µi Bµi ··· = Mµiµi+1 MµL . (1.21) ··· − ···

One can see that the dimension of the contracted index µi (bond dimension di) gives an upper bound of the entanglement

S ln d . (1.22) ≤ i

A tensor network state is a generalization of the matrix product state, where the matrix M is replaced by tensors contacted with neighboring sites in higher dimensions. One notable example is the projected entangled pair state (PEPS)[27, 28] which has wide application in the numerical investigation of many- body wavefunction in two dimensions. PEPS interprets the contraction of the tensor indices as a projective measurement, which motivates the random tensor network construction of the holographic bulk in [20].

We here represent (and motivate) the unitary random tensor network state in the language of quantum computation. A product state is the tensor product of states sitting on diﬀerent local Hilbert spaces. So an n-site product state is a disconnected collection of 1-leg tensors, which we draw in Fig. 1.6. A one-site unitary operator (or unitary gate) is a rank-2 tensor with 2 legs. Acting a one-site unitary gate to a single site amounts to contracting the one leg of the gate with the leg of the state. The resulting tensor is a new state evolving from the original state by the unitary gate.

A one-site gate will not change the entanglement. Therefore we use two-site gate as the building blocks of the network as shown in the middle of Fig. 1.6. In Chap.5, we apply the two-site gate to any neighboring sites, building a network as the rightmost one in Fig. 1.6.

The two-site gate and the way we put them on top of each other generate and propagate the entanglement within the apparent light cone. This is the generic feature we need to model the quench with local interactions. By taking the gate to be a random unitary matrix, the interaction is chaotic enough to model the non-integrable evolution.

The structure of the network itself enables a geometric construction of the entanglement. We take an arbitrary cut where there are nb sites on the left and na sites on the right. We may extend this spatial cut at bond x into the bulk of the tensor network, which will pass through nd contracted legs. Then to the left

12 Figure 1.6: Building blocks of tensor network state. From left to right: a one-site state, a one-site unitary operator, a one-site unitary operator acting on a state, a two-site unitary operator acting on a state, a sample tensor network state. The red line in the tensor network state is a cut extended into the bulk. It separates the tensor network into left and right parts. In much the same way as MPS, the bond dimension of the link the red passes through provides an upper bound of entanglement. The minimal cut gives the tightest upper bound from this geometry.

of the network, we have a rank-nb + nd tensor and to the right we have a rank-na + nd tensor. They are contracted by the nd indices between them to form the state. Like the case of the matrix product state, this gives a natural decomposition of the state into basis in subsystem A and B (albeit not orthonormal). The dimension of these nd contracted legs constrain the maximal entanglement

nd S ln d . (1.23) ≤ i i=1 X Since this is true for any cut, we must have

nd S min ln di. (1.24) ≤ cut i=1 X This inequality is saturated in the large local Hilbert space dimension of random tensor network considered in Chap.5. There the minimal cut exempliﬁes itself as domain walls in the emergent statistical mechanical problem, providing an eﬀective description of entanglement in chaotic quantum quench.

13 Chapter 2

Entanglement Entropy of Local Operators in Quantum Lifshitz Theory

2.1 Introduction

In this chapter, we study the behavior of EE after a local operator quench. Let us ﬁrst recall that this type

of quench acts a local operator on the initial ground state and then evolves it with the Hamiltonian. A

meaningful measure here should be the excess EE compared to that of the ground state, which is expected

to reﬂect the strength and spreading of excitation created by the local operators.

In critical systems described by a rational CFT (CFT with ﬁnite number of primary ﬁelds), local primary

ﬁeld excitations are studied in [9] and the excess EE increases for a long time to a limiting value equal to

the logarithm of the quantum dimension[9]. The growth of the excess EE after the local operator excitation

is constrained by causality: the excess EE is zero until the signal traveling at the speed of light reaches

the entanglement cut. The saturation values are further studied in higher dimensions[29], for descendant

ﬁelds[30, 31], for the thermal and boundary eﬀects[32], where in some cases it is still the logarithm of the

quantum dimension. However in strongly coupled large N CFT[33], the excess EE grows logarithmically

with time that breaks the saturation behavior.

As far as we know, there is yet no analytic result of EE of local operator excitations in a non-relativistic

system. In this chapter, we study the excess EE in such a system. We study the quantum Lifshitz model

whose dynamical exponent z is equal to 2 (while CFT has z = 1) in the presence of the local vertex operator excitation. The model describes a critical line of the quantum eight-vertex model with one special point corresponding to the quantum dimer model on bipartite lattice. The scale invariance of the ground state wavefunction is what makes analytic calculation possible.

We take two types of subsystems on an infinite plane, one the upper half plane, the other a disk and find that the excess EE will grow immediately after the local excitation and reach a limiting value of order the scaling dimension of the vertex operator. The typical time scale when the excess EE is considerable, i.e. of order of the maximal value, is the distance from excitation to the entanglement cut squared. This is when the quasi-particles diffuse to the entanglement cut, consistent with z = 2. We also find small plateau structures

14 in the short time dynamics of the excess EE and conjecture that it reveals quasi-particle density of states

and possible dispersion of diﬀerent species during the propagation. In summary, the quasi-particle picture

can still qualitatively interpret the results with a slight modiﬁcation that replaces causality constraint with

a diﬀusive light cone.

The structure of this chapter is as follows. In Sec. 2.2, we introduce the quantum Lifshitz model and the

vertex operator excitation. We deﬁne the excess EE in Sec. 2.3 and evaluate it for the upper half plane and

disk in Sec. 2.4. We summarize our results in Sec. 2.5.

2.2 Introduction to Quantum Lifshitz Model and Its Dynamics

2.2.1 Quantum Hamiltonian

The Quantum Lifshitz model is a compact boson theory that describes the critical behavior of the quantum

eight-vertex model[34, 35]. The quantum Lifshitz model has Hamiltonian

1 2 H = d2x Π2 + g2 2φ , (2.1) 2 ∇ Z n o where φ is a compact boson ﬁeld φ φ + 2πR and Π = φ˙ is its conjugate momentum. Due to the absence ∼ c of the regular stiﬀness term ( φ)2, this theory does not have Lorentz symmetry. The dynamic exponent z ∇ is 2.

By varying values of g, the Hamiltonian in Eq. (2.1) can in general model a critical line of the quantum eight vertex model[34]. What we have in mind however, is the Rokhsar-Kivelson(RK) critical point of the

1 square lattice quantum dimer model[36, 37, 38, 34] which is at g = 8π . There, the compact boson field is naturally identified as the coarse grained height field on square lattice[34, 36].

It is generally believed that the Hamiltonian (2.1) gives the correct time evolution of the quantum dimer

model. We here present two heuristic ways to justify this point.

One of them is a Ginzburg-Landau type argument that keeps the lowest order possible terms that

consistent with the required symmetry[35]. In the dimer problem, translational and rotational symmetries

enforce the Hamiltonian to have the following form

1 2 H = d2x Π2 + A( φ)2 + g2 2φ . (2.2) 0 2 ∇ ∇ Z n o When A > 0, the system will flow to a phase that pin the φ field to fixed value, which is identified to be the

columnar phase away from the RK point. On the other hand, A < 0 corresponds to an unstable Hamiltonian

15 that is not semi-positive deﬁnite. At A = 0, H0 reduces to the Hamiltonian (2.1) and can be diagonalized using

δ 2 δ 2 Q(x) = g φ(x) Q†(x) = g φ(x), (2.3) δφ(x) − ∇ −δφ(x) − ∇

as a series of harmonic oscillators (normal ordered), such that

2 H = d x Q†(x)Q(x). (2.4) Z

The ground state wavefunction is thus annihilated by Q(x) and has a Gaussian form

1 g gnd = [dφ] exp d2x φ φ φ , (2.5) | i √ − 2 ∇ · ∇ | i Z Z Z where is partition function of the free compact boson Z

= [dφ] exp g d2x φ φ . (2.6) Z − ∇ · ∇ Z Z This reproduces the fact that at Rokhsar-Kivelson critical point, the dimer density operator (derivative of the boson)[39, 40] has a power law correlation function.

Another independent derivation is proposed by Nienhuis[41] and Henley[38] who map the quantum Hamil- tonian of “flipping” dynamics to a Monte Carlo process. The classical Monte Carlo process gives exactly the same probability distribution as the ground state wavefunction on the dimer basis. And the relaxation to the equilibrium is the imaginary time quantum evolution. To obtain a continuous description, the stochastic process is heuristically written as a Langevin equation with a Gaussian noise. The Hamiltonian (2.4) which governs the corresponding master equation is then identified as the effective Hamiltonian.

iαφ The vertex operator e (α2πRc Z) is a sensible set of local operators in the compact boson theory ∈ which creates boson coherent state. It is also the electric operator in the quantum dimer model context[35].

We consider the excess EE generated by this local excitation. Speciﬁcally, we act the vertex operator eiαφ on the ground state and evolve for time t to reach a state

iHt iαφ(x) x, t = e− e gnd . (2.7) | i | i

The excess EE is defined to be the difference of EE between x, t and x, 0 . As a function of time, it should | i | i reflect the spreading of the local excitation.

16 2.2.2 Time Evolution of the Vertex Operator

In this subsection, we solve the time evolution equation and express x, t in terms of the ground state boson | i operators. For clarity, in this subsection we temporarily turn to the hatted notation φˆ to denote the boson operator and reserve the unhatted φ for the eigenvalue of the φ basis. | i Since the ground state is annihilated by H, we rewrite the state x, t as | i

iHtˆ iαφˆ(x) iHtˆ iαφˆ(x, t) x, t = e− e e gnd = e − gnd , (2.8) | i | i | i where the time dependent boson operator φˆ(x, t) is the solution of the Heisenberg equation −

ˆ ∂φ(x, t) iHtˆ iHtˆ i − = [H,ˆ φˆ(x, t)] = e− [H,ˆ φˆ(x)]e = [Q†(x, t) Q(x, t)]. (2.9) ∂t − − − −

The sign in t indicates another equivalent convention used in [9] that interprets the t as the time of local − − excitation and 0 as the time of measurement.

The Q† and Q are the non-standard creation/annihilation operators; they have the commutation relation

2 [Q(x),Q†(y)] = 2g δ(x y). (2.10) − ∇x −

We can then solve their Heisenberg equations

2 ∂tQ† = i[H,Q†] = i2g Q† − ∇ (2.11) i2gt 2 = Q†(x, t) = e− ∇ Q†(x), ⇒

which gives the time evolution of the boson operator

t 2igs 2 2igs 2 φˆ(x, t) = i e ∇ Q†(x) e− ∇ Q(x) ds. (2.12) − − − Z Now consider acting the operator Q on the ground state. We have

2 Q(x) gnd = 0 Q†(x) gnd = 2g φˆ(x) gnd (2.13) | i | i − ∇ | i

and hence t 2igs 2 2 2igt 2 φˆ(x, t) gnd = ds e ∇ (2ig )φˆ(x) gnd = e ∇ φˆ(x) gnd . (2.14) − | i ∇ | i | i Z 2igt 2 One should interpret e ∇ as a Sch¨odingertime evolution. We add a small real positive constant to

17 the imaginary time

τ = + 2igt (2.15) to control the ultraviolet divergence; this is the damping term that screens out the high energy mode[22].

τ 2 The operator e ∇ φˆ(x) obeys

τ 2 2 τ 2 ∂ e ∇ φˆ(x) = e ∇ φˆ(x) , (2.16) τ ∇ whose solution in free space is given by

τ 2 2 e ∇ φˆ(x) = φˆ(x, τ) = d x H(x, τ; x0)φˆ(x0), (2.17) Z where H is the standard heat kernel in 2d

2 1 (x x0) H(x, τ; x0) = exp − . (2.18) 4πτ − 4τ

The time evolved vertex operator is thus

iαφˆ(x, t) τ 2 e − = exp iαe ∇ φˆ(x) (2.19) on the ground state.

2.3 Excess Entanglement Entropy

In this section, we deﬁne and derive the replica formula for the excess EE.

The time dependent EE after the local excitation at t = 0 is the von Neumann entropy

S(t) = tr[ρ (t) ln ρ (t)] (2.20) − A A with respect to the time dependent density matrix

ρ (t) = tr [ρ] = tr x, t x, t (2.21) A B B | ih | associated with the state x, t . | i The excess EE is deﬁned to be

∆S(t) = S(t) S(0), (2.22) −

18 where the ∆ symbol in this chapter always denote the diﬀerence between time t and time 0.

The way to compute EE is to take the analytic continuation of the R´enyi EE

1 S = ln trρn (2.23) n −n 1 A −

n at n = 1. In ﬁeld theory setting, the quantity trρA has n replicated ﬁelds properly glued together[17], thus the name “replica trick”.

We evaluate this time dependent EE by using replica trick at each time slice. In fact, the EE of static

ground state EE has been evaluated and reﬁned by many groups [37, 42, 43, 44, 45]. Here we extend the

trick used in [37, 45] to an excited state.

The extension is diﬀerent from the replica trick commonly used in CFT calculation[17], so we give a brief

review of its derivation. We use discrete notation to derive tr(ρn ). Denoting a and b as sets of complete A | i | i orthonormal basis on subsystem A and B, for the ground state density matrix ρ = gnd gnd , we have | ih |

n tr(ρA) = tr b1 ρ b2 δb1b2 b2n 1 ρ b2n δb2n 1b2n h | | i · · · h − | | i − b Xi = a1 b1 ρ b2 a2 a2n 1 b2n 1 ρ b2n a2n h |h | | i| i · · · h − |h − | | i| i aXi,bi n 1 n (2.24) − δa1a2n δa2ia2i+1 δb2i 1b2i − i=1 i=1 Y Y 2n n 1 n 1 − exp S[φi] δa1a2n δa2ia2i+1 δb2i 1b2i . ∝ − 2 − i=1 i=1 i=1 aXi,bi n X o Y Y

This expression has 2n copies of ﬁelds, while the delta functions enforce the constraint that the ﬁelds of

odd indices are created by concatenating parts from adjacent ﬁelds of even indices. It is then equivalent to

remove those odd ﬁelds; meanwhile duplicate the even ﬁelds and require them to have the same value on the

cut. The later condition on cut ensures the possibility of stitching two parts from two even ﬁelds to create

the odd ﬁeld between them. This procedure is depicted in Fig. 2.1.

The derivation for the excited state is almost the same except for the insertion of operators O(φ, t) =

τ 2 τ 2 exp(iαe ∇ φ(x)) and O†(φ, t) = exp( iαe− ∇ φ†(x)) in front of the partition function. −

n 2n n 1 n n 1 − tr(ρA) O(φ2i 1, t)O†(φ2i, t) exp S[φi] δa1a2n δa2ia2i+1 δb2i 1b2i . (2.25) ∝ − − 2 − i=1 i=1 i=1 i=1 aXi,bi Y n X o Y Y

19 2n

2n-1 n ...... = . . 3 3 ⇒ 2

2 1

Figure 2.1: The gluing conditions for the fields of even indices. The left figure shows a cyclic relation that neighboring copies have the same value (same color in the figure) in one of the subsystems. The right figure shows the collapsing of odd fields to form n independent copies that share the same value only on the entanglement cut.

So relabeling the surviving even ﬁeld from 1 to n, we have

n O(φ , t)O†(φ , t) tr(ρn ) = h i=1 i i iglue , (2.26) A O(φ, t)O (φ, t) n Q h † iFree

where the 2n point function in the numerator is evaluated on the manifold in Fig. 2.1. This formula has

similar structure as the CFT calculation in [9], where the 2n point function in the numerator is evaluated

on the n-sheeted Riemann surface.

We use target space rotation to deal with the gluing condition. First we separate the ﬁeld into classical

and quantum part

φ = ϕ + φcl (2.27)

such that φ (x) = φ(x) and 2φ = 0, then the quantum ﬂuctuation has Dirichlet boundary cl cut cut ∇ cl condition ϕ(x) = 0 and the action separates cut

S[φ] = S[ϕ] + S[φcl]. (2.28)

Notice that the classical part does not evolve with time

τ 2φ(x) τ 2ϕ(x) O(φ, t) = exp iαe ∇ = exp iαe ∇ exp(iαφcl) = O(ϕ, t) exp(iαφcl), (2.29)

20 so in the average on the glued manifold , the real valued classical ﬁeld on the vertex operator cancels, h· · · iglue

n n j j i = exp iα (φ φ †) S[φ ] h· · · iglue h· · · iϕ cl − cl − cl φi j=1 i=1 Xcl h X X i n (2.30) = exp S[φi ] , h· · · iϕ − cl φi i=1 Xcl h X i where the summation is subject to the boundary condition

φ1 (x) = φ2 (x) = = φn (x) = φ (x) mod 2πR on cut. (2.31) cl cl ··· cl cut c

Since the ϕ ﬁeld satisﬁes Dirichlet boundary condition on the entanglement cut, we have

n = O(ϕ, t)O†(ϕ, t) . (2.32) h· · · iϕ h iDirichlet

As a result n n i = O(ϕ, t)O†(ϕ, t) exp S[φ ] . (2.33) h· · · iglue h iDirichlet − cl φi i=1 Xcl h X i The summation of the classical modes is the same as the ground state, so that rotational trick used there works in the same way. We rotate these classical ﬁelds by the following unitary matrix

1 1 1 1 φ¯ − φ cl √2 √2 cl  2   1 1 2   2  φ¯ − φ cl √6 √6 √6 cl        ¯3   1 1 1 3   3   φcl   √ √ √ √−   φcl    =  12 12 12 12    , (2.34)              ···   ···············   ···   n 1  1 1 (n 1)   n 1 φ¯ −   − −  φ −   cl   √n2 n √n2 n √n2 n   cl     − − ······ −     ¯n   1 1 1 1   n   φcl   √n √n √n √n   φcl     ···          such that after the rotation

¯j φ (x) = 2πwjRc on cut for j < n, wj Z. (2.35) cl ∈

It is noted that the n-th ﬁeld is the center of mass mode that is the only one dependent on the value on the cut ¯n x √ x φcl( ) cut = nφcut( ), (2.36)

21 while the rest of the ﬁelds decouple

n n 1 i ¯n − ¯i exp S[φcl] = exp S[φcl] exp S[φcl] . (2.37) i − − n 1 − φ i=1 φcut w − i=1 Xcl h X i X h i ∈XZ h X i

¯n The degrees of freedom on the entanglement cut determine φcl, while the rest determines the Dirichlet two point function, hence if we combine the two, we should collect all the degrees of freedom in this region and end up with a free two point function

n O(ϕ , t)O†(ϕ , t) exp[ S[φ ]] = O(ϕ , t)O†(ϕ , t) h n n iDirichlet − cl h n n iFree φXcut (2.38)

= O(φ , t)O†(φ , t) . h n n iFree

In fact, we can do this with the identiﬁcation of the ﬁeld φn

¯n φn = ϕn + φcl. (2.39)

This free two point function will cancel one of the two point functions in the denominator. We therefore have n 1 n 1 n O(ϕ, t)O†(ϕ, t) Dirichlet − − i tr(ρA) = h i exp S[φcl] . (2.40) O(ϕ, t)O†(ϕ, t) Free n 1 − w − i=1 h i ∈XZ h X i

Note that the sum over φcl is time independent and as a result will be canceled in the excess EE. The excess R´enyi entropy is therefore

∆S = ∆ ln O(φ, t)O†(φ, t) ln O(φ, t)O†(φ, t) , (2.41) n h iFree − h iDirichlet n o

where φ now denotes the non-compact free boson and ∆ denotes the diﬀerence between time t and 0.

2.3.1 Green Function

The two point function of the vertex operators ultimately will be reduced to the Green function of the boson

ﬁeld. In this subsection, we deﬁne and calculate the free space Green function in space and time directions.

The equal time Green function on the ground state

G(x , x ) = φ(x )φ(x ) (2.42) 1 2 h 1 2 i

22 satisﬁes Laplace’s equation

2g 2G(x , x ) = δ(x x ). (2.43) − ∇1 1 2 1 − 2

In free space(R2 plane), the solution is well known

1 G(x , x ) = ln x x . (2.44) 1 2 −4πg | 1 − 2|

We deﬁne the equal space Green function, which is useful later, to be the Green function evaluated at the same position but diﬀerent imaginary time

2 2 τ1 τ2 G(τ1, τ2) = e ∇ φ(x)e ∇ φ†(x) h i (2.45) 2 2 = d x1d x2 H(x, τ1; x1)G(x1, x2)H(x, τ2; x2). Z

By taking the τ1 derivative and using the property of the heat kernel,

2g∂ G(τ , τ ) = d2x d2x H(x, τ ; x )( 2g 2)G(x , x )H(x, τ ; x ) − τ1 1 2 1 2 1 1 − ∇1 1 2 2 2 Z 2 2 = d x1d x2 H(x, τ1; x1)δ(x1, x2)H(x, τ2; x2) (2.46) Z 1 = . 4π(τ1 + τ2)

Convergence of the integral requires Re(τ1 + τ2) > 0, which is satisﬁed by adding a damping parameter to both imaginary time τ1 and τ2. Up to a constant

1 G(τ , τ ) = ln τ + τ . (2.47) 1 2 −8πg | 1 2|

1 1 The 8πg rather than 4πg factor is a manifestation of z = 2. These two limits of the Green function agree with the general Green function expression in [35].

The Green function in this case is associated with the operator 2g 2. In the following we will instead − ∇ calculate in terms of the Green function G (x , x ) of the standard Laplacian operator 2, and relate it ∆ 1 2 −∇ to the two point function via

G∆(x1, x2) = 2gG(x1, x2) G∆(τ1, τ2) = 2gG(τ1, τ2). (2.48)

23 2.3.2 Excess Entanglement Entropy in terms of the Green Function

In Eq. (2.41), we need the two point function of the diﬀused vertex operator

exp iαφ(x, τ ) exp iαφ†(x, τ ) = exp iα[φ(x, τ ) φ†(x, τ )] , (2.49) 1 2 1 − 2 D n o n oE D n oE where

τ = + i2gt τ = i2gt. (2.50) 1 2 −

The exponent is a source term in the Gaussian path integral

i2gt 2 i2gt 2 iα[φ(x, τ ) φ†(x, τ )] = iα e ∇ e− ∇ φ(x) 1 − 2 − 2 = iα d x0 H(x, τ1; x0) H(x, τ2; x0) φ(x0) (2.51) − Z 2 = d x0J(x0)φ(x0). Z

Here J(x0) is a real operator because

i2gt 2 i2gt 2 2 J iα e ∇ e− ∇ = 2α sin( 2gt ). (2.52) ∼ − − ∇ We thus have the standard results for Gaussian integral

2 1 2 2 exp d x0J(x0)φ(x0) = exp d x d x J(x )G(x x )J(x ) (2.53) 2 1 2 1 1 − 2 2 Z n Z o and therefore A B 1 2 2 ∪ ∆Sn = ∆ d x1d x2J(x1)G(x1 x2)J(x2) . (2.54) 2 − Dirichlet Z

We expand the current function by using Eq. (2.51), and ﬁnd that the integral consists of four convolutions of heat kernel with the equal time Green function

d2x d2x J(x )G(x x )J(x ) = α2 d2x d2x [H(x, τ ; x ) H(x, τ ; x )] 1 2 1 1 − 2 2 − 1 2 1 1 − 2 1 Z Z (2.55) G(x x )[H(x, τ ; x ) H(x, τ ; x )] , 1 − 2 1 2 − 2 2 where each one of them is an equal space Green function deﬁned in (2.45). Hence

d2x d2x J(x )G(x x )J(x ) = α2 G(τ , τ ) G(τ , τ ) G(τ , τ ) + G(τ , τ ) . (2.56) 1 2 1 1 − 2 2 − 1 1 − 1 2 − 2 1 2 2 Z

24 Deﬁne the cross Green function to be

G (τ , τ , ) = G (τ , τ ) G (τ , τ ) G (τ , τ ) + G (τ , τ ) . (2.57) ∆ 1 2 × ∆ 1 1 − ∆ 1 2 − ∆ 2 1 ∆ 2 2 The excess R´enyi entropy is 2 A B α ∪ ∆Sn = ∆ G∆(τ1, τ2, ) . (2.58) − 4g × Dirichlet h i

2.4 Results and Discussion

We have obtained the free space Green function in Sec. 2.3.1, where the equal space Green function is evolved from the equal time Green function. When imposing Dirichlet boundary condition on the cut, we can solve

Dirichlet problems in A (and similarly in B)

2 A A 1G∆(x1, x2) = δ(x1 x2) G∆(x1, x2) = 0 (2.59) − ∇ − ∂A

and then construct the Green function on the whole plane as

Dirichlet A G∆ (x1, x2) =G∆(x1, x2)[θ(x1 A)θ(x2 A)] ∈ ∈ (2.60) + GB (x , x )[θ(x B)θ(x B)]. ∆ 1 2 1 ∈ 2 ∈

The step function θ in Eq. (2.60) implements the fact that the Dirichlet boundary condition destroys the correlation between two regions while modifying it within each region through “boundary charge”.

Then the equal space Green function is constructed from the equal time Green function through

Dirichlet 2 2 Dirichlet G∆ (τ1, τ2) = d x1d x2 H(x, τ1; x1)G∆ (x1, x2)H(x, τ2; x2) A B A B Z ∪ × ∪ 2 2 A = d x1d x2 H(x, τ1; x1)G∆(x1, x2)H(x, τ2; x2) (2.61) A A Z × 2 2 B + d x1d x2 H(x, τ1; x1)G∆(x1, x2)H(x, τ2; x2). B B Z ×

In electrostatic language, the free space G∆(τ1, τ2) is the potential energy between two Gaussian charge

Dirichlet distributions (albeit being imaginary), while G∆ (τ1, τ2) is the same thing in the presence of induced boundary charge on the entanglement cut. Hence the diﬀerence, about which the EE is concerned, only depends on the boundary charge.

In App. A.2, we showed that (the double derivative of) the Dirichlet Green function is solely determined

25 by a boundary integral,

no cut A A ∂τ2 ∂τ1 G∆(τ1, τ2) = H(x, τ1; x1)∂n1 ∂n2 G∆(x1, x2)H(x, τ2; x2) dl1 dl2. (2.62) Dirichlet − ∂A ∂A Z ×

We can integrate the heat kernel once to deﬁne

τ1 f(x) = H(0, τ; x1) dτ (2.63) Z0 and recognize that τ2 f¯(x) = H(0, τ; x1)dτ. (2.64) Z0 Using this, the cross Green function becomes

A B ∪ G∆(τ1, τ2, ) = 4 Imf(x1)K(x1, x2)Imf(x2) dl1 dl2, (2.65) × Dirichlet ∂A ∂A Z ×

where K(x1, x2) is the kernel on the boundary

A K(x1, x2) = ∂n1 ∂n2 G∆(x1, x2), (2.66)

which is singular when x1 is approaching x2. In general we can regularize it as

A B 2 ∪ G∆(τ1, τ2, ) = 2 Imf(x1) Imf(x2) K(x1, x2) dl1 dl2. (2.67) × Dirichlet − ∂A ∂A − Z × h i

The excess R´enyi EE is therefore

α2 2 ∆Sn = Imf(x1) Imf(x2) K(x1, x2) dl1 dl2. (2.68) 2g ∂A ∂A − Z × h i

In the following, we analyze two cases where the Green function can be easily ﬁgured out by methods of images.

26 A

exp iαφ(x, t) −

Figure 2.2: The system is an inﬁnite plane. Local operator is placed at (0, y).

2.4.1 Inﬁnite Plane

Excess EE

We consider the geometry of inﬁnite plane with entanglement cut on the x-axis. In complex coordinate, the

equal time Green function in region A can be easily written down

1 1 GA (x , x ) = ln z z + ln z z¯ x , x A. (2.69) ∆ 1 2 −2π | 1 − 2| 2π | 1 − 2| 1 2 ∈

The kernel 1 y x x x x 2 K( 1, 2) = lim ∂y2 ∂y1 G∆( 1, 2) = lim ∂y2 2 2 (2.70) ∂A y2 0 y1=0 π y2 0 (x1 x2) + y2 → → −

is indeed singular at x1 = x2. In App. A.5 we provide a way to interpret the distributional integral. The resulting recipe is the same as the general formula (2.67).

The integral of the heat kernel

1 1 (x x )2 Imf(x ) = Si[ ] + Si − 1 (2.71) 1 −4π ∞ 4π 4t h i

is given by the sine integral function x Si(x) = sin t2 dt. (2.72) Z0 By Eq. (2.68), the excess EE is

2 x2+y2 x2+y2 2 2 Si 1 Si 2 α A α 4t − 4t ∆Sn = G∆(τ1, τ2, ) = 2 2 dx1 dx2 − 4g × 32π g ∂A ∂A n π(x1 x2) o Z × − 2 (2.73) 2 y2 2 y2 2 Si x + Si x + α 1 4t − 2 4t = 2 2 dx1 dx2. 32π g ∂A ∂A n π(x1 x2) o Z × −

27 We can also write the integral in Fourier space through the standard Hilbert transform,

2 2 2 α ∞ 2 y 2 y ∆Sn = = Si x + ∂x(Si x + ) dx 16π2g 4t 4t H Z−∞ 2 2 (2.74) α ∞ dk y 2 = k (Si[x2 + ])(k) . 16π2g 2π | | F 4t Z−∞

App. A.3 gives an alternative route for the calculation and reaches a simpler expression

2 α ∞ 1 2 ∆Sn = 4 q dλ C(λ) S(λ) , (2.75) 8πg y2 λ − Z 4πgt n o where C and S are the Fresnel cos/sin integrals

z π z π C[z] = cos( x2)dx S[z] = sin( x2)dx. (2.76) 2 2 Z0 Z0

Quasi-Particle Interpretation

We plot the expression of inﬁnite plane excess EE in Eq. (2.75) as Figs. 2.3 and 2.4

EE of upper half plane EE of upper half plane 1.2 10

1 1 0.8 ∆Sn 0.6 ∆Sn 0.1

0.4 0.01 0.2

0 0.001 0 5 10 15 20 0.01 0.1 1 10 100 1000 t t

R 1 2 q∞ Figure 2.3: Plot of 4 1 dλ λ C(λ) S(λ) . EE satu- Figure 2.4: This log-log plot exaggerates the t − rates to constant value in the long time. plateau in the short time regime. A linear ﬁt in the regime where plateau are invisible gives the slope to be 1, hence there is a linear increase of excess EE in the short time regime.

Despite some minor modiﬁcations, the quasi-particle picture is still able to interpret the growth of EE in this non-relativistic model.

First of all, the excess EE is a monotonically increasing function of time and eventually saturates to a constant value. The maximal value is proportional to the scaling dimension of the vertex operator, which can be regarded as a dimensionless measure of the strength of the operator. The saturation indicates the

28 exhaustion of quasi-particles, in other words, almost all the downward travelling quasi-particles are in the

lower half plane. It is comforting to conﬁrm the fact that the local vertex operator excitation only inject

small energies to the system.

The causality constraint in CFT is superﬁcially violated. The excess EE grows almost immediately after

y t = 0. In fact, the horizon effect is only visible in the regime where t < c , where c is speed of light (or the equivalent threshold speed in the condensed matter system). While in this non-relativistic theory, the quasi-particle speed is far less than the speed of light, so that we can essentially take the c limit, → ∞ y squeezing the zoom 0 < t < c to empty. Instead, the typical z = 2 diffusive behavior is taking place of the causality constraint. The excess EE grows to (1) at the time scale t y2, when the majority of the O ∼ quasi-particle diffuses to the entanglement cut.

It is interesting to zoom out the small time regime shown in Fig. 2.4. A linear ﬁt in the log-log plot shows

the slope to be 1 and hence there is a linear increase of excess EE ∆S const t in the short time regime. ∼ × y2 There are several staircase-like plateau of increasing sizes appear in the growing region, with the last one

stacked on top saturating to a limiting value. It is tempting to assume that the quasi-particles disperse:

phenomenologically, we see those separated groups of particles arriving sequentially on the entanglement

cut. We examine this idea by using a disk probe in the next section.

2.4.2 Disk

Excitation in the Center

If the vertex operator is placed in the center, then the boundary integral kernel function K(x1, x2) will only be a function of the angle θ. On the other hand, Imf(x), which is the integral of Gaussian, is only a function of the radius. Thus the regularized boundary integral (2.67) is identically zero.

The vanishing of excess EE in this geometry indicates that the quasi-particle are distributed and travelling with spherical symmetry. Points of excitation away from the center is thus a possible way to probe and decompose the quasi-particle distribution.

29 Using Small Disk as a Probe

Now we place a disk of radius R centered at the origin, and the excitation at distance r away from the center. The equal time Green function becomes

2 A 1 1 R G∆(x1, x2) = ln z1 z2 + ln z1 −2π | − | 2π | − z¯2 | 1 = ln r2 + r2 2r r cos(θ θ ) (2.77) −4π 1 2 − 1 2 1 − 2 4 2 1 2 R R + ln r1 + 2 2r1 cos(θ1 θ2) . 4π r2 − r2 − The kernel on the circle is

A 1 ∂r ∂r G (x1, x2) = . (2.78) 1 2 ∆ 2 2 θ1 θ2 4πR sin −2 Hence no cut π π A Imf(x1)Imf(x2) G (τ1, τ2, ) = 4 dθ1 dθ2 ∆ 2 θ1 θ2 × Dirichlet π π 4π sin − Z− Z− 2 π

= 4 f(x)∂θf(x) dθ (2.79) − π H Z− 1 ∞ = 4 n fnf n, − 2π | | − n= X−∞ where is the Hilbert transform on circle, and H

1 r2 + R2 2rR cos θ Imf(x) = Si − x = r. (2.80) 4π 4t | | h i

Rescaling R = 1, we have

α2 1 1 ∞ α2 1 ∞ ∆Sn = 4 n SinSi n = n SinSi n, (2.81) 4g 2π (4π)2 | | − 8πg 4π2 | | − n= n= X−∞ X−∞ where Sin is the Fourier transform

π 2 r + 1 2r cos θ inθ Sin = Si − e dθ. (2.82) π 4t Z− h i

Discussion of the Probed Excess EE

We plot the results in Eq. (2.81) in diﬀerent length scales of distance r to the unit disk probe (R = 1).

In all cases, the excess EE drops down to zero in the asymptotic region when almost all the quasi-particles have passed away. The larger r ﬁgure shows larger separations of peaks. There is a largest peak both in height and width that represents the majority of quasi-particles, which should also be responsible for the

30 Excess EE of Disk of radius 1 0.12 r = 4 r = 5 0.1 r = 6 r = 7 0.08 r = 8 ∆S n 0.02 0.06 0.01 0.04 0 40 50 60 70 80 0.02 t 0 0 5 10 15 20 25 30 35 40 t

Figure 2.5: Excess EE of disk of radius 1 with different distances to the point of excitation. We can see clearly that the quasi-particle densities are not the same and disperse as time goes on. largest plateau in the upper half plane configuration. Smaller peaks travel faster and arrive earlier to the entanglement cut. This verifies our assumption that the quasi-particles “wave” disperse in this diffusion.

The pattern is similar to chromatography in chemical species separation which also takes advantage of their diﬀerent diﬀusive “speeds”.

2.5 Summary

In this chapter we investigate the excess EE created by the vertex operator in the quantum Lifshitz model ground state. We develop the replica trick to derive a formula that relate excess EE to the diﬀerences of the vertex two point function with Dirichlet boundary and free space. It turns out that excess EE can be completely written in terms of boundary integral on the entanglement cut, and in some sense reﬂects the fact that the change of EE only happens in the vicinity of the entanglement cut. This is in compliant with the local interaction nature of the original quantum dimer model.

We pay attention to the upper half plane and disk geometries. We show that the quasi-particle picture can still can interpret the growth of excess EE in this non-relativistic model. In particular, strict causality is replaced by diﬀusive behavior in the sense that the excess EE will reach an (1) scale only when the O majority of quasi-particles arrive at the entanglement cut. Zooming out the small time regime in the upper

31 half plane geometry show plateaus in different scales. We ascribe this to the different density of states for quasi-particles of different speeds. By placing a disk probe away from the excitation point, we are able to see the chromatography pattern in the excess EE, which demonstrates possible dispersion and different density of states for particle species.

Further work can be done to understand the new features discovered in this chapter. In the cases we considered, more evidences are needed to account for the plateau structure we ﬁnd in the short time dynamics of the inﬁnite plane case. We only calculate the single vertex operator excitation and shows that the excess EE is independent of the winding sector, which otherwise plays an important role in the ground

iφ iφ state EE. The excess EE of similar operators like e + e− will have dependence on the compactiﬁcation radius, from which we would expect to obtain universal information. There are general questions like the way to obtain Rieb-Robinson bound for the initial development of excess EE for z = 2. A holographic picture[46, 47, 48, 49, 50] in the dual Lifshitz gravity (with Hoˇrava-Lifshitz gravity[51] being one of the candidates) would also be helpful in understanding the quench behavior here.

32 Chapter 3

Bipartite Fidelity and Loschmidt Echo of the Bosonic Conformal Interface

3.1 Introduction

In this chapter, we will focus on a type of local quench induced by the physical boundary or impurity. Their presence in (one-dimensional) quantum critical systems weakly break the conformal symmetry. Simply put, the interface scatters the otherwise independent modes and therefore demonstrates novel boundary critical phenomena[52]. Operators close to the boundary are interpreted as boundary condition changing (bcc) operators[53, 54] in the boundary conformal ﬁeld theory (CFT). Their correlation functions can exhibit diﬀerent critical exponents from their bulk counterparts[55]. One example is the “Anderson orthogonality catastrophe”, where the core hole creates a potential that acts as an impurity to the conduction band. The

X-ray absorption rate will then have a power law singularity of a boundary exponent[54] at the resonance frequency. There are numerous impurity problems of this kind that have been studied in the last few decades, such as the magnetic impurity in the spin chain[56], boundary and impurity eﬀects in Luttinger liquid[57], the entanglement of the defects[58, 59, 60] etc.

Recently, more attention has been paid to the non-equilibrium dynamics of quantum impurity[61, 62,

63, 64, 65, 66, 67, 68]. The “cut-and-join” quench protocol is a popular framework for investigating the spreading of the inﬂuence from the localized impurity (or boundary) across the system. As shown in the left panel of Fig. 3.1, the system consists of two critical chains A and B, which were prepared in the ground states. They will be joined at t = 0 and evolve. Various quantities can be used to detect the information in the quench process. For instance, Refs. [21, 23] ﬁnd a logarithmic increase of entanglement entropy in subsystem A, when both A and B are identical critical systems. The authors ascribe such increase to the proliferation and propagation of the quasi-particle excitations emitted at the joint. Ref. [69] takes A to be a normal lead and B to be a topological superconductor in the topological phase. In this model, the Majorana zero mode acts as a bcc operator and its conformal dimension appears in the exponent of the power law decay of the Loschmidt echo.

In the path integral language, the “cut-and-join” protocol corresponds to a spacetime diagram as shown

33 iH t e− join

A B A B

A B

Figure 3.1: Cut-and-join quench protocol. Left panel: Prepare the ground states of the two separated chains and join them at t = 0, then time evolve with the whole chain Hamiltonian. Right panel: Spacetime diagram of the cut-and-join protocol. The solid line represents the boundaries of the two disconnected chains. It is totally reﬂective for the incident particles on both sides. The dashed line is the world line of the junction, which we will call interface. It could either be totally transparent or partially permeable, depending on the types of theories of A and B.

in the right panel of Fig. 3.1. The separating ground states prepared before t = 0 are joined to form a new type of interface between them. Before the quench, the slit represents boundaries that are completely reﬂective to the injecting particles. During the quench, the joining turns on the transmission from one side to the other. In the entanglement entropy and Loschmidt echo examples cited above[21, 23, 69], the two sides of the CFTs are the same (chiral fermion CFT in the case of Ref. [69]) and the boundary becomes totally transparent after the joining.

In this chapter, we generalize these ideas to an interface that interpolates between the totally reflective and complete transparent ones. This kind of interface can have many realizations. As discussed in Ref. [70], one can connect two different bosonic CFTs in the “cut-and-join” protocol, and the interface is a domain wall between two free compact boson theories with different compactification radii. Such permeable interface can also be implemented by non-compact free boson/fermion on a lattice with a fine-tuned bond interaction between the boundary sites (see [71, 72] for their entanglement property studies). In these models, there is a parameter λ that is directly related to the transmission coefficient. In the case of the compact boson, λ is controlled by the ratio of the compactification radii, while for the free lattice boson it is controlled by the ratio of masses. We expect it to be tunable in a realistic experimental setting.

We compute the Loschmidt echo to extract information in the dynamics of the quench process of these models. The Loschmidt echo is the (square of the) overlap of the wavefunctions before the quench and the

α wavefunction evolved for some time t. It decays with a power law t− for the lack of length scale in the t limit. The decay exponent α has been calculated for various geometries and combinations of normal → ∞ boundary conditions of the same CFTs in [73, 74]. We extend the analysis to the aforementioned parametric

34 interface of (possibly) diﬀerent CFTs. We will see that there are two categories of the scattering matrices

S(θ) of the interfaces, whose scattering angle parameter θ is determined by the transmission coeﬃcients.

Our analytic and numerical results show that α has a quadratic dependence on the change of θ if the prior

1 and post quench boundary conditions are in the same type of S, while remaining 4 otherwise. The ﬁnite size ﬁdelity calculation further supports these results.

The rest of the chapter is organized as follows. In Sec. 3.2, we introduce the general formalism for the

permeable bosonic conformal interface and its lattice realization. In Sec. 3.3, we analytically evaluate the free

energy associated with the ﬁdelity and Loschmidt echo, and present the numerical results for comparison.

We discuss our results and related experimental works in Sec. 3.4. Finally, we conclude in Sec. 4.7.

This chapter includes several appendices for technical details. In App. B.1, we present the leading order

analytical calculation of the free energy for the setups in Sec. 3.3.3. In App. B.2, we illustrate an alternative

approach with one setup as an example. In App. B.3, we point out two corrections to the free energy, which

are complementary to the argument made in the main text. Up to this point, we work exclusively with the

oscillator modes of the free bosons. In App. B.4, it is shown that the winding modes of the compactiﬁed

bosons will not contribute to the free energy at the leading order. Therefore, the results remain valid in

the physical situation of connecting two compactiﬁed bosons of diﬀerent radii. In App. B.6, we prove one

identity that will be used repeatedly in the analytical evaluation. We derive the scale invariant interface for

the free bosonic lattice in App. B.5. The details of the numerical simulation are presented in App. B.7.

3.2 Bosonic Conformal Interface

3.2.1 General Formulation

The general constraint on an interface is the continuity of the momentum ﬂow across it. If we fold one side

of the system on top of the other, then the resulting interface located on the boundary of the tensor theory

(the crease of the folding) becomes impenetrable and the momentum ﬂow should vanish there. This interface

is naturally a conformal invariant boundary state[52, 55]. The interfaces in this chapter are boundary states

living in the c = 2 boundary CFT.

Although the general classiﬁcation of the boundary states is still an open question[75], there are many

successful attempts to construct a subset of those bosonic boundary states. For example, one may use the

current operator rather than the Virasoro generator to solve the zero-momentum ﬂow condition. This idea

dates back to the discovery of the Ishibashi state[76] and has been applied to the multi-component boson

with a general compactiﬁcation lattice[75, 42, 77]. Additionally, the fusion algebra has also been used to

35 generate new boundary states from the known ones, as shown in Ref. [75, 78].

We here follow the presentation in Ref. [70], which imposes the conformal invariant boundary condition

on the classical scalar ﬁelds and then quantize it to obtain the boundary state. The interface obtained is the

same as the one by using the current algebra[75, 42, 77], but this viewpoint gives a more intuitive scattering

picture and has more transparent relation to the discrete lattice model in Sec. 3.2.2.

Assuming two free boson ﬁelds φ1 and φ2 living on the left and right half planes respectively, the interface

located at x = 0 is characterized by the “gluing condition”

1 2 ∂tφ ∂tφ = M . (3.1)  1  2 ∂xφ ∂xφ         1 The derivatives here should be understood in the appropriate left and right limits, for example ∂xφ is evaluated at x = 0−. As argued before, the momentum components of the stress tensor is continuous across the interface. As a consequence M is an element of the Lorentz group O(1, 1) and can be parameterized as

1 λ− 0 0 λ M1(θ) = ,M2(θ) = , (3.2) ±   ±  1  0 λ λ− 0         where λ = tan θ for θ π , π . ∈ − 2 2 Several special choices of θ need to be noted.

π 1 1. θ = 0, . In this case, λ (or λ− ) appears to be singular and the ﬁeld on either side of the interface ± 2 cannot penetrate. The interface reduces to individual boundary conditions for the boson on the left

and right half planes: they are a combination of the Dirichlet and Neumann boundary conditions. For

1 2 example, λ = 0 for M1 implies ∂xφ = ∂tφ = 0, which means that the Dirichlet boundary condition is imposed on the right and the Neumann boundary condition on the left. Hereafter we shall denote this

combination as ‘DN’. Similarly M ( π ),M (0),M ( π ) correspond to ‘ND’, ‘DD’, ‘NN’ respectively. 1 ± 2 2 2 ± 2 2. θ = π . In this case, M (θ) characterizes a perfectly transmitting interface. For example, there ± 4 1 π is eﬀectively no interface in the case of M1( 4 ). We will denote it as “P” as it corresponds to the traditional periodic boundary condition. For the other three cases, despite picking up a phase, the two

counter propagating modes are still fully transmitted across the interface.

The physical signiﬁcance of θ will be clear in the scattering process described below. We rewrite Eq. (3.1)

2 in the coordinates t x and use ∂ = ∂t ∂x to extract the left and right going modes. For example, ∂ φ ± ± ± − will be a function of t x and hence represents a right going mode on the right half plane. This mode is − 36 1 2 one of the scattering modes that leave the interface. On the other hand, ∂ φ and ∂+φ are modes that − approach the interface from their respective domains. We can therefore establish the scattering relation

1 1 ∂+φ ∂ φ = S − , (3.3)  2  2 ∂ φ ∂+φ  −        and solve the S matrix for the two cases of M1 and M2

cos 2θ sin 2θ cos 2θ sin 2θ − S1(θ) =   ,S2(θ) =   . (3.4) sin 2θ cos 2θ sin 2θ cos 2θ  −  − −      For generic values of θ, the interface is partially-transmitting, whose transmission coeﬃcient is sin2 2θ.

We note that the S-matrices are independent of the wavelength, which agrees with the fact that the interface is scale invariant.

folding

∂+φ1 ∂ φ2 −

∂ φ1 ∂+φ2 −

Figure 3.2: Folding picture for the penetrable interface. Left panel: World line of the penetrable interface. ∂ φ1,2 denote the left and right going modes in their respective domains. Right panel: Folding operation that± sends φ2(x) to φ2( x). The dashline represents the impenetrable boundary for the resulting tensor theory. The arrow represents− the incoming and outgoing particles scattered by the interface.

We now work in the folding picture as shown in Fig. 3.2. The boundary at x = 0 becomes impenetrable for the folded system, and the resulting tensor theory admits a conformal invariant boundary state. The folding sends φ2(x) to φ2( x) and hence the gluing condition becomes −

∂ (sin θφ1 cos θφ2) = 0, ∂ (cos θφ1 + sin θφ2) = 0, (3.5) t − x

for the case M = M1(θ). If we quantize the boson theory on the interface line x = 0, these gluing conditions become an identity for the boson creation and annihilation operators. We shall interpret these identities to be valid only when

37 acting on the boundary states. The mode expansion of free boson at x = 0[79] is

i i 1 n n φ(z, z¯) = φ π ln zz¯ + a z− +a ¯ z¯− , 0 − 4πg 0 √4πg n n n (3.6) n=0 X6

where we take the following choice of the holomorphic and anti-holomorphic coordinates

2πi(x t) 2πi(x+t) z = e T − , z¯ = e T , (3.7)

with T being the time period. We end up with a set of operator identities for each mode

1 2 1 2 sin θan cos θan = + sin θa¯ n cos θa¯ n , − − − − (3.8) 1 2 1 2 cos θan + sin θan = cos θa¯ n + sin θa¯ n , − − − which is valid for the following boundary state

1 1 a n B = exp − S (θ) 1 2 0 , (3.9)   1 a¯ na¯ n | i − n 2 − − | i n n>0 a n o X  −   

where S1(θ) is precisely the scattering matrix in Eq. (3.3). The calculation for the case M = M2(θ) is completely analogous and we just have a replacement of S1(θ) by S2(θ) in Eq. (3.3). The boundary state expression in Eq. (3.9) will be used extensively in the ﬁdelity and Loschmidt echo calculation in Sec. 3.3.

So far the derivation is only for the non-compact bosons, where the interface is determined by the “gluing conditions”. For the case of connecting compact bosons of diﬀerent radii, we will need to generalize the relation in Eq. (3.8) to the winding mode operator a0. Because the winding modes live on a compactiﬁcation lattice, not all θ can satisfy Eq. (3.8) for a0. App. B.4 reviews the derivation about how the winding modes constrain the choice of θ. For example, the S1(θ) interface should satisfy

n R λ = tan θ = 2 1 (3.10) n1R2

for coprime integer n1 and n2 and compactification radii R1 and R2 for the bosons on the two sides. This also suggests that connecting two different CFTs will generate an interface whose transmission coefficient are determined by the universal parameters of the CFTs on both sides.

The winding modes however do not contribute to the ﬁdelity and echo exponent to the leading order, as

38 shown in App. B.4. So the derivations with the non-compact boson boundary state in Eq. (3.9) holds true

for the compact bosons.

3.2.2 A Free Boson Lattice Model

In this section, we consider a lattice model with bosonic interface at the center[71, 60], which reduces to the

one considered in Sec. 3.2.1 in the continuum limit[72]. Therefore, it serves as a numerical tool to check our

analytic results in Sec. 3.3.

We consider two harmonic chains with bosonic ﬁeld φi and conjugate momentum πi at the lattice site i. The left and right chains are connected between site 0 and 1 with the Hamiltonian

1 + Σ Σ φ 1 2 1 2 1 11 12 0 H = π + (φi φi+1) + φ , φ , (3.11) 2 i 2 − 2 0 1     i i=0 Σ21 1 + Σ22 φ1 X X6         where the 2 2 matrix Σ parameterizes the two-site interaction. We performed the standard scattering × analysis in App. B.5. For modes with momentum k, the only possible scale invariant S-matrix is

S = eikaΣ, (3.12) −

where a is the lattice constant. In the continuum limit a 0, the matrix Σ can be parameterized as →

λ2 1 2λ −2 − 2 Σ = lim S = 1+λ 1+λ , (3.13) a 0  2λ 1 λ2  − → 1+− λ2 1+−λ2     where λ R. ∈ The lattice model in Eq. (3.11) with the bond interaction deﬁned in Eq. (3.13) will be used to check our

analytic results for both the Loschmidt echo and the ﬁdelity.

3.3 Bipartite Fidelity and Loschmidt Echo

3.3.1 Deﬁnition

In this section, we deﬁne the ﬁdelity and Loschmidt echo and present their corresponding imaginary time

path integral diagrams. We will see that these path integrals are just the free energy of boson with conformal

interfaces (or boundaries).

39 Fidelity is the square of the overlap of the groundstates of the two Hamiltonians,

ﬁdelity ψ ψ 2. (3.14) ≡ |h 1| 2i|

For the systems we considered, ψ is the groundstate of the two disconnected chains (of equal length L, | 1i hence “bipartite”) and ψ is that of the connected chains with the conformal interface. Both of them can | 2i be produced by an imaginary time evolution. Taking the horizontal axis as imaginary time direction, the

ﬁdelity can be diagrammatically represented in Fig. 3.3, where the slits represent the disconnected boundary

conditions, such as Dirichlet(D) and Neumann(N), and the dashed line represents the conformal interface

parameterized by λ. The logarithmic ﬁdelity is then (twice) the free energy of this diagram

(ﬁdelity) = ln ψ ψ 2 = 2 ln Z . (3.15) F − h 1| 2i − | |

N DN N folding L D λ −−−→ DN λ D

Figure 3.3: Fidelity of connecting two CFTs. The horizontal axis is the imaginary time. Evolution along the two semi-inﬁnite stripes produces the groundstates of the disconnected and connected chain Hamiltonians. The right diagram is the result of folding the lower part of the diagram up, so that all the boundaries are now boundary states. The solid dot represents boundary condition changing (bcc) operator. Here D(Dirichlet), N(Neumman) and λ(permeable interface parameterized by λ) are possible choices of boundary conditions.

The Loschmidt echo is also (square of) the overlap of the two wavefunctions. One of them is the groundstate of the disconnected chains and the other is the groundstate evolved by the Hamiltonian of the connected chains

iHt 2 (t) ψ e− ψ . (3.16) L ≡ |h gnd| | gndi|

Hτ 2 The imaginary time version (τ) = ψ e− ψ has a path integral deﬁnition similar to Fig. 3.1, but L |h gnd| | gndi| to be consistent with the ﬁdelity diagram, we take the horizontal axis as imaginary time and present it in

Fig. 3.4. Viewing the diagram as a partition function subject to the switching of boundary conditions, the logarithmic Loschmidt echo is also the associated free energy. After obtaining the free energy in imaginary time, we can analytically continue back to real time to get the t dependence. For simplicity and comparison with the ﬁdelity result, we will take the length of both chains to be L and set L t, leaving t the only

40 t length scale in the echo calculation (Fig. 3.6). The dependence on nonzero L and the asymmetry of the lengths of the chains will not be discussed here (see the treatment in Ref. [74] for special values of λ).

N c folding N N τ L D λ D −−−→ a b a D

Figure 3.4: Loschmidt echo of connecting two CFTs. Evolution along the two inﬁnitely extended sides produces the groundstate of the disconnected chain Hamiltonians. They sandwich the evolution of the connected chains. In the folding picture on the right, a, b, c represent the most general boundary conditions of the chains (for example, a and c are DN according to the left ﬁgure).

If the interface is completely transparent, i.e. at the special point of λ = 1, the tip of the slit can be regarded as a corner singularity. According to Cardy and Peschel[80], the singularity will contribute a term that is logarithmic of the corner’s characteristic size, which is ln L in the ﬁdelity and ln τ in the Loschmidt echo. One would expect the ﬁdelity and echo to have power law decay with respect to these scales in the long wavelength limit. In fact, the computations have been done in Ref. [73, 74, 69, 81, 82] using either the

Cardy-Peschel formula or the integral version of the Ward identity. If the slit boundary conditions are taken to be Dirichlet, we have the universal behavior for the leading term [73, 74]

c (fidelity) = ln L, F 8 (3.17) c c c (echo) = ln τ ln it + ln t, F 4 | | → 4 | | ∼ 4 where we have performed analytic continuation τ it + with 0 for the echo. → → With the presence of the conformal interface, the tip of the slit is no longer a corner singularity[80] . Its nature is clearer in the folding picture shown in Fig. 3.3 and Fig. 3.4 where the lower half plane is flipped up on top of the upper half plane on both the fidelity and echo diagrams. From the boundary CFT point of view, the change of boundary conditions can be regarded as inserting a bcc operator. The diagrams for the

ﬁdelity and echo then become the one point or two point functions of the bcc operators respectively, and the free energy’s leading logarithmic term extracts their scaling dimensions.

3.3.2 Notation of Boundary Conditions

We use chemical reaction style to represent the change of boundary conditions. Taking the example of the echo diagrams in Fig. 3.4, there are three boundary conditions a, b, c in the folding picture, which represents

41 the status of the two ends of the chain before and after the quench. The choice of a uniform c boundary condition on the far end of the chain is to isolate the eﬀect coming from the bcc on the ab interface. The process a + c b + c represents the change of boundary condition from the combination a/c prior to the → quench to b/c after the quench. Since each letter can take a general conformal interface deﬁned by the S matrix in Eq. (3.4), we denote it as

S (θ ) + S (θ ) S (θ ) + S (θ ). (3.18) a a c c → b b c c

In most cases of the following, we will consider taking a = c to remove the bcc operator from a to c at inﬁnity. And we will use the shorthand notation

S (θ ) S (θ ) (3.19) a a → b b to remind ourselves that we are isolating the boundary condition change only on the joint of the two chains.

In the “cut-and-join” protocol we considered, a should be one of ‘DD’, ‘DN’, ‘ND’, ‘NN’, b is taken to be S1(θ) or S2(θ). The physical situation of connecting two compact bosons (and our numerical simulation) corresponds to the choice of S1(θ), and we reserve the notation λ for this type of the boundary condition. For instance, the notation for the process presented in Fig. 3.4 is

DN λ. (3.20) →

π Another interesting case is to take a or c to be a completely transmitting interface, i.e. S2( 4 ). This S-matrix corresponds to the traditional periodic boundary condition and we use symbol ’P’ to denote it.

3.3.3 Analytic Evaluation

In this subsection, we relate the free energy to the amplitudes between the boundary states, and present the analytic results.

We notice that there is only one apparent length scale in these diagrams – the ﬁnite size L for ﬁdelity and imaginary time τ for the Loschmidt echo. These are the characteristic size of the corners at the tip of the slits. Regulators are necessary in keeping track of the scale dependence, otherwise a dilation transformation can rescale both L and τ to 1 and drop those scales. The introduction of the regulators is also physically sensible when considering the lattice realization of the systems.

We thus add small semi-circles around the points where the bcc operators reside, and then apply a series

42 of conformal mappings.

For the ﬁdelity case, the regulators as well as the conformal maps are depicted in Fig. 3.5. We add a

c a(= c) z ξ w L

a b c a b b

πz Figure 3.5: Mapping from a strip to the upper half plane ξ = exp( L ). The two black dots represent possible locations of the boundary condition changing (bcc) operators. The dot inside the blue semi-circle has coordinate ξ = 1, which is the image of the point connecting a and b boundaries. The other dot ξ = 0 corresponds to the connection between a and c boundaries at . To evaluate the diagram, we add the −∞ outer blue semi-circle centered at ξ = 1 with radius Rξ to be the IR cut-oﬀ and map it to the cylinder with w = ln(ξ 1) − small blue semi-circle to the folded strip in Fig. 3.3 as the UV regulator and map it to the upper half plane

πz using ξ = exp( L ). Then both ξ = 0 and 1 can host bcc operators. We assume a = c such that the only bcc operator on the real axis is the one enclosed by the blue semi-circle around ξ = 1. In order to evaluate this

diagram, we add another semi-circle centered around ξ = 1 with radius Rξ (this will introduce a correction as explained in App. B.3), and map it to a cylinder of height π on the right by w = ln(ξ 1). Finally the − cylinder diagram can be viewed as an imaginary time path integral amplitude between the boundary states

b and a

πH Z = a e− b . (3.21) ab h | | i

The two end points of the radius semi-circle on the z plane are mapped to

exp( π ) 1 π . (3.22) ± L ∼ ± L

The bigger blue semi-circle intersects the real axis at 1 R and so the width of the cylinder is ± ξ

π ln R ln = ln L + constant. (3.23) ξ − L

The Loschmidt echo can be evaluated in the same way. Again, we introduce two semi-circles (blue in

Fig. 3.6) as regulators and then perform the conformal transformation shown in Fig. 3.6. From the z plane

z to the ξ plane, we use ξ = τ z to map the two slits to half of an annulus, which is the same as the ﬁdelity − case. With one more conformal mapping w = ln ξ, the diagram again becomes the cylinder partition function

43 between the two boundary states.

The height of the cylinder is still π. In the ξ plane, the coordinates of the two end points of the small

τ τ semi-circle are τ± ±τ , while those of the larger semi-circle are ± ∓ . Hence the width of the ∓ ∼ ∓ ∼ cylinder is τ ln ln = 2 ln τ + constant. (3.24) − τ

z ξ w

0 τ 0

Figure 3.6: The dashed (solid) lines are gluing (completely reflective) boundary conditions. Red arrows are the directions of Hamiltonian flow that propagates the dashed line boundary state to the solid line boundary state. Left: Diagram of the Loschmidt echo that reduces to a partition function with imaginary time in the horizontal direction. The blue semi-circles of radius are the UV regulators and they are identified as periodic boundaries in the direction perpendicular to the red arrow (equal time slice). Middle: Image of z 1 the map ξ = τ z . The two semi-circles have radii (τ/)± respectively. Right: Image of w = ln ξ. It is a cylinder by identifying− the blue lines and the standard radial quantization procedure can be applied.

One subtlety of the above description is that the two semi-circles in the center diagrams of Fig. 3.5 and

Fig. 3.6 are not precisely concentric. This can be resolved by the following observation. There exists a

conformal map ζ(ξ) that maps the non-concentric circles to the two standard concentric circles of radii 1

and R (R > 1) on the ζ plane[83]. Then the logarithmic map w = ln ζ produces a cylinder of width ln R. In

our case, since the height of the cylinder is always π, the width of the cylinder is a conformal invariant that

only depends on the cross ratio of the half annulus. The four intersection points of two standard concentric

circles on the ζ plane are ( 1, 0) and ( R, 0), whose cross ratio is ± ±

(1 + R)2 η = . (3.25) (1 R)2 −

√η 1 − Hence the width of the cylinder is ln √η+1 . Since conformal transformation preserves the cross ratio, the result is the same if we use the cross ratio of the slightly non-concentric diagrams of Fig. 3.5 and Fig. 3.6.

The calculation in Eq. (3.23) and Eq. (3.24) equivalently use the leading order approximation to η in the respective geometries and thus get the leading order term in the width of the cylinders. The slight deviation

to the precise concentric geometry will only bring in L , τ corrections to η and the width parameter, which will not aﬀect the ﬁdelity and echo exponents.

For the rest of this section, we should denote the width of the cylinder as β. After obtaining the partition

44 function on it, we should set β = 2 ln L or 4 ln τ because the ﬁdelity and Loschmidt echo are both square of

the amplitudes.

The actual boundary conditions on the blue lines, which are the regulators in Fig. 3.5 and Fig. 3.6,

are not important in the leading order. Taking Fig. 3.5 for example, rather than using Eq. (3.21), we can

alternatively view the right panel as the amplitude between the two blue boundary states 1 and 2 | i | i

βH Z = 1 e− ab 2 , (3.26) ab h | | i

where Hab is the Hamiltonian with boundary condition a and b. Since β is taken to be large, we expect the imaginary time evolution (which is horizontal in this case) to project out only the groundstate 0 . Hence | abi the free energy is

βHab F = ln Zab ln 1 0ab 0ab e− 0ab 0ab 2 − ∼ − h | ih | | ih | i (3.27) = βE ln 1 0 ln 0 2 , c − h | abi − h ab| i where Ec is the groundstate/Casimir energy of Hab. We see that diﬀerent choices of the boundary conditions only change the term independent of β. Thus in the leading order we can choose any boundary conditions.

The one we pick is the simplest one: the periodic boundary condition that identiﬁes the two blue lines.

With these simpliﬁcations, we now set up the partition function calculation of the general process

S (θ ) S (θ ). We deﬁne a set of bosonic operators related to the ai s in Eq. (3.6) through a 1 → b 2 n

i i i an i a n b = (b )† = − n √n n √n (3.28) i i i a¯n i a¯ n ¯b = (¯b )† = − n √n n √n

for n > 0, i = 1, 2, and group them compactly with the vector notation

i i ¯ ¯i ¯i bi = (b1, b2, )>, bi = (b1, b2, )> ··· ··· (3.29) i i i i b† = ((b )†, (b )†, )>, ¯b† = ((¯b )†, (¯b )†, )>. i 1 2 ··· i 1 2 ···

The boundary state in Eq. (3.9) is then

¯† b1 b†b† exp ( 1 2)Ra(θ)   0 , (3.30) ¯b† | i n 2 o    

45 where Ra(θ) = Sa I. Using even a lazier notation b = (b1, b2)>, we have − ⊗

a = exp b†R (θ)¯b† 0 . (3.31) | i a | i n o

The matrix notation here should be understood as a bilinear expression. For example, b†Rb¯† actually means ¯ ij bi†Rijbj† where the dagger does not transpose the vector.

P The Hamiltonian of the folding picture has the mode expansion in terms of the bn (with periodic boundary conditions) 2π 4π 4π i i H = (L + L¯ ) = L = n(b )†b β 0 0 β 0 β n n n>0 iX=1,2 (3.32) 1 b1 1 = (b†b†)(I2 M) = b†(I2 M)b, π 1 2 ⊗   π ⊗ b2     where L0 + L¯0 are the dilation operator in CFT and we have used the condition L0 = L¯0 when restricted to the space of the boundary states. The inﬁnite dimensional matrix M is

4π2 M = diag(1, 2, ). (3.33) β ···

The partition function in Eq. (3.21) becomes

πH ¯ ¯ Zab = b e− a = 0 exp bRb(θ)b exp b†(I2 M)b exp b†Ra(θ)b† 0 . (3.34) h | | i h | − ⊗ | i n o n o n o In App. B.1, we obtained the leading order term in the free energy associated with Eq. (3.34). This expression is also obtained by an alternative Casimir energy calculation in App. B.2 for one set of the boundary conditions. A na¨ıve application of the result however will lead to an apparent contradiction.

One notable example is that when a = b = P, the free energy given by App. B.1 is 1 β, which should − 12 actually be zero because this is the (regularized) free energy on a plane without any interface. Physically this corresponds to the situation that the boundary condition does not change after joining the two chains.

Hence the Loschmidt echo will stay at 1 and the free energy is 0. This motivates a shift to the free energy

1 = ln Z (β) + β, (3.35) F − ab 12

1 where 12 β is the value of ln Zab(β) when a = b = P. A more careful inspection in App. B.3 shows the origin of the shift: part of it comes from the outer semi-circles in the middle panel of Fig. 3.5 and Fig. 3.6, and

46 another part comes from the non-homogeneous term in the conformal transformation of the stress tensor from annulus to cylinder.

After incorporating this shift, for the process (c is assumed to be the same as a)

S (θ ) S (θ ), (3.36) i 1 → j 2 the free energy is 1 2 ( x x )β i = j θ θ (β) = 2 | | − x = 2 − 1 . (3.37) F  1 π  β i = j, 16 6  We can then set β = 2 ln L and 4 ln t(after analytic continuing to real time) to get the ﬁdelity and echo exponent.

As analyzed in Sec. 3.3, S2(θ) interpolates between DD and NN, and S1(θ) interpolates between DN and ND. In the region accessible to the numerical calculation in the lattice model, we choose the process

DD λ to verify → 1 ln L fidelity = 8 . (3.38) F  1  ln t echo  4  The same results have already been obtained forλ = P[73, 74, 69, 81, 82]. Another process DN λ is used → to verify (x x2) ln L fidelity = − , (3.39) F   2(x x2) ln t Loschmidt echo  − θ  where λ = tan θ and x = π . We also use a more artificial process P λ to check the shift of the curve → 1 1 x (x )2 ln L fidelity | − 4| − − 4 =  . (3.40) F 1 1 2  2 x (x ) ln t Loschmidt echo | − 4| − − 4  3.3.4 Numerical Results and Comparison

We use the lattice model introduced in Sec. 3.2.2 to check the analytic results. Our numerical calculations are based on a boson Bogoliubov transformation and the explicit form of the groundstates. The readers are referred to App. B.7 for the technical details. In all the figures, we present the coefficients of the logarithmic terms lnFL and lnFt and call them fidelity and echo exponents respectively. We first consider the process DD λ and show its Loschmidt echo of system size 30000 sites in Fig. 3.7. →

47 0.30 analytical numerical 0.25

0.20 100

0.15

Echo Exponent 0.10 0.25 t− Loschmidt Echo 0.05 1 10− 2 1 0 t 1 2 3 10− 10− 10 10 10 10 0.00 0.0 0.1 0.2 0.3 0.4 0.5 θ π

Figure 3.7: The Loschmidt echo decay exponent of the process in DD λ , with gluing condition S1(θ). We → 8 work with the total system size N = 30000 sites, and parameters m = 10− , k = 1. The lattice constant is set to unity. The blue dots representing the numerical results lie on the red analytic line. As predicted, the echo exponents are all equal for different values of θ. Inset: An example of Loschmidt echo with θ = 0.02π 0.25 shown in log-log scale. The dashline denotes the expected power law of t− . Finite size effect does not emerge before t = 103, which sets the right boundary of the range we fit. See main text for the curve fitting method.

The inset is a typical Loschmidt echo diagram, whose linearly decreasing behavior in the log-log scale

0.25 indicates the expected power law decay. We also provide the analytic prediction (t) t− (cf. Eq. (3.38) L ∼ as contrast). The exponent (negative of the slope of the line in the log-log plot) is calculated by ﬁtting such

diagrams for θ = 0.01nπ, n = 1, ..., 50. The ﬁtting is performed before the ﬁnite size revival surges and error

is estimated by assuming independent and identical Gaussian distribution for each point. We see that the

1 exponents all match with the 4 theoretical line within error. We also calculate the companion process NN λ and obtain identical exponents as in Fig. 3.7. We → 8 avoid the technical subtlety of the zero mode by adding a small mass regulator m = 10− . While the short time decay pattern is diﬀerent from the DD case, the long-time behavior and exponents remain the same for both echo and ﬁdelity. We therefore do not present the result here.

Next, we analyze the more interesting θ dependent process DN λ in which the boundary condition → after joining is determined by S1(θ). We work with a system containing 35000 sites. A direct calculation with the mass regulator does not perform very well in the small θ regime: the exponent is slightly larger than the theoretical prediction. We therefore turn to another regulator that shifts the far end boundary

48 0.6 0.30 (a) 2 (b) 2 analytical 2 θ θ analytical θ θ π − π π − π 0.5 0.25 numerical numerical

0.4 0.20

100 1 0.3 0.15 10− Echo

Echo Exponent 0.2 0.10 Fidelity Fidelity Exponent 3 3 10− 10− 0.1 101 102 103 0.05 102 103 104 t L 0.0 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 θ θ π π

Figure 3.8: The slope of the free energy for (a) the Loschmidt echo and (b) the bipartite fidelity of the process DN λ. The total system size is N = 35000 sites with the same parameters as in Fig. 3.7. The numerical value→ of exponents follow a quadratic relation as predicted. There is still visible deviation from the analytic results in (a) due to the subtlety of zero mode, see the discussion in the main text. Inset in (a): From the top to bottom, we show the power law decay of the Loschmidt echo with θ = 0.02π, 0.12π and 0.24π. Finite size effect does not emerge before t = 104. We use the same curve fitting method as described in Fig. 3.7.

condition DN to S1(δθ) and consider the following process (see the full notation in Eq. (3.18))

S (0) + S (δθ) S (θ) + S (δθ). (3.41) 1 1 → 1 1

Since DN = S1(0), taking smaller and smaller δθ should correspond to the original process. This “shift” regulator works very well for the ﬁdelity calculation, where δθ = 0.001π, while moderately good for the

Loschmidt echo, where δθ = 0.003π, see Fig. 3.8. The inset shows the θ-dependence of the power law decay, and the corresponding exponents follow the quadratic relation as predicted in Eq. (3.39).

We ﬁnally consider the process P λ in Fig. 3.9. Since P = S ( π ), the zero mode now occurs at θ = π . → 1 4 4 We therefore apply the shift regulator there

π π π S + S + δθ S (θ) + S + δθ , (3.42) 2 4 2 4 → 2 2 4

π where δθ = 0.003π. The θ dependent exponents are now symmetric about θ = 4 and quadratic on each side, in accordance with Eq. (3.40).

Finally, we also provide the data for the process

DN + P λ + P, (3.43) →

49 0.6 0.30 (a) 2 (b) 2 analytical 2 θ 1 θ 1 analytical θ 1 θ 1 π − 4 − π − 4 π − 4 − π − 4 0.5 0.25 numerical numerical 1 10− 1 0.4 0.20 10− Echo

0.3 0.15 Fidelity 3 2 10− 10− 101 102 103 102 103 104

Echo Exponent 0.2 0.10

t Fidelity Exponent L

0.1 0.05

0.0 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 θ θ π π

Figure 3.9: The decay exponent of (a) the Loschmidt echo and (b) the bipartite fidelity of the process P λ. The parameters are the same as those in Fig. 3.8. The plot is symmetric with respect to θ = 0.25π → π as predicted. The deviation around 4 in (a) is small but visible, see the discussion in the main text. Inset in (a): From top to bottom, we show the power law decay of the Loschmidt echo with θ = 0.04π, 0.08π and 0.18π. Finite size effect does not emerge before t = 104. We use the same curve fitting method as described for Fig. 3.7.

to evaluate the inﬂuence of the boundary condition c. It does inﬂuence the scaling dimension, which is not

captured by our analytic calculation. We note that the exponent still follows the quadratic relation with a

1 deﬁcit of 8 at large value of θ. The deﬁcit approaches zero at θ = 0 where the setups are the same before and after the quench.

3.4 Discussion

In the computations we have done, the results of the fidelity can be converted to that of the Loschmidt echo by the replacement recipe ln L 2 ln τ. The numerical factor of 2 comes from the fact that the Loschmidt → echo has two slit tips. Other than that, we see that they probe the same finite size effect of the free energy associated with the new interfaces. The computation of the (simpler) fidelity is diagnostic and so in the following we will mainly discuss the echo properties.

In Sec. 3.3.3, we have presented the analytic results for the general process S (θ ) S (θ ) (assuming i 1 → j 2 the far end boundary condition c is the same as prior-quench condition a).

We ﬁnd that if the conformal interfaces are of diﬀerent types, i.e. i = j, the (long time) free energy is 6 1 always 4 ln t, regardless of the theta angles. The two types of conformal interface do not talk to each other

because they are imposed on diﬀerent ﬁelds. If we treat S1 as a combination of the Dirichlet and Neumann

boundary conditions on the rotated φ ﬁelds as in Eq. (3.8), then S2 imposes one of them on the dual ﬁeld of φ. In the derivation of the M matrix, these two correspond to the parts of the Lorentz group that can

50 0.6 2 analytical θ θ π − π 0.5 2 θ θ 1 π − π − 8 0.4 numerical

0.3

Echo Exponent 0.2

0.1

0.0 0.0 0.1 0.2 0.3 0.4 0.5 θ π

Figure 3.10: The decay exponent of the Loschmidt echo for the process in Eq. (3.43). The boundary condition c, which is now different from a, changes the scaling completely from the analytic quadratic curve. 1 π The deficit to the analytic value is roughly 8 for θ > 4 , It becomes smaller and approaches zero for small θ. not be connected even by taking singular values of λ. It is then reasonable to find a universal echo between them. The special value of DD P also agrees with the existing general CFT result of the completely → transparent interface[73, 74, 69, 81, 82].

For the more interesting case where the boundary conditions are of the same type, we have veriﬁed the quadratic angle dependence numerically for DN λ . We can ﬁrst understand the values of several special → points on this curve.

• θ = 0: This is where the boundary condition does not change before and after the quench, so the Loschmidt echo stays at 1 and hence the exponent is 0.

• θ = π : This is the process DN ND. The chain is still disconnected after the change of the boundary 2 → conditions. We can thus view the problem as changing the boundary conditions for two independent

chains in the left panel of Fig. 3.4, one from D to N and the other from N to D. The Loschmidt echo

can then be viewed as the product of the boundary two point correlation functions of the associated

1 bcc operators φDN and φND, whose dimensions are both ∆ = 16 . From this,

1 we get the exponent to be 2 , which agrees with Fig. 3.8.

51 • θ = π : This is the process DN P. The exponent 3 agrees with Ref. [82, 73], where the diﬀerence 4 → 8 with the exponent 1 of DD P is interpreted as twice the dimension of the bcc operator φ (∆ = 1 ) 4 → DN 16 that transforms D to N.

In general, the result gives the full spectrum of operator dimensions in the DN λ transition. In the bosonic → CFT we consider, the primary ﬁelds are the vertex operators exp(νφ). Depending on the convention, its

ν2 dimension is a numerical constant times 8π . So if ν depends linearly on θ (or x), then we will end up with a quadratic relation whose expression can already be ﬁxed by the three special points above. In a rational

CFT theory, the number of primary ﬁelds is ﬁnite. It requires further exploration to identify these bcc

operators with the existing primary ﬁelds and their physical signiﬁcance.

On the other hand, in our lattice boson model, θ parameterizes the bond interaction between the boundary

sites of the chains. Consequently it characterizes the strength of the local perturbation to the Hamiltonian:

smaller θ means smaller change of the bond interaction matrix Σ thus a smaller perturbation and vise versa.

Therefore, we expect that a larger perturbation will result in a faster decay of the Loschmidt echo, which is

reﬂected by the monotonically increasing decay exponent (the absolute value of the exponent) in the ﬁgures.

Our numerical study also shows that the far end boundary condition c, which in the large system size

limit should not impact the system, does change the scaling dimension in a way that is not captured by our analytic computation. The reason is that the boundary condition on the far end may introduce additional bcc operators and thus change the free energy. It would be interesting to have a CFT calculation that reproduces the better numerical result in Fig. 3.10 for the process in Eq. (3.43).

This set of the boundary conditions can be realized by connecting two compact bosons. There are already numerous theoretical and experimental works on the boundary conditions of a Luttinger liquid[84, 85, 86,

57, 87], which is the universal compact boson theory of the (bosonized) one-dimensional electron gas[88]. For example, gate voltage [87] may be used to twist the left and right modes of the boson to create a boundary condition interpolating between the normal open and ﬁxed boundary conditions. The interface studied in this chapter is a generalization which (in the folding picture) twists the two independent bosons (two left modes plus two right modes) on their connecting ends. An X-ray edge singularity experiment in a quantum wire system, which uses ions to switch on and oﬀ the boson interfaces should be plausible to detect the exponents found in this chapter.

52 3.5 Conclusion

In this chapter, we analyze a class of boson conformal interfaces by computing the Loschmidt echo and the

bipartite ﬁdelity.

We begin by classifying the boundary states by two types of S matrices S1(θ) and S2(θ), where the parameter θ – the scattering angle – is determined by the transmission coeﬃcient of the interface. The conventional ‘DD’, ‘NN’ boundary conditions are among the special choices of θ in S1, and ‘DN’ and P are among the special choices of S2. Generic value of θ then interpolates between those conventional boundary conditions. A harmonic chain model allows us to realize part of these partially-transmitive boundary conditions in a concrete lattice setting.

The dynamical behavior of the Loschmidt echo reﬂects the change of the conformal interfaces during the process described in Eq. (3.36). Its power law decay exponent is related to the scaling dimension of the bcc

1 operator that mediates the interfaces. Analytic computation shows that the exponent is always 4 when the change of boundary conditions is made between diﬀerent types of S matrices (i = j), regardless of the choice 6 of θ. On the other hand, the exponent depends on the diﬀerence of angles θ θ as a quadratic relation 1 − 2 when the change is made between the same type of S matrices (i = j).

These two features are tested in three typical processes in the numerical calculation of the harmonic chains. After using suitable regulators for the zero-mode problem, the numerical results agree with the analytic calculation within error. Although tangential to the non-equilibrium dynamics, the ﬁdelity calculation is used as a diagnostic tool and shows better agreement of the exponent, providing more conﬁdence about our analytic results.

We proposed that the Loschmidt echo exponent in principle should be detectable in an X-ray edge singularity type experiment on the quantum wire systems.

53 Chapter 4

Operator Entanglement Entropy of the Time Evolution Operator

4.1 Introduction

Chaotic behavior is usually associated with a rapid loss of information about the initial state of a system.

In quantum systems, this can be quantiﬁed by studying the time dependence of measures for quantum

information, most notably for the entanglement of two subsystems. Typically, chaotic systems will quickly

entangle the subsystems over time, even if they are initially in a product state and the spread of entanglement

is usually faster than the transport of particles. The notion of quantum chaos is now usually connected to an

eﬀective random matrix theory, which is argued to be responsible for the mechanism of thermalization[89,

90, 91].

The dynamical process of thermalization can be studied by a quench, where the initial state is prepared

for example as the groundstate of a local Hamiltonian H0 and the Hamiltonian is suddenly changed to another Hamiltonian H of interest, governing the time evolution of the wave function. Thermalization can then be monitored by various quantities, among which the entanglement entropy (EE) provides a particularly appealing measure, since it encodes the scrambling of information about the initial state. In generic quantum systems, it grows very fast (a power law[8], except for the logarithmic growth in many-body localized (MBL) systems [92, 93, 94, 95,8]) until it saturates to a large value which scales as the volume of the system and is determined by the initial state (which itself is usually constructed to have low entanglement).

While this scenario is very well studied, it seems clear that the scrambling of information about the initial state is not a property of the wave function, but rather that of the Hamiltonian. This is particularly plausible when thinking of two extreme cases, a generic system with diﬀusive transport, which exhibits a ballistic growth of the quench EE[96,8], compared to an MBL system, where the growth of the quench

EE is logarithmic in time and thus very slow, while using the same initial product state in both cases.A multitude of previous works has established the diﬀerences of these two classes of systems regarding aspects of the eigenvalues and eigenvectors of the Hamiltonian, exhibiting e.g. volume- vs area-law entanglement entropy[97, 98, 99, 100, 101, 102, 103] and the validity or violation[104, 102, 105] of the eigenstate thermal-

54 ization hypothesis[106, 107, 108], all revealing properties of the Hamiltonian.

Motivated by the success of the study of the quench EE, in the chapter we propose to study the operator

entanglement entropy (opEE)[109, 110, 111, 112, 113] of the time evolution operator, which is one of those state independent measures[114, 115] for a quantum operation (in some references[116, 113] termed as

Shannon entropy of the reshuﬄed matrix). It is probably simplest to be described in the matrix product operator (MPO) language[117]. As shown in Fig. 4.1, an operator can be viewed as a matrix product state

σ1 σ2 σ3 σ4 σ5 σ6

A B

σ1 σ2 σ3 σ4 σ5 σ6

σ10 σ20 σ30 σ40 σ50 σ60 A B

Figure 4.1: Matrix product state (MPS) and matrix product operator (MPO). Upper panel: diagrammatic representation of matrix product state. Each tip of the vertical bond is a physical index; horizontal bonds contract auxillary matrix indices. After performing a left and right canonicalization up to one bond, the chosen bond will contain all the Schmidt eigenvalues that determine the entanglement entropy. Bottom panel: the matrix product operator can be viewed as a matrix product state with two copies of physical indices on each site, then its entanglement entropy can be similarly deﬁned and calculated by a matrix product algorithm.

with two copies of physical indices. Then its entanglement entropy can be similarly deﬁned as for the state.

Since the time evolution operator of a local Hamiltonian clearly contains all the information for all possible

initial states and correlates distant parts of the system increasingly with time, we expect that its opEE will

grow with time and possibly saturate close to its maximal value.

We note that the operator entanglement entropy is the relevant quantity for the eﬃciency of encoding

operators as an MPO[93, 111], since it governs the required bond dimension. In the context of MBL, it was

shown that the Hamiltonian is diagonalized by a unitary operator that can be represented by an MPO with

ﬁnite bond dimension and therefore low opEE[118, 119], compared to the case of a chaotic system, where

we expect the opEE of the diagonalizing operator to be a volume law.

In this chapter, we present the concept of the opEE of the time evolution operator and demonstrate

that it grows as a power law in time for generic quantum systems up to a saturation value which scales

as the volume of the system. In Sec. 4.2, we deﬁne the opEE of time evolution operator. In Sec. 4.3,

we map the opEE to the state EE in a quench problem as a way to understand the general growth and

55 saturation behaviors. Then in Sec. 4.4 we introduce the spin chain models and discuss in detail the growth and saturation of their opEE in Sec. 4.6 and 4.5. Finally we conclude in Sec. 4.7. App. C.1 introduces the channel-state duality to understand opEE. App. C.2 contains a technical calculation of the average opEE of random unitary operators. App. C.3 contains the numerical technique we used for a conserved sector of the

Heisenberg spin chain.

4.2 Operator Entanglement Entropy (opEE) of the Time

Evolution Operator

4.2.1 Deﬁnition

We begin by reviewing the deﬁnition of entanglement entropy for a pure state. Then by assigning a Hilbert space structure for linear operators, we show that the time evolution operator is a normalized “wave function” in the operator space and therefore the entanglement entropy for it can be naturally deﬁned.

Wave function entanglement entropy

For a real space bipartition into subsystems A and B, a pure state ψ in the Hilbert space can be | i H represented in a basis which is the tensor product of the orthonormal bases of the two subsystems i {| iA} and j : {| iB} ψ = ψ i j , (4.1) | i ij| iA ⊗ | iB ij X where the coeﬃcients are given by the inner product

ψ = i j ψ . (4.2) ij Ah | ⊗ Bh | | i

The reduced density matrix of subsystem A is then obtained by performing the partial trace of the pure density matrix ψ ψ over the subsystem B: | ih |

A ρ = Tr ψ ψ ρ = ψ ψ∗ (4.3) A B| ih | ⇒ ij ik jk Xk and the von Neumann entanglement entropy is the Shannon entropy of its eigenvalues λn:

S = Tr(ρ ln ρ ) = λ ln λ . (4.4) A − A A − n n n X

56 Operator entanglement entropy

The concept of the entanglement entropy introduced above has been generalized to the space of operators[110],

which is also a Hilbert space O, ( , ) with the inner product ( , ): O O C on the space of linear { · · } · · × → operators O on . The inner product for two operators Oˆ , Oˆ O is deﬁned by H 1 2 ∈

1 (Oˆ1, Oˆ2) = Tr(Oˆ†Oˆ2). (4.5) dim(O) 1

Here dim(O) = dim( )2 is the dimension of thep operator space. We can now construct two complete H basis sets Aˆ and Bˆ which are orthonormal with respect to the inner product (4.5) and consist only { i} { i} of operators with a support on the subsystems A and B respectively. The two bases span operator spaces

O = span( Aˆ ) on the subsystem A (and B respectively) and their tensor product is the full operator A { i} space O = O O . A ⊗ B For any linear operator Oˆ O, we then have a unique decomposition ∈

Oˆ = O Aˆ Bˆ (4.6) ij i ⊗ j ij X with coeﬃcients Oij C obtained by the inner product ∈

O = (Aˆ Bˆ , Oˆ). (4.7) ij i ⊗ j

In particular, we consider the unitary evolution operator

R t i H(t0)dt0 Uˆ(t) = e− 0 (4.8) T given in general by a time ordered exponential. It propagates the wave function from time zero to time t and satisﬁes the unitary condition at all times t

1 Uˆ(t), Uˆ(t) = Tr Uˆ(t)†Uˆ(t) = 1. (4.9) dim( ) H h i

As a result, it is a normalized element of the operator space O in the same way as a normalized wave function

in the Hilbert space . H With these ingredients, we can deﬁne the operator entanglement entropy (opEE) as the Shannon entropy

57 of the eigenvalues of the reduced operator density matrix

S = Tr(ρA ln ρA ), (4.10) − op op

where the operator reduced density matrix in this basis is

A (ρop)ij = Uik(U †)kj, (4.11) Xk

with U = (Aˆ Bˆ , Uˆ(t)). ij i ⊗ j When t = 0, the evolution operator is the identity operator and hence has zero initial opEE. As t increases,

the operator becomes more and more complicated and we expect the opEE to reﬂect the complexity of the

time evolution. To ease the notation, we will drop the hat if its operator nature is clear in the context, but

will reserve it in ﬁgures to make the diﬀerence to the usual “state” EE explicit.

4.3 General Behavior of opEE

4.3.1 Mapping to Quench

It is useful to map the opEE of the time evolution operator to a quench problem in a larger Hilbert space,

since this allows us to connect to known features of the wave function entanglement entropy. In App. C.1,

we introduce one possible mapping via the channel-state duality, but this is not the only possibility (for

example see [120] for one using swap operation and [110] for another map in Majorana representation, etc.).

Let us introduce below a diﬀerent mapping preserving locality, which makes it easier to identify the general

features of the opEE of the time evolution operator as a function of time in ﬁnite systems. Speciﬁcally,

iHt for an evolution operator (of a time independent Hamiltonian) U(t) = e− , our goal is to construct a

iHt corresponding H and state ψ , such that state EE of e− ψ is the same as opEE of U(t) under the same | i | i real space partition:

iHt iHt U(t) = e− e− ψ . (4.12) ↔ | i

We will see in the next two subsections that in the MBL phase, a simple choice is

H = H, ψ = L , (4.13) | i ⊗i=1|↑i

58 where L is the number of sites. Whereas for a generic Hamiltonian, ψ can be taken as | i

ψ = 2L , (4.14) | i ⊗i=1|↑i

which is in a two-copy Hilbert space that can represent all possible operators in O. H in this case H ⊗ H will be an operator acting on , which we will construct explicitly below. H ⊗ H

MBL Hamiltonian

In the MBL phase the Hamiltonian is eﬀectively given by multi-spin interaction terms between the “l-bits”

x σi [121, 95, 122, 123, 124, 125, 126]

H = hi σx + J 0 σxσx + J 0 σxσxσx + . (4.15) MBL 0 i ij i j ijk i j k ··· i ij X X Xijk

The “l-bits” are local integrals of motion, and the exponentially (with distance) decaying interactions between

them are responsible for the slow entanglement dynamics in MBL systems[92, 93, 95, 94,8, 127]. For this

particular type of Hamiltonian, the time evolution operator U(t) will only consist of products of the identity

operator I σ0 and σx. In contrast, the local basis of the complete operator space O consists of the 4 ≡ x y z µ elements Ii, σi , σi and σi on site i of the system (compactly denoted as σi with µ = 0, 1, 2, 3 for later convenience). This means that the MBL time evolution operator is only contained in a subspace of dimension

2L of the total operator Hilbert space O which has the dimension dim (O) = 4L. This allows us to map the evolution operator to a state Hilbert space of dimension 2L without doubling the number of degrees of freedom.

Note that the states = I and = σx are orthonormal in the single-site Hilbert space and we |↑i |↑i |↓i | ↑i can therefore map I I and σx σx . In the multiple-site situation, the basis which only consists of → |↑i → |↓i products of I σ0 and σx ≡ σµ1 σµ2 σµL µ = 0 or 1 (4.16) 1 1 ··· L i can be mapped to orthonormal basis in as H

σµ1 σµ2 σµL . (4.17) 1 1 ··· L |↑ · · · ↑i

Then the decomposition of U(t) ... in the state basis is identical to the decomposition of U(t) in | ↑ ↑i the operator basis and the state EE of the wave function after a global quench from the state given |↑ · · · ↑i by ψ(t) = U(t) is identical to the opEE. Therefore in this case the bar transformation is the trivial | i |↑ · · · ↑i 59 identity map and ψ = L . | i ⊗i=1|↑i Applying well known results on the logarithmic EE growth after a quench from a product state in MBL

systems[92, 93, 94, 95,8, 127], this mapping immediately implies a logarithmic long time growth and an

extensive submaximal saturation value of the opEE in MBL systems and therefore gives an initial state

independent description of the information propagation in MBL.

Generic Hamiltonian

A completely generic (and possibly non-local) Hamiltonian can be composed of all possible spin interactions

3 µ1 µ1 µ1 H = Jµ1µ2 µL σ1 σ2 σ , (4.18) ··· ··· L µ ,µ , ,µ =0 1 2X··· L

0 where σ I is understood as identity operator and Jµ µ µ are complex interaction coeﬃcients (this is ≡ 1 2··· L in fact an operator basis decomposition of the Hamiltonian). It occupies the full operator Hilbert space O,

and so by dimensional counting, it is only possible to map the operator to a double-site state Hilbert space

. In order to do so, we equip each site with an auxiliary site and upgrade the Hamiltonian H to H × H

3 µ1 µ1 µ1 H = Jµ1µ2 µL σ1 σ2 σ , (4.19) ··· ··· L µ ,µ , ,µ =0 1 2X··· L

where each σ is acting on both the physical site and the nearby auxiliary site. For one dimensional systems,

it is helpful to think of the new Hamiltonian H as a ladder system (bilayer for 2d), which implies that the locality features of the initial system are preserved. The “bar” transformation is deﬁned by the mapping

I = I I, σx = σx σx, ⊗ ⊗ (4.20) σy = σy I, σz = σz σx, ⊗ ⊗

where we introduce a σx operator on auxiliary degrees of freedom for every operator except for the identity

(producing an identity on the auxiliary site).

The bar transformation is chosen such that

σµ (4.21) { | ↑↑i}

corresponds to an orthonormal basis of the local Hilbert space corresponding to one site and its auxiliary

site. Therefore by the same generalization to multiple sites, the opEE of U(t) will be identical to the wave

60 function entanglement entropy of ψ(t) = U(t) . | i | ↑ · · · ↑i On the other hand, the bar transformation is an operator algebra isomorphism, which means for operators

O ,O O, we have 1 2 ∈ O1O2 = O1O2. (4.22)

For the time evolution operator of a time independent Hamiltonian (or each inﬁnitesimal time evolution in the time ordered product), we have

iHt t n e− = lim (1 iH ) n →∞ − n t t = lim (1 iH )n = lim (1 iH )n (4.23) n n →∞ − n →∞ − n = exp( iHt). −

As a result, the opEE of U(t) is equal to the state EE of exp( iHt) . − | ↑ · · · ↑i As argued above, the barred Hamiltonian will preserve the locality of interaction, (non-)integrability of the model and the spatial disorder distributions across the system. As a result, our knowledge of the global quench EE of those systems can be carried over.

We therefore expect that the opEE of the evolution operator will in general have three domains in its evolution with time. The behavior for times less than propagating over one lattice spacing should not be taken seriously, because of its regulator dependence. In the intermediate region when the EE is propagating through the system, the opEE will increase in a certain manner that is described by some scaling function (e.g. linear growth in CFT[22], power law in thermal phase[96,8], logarithmic growth in

MBL phase[92, 93, 94, 95,8, 127]). In the thermal phase, the opEE will reach a saturation value which is extensive, however for integrable systems this may not be true due to the possible recurrence behavior[128].

4.3.2 The Page Value

Before analyzing the unitary time evolution of a physical Hamiltonian, let us ﬁrst consider the average opEE of a random unitary operator, which will give us a guideline for the saturation behavior of random systems.

The average EE of a random wave function (with a measure that is invariant under unitary transformations, i.e. Haar measure) was derived by Page[129] to be

mn 1 m 1 SPage = − k ! − 2n k=Xn+1 (4.24) m 1 = ln m + ( ), − 2n O mn

61 where m and n are the dimensions of the Hilbert space of the two subsystems and m n. ≤ 1 To ﬁx the notation for clarity, let us consider L sites hosted with j = 2 spins in a chain with a bipartition in a smaller system (A) composed of ` spins (its complement B then has ` = L ` spins), then A B − A

` ` 1 L S [ψ] = ` ln 2 2 A− B − + (2− ). (4.25) Page A − O

For an equal partition (`A = `B)

L 1 L S [ψ] = ln 2 + (2− ). (4.26) Page 2 − 2 O

1 The deﬁcit 2 from the maximal possible EE here suggests a deviation from a maximally entangled state on average.

In App. C.2, we perform an analytic calculation based on an integration technique on the unitary group to show that the average opEE of random gate (unitary operator) is given by the Page value of a doubled system

2` 2` 1 L S [Uˆ] = 2` ln 2 2 A− B − + (4− ). (4.27) Page A − O

1 The deﬁcit for the even partition is again 2 . In the analysis of our numerical results for the opEE of the evolution operator, we will compare the saturation values to the Page value.

4.4 Models

We study various spin models to investigate the behavior of the opEE as a function of time. One of the simplest models is the Ising model in a tilted ﬁeld

z z x x z z H = σi σi+1 + hi σi + hi σi (4.28) i X

x z x z with the Pauli matrix σi and σi on site i. Here, hi (hi ) are the transverse (longitudinal) magnetic fields and we choose a homogeneous configuration without disorder. This model is integrable in the clean case if either hx or hz vanish, and non-integrable if both fields are non-zero with generic parameter choices. We adopt a popular set

hx = 0.9045 hz = 0.8090 (4.29) to compare it with existing literature where it was argued that the system becomes robustly nonintegrable with these parameters[130, 96, 131, 132, 133, 134].

62 10

)] 6 t ( ˆ

U 1

[ random operator (L ln 2 ) 4 − 2 S clean Floquet 2 clean Ising (hz = 0.8090, hx = 0.9045) random Heisenberg (W = 1.0) 0 L =14

6 )] t ( ˆ U

[ 4 S integrable clean Ising (hx = 0, hz = 0.5) 2 integrable clean XXX MBL disordered XXX (W = 10) 0 L =14 0 20 40 60 80 100 120 140 t

Figure 4.2: Operator entanglement entropy S[Uˆ(t)] of the time evolution operator Uˆ(t) for the models introduced in Sec. 4.4 as a function of time t. Top panel: Chaotic models. Bottom panel: Integrable (MBL) models. For all models, the opEE grows fast for short times and for the considered nonintegrable models, 1 the opEE reaches a large value close to that of a random operator (L ln 2 2 ). For the clean integrable cases, the opEE ﬂuctuates at large times due to commensurate periods of integrals− of motion. The MBL model exhibits the slowest growth of the opEE with time. For the Floquet model, the opEE grows nearly linearly and saturates at the limit of a random unitary operator. The results for disordered models are averaged over 100 to 1000 realizations.

We also study the standard model of MBL, the random ﬁeld Heisenberg chain

1 hz H = σxσx + σyσy + σzσz + i σz, (4.30) 4 i i+1 i i+1 i i+1 2 i i X

where hi is the random ﬁeld on site i. We take it to be drawn from a uniform distribution on the interval [ W, W ], where W is the disorder strength. This model has an MBL transition at a disorder strength − of W 3.7 [104, 135] and recent numerical evidence points to slow dynamics and subdiﬀusion at weaker ≈ z disorder [136, 137, 138,8, 139, 91]. It conserves the total magnetization Sz = i Si , and allows therefore to

study only one magnetization sector. We project to the largest sector with SPz = 0, having a Hilbert space

L dimension of L/2 . In the clean case W = 0, it is the integrable XXX chain. While the Ising model in a tilted random ﬁeld does not conserve magnetization, it still conserves energy.

In order to have a completely generic quantum system, one can even break energy conservation by introducing

periodic driving. We study a Floquet system with driving period τ given by the time evolution operator

over one period

i2H τ i2H τ U(τ) = e− xy 2 e− z 2 , (4.31)

63 where i x i y i Hxy = hxσi + hyσi , hx = 0.9045, i X (4.32) z z i z i Hz = σi σi+1 + hzσi , hz = 0.8090. i X i We set τ = 0.8 and make two choices of hy.

i 1. hy = 0. This case has the same time averaged Hamiltonian as the chaotic Ising model evolution[132]. However the system is time-reversal invariant (by shifting half of the period, the unitary matrix is

symmetric) and therefore corresponds to circular orthogonal ensemble (COE).

i y 2. hy = 0.3457. The σ term breaks the time reversal symmetry and thus U(τ) should be in the circular unitary ensemble (CUE).

Both choices lead to essentially identical behavior of the opEE, although in the CUE case, the dynamics

seems to be slightly more chaotic, as we discuss in Fig. 4.8. With this exception, we study the COE model

in the rest of this chapter. We employ open boundary conditions throughout this work.

Fig. 4.2 gives an overview of the results for all these models with system size L = 14. In accordance with

our state-quench mapping, the opEE has a fast growth at short times (except for the MBL case) and then

saturates to a constant value for thermal phase, possibly oscillating in the integrable models. In the next

few sections, we will address in detail the saturation value and the scaling of the growth respectively.

4.5 Saturation Value

Let us ﬁrst focus on the long time behavior of opEE S[Uˆ(t)] for various models. In nearly all systems we

considered, the opEE saturates to a constant value in Fig. 4.2 at suﬃciently long times, and we classify

them as follows: Maximally scrambling behavior is found in the Floquet system, where S[Uˆ(t)] saturates to the Page value corresponding to a random unitary operator sampled from the Haar measure. Chaotic systems with conservation laws (energy, magnetization) also exhibit a large saturation value close to the

Page value but with a small deficit which seems to be independent of system size and more conservation laws seem to lead to a larger deficit. In the MBL system with local integrals of motion the opEE saturates after a very long time at a value much smaller than the Page value, whereas clean integrable models with nonlocal conservation laws show no saturation of the opEE, but rather fluctuate more or less strongly around a value that is smaller than the Page value. From this observation, we speculate that the specific nature of the integrals of motion and presumably their incompatibility with the real space partition causes these

ﬂuctuations.

64 We will devote the rest of this section to the details of the two chaotic classes.

4.5.1 Floquet Spin Model

We study the Floquet spin model (4.31) introduced in Sec. 4.4 as a typical model with no conservation laws.

In previous studies, the average global quench EE of an initial product state after a long time evolution with

the Floquet Hamiltonian was found to be given by the Page value L ln 2 1 [131, 132]. 2 − 2 Fig. 4.2 illustrates that the numerically calculated opEE for the equal bipartition (`A = `B) saturates to L ln 2 1 in the long time limit. In App. C.2, we show that this is the average opEE of a random unitary − 2 operator by partly using Page’s result for a random state[129]. This saturation value is in agreement with

the consensus[140] that the Floquet evolution operator (without time reversal symmetry) is indeed a physical

realization of the circular unitary ensemble.

8 Page L=6 Page L=8 6 Page L=10 ] Page L=12 ˆ U [ Page L=14 S 4 S[Uˆ ( )], L=6 ∞ S[Uˆ ( )], L=8 ∞ S[Uˆ ( )], L=10 2 ∞ S[Uˆ ( )], L=12 ∞ S[Uˆ ( )], L=14 ∞ 0 0 2 4 6 8 10 `

Figure 4.3: Comparison of the saturation value of the opEE of the time evolution operator S[Uˆ( )] (crosses, error bars illustrate the size of ﬂuctuations around the saturation value) of the Floquet model with∞ the result for a random unitary operator (Page value, given by full lines) for all possible smaller subsystem sizes `. The opEE for the Floquet system matches the Page result for all system sizes and partitions perfectly.

In order to conﬁrm this, we calculate the long time opEE S[Uˆ( )] of the Floquet evolution operator for ∞ all possible bipartitions of the system and compare the results in Fig. 4.3 to

2` 2` 1 S = 2` ln 2 2 A− B − , (4.33) Page A −

which is essentially the average opEE of a random unitary operator in the corresponding partition, where `A is the length of subsystem A and ` = (L ` ) correspondingly the length of its complement. The Floquet B − A evolution operator opEE matches to the Page value for all possible partitions even in very small systems.

65 4.5.2 Chaotic systems with conservation laws

The next set of examples we consider in Fig. 4.2 are generically nonintegrable systems with conservation laws,

in particular the random ﬁeld Heisenberg chain at weak disorder (such that it does not exhibit MBL[135])

and the tilted Ising chain. The former conserves energy and total magnetization, while the later was shown

to be generically nonintegrable in Ref. [96] and conserves energy and parity under reﬂection.

We choose to present in detail the results of the random ﬁeld Heisenberg model to show the diﬀerent

behaviors owing to the conservation laws and the locality of interactions.

12 L ln 2 0.8160 W = 1.5 − 0.3292L+ 0.0577 W = 2.0 10 W = 0.5 W = 5.0 W = 1.0 W = 10.0

)] 8 ∞ ( ˆ U [

S 6

2 6 8 10 12 14 16 18 L

Figure 4.4: Saturation value of the disorder averaged opEE for the equal bipartition of the random field Heisenberg model as a function of system size for different disorder strengths. For strong disorder W & 3.7, the system is in the MBL phase and we observe a suppressed but extensive saturation value. At weak disorder, the saturation value scales as L ln 2 but has a constant deficit.

The saturation behaviors for various disorder strengths and diﬀerent system sizes are shown in Fig. 4.4.

1 For weak disorders, the saturation values is still around L ln 2, but the deﬁcit value is larger than 2 . When the disorder strength is so large that the system is in the MBL phase, the saturation value is extensive but is

only a fraction of L ln 2. Note that at weak disorder, there are visible ﬁnite size eﬀects, as it seems that for

larger system sizes, the saturation values for diﬀerent disorder strengths approach each other. We suspect

that for much larger system sizes, the deﬁcit for disorder strengths below the MBL critical disorder becomes

1 in fact equal, but will remain larger than 2 . The MBL behavior is easy to be interpreted from the state-quench mapping discussed in Sec. 4.3.1. In

fact, for systems deep inside MBL phase, the opEE can be directly mapped to the global quench of the same

L Hamiltonian, and so an upper bound for the saturation value is 2 ln 2, which is indeed far less than the value in large chaotic systems.

In the next section, we ascribe the opEE deﬁcit in the thermal phase to the block structure of the

reduced density matrix, which ultimately is a result of the conserved total magnetization. We believe that a

66 similar reasoning can also be applied to other thermal phase models with conservation laws, but an explicit

demonstration is lacking.

4.5.3 Deﬁcit Value of Random Field Heisenberg Model

The thermal phase saturation value of the chaotic models we studied is close to the maximal value, but the

deﬁcit is greater than that of the Floquet systems. Here we present an argument to explicitly show how

the conservation law is responsible for this fact in a ﬁxed magnetization sector (Sz = 0) of the random ﬁeld Heisenberg model.

For simplicity, we will present this argument for the EE of a wave function and explain the generalization to the opEE in the end of this subsection.

The constraint Sz = 0 tells us that the Sz bases for part A and B have to be complementary, i.e. only states with n up spins in A and N n up spins in B can be paired to form the basis of Sz = 0 sector of ↑ − ↑ 2N sites (other combinations will yield a vanishing wave function coeﬃcient).

Thus for any given state ψ with ﬁxed magnetization S = 0, we can do a decomposition | i z

N ij ψ = ψn n , i A N n , j B, (4.34) | i ↑ | ↑ i | − ↑ i n =0 ij X↑ X

ij N where ψn is a block matrix with n rows and columns and ij are the row and column indices. This implies ↑ ↑ N that the reduced density matrix is also block diagonal with size n for each block. The operator version ↑ of this decomposition is just the one with a two-copy block size N N . App. C.3 gives a detailed account n n0 ↑ ↑ of how to utilize this structure in numerical computations.

piρi

ρA =

Figure 4.5: Block diagonal form of the reduced density matrix.

67 (i) Let the eigenvalues of the i-th block be pj (the index i is the same as n ), and the trace of block i be ↑ (i) pi = j pj , then the EE is P S = p(i) ln p(i) = p ln p + p S , (4.35) − j j − i i i i i,j i i X X X where Si is the EE of the i-th block p(i) p(i) S = j ln j . (4.36) i − p p j i i X

If we regard each block rescaled by pi to be a density matrix ρi, the total density matrix is then a classical mixture

ρ = piρi [ρi, ρj] = 0 (4.37) i X and the EE is the sum of the occupation entropy plus the average EE of all the blocks.

In principle, one should average the expression of S over a probability distribution of the occupation

probability pi and entropy distribution of Si. For the sake of obtaining an upper bound, we can avoid this complication by just calculating the maximal value. The optimization problem

max S[ p ] subject to p = 1 (4.38) pi { i} i i X

eSi is easily solved by the Lagrangian multiplier technique and the maximal EE taking place at pi = P Si is i e

Si Smax = ln e . (4.39) i X

If the blocks are independent and random, Si will take the corresponding Page value of the corresponding

N L block size. In fact, for almost all blocks, Si will be bounded by the Page value of block size i (N = 2 ) N 1 1 S < Si = ln + ( ) i > 0. (4.40) i Page i − 2 O N

This is verified numerically in Fig. 4.6 for different blocks of different sizes. There are only 4 exceptional

Si 1 i = 0 blocks, which only give a o(1) correction to the e , becoming an o( N ) correction to the total P

68 block (0,3), dim=56 block (1,2), dim=224 102 block (1,4), dim=560 block (2,3), dim=1568 block (4,4), dim=4900 101 ) i S ( p

100

1 10−

1 2 3 4 5 6 7 8 9

Figure 4.6: Probability distribution of the entropy Si of diﬀerent blocks of the operator reduced density matrix ρ. We take L = 16 disordered Heisenberg model at W = 0.5. The blocks are labelled by the number of up spins on subsystem A in the ﬁrst and second index of ρ and the Page value for each block is indicated by a dashed line, clearly showing that it is an upper bound for the block entropy.

100

1 10−

2 10−

3 10−

i 4

p 10−

5 10−

6 10−

7 10−

8 10− 0 1 2 3 4 5 6 7 8

Figure 4.7: Average weight pi of diﬀerent blocks of the operator reduced density matrix ρ vs. the value of the block entropy Si. We use the same parameters (L = 16, W = 0.5) as in Fig. 4.6.

entropy. Therefore the maximal value of the opEE

Si Smax < ln e Page + o(1) i ! X Si 1 = ln e Page + ( ) (4.41) O N i X 1 1 = N ln 2 + ( ) − 2 O N

is bounded by the Page value. When N is large, we conclude

1 1 S < S N ln 2 + ( ). (4.42) max ≤ − 2 O N

69 N N For the operator version, the only change is the block size which becomes i j , and hence N N 1 1 1 S < ln = 2N ln 2 = L ln 2 . (4.43) op i j − 2 − 2 − 2 ij X

For large blocks, the numerically calculated values for Si concentrate on the average as illustrated in

eSi Fig. 4.6. Similarly, the average values of pi obey the optimal distribution P Si as shown in Fig. 4.7. Hence i e the distribution of pi gives the largest opEE it can support for a given subblock entropy Si. We conclude from our numerical analysis that the total deﬁcit probably stems from the deﬁcit observed in each block.

4.6 Growth

In the previous section, we studied the behavior of the opEE of the evolution operator at very long times in

ﬁnite systems and found that a saturation value is reached. Since it is clear that at the initial time t = 0 the opEE is zero, we will now consider how the saturation value is reached in several example systems.

4.6.1 Growth of S[Uˆ(t)] for the Floquet model

COE CUE 10 0.8840t0.9641 0.8682t0.9967 L =6 L =6 8 L =8 L =8 L =10 L =10 L =12 L =12 6 )]

t L =14 L =14 ( ˆ U [

S 4

0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t t

Figure 4.8: Growth of the operator entanglement entropy in the Floquet model (4.31). The thick black lines i correspond to a power law ﬁt. Left panel: Floquet model with time reversal symmetry (hy = 0), a small deviation from a linear growth is visible in the exponent. Right panel: Floquet model without time reversal i symmetry (hy = 0.3457), the linear growth is almost perfect.

In Fig. 4.8 we show the short time behavior of the opEE (equal bipartition) for the Floquet model (4.31) of different system sizes in both COE and CUE parameter choices. Clearly, the opEE grows very fast at short times and for different system sizes there are almost no visible finite size effects. We determine a fit to a power law growth at short times according to the form S[Uˆ(t)] = atα and obtain an exponent of α = 0.9641

70 for COE system and α = 0.9967 for CUE system. With the available system sizes it is difficult to determine whether the discrepancy in COE system from a perfect linear growth (which is for example observed in the growth of the quench EE in this model, starting from a product state[96]) is a finite size effect or prevails in the thermodynamic limit, but from the robustness of the result in Fig. 4.8, it is likely that the remaining time reversal symmetry leads to the deviation. After this initial almost ballistic growth, the opEE saturates to a value very close to the Page value as discussed in Sec. 4.5.1.

These two results together are consistent with the expectation that the Floquet Hamiltonian (without conservation) can be considered as an almost perfect scrambler.

4.6.2 Growth of S[Uˆ(t)] in the random ﬁeld Heisenberg chain

Let us now consider the growth of the opEE in the random ﬁeld Heisenberg chain (4.30). We have already

seen that the opEE at long times saturates to a value close to the Page limit, oﬀset by a system size

independent deﬁcit, which we attributed to the constraints caused by conservation laws.

Here, we restrict ourselves to the case of weak disorder, where the Heisenberg chain is not fully many-body

localized.

In Fig. 4.9, we show the growth of the disorder averaged opEE for diﬀerent system sizes and disorder

strengths. The saturation values obtained for very long times are displayed by dashed lines. It is obvious that

the opEE of the evolution operator reaches the saturation value at much later times at stronger disorder

(W = 2), compared to weak disorder (W = 1). This is consistent with the numerical evidence for slow

dynamics in this region of the phase diagram, leading to a subballistic growth of the (state) entanglement

entropy after a quench[8], power law information transport as quantiﬁed by the out of time order correlation

function[141] and subdiﬀusive transport[136, 137, 138,8, 139, 105] (see Ref. [91] for a recent review of the

numerical evidence).

To analyze the ﬁnite size scaling of the opEE of the evolution operator, we conjecture that the opEE

will grow as a power law in time up to saturation, as was observed for the EE after a quench[8]. Then,

we can make the scaling assumption that for hydrodynamic times (after an initial transient, but before the

saturation), the opEE grows like S[Uˆ(t)] = βtα and thus we can estimate the time t after which the opEE ∞ saturates:

1 t S( ) α , (4.44) ∞ ∝ ∞

where S is the saturation value S[Uˆ( )]. ∞ ∞ With this natural timescale in the problem, we propose the scaling hypothesis

71 14 14 14 L = 8 L = 14 L = 8 L = 14 L = 8 L = 14 12 L = 10 L = 16 12 L = 10 L = 16 12 L = 10 L = 16 L = 12 L = 12 L = 12 10 10 10 )] )] )] t t t ( 8 ( 8 ( 8 ˆ ˆ ˆ U U U [ [ [ S S S 6 6 6

4 4 4 W =1.0 W =1.5 W =2.0 2 2 2 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 t t t

100 100 100 )] )] )] ∞ ∞ ∞ ( ( ( ˆ ˆ ˆ U U U [ [ [ /S /S /S )] )] )] t t t

( W =1.0 ( W =1.5 ( W =2.0 ˆ ˆ ˆ U U U [ [ [

S α =0.534 0.018 S α =0.483 0.028 S α =0.454 0.041 0.5 ± 0.5 ± 0.5 ± αfit =0.498 0.003 αfit =0.452 0.004 αfit =0.381 0.008 1 ± 1 ± 1 ± 10− 10− 10− 1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4 10− 10 10 10 10 10 10− 10 10 10 10 10 10− 10 10 10 10 10

t/t0.5 t/t0.5 t/t0.5

Figure 4.9: (Top) Growth of the disorder averaged operator entanglement entropy of the evolution operator in the random field Heisenberg model (Sz = 0) at short times for various disorder strengths. (Bottom) Growth of the opEE in units of the saturation value S (dashed horizontal lines) as a function of time in ∞ units of the half saturation time t0.5. We observe a power law growth at weak disorder with an exponent α < 1, which is valid for intermediate times, when t t0.5. At stronger disorder, the exponent decreases and finite size effects are stronger, seemingly leading to≈ a long domain of sup-power law growth of the opEE.

S[Uˆ(t)]/S( ) = f(t/t ). (4.45) ∞ ∞

Numerically, it is diﬃcult to determine the saturation time accurately, as in its proximity the power law

growth seems to be violated. Therefore, we deﬁne instead the time t0.5 at which the opEE reaches half of the saturation value by S[Uˆ( )] S[Uˆ(t )] = ∞ (4.46) 0.5 2

and use t0.5 as the natural timescale. We determine this time by interpolating the time evolution of the

opEE and solving Eq. (4.46) for t0.5 numerically for each system size and disorder strength. The associated error bar of t0.5 is estimated by the error of the opEE divided by the derivative of the opEE with respect to time.

In the right panels of Fig. 4.9, we display the opEE divided by the saturation value S[Uˆ( )] as a function ∞ of time in units of the half saturation time t0.5 for diﬀerent system sizes on a log-log scale. At weak disorder, all curves collapse almost perfectly to one universal curve, displaying a clear power law for t t . For ∼ 0.5

72 stronger disorder, finite size effects become more visible but it seems that the curves still converge to a universal curve for larger system sizes. At a disorder strength of W = 2 (bottom panel), the power law regime is shorter than at weaker disorder and at intermediate times an extended regime of slow growth of the opEE is visible in the curvature of the curve. Currently, it is unclear where this regime comes from, but we may speculate that it is due to the fact that at this disorder strength for finite systems a fraction of the eigenstates of the Hamiltonian are already many-body localized[135]. Although there is no direct connection to the behavior of the eigenstates of the Hamiltonian, to the growth of the opEE of the evolution operator can be influenced by this fact and therefore exhibit slower dynamics. At weaker disorder, the fraction of localized states in the spectrum of the Hamiltonian is much smaller and therefore the effect of slower dynamics can be expected to be less visible, which is the case in our results.

Let us finally address the power law growth of the opEE at short times. In the previous subsection, we have shown that for a Floquet system, the growth is almost linear, however in the random Heisenberg chain slower dynamics can be expected. We use two methods to determine the exponent of the power law growth: First, we fit a power law to the growth of the opEE in time for the largest system size over a time window where it appears to be linear on a log-log scale, yielding an exponent αfit. The fit and the value of the exponent is reported in Fig. 4.9. The second approach relies on the scaling ansatz, since according to the scaling arguments explained above, we expect that

1 t L α , (4.47) 0.5 ∝ and we can use this ansatz to ﬁt a power law to t0.5(L), yielding α0.5. Both exponents agree reasonably well with errorbars. We also show the corresponding power law curves together with the data collapse for comparison.

4.6.3 Growth of S[Uˆ(t)] in the MBL phase

It is known that the entanglement dynamics in MBL systems is much slower than in ergodic systems, in fact after a quench, the (state) entanglement entropy grows logarithmically and saturates to a volume law value with a suppressed prefactor[93, 94,8, 127].

In Fig. 4.10, we show the disorder averaged opEE of the evolution operator for diﬀerent system sizes of the random ﬁeld Heisenberg chain (4.30) at strong disorder W = 10, where the system is surely in the MBL phase. On a logarithmic scale in time, it is visible that the saturation value is approached extremely slowly and our results are consistent with a logarithmic growth of the opEE, although larger system sizes would be required to test this hypothesis thoroughly.

73 6

4 )] t ( 3 ˆ U [ S 2 L = 8 L = 10 MBL, W =10.0 L = 12 1 L = 14 L = 16 0 1 0 1 2 3 4 5 6 7 8 10− 10 10 10 10 10 10 10 10 10 t

Figure 4.10: Growth of the operator entanglement entropy in the MBL phase of the random ﬁeld Heisenberg model (Sz = 0).

4.7 Conclusion

We have deﬁned the opEE of the time evolution operator and analyzed the saturation and growth patterns in various spin systems.

The Floquet system is the most chaotic among the models we studied. It has a linear initial growth of the opEE, saturating at the Page value: the average EE of a random unitary operator. We note that the linear growth is also observed in other Floquet models[142] and quenches under a random unitary gate[143].

It would be interesting to use the hydrodyanmic theory and surface growing model in Ref. [143] to explain the linear growth in our opEE.

We also consider another chaotic system with global conservation laws: the Heisenberg model with disorder ﬁeld. There, we ﬁnd a power law growth with an almost perfect data collapse in the weak disorder regime and a saturation value only less than the Page value by a non-extensive amount. We believe that the conservation law and locality of the interaction is responsible for the slower growth and smaller saturation value (compared to Floquet model). The opEE in the MBL phase exhibits a logarithmic growth in time and saturates to an extensive value which is given by a fraction of the saturation value in chaotic models.

Due to the mapping to a quench problem in Sec. 4.3.1, we understand the behavior of opEE through the knowledge of the wave function EE after a global quench. Yet one advantage of the opEE is its initial state independence. It is therefore useful to characterize the scrambling properties of the time evolution operator itself.

The channel-state duality in App. C.1 gives us a double-system picture that also emerges in the ther- moﬁeld double state in the study of the holography with the presence of the eternal black hole[144]. The opEE is the state EE of the global quenched dual state, and in the gravity dual, the Ryu-Takayanagi

74 surface[18] will probe behind the horizon of the black hole[145]. It would be interesting to reproduce the scaling and saturation in a holographic calculation.

Note: Shortly after the submission of this manuscript, we became aware of a preprint[146] that investigates the opEE mainly from the CFT perspective. Section 4 of Ref. [146] has an overlap with some of our conclusions.

75 Chapter 5

Entanglement of Quantum Quench by Random Unitary Circuits

5.1 Introduction

To understand nonequilibrium dynamics in generic quantum many-body systems, we need models that are analytically tractable but which are not integrable. Randomness is a key tool for constructing such models, even if our aim is ultimately to learn about systems that are not random. This philosophy is familiar from random matrix theory [147, 148, 149]. Random unitary circuits [152, 153, 154, 155, 156, 157, 10, 11, 158] are minimal models for chaotic quantum evolution. They retain two fundamental features of realistic systems, namely unitarity and spatial locality, while dispensing with any other structure: The interactions (between spins or qubits) are taken to be random in both space and time. Randomizing the interactions yields models that are analytically tractable to a large extent despite being nonintegrable. They therefore oﬀer the hope of revealing universal ‘hydrodynamic’ structures that are shared by a broad class of many-body systems.

In this chapter, we will use random tensor network to study the entanglement propagation in quantum chaotic systems. Spatially local random circuits are powerful tools that have led to long-wavelength dynamical equations for entanglement production [157, 169] and also for operator spreading [10, 11], i.e. for the

‘quantum butterfly effect’ [171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184] in spatially local systems. They have also elucidated effects of conserved quantities [185, 186] and quenched disorder

[187] on the spreading of quantum information. Very recently exact results have also been obtained for the dynamics of random Floquet circuits and related models [188, 189, 190]. Here we will be interested in universality associated with the propagation of information through space, so we consider spatially local circuits, but we note that interesting lessons have been learned even from ‘zero-dimensional’ random circuits in which any qubit can couple with any other [191, 192, 193, 194, 195, 196].

So far, exact calculations in random circuits have been restricted either to quantities that can be expressed in terms of a low power of the time evolution operator U(t) (such as the averaged out-of-time-order correlation function) or to simplifying limits (such as the limit q in a circuit with local Hilbert space dimension → ∞ q for the ‘spins’). In this chapter we construct mappings that allow more quantities to be calculated, and

76 t A A

Figure 5.1: The structure of the random circuit. Each two-site random gate is shown as a four-leg block. x > 0 corresponds to region A. Time evolution is going upward. which allow other calculations to be extended away from q = . We focus on the R´enyi entanglement ∞ entropies for a 1D system after a quench. However the mappings we develop can be applied to many other quantities of interest.

An appealing feature of random circuits is that various observables for the real time quantum evolution can be related to the statistical mechanics of eﬀective classical degrees of freedom. In particular the coarse- grained picture for the growth of entanglement of [157, 169] involves a ‘minimal membrane’ in spacetime

(this picture has now been shown to apply in holographic conformal ﬁeld theories [170]). In 1+1D, the membrane is a one-dimensional minimal curve or ‘polymer’ embedded in two-dimensional spacetime. At leading order in time (at late times), the computation of S(t), for a given initial condition, can be related to a classical free energy minimization problem for this polymer. The polymer is characterized by a ‘line tension’ (v) which depends on its speed v, i.e. its slope in spacetime. E In the simplifying limit q for the local Hilbert space dimension q, this polymer can be identiﬁed [157] → ∞ with a coarse-grained ‘minimal cut’ [197, 198, 164, 199] through the unitary circuit generating the dynamics.

S For ﬁnite q, the computation of the averaged purity, e− 2 , leads to a related directed walk problem [10].

But for ﬁnite q the minimal cut formula is no longer accurate, and the calculation of Sn is complicated by the need for a replica limit to handle the logarithm in the deﬁnition of the entropy.1

Here we show explicitly how the minimal curve emerges in 1+1D random circuits by analyzing Sn for n 2. This reveals a mapping to an inﬁnite hierarchy of classical statistical mechanics models involving ≥ directed random walks, extending [10] where the lowest member of this hierarchy was studied. These models have a rich structure. Many properties, both universal and nonuniversal, can be obtained exactly.

Concretely, we take the time evolution operator (quantum circuit) to be a regular array of Haar-random two-site unitaries as shown in Fig. 5.1. This is quantum evolution with no conserved quantities, which equilibrates locally to the inﬁnite temperature state. We obtain the time-dependence of the entropies in

1 S S2 cannot be obtained from the averaged purity as S2 = ln e 2 (the overline is the average over the random circuit). 6 − −

77 systematic expansions in 1/q (accounting for both the mean behaviour and ﬂuctuations) and we show how

the late-time saturation can be understood in the minimal curve picture. We also brieﬂy discuss the operator

entanglement of the time evolution operator itself [114, 200, 110, 146].

In our mappings the minimal membrane arises from domain walls between two kinds of permutations.

These permutations appear in the average over the random unitaries in the circuit: similar degrees of

freedom appear for random tensor networks that are not made of unitaries [20, 201]. Mathematically the

permutations represent diﬀerent patterns of index contractions. For unitary dynamics these permutations

can be understood more physically as distinct ways of pairing spacetime trajectories in the ‘path integral’

for the R´enyi entropy, which involves multiple copies of the system [10]. We expect this idea to be more

generally applicable.

The domain walls between permutations can be viewed as a collection of interacting, directed random

walks, with interactions of several kinds. The entanglement is related to the free energy of these walks (in

the language of the classical problem) and has both energetic and entropic contributions.

S In more detail: For computing e− 2 it is suﬃcient to consider a single walk which represents an ‘elementary’ domain wall (Fig. 5.2, Left). Collections of multiple interacting walks appear for two reasons.

First, if we consider a higher R´enyi entropy, the relevant domain wall is in fact a composite of (n 1) − ‘elementary’ domain walls, i.e. (n 1) walks (see also a similar picture in a Floquet circuit [188]). These − attract each other strongly through a combinatorial mechanism and can form a ‘bound state’ (Fig. 5.2,

Right). In the continuum this bound state forms the minimal membrane. There is also an unbinding phase transition for these walks as a function of their velocity: this unbinding is important in allowing general constraints conjectured in [169], relating entanglement growth to the butterﬂy velocity vB, to be satisﬁed. B A B A

Figure 5.2: In the calculation of S2 for n = 2 (cartoon on Left) each replica contains a single elementary walk (domain wall). For n > 2, e.g. n = 3 (Right), each replica contains multiple walks, which can form a ‘bound state’ with a ﬁnite typical width. To calculate averages involving Sn we must use k replicas (which multiplies the number of elementary walks by k) and take k 0 at the end of the calculation. →

Second, to compute, say, Sn or the ﬂuctuations in Sn, we must employ the replica trick. We consider a k-fold replicated system and take the limit k 0 at the end of the calculation. There are then k sets →

78 of domain walls, one for each replica. Distinct sets interact with a weak interaction that we compute by

expanding in 1/q.

This replica treatment allows us to pin down universal ﬂuctuations in the entanglement that are due to

randomness in the circuit. It was argued that for dynamics that is random in time these ﬂuctuations are

governed by the Kardar–Parisi–Zhang (KPZ) universality class [202, 203], and have a magnitude that grows

in time as t1/3 [157]. (These fluctuations are therefore subleading at large time compared to the leading order deterministic growth.) We confirm these universal properties by an explicit mapping between the dynamics of the Rényi entropies and a problem that is equivalent to KPZ, namely the problem of a directed polymer in a random medium (at finite temperature [204, 205, 203, 202, 206]).

Strikingly, the replica limit needed to handle the logarithm in the deﬁnition of the entanglement entropy is transmuted by this mapping into the replica limit associated with the disorder in the classical polymer problem. As a result, the mapping to the directed polymer in a random medium can be carried through exactly on the lattice when q is large but ﬁnite. This polymer can be coarse-grained to give the exact leading q dependence of the constants in the continuum KPZ equation describing the entanglement growth. At large q there are several early-time crossovers in the entanglement growth. In fact the timescale required to see

KPZ ﬂuctuations is numerically large even at q = 2: we suggest that this is why quantum simulations of this model at short times did not show signatures of KPZ [11], resolving an apparent paradox.

5.2 Overview of Results

Let us ﬁrst discuss the generation of entanglement after a ‘quench’ from an initial product state (we will discuss some other setups later on). For simplicity, take the chain to be inﬁnite. The dynamics is generated by a random circuit. The time evolution operator U(t) is made up t ‘layers’ of two-site random unitaries,

each independently Haar-random, with unitaries applied to even bonds in even layers and odd bonds in odd

layers (Fig. 5.1).

Let A denote the half-chain with x > 0, and let Sn(t) be the nth R´enyi entropy of this region at time t. We deﬁne

n (n 1)S Z Tr ρ = e− − n , (5.1) n ≡ A

where the t dependence is implicit.

The physical quantities of interest to us are averages such as Sn — where the average is over the random

unitaries in the circuit — and also ﬂuctuations around these averages. To obtain Sn we must average the logarithm of Z , since in general exp( αS ) = exp( αS ). For this we will use the replica trick, studying n − n 6 − n

79 the average of the kth power of Zn for an arbitrary integer number of ‘replicas’ k, and then taking the formal limit k 0. → The average entanglement is given by

k 1 ∂Zn Sn = (5.2) −n 1 ∂k − k=0

and higher terms in the expansion about k = 0 yield higher cumulants which quantify the ﬂuctuations in

the entanglement, 2 2 k (n 1) 2 ln Z k = k(n 1) S + − S S + ... (5.3) n − − n 2 n − n We will give a brief overview of the general features of this replica calculation of the entropies in Sec. 5.2.1.

Then in the remainder of this section we summarize our basic results for the entanglement. We divide these

into two classes. First the leading order dynamics of the entanglement entropy at large times (Sec. 5.2.2).

This leading order dynamics is deterministic, despite the randomness in the circuit. Second, subleading

ﬂuctuations arising from randomness in the circuit (Sec. 5.2.3). Although these ﬂuctuations are subleading

at large time, they have interesting universal structure.

Here to clarify the distinction, consider the above example of Sn(t) for an initial product state. When t 1 the leading order behaviour is the deterministic growth at a rate set by an ‘entanglement speed’ v n (which we ﬁnd to be n-dependent). Randomness consists in subleading ﬂuctuations, which obey KPZ scaling

[157], and are on the parametrically smaller scale t1/3.

We may write Sn(t) as S (t) s v t + B t1/3χ(t) . (5.4) n ' eq n n The ﬁrst factor is the equilibrium entropy density seq: since the models we study have no conservation laws,

they equilibrate locally to the inﬁnite temperature state, and seq is set by the local Hilbert space dimension,

seq = ln q. (5.5)

The ﬁrst term inside the brackets in (5.4) is the deterministic leading order growth (the deterministic growth

will have nontrivial time-dependence for example a more general initial state, or for the entanglement of a

1/3 ﬁnite region). The second term includes the KPZ ﬂuctuations of size t . Bn is the nonuniversal constant governing their strength. χ(t) is a random variable whose magnitude is of order 1 at late times, whose

probability distribution is universal and given by the Tracy-Widom distribution F1 [207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220].

80 We will obtain Eq. (5.4) by explicit calculations at large but ﬁnite q (therefore, modulo the fact that the

replica calculation is nonrigorous, we conﬁrm the conjectures of Ref. [157] for the universal properties, also

ﬁxing various nonuniversal constants) and we will discuss various extensions.

5.2.1 General features of the mappings

k We map Zn to an eﬀective classical statistical mechanics problem involving interacting random walks which

has some striking features. This signiﬁcantly extends the mapping of Ref. [10] for the case Z2. We will use an expansion in powers of 1/q to obtain analytical control on the interaction constants in the eﬀective

classical problem, but we expect the resulting universal results to hold for all q, including the minimal value

q = 2. In fact even our large q results for various nonuniversal constants should be reasonably close for small q, because of the numerical smallness of various constants.

k The mapping involves several steps. We ﬁrst map Zn to a partition function for a ‘classical magnet’. The ‘spins’ in this magnet take values in a permutation group, as in work on random tensor networks [20].

The group relevant to us is S for N = n k, as a result of the replica limit. Eventually we must consider N × the limit k 0. → The interactions in the classical magnet are initially rather complicated but permit simpliﬁcations. In

Ref. [10] it was shown that in the special case n = 2, k = 0 the partition function could be radically simpliﬁed by integrating out half of the spins. We extend this idea to general n and k. This allows a much richer set of

configurations, and the Boltzmann weight for a general configuration in the effective classical model remains

complicated. However these Boltzmann weights obey crucial simplifying constraints, due to the unitarity

of the underlying quantum dynamics, which imply that many spin conﬁgurations do not contribute to the

partition function. We exploit these constraints, together with a large q expansion, to reduce the partition

function to one for multiple directed paths with interactions of various kinds.

These paths arise as domain walls in the classical magnet. They may be viewed as living on a rotated

square lattice, and they are directed in the time direction. Each domain wall carries a label, analogous to a

particle type. This label is an element of SN , which in the simplest case (an ‘elementary’ domain wall) is a transposition such as (12). We explain this structure in Sec. 5.3.

k For Z2 , which yields the second Rényi entropy, we obtain a partition function for k directed paths, one for each replica: see Fig. 5.3. There is an effective local attractive interaction between different replicas

4 (diﬀerent paths). This attraction is parametrically small when q is large (of order q− ). The problem of k directed paths or ‘polymers’ with attractive interactions, in the replica limit k 0, is a → well-known one [206, 221, 222, 223, 224]. It is the replica description of a single directed polymer in a random

81 ?

k Figure 5.3: Z2 mapped to k domain walls on tilted square lattice (vertical direction is physical time). The figure has k = 2. Left: Two domain walls going downwards. There are no interaction between them at order 1/q4. Right: The two domain walls have local attractive interaction in the square with star, see detailed analysis in Sec. 5.6 potential [204, 205, 203], a model which can be mapped to KPZ. Therefore this sequence of mappings relates the universal properties of the entanglement to those of the directed polymer in a random potential. At large but finite q it is even possible to make an explicit microscopic correspondence with a specific lattice model for directed polymer in a random medium.

The entanglement S2(t) is the free energy of the polymer: the growth rate of the entanglement has both an ‘energetic’ and an ‘entropic’ contribution. In addition to addressing the universal properties, we calculate some nonuniversal growth rates associated with S2 by applying exact Bethe ansatz results for directed polymers in the continuum [206, 224].

The statistical mechanics problem becomes more intricate when n > 2. Each replica now contributes n 1 ‘polymers’. Within each replica there are interactions which are not small at large q. These interactions − have an appealing combinatorial origin. We give some exact results for n = 3 and a schematic picture for general n. For n > 2, the interactions lead to the formation of a ‘bound state’ of multiple walks, see Fig. 5.2 for a cartoon.

Above we have focused on the mapping for the entanglement of quenched state. However many other quantities can be studied using our lattice magnet mapping. We brieﬂy discuss the operator entanglement entropy [225, 109, 110, 111, 112, 200, 146] of the time-evolution operator (i.e. of the whole unitary random tensor network). This is obtained from the same lattice magnet partition function, just with slightly diﬀerent boundary conditions. This quantity may also be used to obtain the entanglement line tension Sec. 5.5.

5.2.2 Leading scaling at large times

The leading order dynamics of Sn(t) at large t is deterministic, regardless of the value of q — this is why random circuits are reasonable minimal models for entanglement dynamics in realistic many-body systems with non-random Hamiltonians. This deterministic dynamics extends beyond the simple linear-in-t

82 growth in Eq. (5.4): for example we also expect universal deterministic scaling forms for the saturation of

the entanglement of a large ﬁnite region [157], and for entanglement growth starting from a state with a

nontrivial entanglement pattern [169].

The entanglement speeds vn may be calculated in a large q expansion for n = 2 and n = 3. For v2 we ﬁnd 1 q2 + 1 1 1 v = ln + + ... 2 ln q 2q ln q 384q8 (5.6) 1 1 1 1 95 1 = 1 + ln 2 + + + . ln q − q2 − 2q4 3q6 − 384q8 O q10 !

For v3 we are only able to go to a lower order:

ln 2 + 3 √2 3√2 1 1 v = 1 + + . (5.7) 3 − 2 ln q 4 q2 ln q O q4 ln q

This shows that the entanglement speed depends in general on n, which resolves a question left open in

Ref. [157]; see also Ref. [188].

The above entanglement speeds may be compared with the growth rates deﬁned by averaging Zn instead

of its logarithm, which we denote vn:

e 1 ln Z s v t. (5.8) − n 1 n ∼ eq n − e vn does not include eﬀects due to interactions between replicas. These interactions are suppressed at large q, so the diﬀerence between v and v is parametrically small at large q, and for v is numerically quite small e 2 even when q = 2. The average of the purity Z gives the ‘purity speed’ v (called v in Ref. [10]) which has e 2 2 P been obtained previously in a variety of ways [226, 157, 10, 11] e

1 q2 + 1 v = ln ln 2 . (5.9) 2 ln q q − e In the random walk picture, Z2 is the partition function for a single random walk, and seqv2 is its free energy per unit ‘length’ in the time direction [10]: in the above expression the ﬁrst term is the energetic e contribution to this free energy and the second term is the entropic one. The diﬀerence between v2 and v2 is e 1 1 v v + ( ) (5.10) 2 − 2 ' 384q8 ln q O q10 ln q

4 at large q, and arises from interactions betweene replicas that are of order q− .

When we consider S3, each replica contributes two walks, and these walks have eﬀective interactions that

83 are diﬀerent from the interactions between replicas. At the leading nontrivial order in 1/q, the interaction

arises for combinatorial reasons. In the simplest case, Z3, the walks are labeled by transpositions in the permutation group S3: either (12), (23), or (13). Further, the labels on the two walks must multiply to give the 3-cycle (123). This leaves three possibilities for how the walks are labeled, corresponding to the three ways to decompose the three-cycle:

(123) = (12) (23) = (23) (13) = (13) (12). (5.11) × × ×

Figure 5.4: Outcome of a splitting event. Left: Two commutative domain walls have two ways to split: exchange or passing through. Right: Two noncommutative domain walls have three ways to split. Red represents (12), blue represents (23), green represents (13).

Each time the two walks meet, the labeling can change, as in the cartoon in Fig. 5.4. For a given

spatiotemporal conﬁguration of the domain walls we must therefore sum over all the consistent labelings.

The resulting factor in the partition sum may be reinterpreted as a local attractive interaction between the

two walks. This causes them to form a bound state: see Fig. 5.2 for a cartoon. (This phenomenon is similar to

one appearing in the replica treatment of directed paths with random-sign weights [221, 227].) The attraction

means that the constant ln 2 appearing in Eq. (5.9) is replaced with the constant (1/2) ln(2 + 3/√2) in

Eq. (5.7). As a result, the growth rate of S3 is not the same as that of S2. This combinatorial factor has also been obtained in the Floquet model of Ref. [188].

In the next order, ( 1 ), there is a weaker interaction arising from the non-commutativity of the con- O q2 stituents of the composite walk (123). Taking it into consideration gives Eq. (5.7).

The entanglement speeds v are in fact special cases of the more general quantities (v) which determine n En the entanglement dynamics for more general initial states, in which the entanglement is diﬀerent across spatial

cuts at diﬀerent positions x [169]. For a detailed discussion of (v) see Ref. [169], where it is argued that E for typical initial states with a given initial entanglement proﬁle S(x, 0), the leading order dynamics is

Sn(x, t) = min (Sn (x vt, 0) + seqt n(v)) . (5.12) v − × E

The quantity (v) has a transparent meaning in the present approach. s (v) is the coarse-grained free En eqE2 energy (per unit length) of a walk when its coarse-grained ‘speed’ is equal to v (in the replica limit). For

84 higher n,(n 1) (v) is the analogous free energy for a ‘bound state’ of n 1 walks. We refer to as − × En − En the slope-dependent line tension.

The line tension of a walk which is vertical in the coarse-grained sense is (0), which is simply v . As En n noted in Ref. [169], the random walk picture gives an explicit form for at large q. We quantify the size of E2 the corrections to this large q result, and we also give explicit formulae for (v). These forms are consistent E3 with the general constraints [169] (v ) = v , 0(v ) = 1. En B B E B We also find an interesting phenomenon for the Rényi entropies Sn with n > 2; there is a nonanalyticity in the line tension (v) at the value v = v . This is associated with an ‘unbinding’ phase transition for the En B (n 1) walks that appear in the calculation of S . This phase transition is crucial in allowing the constraint − n (v ) = v to be satisfied. En B B So far we have discussed the entanglement of an infinite subsystem, which grows indefinitely. We also consider the saturation of the entanglement for a finite subsystem (Sec. 5.8). At asymptotically late times the entanglement saturates to a value given by Page’s formula and its generalizations[129, 228]. We show that the universal constants appearing in this formula have an appealing combinatorial interpretation in terms of domain walls. We also confirm conjectured scaling forms for entanglement saturation [157], and we show that at the moment of saturation there are subleading corrections to these scaling forms with a similar combinatorial origin to the constant in the Page formula.

5.2.3 Randomness and KPZ ﬂuctuations

The statistical mechanics problem becomes more interesting when interactions between replicas are considered. These interactions encode the ﬂuctuations in the entanglement,2 and they also determine subleading

(in q) corrections to vn and other constants. Above we reviewed the basic features of KPZ scaling of fluctuations (Eq. (5.4)) with their characteristic t1/3 growth. It was conjectured, on the basis of analytic results in particular limits3 and numerical results on some more generic circuits, that KPZ scaling should hold in any generic random circuit [157]. However until now there has not been an explicit analytic derivation of KPZ scaling in a generic circuit that does not have the simplifying feature of either q = or Clifford structure. ∞ The need for such a demonstration is pressing in the light of the recent numerics for the second Rényi entropy in regular Haar random circuits. Numerical results for q = 2, for times up to t = 20 in Ref. [11]

showed no obvious sign of ﬂuctuations growing with time, and it was conjectured there that KPZ ﬂuctuations

2 k k In the absence of interactions between replicas Zn would be equal to (Zn) , so that the generating function exp ( k(n 1)Sn) would be trivial and equal to exp k(n 1)Sn . 3Haar-random− − circuits with a random geometrical structure− − at q = , and Cliﬀord circuits at q = 2. ∞

85 were absent. Here we ﬁnd that KPZ ﬂuctuations are indeed present, and the reason for their apparent absence at small times is that they are (numerically) surprisingly small.

We focus on S2, where a quantitative calculation is possible for large but finite q. We argue that a similar logic implies KPZ fluctuations also for the higher Rényi entropies, but with an additional coarse-graining step (we comment briefly on S1). This gives for the first time an analytic demonstration of KPZ scaling in a random circuit that is not Clifford and which has finite (albeit large) q.

For S2, we ﬁnd that the prefactor B2 in Eq. (5.4) governing the strength of ﬂuctuations at asymptotically late times is (at large q) 1 B . (5.13) 2 ' −4 21/3 q8/3 × × We obtain this using the directed polymer mapping together with the fact that at large q the weakness of the interactions between replicas can be used to justify a continuum treatment. In this continuum treatment each polymer is interpreted as the worldline of a boson, so we have a problem of k 0 interacting bosons[206]. → The constants in their Hamiltonian are

q2 + 1 1 k ∂2 1 = k ln δ(x x ), (5.14) H 2q − 2 ∂x2 − 4q4 α − β α=1 α X α<βX where the interaction arises from certain domain wall conﬁgurations that are absent in the q = limit. ∞ We use numerical simulations for small t to check various diagrammatic calculations that go into the directed polymer mapping. While Eq. (5.13) is valid at asymptotically late times, these small t simulations are consistent with the weakness of interactions between replicas being the reason why KPZ growth of

ﬂuctuations cannot be seen, even for q = 2, on timescales accessible using MPS techniques.

So far we have discussed only the Haar random quantum circuit with a fixed regular geometry. Perhaps the simplest modification of this circuit is to draw each local unitary from a modified probability distribution, which returns a Haar random 2-site unitary with probability 1 p, and the 2-site identity with probability − p. That is, we punch a density p of holes in the circuit. In the limit of small 1 p this gives (after an − appropriate rescaling of time) the model in which unitaries are applied in continuous time in a Poissonian fashion [157]. The strength of attraction between replicas — the strength of disorder in the directed polymer language — varies with p. If p is nonzero, there are nontrivial KPZ fluctuations even in the strict q = ∞ limit [157].

86 5.3 Mapping to a ‘Lattice Magnet’

Our starting point is the quantity

k n k Zn = (Tr ρA) , (5.15)

where ρA = ρA(t) is the reduced density matrix for a region A in a chain that is globally in a pure state. Writing the RHS in terms of the circuit, we see that each local unitary U as well as its complex conjugate

U ∗ appear N times, with N nk. (5.16) ≡

Speciﬁcally, each local unitary U gives rise to the tensor product U U ∗ ... U ∗. This tensor product ⊗ ⊗ ⊗ is shown graphically in Fig. 5.6, Left (there is one such ‘block’ for each unitary in the circuit). Next we perform the Haar average over the unitaries to obtain

k n k Zn = (Tr ρA) , (5.17) where each unitary is averaged independently. Taking the replica limit k 0 in this quantity gives averages → of the nth R´enyi entropy, as described in Sec. 5.2. In this section we set up the necessary machinery and in the following sections we use it to calculate the entropies in various regimes.

A standard result gives this single-unitary average in terms of a sum over two elements, σ and τ, of the permutation group on N elements [229, 230]:

1 U U ... U = Wg(τσ− ) ττ σσ . (5.18) ⊗ ∗ ⊗ ⊗ ∗ | ih | σ,τ S X∈ N

Because we have N nk copies of the circuit and N = nk copies of its conjugate, at each physical site we ≡ now have the tensor product of N factors of the physical Hilbert space and N factors of its dual. The state

σσ = σσ is a product of identical states σ and σ on each of the two sites that the unitary acts | i | ii,i+1 | ii | ii+1 on. The state σ is labeled by the permutation σ S . In the natural basis, its components are | i ∈ N

a , a¯ , . . . , a , a¯ σ = δ(a , a ). (5.19) h 1 1 N N | i j σ(j) j Y

Two examples of such states and their inner products are shown in Fig. 5.5.

1 Finally, Eq. (5.18) contains the Weingarten function, Wg(τσ− ). At this point, we only need to know

1 that it is a function of the cycle structure of the permutation τσ− . We reserve App. D.4 to discuss its

87 δ δ δ I i1i1 i2i2 i3i3 ∼ h |

δ δ δ 123 i1i2 i2i3 i3i1 ∼ | i

I 123 h | i

Figure 5.5: Contraction of the permutation states by counting the number of cycles (loops). properties and perturbative expansion in 1/q.

Graphically, Eq. (5.18) can be represented as in Fig. 5.6. We will refer to σ and τ as ‘spins’. Each unitary gives rise to an independent σ spin and an independent τ spin living on the vertices connecting the vertical link.

τ 1 = τ,σ Wg(τσ− ) σ P

Figure 5.6: Graphical representation of the unitary averages in Eq. (5.18). Each blue square with four legs is the two site local unitary gate and each red square is its complex conjugate. The ellipsis represents a total of N = nk copies of each. On the right, the two top legs are τ i τ i+1 and the two bottom legs are σ σ . We associate ‘spins’ σ and τ with the vertices. | i | i ih |i+1h |

k The full expression for Zn is obtained by contracting the tensors (‘blocks’) deﬁned in Eq. (5.18) in accordance with the spatial structure of the circuit. Each non-vertical link connecting two blocks yields a power of q:

1 N τσ− σ τ = q −| |. (5.20) h | i

1 1 The exponent N τσ− is simply the number of cycles in the permutation τσ− , which is at most N. The − | | 1 term τσ− is the distance between σ and τ, which is minimized when σ = τ. It is given by the minimal | | 1 number of transpositions required to construct τσ− .

k After including these inner products, Zn becomes a partition function for the σ and τ degrees of freedom, with one σ and one τ associated with each unitary in the circuit. This structure is shown in Fig. 5.7(left).

At the time 0 (bottom) boundary, we obtain contractions with the initial density matrix. If the initial physical state is taken to be a product state e , then at the time 0 boundary we have N copies of this i | ii state and N copies of its dual, giving at eachQ site e e¯ ... e¯ . This is then contracted with a state σ | ⊗ ⊗ ⊗ ii | i

88 I I I τ˜ τ˜ τ˜ h | h | h | h | h | h |

n Figure 5.7: Left: Tr(ρA) represented as a lattice magnet. The upper boundary is contracted with the boundary state τ˜ = τn,k for region A and I for region B, the bottom boundary is identical to the top for the operator entanglementh | h | and free for the stateh | entanglement. Each 4-leg block is the tensor in Eq. (5.18). Right: The domain wall representation on the triangular lattice after integrating out the τ spins in each center of the down-pointing triangle (green).

associated with one of the unitaries in the lowest layer. This gives

σ e e¯ ... e¯ = e e 2N = 1. (5.21) h | ⊗ ⊗ ⊗ ii |h | ii|

At the ﬁnal time (top) boundary there are contractions which come from the traces in Eq. (5.15). This

gives a weight which depends on the τs for the top row of unitaries. For each link, this is the inner product

between that τ and another permutation which is determined by the structure of the trace. Outside region

A we contract row and column indices of ρ, which corresponds to contracting with the state I . Inside h | region A we ﬁrst take the product of n copies of ρ before taking the trace. This corresponds to contracting

with the state

τ (1, 2, 3, . . . , n)(n + 1,..., 2n) ... , (5.22) h n,k| ≡ h |

k given by a product of n-cycles, one for each power of Zn in Zn.

k Thus we have converted Zn to a partition function for τ and σ spins on the honeycomb lattice. Each non-vertical link has weight speciﬁed by the inner product in Eq. (5.20), and each vertical link carries a

Weingarten function. The boundary conditions are illustrated in Fig. 5.7(left). We will discuss the Boltzmann weight in terms of the domain wall picture in the following subsection.

5.3.1 Domain walls on triangular lattice

Now let us discuss the weight for a given spin conﬁguration. At this point the weight is complicated because the Weingarten function in Eq. (5.18) leads to a profusion of nonzero and also negative weights. (For example, if δ is an elementary transposition of two elements, Wg(δ) is negative, while if δ = I it is positive.) Remarkably, the partition function simpliﬁes if we sum over the τ degrees of freedom [10] associated with

every unitary, giving a partition function for the σs only. Each τ couples to three σs, as can be seen in the

89 green down-pointing triangles in Fig. 5.7, so integrating it out gives a three-spin interaction. These σs form

down-pointing triangles. We denote the weight for this triangle by

σb σc = J(σ , σ ; σ ) (with σ , σ , σ S ). b c a a b c ∈ N (5.23) σa

We will specify J below for the cases of interest. For many values of σ , σ , σ the weight J(σ , σ ; σ ) { a b c} b c a vanishes, and this leads to considerable simpliﬁcations. Formally, we have

1 1 1 2N σ− τ τ − σc J(σb, σc; σa) = Wg(σaτ − )q −| b |−| |. (5.24) τ X This weight deﬁnes a partition function for spins on the vertices of the triangular lattice. At the top boundary

we have triangles whose upper spins are ﬁxed to be τn,k inside region A and I outside region A. The rules to obtain the weight in those slightly slimmer down-pointing triangle is the same. At the lower boundary

the spins are free.

It is easiest to visualize the weights in terms of the domain walls. Each domain wall is itself labeled by

a permutation µ, as in Fig. 5.8. To ﬁx the labeling we must assign a direction to the domain wall, either

upgoing or downgoing. This choice is arbitrary: a downgoing domain wall with label µ is equivalent to an

1 upgoing domain wall with label µ− . In our ﬁgures we take domain walls to be upgoing. Our convention is that if an upgoing domain wall labeled by µ has a domain of type σ to the left, then the domain to the right

is σµ, see Fig. 5.8.

(12)(34)

I (12)(34) σ µ σµ

(12) (34) (12)

Figure 5.8: Deﬁnition of domain walls. Left: Domain wall labeling convention. Right: σ spins on the triangular lattice and domain walls on the dual lattice. The ﬁgure shows the splitting of two commutative elementary domain walls.

If a domain wall corresponds to a single transposition, for example (12), we refer to it as an elementary

domain wall. When µ = m, meaning that µ can be written minimally as the product of m transpositions, | | we will refer to a µ domain wall as a composite of m elementary domain walls. However we must be careful

to distinguish between e.g. µ = (123) and µ = (12)(34). Both of these have µ = 2 but they are not | | equivalent as they have diﬀerent cycle structure.

90 For simplicity let us take A to be the region x > 0 in a ﬁnite or inﬁnite chain so that there is a single

k entanglement cut. Then for Zn the top boundary has a single domain wall of type τn,k which enters the system at the link of the entanglement cut.

We will show that Zk can be regarded as a partition function for k sets of (n 1) elementary domain n − walls in τn,k, with nontrivial interactions both within sets and between sets. These domain walls start at the top of the system at the position of the entanglement cut and undergo random walks downwards towards

the bottom, where the boundary condition on the spins is free.

5.3.2 Triangle weights

In Eq. (5.25) we give the exact results for the weights of the simplest conﬁgurations of a triangle, which

involve at most 1 ‘incoming’ elementary domain wall at the top of the triangle.

q = 1, = = q2 + 1 (5.25) = 0 = 0 (µ = I). 1 6 µ µ− 1 µ µ−

For a given triangle, we describe a domain wall at the top of the triangle as ‘incoming’, and domain walls

at the bottom left and right as ‘outgoing’. The formula (5.24) involves a sum over N! elements of the

permutation group, with nontrivial weights. Remarkably, the ﬁnal results for the weights above are N independent. The non-vanishing diagrams are the ones that conserve the number (either 0 or 1) of incoming elementary domain walls. For example, it is not possible for the incoming elementary domain wall (12) to

1 split into (12)µ and µ− with µ = I, despite the fact that this splitting is consistent with the domain wall 6 multiplication rule. Similarly, if the number of incoming domain walls is zero, there are no outgoing domain walls: generation of domain wall pairs out of the “vacuum” is forbidden.

We can summarize these rules algebraically as

J(σb, σb; σa) = δσa,σb q (5.26) J(σ , (12)σ ; σ ) = δ + δ . b b a q2 + 1 σa,σb σa,σb(12) We also give exact weights for the case N = 3 in Appendix D.6. For a general conﬁguration at large N it is

hard to evaluate the exact weights of the diagrams. However, we conjecture that, as in the example above,

J does not depend directly on N: i.e. on the number of ‘additional’ unused elements in permutations σa,

σb, σc. For example, we may evaluate J(I; I, (123)) for any N 3, and we conjecture that the result is ≥

91 independent of N. We have checked the conjectured N-independence of weights by explicitly evaluating all

Js for N up to 4. However, for most of our purposes it will be suﬃcient to evaluate triangle weights in a

large q expansion.

Finally we specify the weights in the presence of spatial boundaries. These involve identical three-spin

weights, but now the corresponding triangles are tilted: see Fig. 5.9.

τ σ

Figure 5.9: Spins on the spatial boundary. Left: The left-most legs of the unitary gates act on the same boundary site, so they are eﬀectively connected, as shown by the dashed line. The τ spin on the boundary still connects to 3 σ spins, and form a tilted triangle. Right: The boundary triangle on the triangular lattice. The red line is the top link, the blue lines are the bottom left and right links of the down-pointing triangle.

5.3.3 The q = limit ∞ The partition function Zk simpliﬁes in the limit q = , and this limit is a useful starting point for thinking n ∞ about ﬁnite q. When q , the terms that survive in the partition function Zk are those with the minimal → ∞ n total length of elementary domain walls. This means that domain walls cannot ‘split’: for each down-pointing triangle, the number of elementary domain walls entering from the top is equal to the number leaving from the bottom.

At leading order in q, the weight of a triangle with m elementary domain walls passing through it is just

m q− : for example (using doubled/tripled lines to represent composite domain walls)

1 . (5.27) ' ' ' ' q3

This simpliﬁcation of the weights means that at q = distinct replicas decouple: ∞

ln Zk k ln Z at leading order in q. (5.28) n ∼ n

This means that in this limit the ﬂuctuations in Sn are negligible (i.e. we must go to higher order to see them) and also that 1 S ln Z at leading order in q. (5.29) n ∼ −n 1 n −

92 Therefore leading order results in q can be obtained by studying the partition function for a single replica,

Zn. In fact, this is sufficient to obtain not just the first term but the first few nontrivial terms in a large q expansion for various quantities such as the growth rate of entanglement after a ‘quench’. We do this in the

next section.

5.4 Entanglement Production Rates

In this section we consider the partition functions Z2 and Zn>2 for a single replica. These suﬃce to obtain

the ﬁrst few orders in a large q expansion of the R´enyi entropy growth rates v2 and v3, as well as the ‘entanglement line tension’ that generalizes these growth rates when the initial state is not a product state

(Sec. 5.5). Later on we will address the eﬀects of interactions between replicas.

Let us ﬁrst consider only the leading order contributions to the partition function at large q. Zn is then the partition function for n 1 elementary domain walls making up the permutation (12 n), and the − ··· entanglement entropy Sn is proportional to the free energy for this random walk problem. Since the number of domain walls is conserved at each time step, each layer in the triangular lattice

(n 1) contributes a factor of q− − to Zn. (Minus) the logarithm of these factors gives the ‘energy’ of each con-

ﬁguration. There is also an entropy term, coming from counting the number Ωn(t) of distinct conﬁgurations:

(n 1)t ln Ωn(t) Z Ω (t)q− − , S t ln q . (5.30) n ∼ n n ∼ − n 1 −

To go beyond this leading order result we use the more detailed weights in Eq. (5.24). Let us now consider

various cases.

5.4.1 Second R´enyi entropy

The case n = 2 has been treated in Ref. [157]. There is only a single domain wall (12) starting from the

entanglement cut at the ﬁnal time. In this case the mapping of Z2 to the partition function for a single simple walk is exact for any q if we replace the approximate energy ln q with the logarithm ln K of the − exact weight for a single triangle in Fig. (5.25),

q K = . (5.31) q2 + 1

93 The number of conﬁgurations is 2t. Therefore

S s v t, v v (5.32) 2 ∼ eq × 2 2 ' 2 e with seq = ln q and the ‘purity speed’ [114, 226, 157, 10, 11]

1 q2 + 1 v = ln . (5.33) 2 ln q 2q e Once interactions between replicas are taken into account this growth rate is corrected at the relatively high order 1/q8 ln q as we discuss in Sec. 5.6.

5.4.2 Higher R´enyi entropies

For general Sn we must consider the composite domain wall (12 . . . n). We may write this as a product of (n 1) elementary domain walls labeled by transpositions. These transitions are non-commuting, which − gives rise to nontrivial combinatorial ‘interactions’.

To see this, consider the case n = 3. There are 3 ways to split a domain wall labeled (123) into a product of two elementary domain walls, one on the left and one on the right:

(123) = (12) (23) = (23) (13) = (13) (12). (5.34) × × ×

This may be contrasted with the 2 ways to split a product of commutative transpositions:

(12)(34) = (12) (34) = (34) (12). (5.35) × ×

The partition function for (123) involves a nontrivial sum over how the elementary walks are labeled.

Each time the walks meet, the labeling can change from one of the possibilities in Eq. (5.34) to another.

At ﬁrst sight we must now keep track of the label on each domain wall, but in fact we can absorb the combinatorial factors associated with the labeling into a simple eﬀective interaction.

If two independent random walks A and B (for example two ‘commuting’ elementary domain walls in the present problem) meet at a given time step, there are two possibilities for the conﬁguration subsequently: either A is on the left or B is on the left (corresponding to the two terms on the RHS of Eq. (5.35)). For noncommuting domain walls such as (12) and (23), there are instead three possibilities for the subsequent conﬁguration. These are listed above in Eq. (5.34). This relative factor of 3/2 means that Z3 maps to a

94 partition function for a pair of (distinguishable) directed random walks with an attractive interaction. In a given conﬁguration, let the number of ‘splitting events’ be the number of times the walks meet and split.

(When they meet, they may either split again immediately, or they may form a section of composite domain wall which extends for a ﬁnite period of time). Then

3 # splitting events Ω (t) = . (5.36) 3 2 conﬁgs of 2X walks

The attraction means that the free energy is smaller than that of a pair of independent random walks. This

2S3 2 2S2 means that Z3 = e− is larger than Z2 = e− , so that the entanglement velocity v3 is smaller than v2. Interestingly, an eﬀective combinatorial interaction between paths also arises in the replica treatment of directed polymers with Boltzmann weights of random signs, by a diﬀerent mechanism [227, 221].

By a combinatorial computation in App. D.1, we obtain the exact asymptotic expression for Ω3(t):

3 t Ω3(t) 2 + . (5.37) ∼ √2

This constant was also obtained independently in a related Floquet model in Ref. [188], where it arises from essentially the same combinatorial mechanism.

The constant Ω3 gives the ﬁrst nontrivial term in the expansion of v3 at large q. We can go to one higher order by taking into account subleading repulsive interactions of strength (1/q2) between the walks O which appear when we go beyond the leading order expression for the triangle weights (exact results are in

App. D.6, App. D.7). For example, if the composite domain wall on the LHS below is (123),

1 1 , (5.38) ' × × − q2 corresponding to a reduction in the weight for non-commutative elementary walks that are on top of each other compared to walks that are separate. The calculation is performed in App. D.1 and yields the growth rate: ln 2 + 3 √2 3√2 1 v3 1 + . (5.39) ' − 2 ln q 4 q2 ln q

For larger n, the combinatorial factors can no longer be absorbed into a simple eﬀective attraction.

It appears to be necessary to keep track of the labeling of the walks explicitly. This is because diﬀerent decompositions of (12 . . . n) can be inequivalent. For example the decomposition (1234) = (14)(13)(12) and the decomposition (1234) = (13)(12)(34) are inequivalent: in the former case none of the adjacent domain

95 wall pairs commute while in the latter case one adjacent pair commutes.

The above shows that different Rényi entropies grow at different rates following a quench (see also [188]).

5.5 The Entanglement Line Tension En

Above, v2 is the coarse-grained line tension, or free energy per unit length of the elementary domain wall

that appears in the calculation for S2 (up to a factor of seq = ln q; ‘length’ here is in the t direction). To be more precise, this is the line tension of a domain wall which is vertical on large scales. As argued in [169] (see also [157]) it is useful to deﬁne a more general line tension (v) which is a function of the coarse-grained E2 ‘velocity’ of the domain wall. The velocity v(t) of the domain wall is its inverse slope, dx(t)/dt, at a given

value of t. The free energy of the domain wall scales as s t (v) if its average velocity is ﬁxed to be v, eq × E2 i.e. if its total displacement over time t is vt.

Here we briefly review the role of the line tension and its generalization . In Sec. 5.5.1 we discuss E2 En the meaning of for higher n in more detail, and calculate . This will introduce the concept of the En E3 ‘bound state’ of domain walls, which will be important to understand nonanlyticities in , and later how En>2 Page’s formula arises (Sec. 5.8) and the fluctuations of the higher Rényi entropies (Sec. 5.6.5).

It was conjectured that the line tension determines the time dependence of the entanglement entropy En 4 Sn, in an appropriate scaling limit, for more general initial states that are not necessarily product states [169]:

Sn(x, t) = min [Sn(x vt, 0) + seq n(v)] . (5.40) v − E

Consider ﬁrst S2. The above formula arises from the random walk picture when we consider only the leading behaviour at large time. In this scaling limit the walk’s ﬂuctuations are negligible, and it forms a straight

line connecting (x, t) to (y, 0). The position y is determined by minimizing the free energy. This gives the

above, if we assume that the initial state at t = 0 simply contributes an ‘energy’ equal to its entanglement

across y. This will not be true for all possible initial states but may hold for states that are ‘typical’ in some

sense.

Ref. [169] conjectured the general constraints

(v ) = v , 0 (v ) = 1, (v) v. (5.41) En B B En B En ≥

In the next section we give a nontrivial check on these constraints. By deﬁnition we also have (0) = v . En n 4This limit is where the length and timescales of interest are parametrically large and of the same order. Since S is also of this order, ∂S/∂x can be order 1 in this regime, but higher derivatives such as ∂2S/∂x2 are subleading.

96 Considering the free energy of a random walk with a ﬁxed slope gives [169]

q2+1 1+v 1+v 1 v 1 v ln 2 + ln + − ln − (v) = 1 + q 2 2 2 2 . (5.42) E2 ln q

This function satisﬁes the relations (5.41) above. If the ‘free energy’ is deﬁned using the replica limit, as is

8 appropriate for calculating S2, then Eq. 5.42 will be modiﬁed at order 1/(q ln q). The function (v) is analytic for all v < 1, i.e. for all speeds up to the lightcone speed, including E2 | | speeds greater than v . In fact, the minimum in Eq. (5.40) is always in the range [ v , v ]. However, the B − B B mapping of [10] shows that the function (v) is relevant to the scaling of the exponentially small tail of the E2 out of time order correlator beyond the lightcone [231].

5.5.1 Higher R´enyi entropies: The ‘bound state’ phase transition

As we saw above, the calculation of S3 yields a pair of elementary domain walls with an attractive interaction. In 1+1D, two walks with an attractive interaction form a bound state: the typical separation between the walks, in the x direction, is of order one even in the limit t . Therefore at large scales the two walks are → ∞ paired and can be regarded as a single composite domain wall. (The ‘bound state’ terminology is natural if we think of the walks as worldlines of ﬁctitious particles.)

The line tension (v) is deﬁned as 1/(2s ) times the free energy per unit length of this composite E3 eq domain wall, when its coarse-grained velocity is ﬁxed to be v. The factor of 1/(2seq) is to compensate the

2seqS3 2seq in Z3 = e− . For higher n the combinatorial interactions between walks are much harder to treat, but we expect that the walks will again form a bound state with a spatial extent of order 1. Then (v) is En 1/[(n 1)s ] times the free energy per unit length of this composite domain wall, when its coarse-grained − eq velocity is v.

We ﬁnd that the line tension for n = 3 has interesting structure that is not present in (v). This is due E2 to a phase transition that is driven by varying v. As v is increased towards a critical value vc, the extent of the bound state (in the x direction) diverges. For v < vc, the ‘binding energy’ of the bound state means that (v) is smaller than (v). But for v v , the walks are unbound and their free energy is simply that E3 E2 ≥ c of two independent walks: this means that in this regime (v) = (v). E3 E2 We conjecture, and check explicitly to leading nontrivial order, that the critical speed associated with this unbinding transition is precisely vB:

vc = vB. (5.43)

97 This mechanism is how the conjectured constraint (v ) = v in Eq. (5.41) is satisﬁed. We conjecture that E3 B B a similar mechanism applies for higher n also, with the (n 1) walks becoming unbound at v . − B We now give explicit formulas for . Firstly, we show in App. D.2 that to order 1/ ln q the line tension E3 for S3 is 1 3 ln(φ− + φ + ) v ln φ (v) = 1 √2 − , (5.44) E3 − 2 ln q

with 3v + √8 + v2 φ = . (5.45) √8(1 v) − The functional form diﬀers nontrivially from that for . However, the bound state phase transition cannot E2 be seen at this order in q. Therefore in App. D.3 we perform a separate expansion for speeds close to the lightcone, writing α v = 1 , with α of order 1. (5.46) − q2

v close to 1 is of course equivalent. − First let us consider how the phase transition can occur in principle. Recall that each time the walks

merge and split, they ‘gain’ a weight 3/2 for combinatorial reasons. This is an eﬀective attraction that

encourages them to bind together.

However, examining the exact weights for the walks (App. D.6), we ﬁnd that there is also a weak repulsion,

of order 1/q2, for time steps in which the two walks are on top of each other (combined into a composite

walk). For generic values of v, this weak repulsion is negligible compared with the (1) attraction arising O from the combinatorial eﬀect. But for walks moving at speeds very close to unity, this repulsion is magniﬁed

as follows.

For a walk moving at the speed in Eq. (5.46), almost every step is to the right: only an (1/q2) fraction O of steps are to the left. This means that when the two walks meet, they typically remain together for a

long time, of order q2 (both taking rightward steps) before one of them takes a leftward step and they split.

Therefore the total repulsion energy for the time interval between the merging and splitting is (1), and O can compete with the (1) combinatorial attraction. For small enough 1 v , the repulsion dominates and O − | | the bound state disappears.

In App. D.3 we ﬁnd that the critical speed for disappearance of the bound state is

2 v = 1 + (5.47) c − q2 ···

98 This is consistent with v = v . For v > v we have (v) = (v). For v < v we ﬁnd c B c E3 E2 c

α A (α) (v) = 1 + 3 + ... (5.48) E3 − q2 q2 ln q

with 2 9 9 + 4α2 2 + 4 + 9 + 2α ln α 2 + 4 + 9 − α2 18 α2 A (α) = r . (5.49) 3 q 8 h q i

In this regime (v) < (v), which is necessary for Eq. (5.40) to be consistent with the general constraint E3 E2 S S . 3 ≤ 2 Note that A (2) = 0 and A0 (2) = 0, showing that the line tension (v) obeys the general constraints in 3 3 E3 Eq. (5.41) at least up to order 1/(q2 ln q).

5.6 Fluctuations and the Replica Limit

k In this section we treat interactions between replicas in the replicated partition function Zn. We study S2 in detail, mapping it explicitly to the free energy of a directed polymer in a classical random medium and to

the height ﬁeld in a continuum KPZ equation. The extension to n > 2 is discussed more brieﬂy in Sec. 5.6.5.

The Kardar-Parisi-Zhang equation was proposed to model universal scaling in classical surface or interface

growth[202]. The time-dependent height function h(x, t) obeys the nonlinear stochastic equation

λ ∂ h = ν∂2h + (∂ h)2 + η(x, t) + const. (5.50) t x 2 x

The ﬁrst term represents diﬀusive relaxation. The non-linear term is a relevant perturbation of the linear

theory. The η term is white noise.

Let us summarize the connection between the domain wall picture and KPZ. First, to avoid confusion,

recall that there are several equivalent ways to think about the universal properties of the KPZ universality

class[202, 206, 203]: (i) in terms of the KPZ equation; (ii) in terms of a directed polymer in a random

medium (of height t in the vertical dimension); (iii) in terms of a system of k 0 interacting bosons. We → will use all of these languages. Heuristically, the relation between (i) and (ii) is seen by writing a recursive

equation for the free energy of the polymer as a function of the height of the medium: this gives the KPZ

equation in the continuum limit. The relation between (ii) and (iii) comes from writing a transfer matrix

for the directed polymer, giving (in the continuum) an eﬀective Hamiltonian for a boson, and then using the

replica trick to treat the disorder, which gives k 0 interacting bosons. See [202, 206] for further details. →

99 k For n = 2, each replica in Z2 gives only one domain wall, so that there are k elementary walks in total. 1 A diagrammatic calculation shows that these k walks have an eﬀective pairwise attraction at order q4 . This 8 1 ultimately leads to KPZ ﬂuctuations of the entanglement of order q− 3 t 3 . We have k 0 directed walks with a pairwise interaction. Because the interaction is parametrically → small at large q, we can make a controlled continuum approximation. This is most convenient if we think of the coarse grained walks as worldlines of bosons in 1+1D Euclidean spacetime, with attractive contact

k potentials between the bosons. In this language the partition function Zn is the imaginary time path integral amplitude for the bosons (and the entanglement growth rate is proportional to their ground state energy).

The resulting boson Hamiltonian is integrable [206, 224] and this is one way to obtain the ﬂuctuations of the entanglement.

Having replicated the quantum system and mapped it to a classical one, we can now undo the replica limit to obtain a classical model with randomness. We discuss this both in the continuum and on the lattice.

In the continuum it is convenient to think in terms of the KPZ equation. Remarkably even the nonuniversal constants in this equation can be fixed. At large but finite q the second Rényi entropy S S ≡ 2 obeys5 1 1 ∂ S = c + ∂2S (∂ S)2 + η(x, t) (5.51) t 2 x − 2 x for a weak Gaussian noise 1 η(x, t)η(x0, t0) = δ(x x0)δ(t t0). (5.52) h i 4q4 − −

Above, c is a constant which contributes to the entanglement growth rate v2 (given in Eq. (5.6)) which we

ﬁx using the boson mapping. The large-time scaling of S2, for a cut across a given bond, may be written in terms of a ﬂuctuating random variable χ(t) whose size is of order 1 at large times (below seq = ln q):

S (t) s v t + B t1/3χ(t) . (5.53) 2 ' eq 2 2 h i

Using the exact results in Ref. [224], the magnitude of the ﬂuctuations are controlled by the constant

1 B . (5.54) 2 ' −4 21/3 q8/3 × ×

The cumulative probability distribution of the random variable χ(t) is the Tracy-Widom distribution F1.

5The sign of the nonlinear term in Eq. (5.51) is opposite to that of Eq. (5.50) because in the correspondence with the directed polymer, S is proportional to the free energy, while h is proportional to minus the free energy. The sign of the nonlinear term can be changed by a change of variable h h so does not aﬀect the exponents. → −

100 The mean and standard deviation are

mean(χ) = 1.20653..., std(χ) = 1.26798... (5.55) −

On the lattice, undoing the replica limit on the classical side of the mapping gives a well-deﬁned lattice

directed polymer problem with a short-range correlated random potential. We make this mapping explicitly

for large but ﬁnite q.

With the bound state concept introduced in Sec. 5.5, we can generalize the above pictures to Sn with

n > 2. The composite domain walls in Sn for n > 2 first form a bound state due to the leading order combinatorial interaction in Sec. 5.5.1. Then by a similar mechanism as for S2, the weak pairwise interaction between the bound states from different replicas gives rise to the KPZ fluctuations, showing that such

ﬂuctuations are present in all R´enyi entropies with n 2. ≥

5.6.1 Interactions between Replicas

In this section, we focus on the n = 2 case where the leading order picture involves k independent commutative elementary domain walls.

First of all, the exact partition function for a single elementary domain wall (k = 1) is

q Z = 2tKt,K = , (5.56) 2 q2 + 1

where 2t is the number of random walk conﬁgurations and Kt is the product of weights of t down-pointing

triangles, c.f. Eq. (5.25). Compared with the leading order result 1/q, we see that the more accurate weight

K for each down-pointing triangle is

1 1 1 = K = 1 + + . (5.57) q − q2 q4 ···

These corrections determine the ﬁnite q corrections to the energy per unit length, or the line tension, of a single walk.

Similarly, we may consider the higher-order corrections to the weights of triangles that host ` > 1 walks.

` At leading order the weight of such a triangle is q− . At ﬁrst sight we might expect corrections to this leading order result to be the dominant source of interactions between the replicas. However we ﬁnd that

101 1 to order q4 , we have 1 = 1 + ( ) × × O q6 (5.58) 1 = 1 + ( ) . × × O q6

This kind of decomposition also holds for all ` 2 up to ( 1 ) order, see the perturbative calculation in ≥ O q4 1 App. D.7. Consequently, if there is an interaction at order q4 it must come from additional domain wall configurations which are absent in the q = limit. This is indeed the case. ∞ What are the lowest order (in 1/q) modifications to the domain wall configurations described above? By

Eq. (5.25), we cannot add isolated ‘bubbles’, i.e. closed domain wall loops that are not attached to any of the k walks: such conﬁgurations have weight zero. Similarly, the last formula in Eq. (5.25) prevents us from modifying an isolated walk. However when two walks meet additional conﬁgurations are possible.

As mentioned in Sec. 5.4, the ‘na¨ıve’ order in 1/q of a down-pointing triangle is equal to the number of elementary domain walls that pass through its lower edges. [The actual order may be higher, as a result of cancellations in the sum defining J(σa; σb, σc).] There are two possibilities allowed by Eq. (5.25) that are na¨ıvely of relative order 1/q4 compared to the leading order configurations. The first corresponds to adding

a hexagonal ‘bubble’ of the transposition α to a conﬁguration of two walks, say (12) and (34):

(12)(34)

. (5.59) (12) (34)

1 α α−

However, the relative order of this conﬁguration is in fact 1/q6, as the result of a cancellation between two

values of the Weingarten function in Eq. (5.25), see App. D.7.6

The second possibility relies on the following decompositions of the product (12)(34),

(12)(34) = (14)(23) (24)(13) = (24)(13) (14)(23). (5.60) × × 6 1 The two values correspond to τ = σa and τ = σaα− in Eq. (5.25).

102 Each of these decompositions leads to a ‘Feynman diagram’:

(12)(34) (12)(34)

(14)(23) (24)(13) (24)(13) (14)(23) . (5.61)

(12)(34) (12)(34)

Each such conﬁguration has relative weight, compared to the dominant conﬁgurations, of 1/q4 (plus higher order corrections).

These ‘special hexagons’ are the only source of interactions in the bulk of the sample at order 1/q4. We may add the weight of these conﬁgurations to the weights of the leading-order conﬁgurations to obtain a

‘dressed’ weight for a pair of walks which both visit a pair of triangles that are vertically adjacent as shown below: . . = + + + . . (5.62)

1 + 2 + ( ). × O q6

The factor of two indicates the two possibilities in Eq. (5.61). This gives the total weight (recall K ≡ q/[q2 + 1])

. . 1 2 1 = 4K4 + + . q4 q4 O q10 . (5.63)

1 1 1 = 4K4 1 + + . 2 q4 O q6 This is an interaction of order 1/q4, and it is attractive, because it increases the Boltzmann weight for conﬁgurations in which two walks collide. Furthermore, it is a pairwise interaction – we can insert a ‘special hexagon’ for any pair of (commutative) domain walls, and to leading order this insertion is not ‘seen’ by any of the other k 2 walks. In Sec. 5.7 we also check these properties of the interaction numerically. − The fact that the attraction is small at large q allows an analytical treatment which we discuss next

(Secs. 5.6.2, 5.6.3, 5.6.4). It should be noted however that for other random circuits, in which the microscopic

103 probability distribution of gates is diﬀerent, the interaction strength can remain of order one even in the limit q . The simplest way to obtain an interaction of (1) strength in the q limit is to allow the → ∞ O → ∞ local unitaries to be equal to the identity with a nonzero probability p.

These identity gates create ‘holes’ in the circuit through which the domain walls can pass without costing any energy at all. Averaging over the locations of these holes gives an effective attractive interaction between replicas. This has similar effects to the attractive interaction described above, but the strength of the attraction remains finite at q = and can be controlled by varying p. This is essentially a model ∞ considered in [157] where KPZ behaviour was obtained in the limit q . → ∞

5.6.2 Mapping to polymer in random medium

For a circuit with regular structure the attractive interaction between replicas is small at large q. This allows both a controlled continuum description and an explicit mapping to a classical disordered model.

Let us simplify the lattice structure. Above, each random walk lives on the honeycomb lattice which is dual to the triangular lattice. Each honeycomb site corresponds to a triangle, either up- or down-pointing.

However it is suﬃcient to draw only the sites corresponding to the down-triangles, as shown in Fig. 5.10.

That is, we can view the walks as living on a square lattice (rotated by 45◦). Adjacent sites of this square lattice diﬀer by (∆x, ∆t) = ( 1, 1). For an isolated walk, each step along a bond of this lattice is weighted ± ± by K.

It is useful to think of pairs sites (x, t) and (x, t + 2) as connected by vertical bonds, even though the walks cannot occupy such bonds. One such vertical bond is illustrated in Fig. 5.10, right.

If two walks both visit both of the sites (x, t) and (x, t + 2), then the associated weight is not 4K4 but rather 4K4 exp 1/2q4 , by Eq. (5.63). When any one walk visits both (x, t) and (x, t + 2), we say that the vertical bond from (x, t) to (x, t + 2) is visited by that walk. Two walks therefore interact if they both visit the same vertical bond.

Figure 5.10: Reduction from triangular lattice to square lattice. Pairs of consecutive steps are combined into a single step on the square lattice. Each blue (red) step on the left corresponds to a blue (red) step on the right. We also refer in the text to ‘visits’ to vertical bonds like the one indicated by the dashed line. According to our deﬁnition, the vertical bond indicated here is only visited by the blue walk and not the red (so there is no special hexagon interaction between these two walks).

104 For each vertical bound b, let the number of walks which visit it be n . If n 2 walks visit bond b, b b ≥ there is an interaction between each of the n (n 1)/2 pairs, as discussed below in Eq. (5.63). The weight b b − associated with the interactions is thus exp 1/2q4 n (n 1)/2 . × b b − Using D to denote the tilted square lattice the eﬀective partition function is:

k kt Z2 = K exp A, (5.64) k directed walksX on D

where, neglecting boundary eﬀects (which are discussed in Sec. 5.6.4)

1 n (n 1) A = b b − . (5.65) 2q4 2 vertical bondsX b

k Remarkably, this form means that we can interpret Z2 as the average of a replicated classical partition function for a single walk or polymer. Let us deﬁne the partition function for this ﬁctitious classical polymer

by:

= Kt exp n V . (5.66) Z − b × b polymer vertical onXD  bondsX b    Since this partition function is for a single polymer, nb is either 0 or 1. On each vertical bond b, the polymer

experiences a Gaussian random potential Vb. We take these random potentials to be independent, with mean and variance

1 1 mean(V ) , var(V ) . (5.67) b ' 4q4 b ' 2q4

With these choices, averaging over the random potentials V yields precisely the expression for Zk in Z b 2 Eq. (5.64). Writing the average over Vb as [...]V :

Zk k . (5.68) 2 ∼ Z V

The identity above implies that the statistics of S2 in the quantum problem map onto the statistics of a classical polymer in a random potential that is speciﬁed by Eqs. (5.66), (5.67). Note that this makes the

dynamics of the entropy eﬃciently simulable (modulo the large q approximation used) for large t that would

be beyond the reach of a direct computation as in Sec. 5.7.

105 5.6.3 Continuum description

Next we discuss the continuum limit. Consider ﬁrst Z2, i.e k = 1. In the continuum the walk becomes a Brownian path characterized only by its free energy per unit time, f = ln 2K, and its diﬀusion constant, − which is easily seen to be D = 1/2.7 Viewing the walk as the Feynman path of a boson in Euclidean spacetime, with spatial coordinate x, the Hamiltonian for this boson is

1 ∂2 = ln 2K . (5.69) H − − 2 ∂x2

The scaling of the partition function is given by the ground state energy E0 of this system of bosons:

E t Z e− 0 . For the above Hamiltonian E is simply ln 2K. 2 ∼ 0 − For k = 1 we must take into account the attractive interactions between bosons. The continuum Hamil- 6 tonian contains only a delta-function interaction and is solvable by Bethe ansatz in the k 0 limit [206]: →

1 k ∂2 = k ln 2K λ δ(x x ). (5.70) H − − 2 ∂x2 − α − β α=1 α X α<βX

Since λ 1 at large q we can ﬁx it explicitly using the lattice results above (see App. D.8):

1 λ . (5.71) ' 4q4

Standard mappings [206] relate the coeﬃcients in Eq. (5.78) to those in the KPZ equation (5.51).

The energy of the system of bosons as k 0 gives the average free energy density f of the polymer in → the random medium, or equivalently the growth rate of the averaged entropy: f = seqv2. Using the result of [206], 1 q2 + 1 1 v = ln + + ... . (5.72) 2 ln q 2q 384 q8 ln q

The first term is the ‘purity speed’ v2 (Sec. 5.2), and the second is a correction from replica interactions. The Bethe ansatz results in [224] also fix the prefactor of the KPZ fluctuations, Eq. (5.54). e The above results apply in the limit of large times. Since at large q the interaction is weak (but

renormalization-group relevant) there is a large crossover scale. The crossover length scale (in the spatial x

direction) is in the notation of Ref. [206]: 2 l = 8q4. (5.73) d λ '

This corresponds to a timescale of order l2. For t l2 the polymer of the previous subsection resembles a d d 7The mean squared displacement in the x direction, for a section of duration t, is simply 2Dt = t.

106 (12)(34)

(14)(23) (24)(13)

Figure 5.11: An example of the half special hexagon interaction at the boundary. (Left) The special hexagon 1 in the bulk gives an interaction of order q4 , while the half hexagon at the bottom boundary gives an 1 interaction of order q2 . Hence the boundary interaction will dominate the early time ﬂuctuation. (Right) A 1 domain wall conﬁguration of the half special hexagon. Since there are only 2 extra legs, it is of order q2 .

random walk, with diﬀusive scaling between x and t. For t l2 its conformation is strongly aﬀected by the d quenched randomness, and KPZ scaling exponents govern its statistics and the statistics of S2. It is notable

2 that in the present model the crossover timescale ld is large even for q = 2. We discuss crossovers in more detail in the next section.

5.6.4 Early-time crossovers for large q

So far we have considered KPZ scaling at asymptotically long times, which we expect to hold for any q.

However when q is large there are interesting early and intermediate time regimes, while ﬂuctuations in S2

2 remain small, i.e. before the onset of KPZ scaling at times of order ld. We first note that when two walks from different replicas meet at the t = 0 boundary, there is an interaction that corresponds to ‘half’ of the special hexagon in Sec. 5.6.1. This is shown in Fig. 5.11. This interaction is of order 1 . Since this is parametrically larger than the (1/q4) bulk interaction, it dominates q2 O at early times. In the polymer language it corresponds to a boundary disorder potential of strength 1/q. ∼ For times 1 t q2, as we show below, this leads to fluctuations which decrease with time as

1 1 2 Var(S2) 1 , 1 t q . (5.74) ∼ (4πt) 4 q p The reason for the decrease of the ﬂuctuations is that a polymer of length t explores, through thermal

fluctuations, a length of the boundary of size t1/2. It is therefore effectively subject to the disorder ∼ potential averaged over this region. The average of (t1/2) local potentials with mean zero and typical O magnitude 1/q gives the 1/(qt1/4) scaling above. ∼ More precisely, an exact combinatorial counting, involving pairs of walks from different replicas which

meet at the t = 0 boundary, gives 2 2(t 1) Var(S ) = − (5.75) 2 q24t (t 1) −

107 and hence Eq. (5.74). We calculate the early time ﬂuctuations of S2 numerically for various q, see Fig. 5.12. The largest q values agree fairly well with the lowest order result in Eq. (5.75) for larger t.

2 S2 S 2 − 2 0.4 q q = 2 0.35 q = 3 q = 4 0.3 q = 5 q = 6 0.25 0.2 0.15 0.1 0.05 0 1 2 3 4 5 6 t

Figure 5.12: Fluctuations of S2 for different q and number of layers of network. Depending on the position of the entanglement cut the entanglement increases either in odd or even time steps. For each t we have placed the cut so that the final layer of unitaries can create entanglement. The lines are the analytic lowest 1 2 2(t 1) order result q 4t t −1 . − q When t q2, the bulk contribution to fluctuations dominates, and fluctuations grow with time. However they are not immediately governed by KPZ exponents. From the continuum description in (5.51), (5.52) the rescaled entropy S = 4q4S satisfies p e 1 1 ∂ S = ∂2S (∂ S)2 + η(x, t) + c (5.76) t 2 x − 4q2 x e e e e e subject to the normalized noise

η(x, t)η(x0, t0) = δ(x x0)δ(t t0). (5.77) h i − − e e At early times the nonlinear term may be neglected. The resulting noisy linear equation (note that all coeﬃcients are order 1) is known as the Edwards-Wilkinson equation and gives ﬂuctuations in S of order

1/4 2 1/4 2 t [232], or of order q− t for S. These dominate the boundary ﬂuctuations at time t q . EW ∼ e

108 Var(S2) t 1 1/4 2 boundary dominated pq− t− t . q ∼ 2 1/4 2 8 Edwards-Wilkinson q− t q . t . q ∼ 8/3 1/3 8 KPZ q− t t q ∼ &

Table 5.1: Three regimes of ﬂuctuations in S2 and the associated time scales.

However the nonlinear term is RG relevant, and it can no longer be neglected at times t & tKPZ, where t q8 [202]. This is also the time scale at which ﬂuctuations are of order one. Since t t 1 KPZ ∼ KPZ EW at large q, there are three regimes, see Tab. 5.1.

5.6.5 KPZ for R´enyi entropies S with n = 2 n 6

S2 is the simplest entropy to calculate because each replica gives rise to only one elementary domain wall. However we can use the concept of the bound state (Sec. 5.4) to outline a generalization to larger n. For concreteness, consider the case n = 3, with q large. This limit simpliﬁes the analysis by giving a clear separation of scales between two kinds of interactions.

First, within each replica there is a pair of walks, or equivalently quantum particles, with an attractive interaction between them of order 1 strength (Eq. (5.36)). Then at a parametrically smaller energy scale of

4 order q− there is the attractive interaction between walks in diﬀerent replicas which we have discussed in the previous section.

Therefore in the first step of RG — at lengthscales of order one — the walks form independent bound states within each replica. On larger scales each bound state can be treated as a walk (or particle) with a single position coordinate xα for α = 1, . . . , k. The bound states have a well-defined coarse-grained line tension and diffusion constant. Finally, there are weak attractive interactions between bound states arising from the weak interactions between the microscopic walks. Therefore the next stage of the RG flow can again be described by a Hamiltonian like Eq. (5.78), but with different numerical constants. As a result we again expect KPZ scaling. We expect that a similar two-step RG picture applies for any n > 2 when q is large.

For each n, the continuum Hamiltonian for the bound states generalizing (5.78) is characterised by three constants, k ∂2 = k D λ δ(x x ), (5.78) H − n − n ∂x2 − n α − β α=1 α X α<βX 2/3 1/3 and the magnitude of the KPZ ﬂuctuations in S is proportional to λn /[Dn (n 1)]. n − If it was possible to compute these constants for arbitrary n, we could hope to analytically continue to

109 n = 1 to compute the ﬂuctuations of the von Neumann entropy.8

5.7 Numerical Checks Using the Operator Entanglement

In this section we perform numerical checks on some of the analytical arguments in Sec. 5.6. We argued that

the dominant interaction between replicas, for large q, arose from a ‘special hexagon’ diagram and that this

is a pairwise interaction between replicas.

Here we check this result for the interactions by comparing numerics with the analytic form for

exp ( kS [U(t)]), (5.79) − 2

where S2[U(t)] is the operator entanglement of the time evolution operator. Recall that we may regard the tensor network deﬁning U(t) as a tensor network state for 2L spins, L at the top boundary and L at the

bottom boundary. S2[U(t)] is then the entanglement of a subsystem containing the L/2 spins on the left part of the bottom boundary together with the L/2 spins on the left part of the top boundary. This may be

mapped to a lattice magnet by a simple extension of the above formulas. The only change compared to the

k calculation of Z2 is the boundary condition at the bottom boundary. Since the top and bottom boundaries are treated on equal footing, the bottom boundary condition will be the same as the top one: there will be

a composite domain wall τn,k at the bond of the entanglement cut. The conﬁgurations of incoming domain walls on the top and outgoing domain walls on the bottom are exactly the same as in Eq. (5.61).

We check the analytic result for t = 1 and t = 3. For t = 1 we must consider a single unitary gate and

the associated special hexagon interaction in Eq. (5.61). This gives

k(k 1) 1 ln Tr(ρ2[U(1)])k k ln(Tr(ρ2[U(1)]) − . (5.80) − ' 2 2q4

The subtraction on the LHS is to isolate the interaction contribution. For t = 3 we must sum over 6

conﬁgurations. At leading order in 1/q this gives

k(k 1) 5 ln Tr(ρ2[U(3)])k k ln(Tr(ρ2[U(3)]) − . (5.81) − ' 2 9q4

In Figs. 5.13 and 5.14 these results are compared against numerical results for k = 2,..., 6 and for various

values of q. (We average over 4000 and 100 realizations for t = 1 and t = 3 respectively.) In both cases

8The scaling as n may be more easily tractable and is also interesting ( S is the logarithm of the largest eigenvalue of the reduced density→ matrix). ∞ − ∞

110 the agreement is good at large q. This conﬁrms that the special hexagon is indeed the interaction between

1 replicas, at order q4 , in the bulk of the system.

ln(tr(ρ2)k) k ln(tr(ρ2)) 0 − 10 4 k = 2 0.5q− 4 1 k = 3 1.5q− 10− 4 k = 4 3q− 4 2 k = 5 5q− 10− 4 k = 6 7.5q− 3 10−

4 10−

5 10−

6 10−

7 10− 0 5 10 15 20 25 30 35 40 q

Figure 5.13: Bipartite operator R´enyi entanglement entropy for 2 site gate. This is the simplest one-layer random tensor network. The domain wall diagrams correspond exactly to those in Eq. (5.62). We veriﬁed 1 the strength 2q4 in Eq. (5.62) as well as the pairwise nature of the interaction.

5.8 Saturation at Late Time and Page’s Formula

For a ﬁnite system of even size L, when t is far greater than the saturation time, we expect the half-chain entanglement to saturate to the value given by the generalized Page formula for R´enyi entropy [129, 228]:

L ln C SPage = ln q n . (5.82) n 2 − n 1 −

L In this formula 2 ln q is the maximal possible entanglement for the half-chain and Cn is the nth Catalan

number: C1 = 1, C2 = 2, C3 = 5, C4 = 14, etc. This formula is valid for all q if L is large, and the corrections are exponentially small in L. Fluctuations about the Page value are also exponentially small in L.

We show that the constants Cn have a simple and appealing explanation in terms of the domain walls. We ﬁrst discuss the limit of large q, then give a sketch (which is partly conjectural) for how the domain wall

picture allows the result to survive when q is not large.

111 ln(tr(ρ2)k) k ln(tr(ρ2)) 1 − 10 4 k = 2 0.5556q− 4 0 k = 3 1.667q− 10 4 k = 4 3.333q− 4 1 k = 5 5.556q− 10− 4 k = 6 8.333q− 2 10−

3 10−

4 10−

5 10−

6 10− 2 4 6 8 10 12 14 16 q

Figure 5.14: Bipartite operator Rényi entanglement entropy of 4 sites. The tensor network has 3 layers: the k(k 1) 5 first and last layers have one gate and middle layer has two gates. The interaction still fits 2− 9q4 , which is predicted by only considering the special hexagon interaction.

Consider the ﬁnite system with two spatial boundaries shown in Fig. 5.15 (Left). For a single elementary

S domain wall, as appears in the calculation for Z2 = e− 2 , there are two possibilities at late time: it must exit the system via either the left or right spatial boundary. These possibilities are shown in red and blue respectively in the ﬁgure. At large q the optimal slope for each domain wall (minimizing its total energy) is approximately unity, and the energy of such a domain wall is (L/2) ln q. The two possibilities lead to a

L/2 factor of two: Z2 = 2q− . Now consider Zk. At leading order in q the replicas decouple, so Zk (Z )k. This is equivalent to the n n ' n statement that at leading order in q there are no ﬂuctuations in the entanglement. The boundary condition for Zn introduces a domain wall of type τn,1 = (12 . . . n) at the top boundary. This domain wall can split into two domain walls µ and ν, satisfying

τ = µ ν, (5.83) n,1 × with the µ domain wall exiting to the left and the ν wall to the right. The blue and red paths in Fig. 5.15

L (n 1) L both cross down-pointing triangles, so Z c q− − 2 . The number of conﬁgurations c is the number 2 n ' n n of ways to factorize τ into a product µ ν. There are precisely C such choices (see App. B of [200]), so n,1 × n cn = Cn. This reproduces Eq. (5.82).

112 1 A τn,1 B B τn,1 A τn,−1 B

Figure 5.15: Domain wall paths for t L/2v2. Left: entanglement of half of a chain with two boundaries. There are two possible paths at the leading order in q. They exit the system through a tilted triangle on the boundary. Right: entanglement of half of a chain with periodic BCs. The two entanglement cuts give two domain walls, τn,1 and its inverse. They meet at a down-pointing triangle either in region A or in B again giving two possibilities.

The case of a ﬁnite interval (thus two cuts) of size L/2 in a chain with periodic boundary conditions is

1 similar. Here we have boundary condition changes which insert domain walls τn,1 and its inverse τn,−1 at the two cuts. The two optimal paths correspond to domain walls that meet either inside A or B (Fig. 5.15, right). Otherwise the discussion is as above. (Note that if the two subsystems contain different numbers of sites the energies of the different domain wall configurations will no longer be degenerate.)

This gives a combinatorial interpretation of the (1) correction in the Page value as an entropy associated O with the large-scale conﬁgurations of the random walks.

For finite q Eq. (5.82) remains true so long as L is large [129, 228]. For this result to emerge from the random circuit, two things must happen. First, at finite q, the replicas must effectively decouple in the configurations that obtain at late time, to ensure that fluctuations about the Page value are parametrically small in L. Second, all of the ways of splitting τ into µ ν must have the same free energy. n,1 × This is closely connected to the conjectured constraints on (v) in Eq. (5.41)[169]. Consider a domain En wall that exits the boundary of the system (as in Fig. 5.15). Approximately speaking, Eq. (5.41) ensures that the preferred velocity of this domain wall, selected by free energy minimization, is v (rather than 1, as ± B ± at q = ), and that the line tension (free energy per unit time) per elementary walk is independent of how ∞ the composite walk τn,1 is split into smaller composites µ and ν. At a more microscopic level, what allows this to happen is the unbinding transition which we demonstrated for the case of two walks in Sec. 5.5.1.

To be more accurate, we must also assume that different replicas, and more generally commuting domain walls, also decouple by a similar mechanism: a vanishing of the effective attractive interaction when the coarse-grained speed is fixed to vB.

113 5.8.1 The moment of saturation

So far in this section we have discussed the asymptotic value of the entanglement at very late times. We may address the moment of saturation in a similar way. For deﬁniteness, consider the entanglement of the

first ` sites in a chain of size L, with ` L/2, so that the saturation time is approximately9 t `/v . ≤ sat,n ' n Let’s assume L/2 ` 1, so that we can neglect walks which travel to the right hand spatial boundary. − The leading order scaling picture for the moment of saturation is a sharp crossover in free energy, as a function of t, between ‘vertical’ domain wall configurations which reach the t = 0 boundary, and domain wall configurations which travel to the left spatial boundary. Let us consider how this sharp transition is rounded out.

We must split τn,1 into µ, a composite walk which travels to the left boundary, and ν, which travels to the t = 0 boundary. We will consider only the cases S2 and S3, i.e. τn,1 = (12) and τn,1 = (123). In the first case we have, if we make the further simplification of neglecting fluctuations due to circuit randomness

(this is controlled at large q): S (t) s ` s v t e− 2 e− eq + e− eq 2 ' (5.84) s ` s v [t t ] =e− eq 1 + e− eq 2 − sat , where the ﬁrst term represents the domain wall exiting at the left boundary and 2nd term is the domain wall going vertically as in the inﬁnite system case.

For S3 we at ﬁrst sight have more terms, because we can choose ν = 1, ν = (123), or take ν to be an elementary domain wall. However, because v2 > v3, the latter option is always exponentially subleading, so we have

2S (t) 2s ` 2s v [t t ] e− 3 e− eq 1 + e− eq 3 − sat . (5.85) '

Saturation is sharper for S3 than for S2. It is straightforward to extend these expressions to similar situations, e.g. to the case ` L/2 by including conﬁgurations with walks that travel to the right spatial boundary. '

5.9 Dynamics without Noise

The microscopic models we have studied here include randomness both in space and time. The corresponding effective directed polymer partition function involves both thermal fluctuations and quenched disorder. It is natural to expect that in realistic models without randomness, a mapping to a coarse-grained directed polymer problem will still be possible, and that this effective description will still include thermal fluctuations

9The time at which the crossover happens will fluctuate by (`1/3), due to KPZ fluctuations in the growth over this period. O However, these fluctuations are between realizations, and should not be confused with the rounding of the Sn(t) profile within a realization which we discuss here.

114 of the polymer. The quenched disorder will of course be absent in that case. In fact this is very similar to

what we have at large q, since as we have seen the eﬀects of randomness are suppressed by a high order in

1/q. This picture is supported by the results of Ref. [188], which derived a domain wall picture for Zn in a model with large q unitaries that are random in space but not in time. At leading order in q this picture

coincides with that for the random circuits here.

The concept of local pairing of spacetime histories that we ﬁnd in random circuits is likely to be useful

also in non-random models. The basic point is that paired histories (one from a U evolution and one from

a U ∗ evolution) contribute cancelling phases to the path integral, i.e. appear with positive weight. This suggests that paired conﬁgurations, with domain walls between pairings that are enforced by boundary

(n 1)S conditions, dominate the path integral for quantities such as e− − n even in the absence of any disorder average. We will discuss this further elsewhere.

It would be interesting to consider models in which the dynamics is time-independent but spatially random in more detail. A coarse-grained description for entanglement growth in such models was discussed in [187], in terms of a directed polymer subjected to randomness that depends on space but not on time. In the replicated language, this corresponds to interactions between replicas that are local in space but nonlocal in time. In the case where the spatial randomness allows for ‘weak link’ locations where the entanglement growth rate is arbitrarily small, a Gaussian average over disorder is not suﬃcient since the weakest links, which are rare events, are important at late times. It would be interesting to search for these phenomena by applying the replica trick to the model of Ref. [188] or extensions thereof.

5.10 Outlook

We have shown that the minimal membrane picture for S makes sense beyond the q = limit, in a n ∞ regime where it can no longer be identiﬁed with a ‘minimal cut’ through the unitary circuit. There is an

emergent statistical mechanics governing the entanglement in which the R´enyi entropy is the free energy of

an emergent domain wall. The interactions and ‘thermal’ ﬂuctuations of the domain walls play an important

role in determining the entanglement velocities (and entanglement line tensions), which diﬀer for diﬀerent

R´enyi entropies. Additional ﬂuctuations associated with disorder in the circuit are responsible for universal

KPZ scaling at late times. A diﬀerent type of large scale ﬂuctuation also governs the Page-like corrections

to the entropies at late times. For random circuits many properties are computable analytically in a power

series expansion in 1/q, where q is the local Hilbert space dimension (and in some cases exactly).

A lacuna in this work is an explicit treatment of the von Neumann entropy, as opposed to the higher

115 Rényi entropies. This requires an additional replica limit (n 1) which is likely to be more complicated → than the one we used here to average the Rényi entropies (k 1). Computing the von Neumann entropy → for finite q is a task for the future. It is important to ask whether by focusing on n > 1 we are missing important phenomena specific to S1.

Another intriguing task is to obtain explicit numerical or analytical results for the entropies Sn with n > 3, extending the schematic picture above in terms of the bound state. This would require us to

understand the combinatorics associated with the labeling of the paths (by transpositions).10 This would

shed light on the structure of the evolving entanglement spectrum. More detailed treatment of Sn>3 would also be interesting in the context of the bound state phase transition which we have argued occurs in (v) E at v = vB.

For S2, our explicit mapping to a lattice directed polymer in a random medium problem means that the dynamics of the entanglement could be simulated, classically, over timescales which interpolate between the

short times accessible in quantum simulations and the large times required to see KPZ in the present model,

at least for reasonably large q (we have argued that a large crossover time is responsible for the apparent

absence of KPZ scaling in short-time simulations).

In future work we will extend these mappings to related phenomena including light-cone eﬀects in cor-

relation functions, as well as entanglement growth for more general initial states.

10The ‘scattering matrix’ for these domain walls does not appear to satisfy a Yang-Baxter equation for n > 3, so perhaps the problem is non-integrable for n > 3.

116 Appendix A

Appendices for the Lifshitz Quench Problem

A.1 Boundary Reproducing Kernel

In this appendix, we show that the normal derivative behaves like a boundary reproducing kernel for harmonic

functions.

A Let A be a simply connected region and G∆(x1, x2) the associated Green function with Dirichlet boundary condition, 2 A x x 2 x x A x x x1 G∆( 1, 2) = δ ( 1 2) G∆( 1, 2) = 0. (A.1) − ∇ − x1 ∂A,x2 A ∈ ∈

Let f(x) to be a function deﬁned on ∂A, then

A f(x2) = lim ∂n1 G∆(x1, x2) f(x1) dl1, (A.2) x2 ∂A ∂A − → Z h i in other words, ∂ GA (x , x ) looks like a delta function on the boundary. − n1 ∆ 1 2 To prove this, we construct a harmonic function φ(x) whose boundary value is f(x),

2φ(x) = 0 φ(x) f(x). (A.3) − ∇ ∂A ≡

Then we express the harmonic function in terms of its boundary value,

φ(x ) = δ2(x x )φ(x )d2x = ( 2 )GA (x , x )φ(x )d2x 2 1 − 2 1 1 −∇x1 ∆ 1 2 1 1 ZA ZA = ∂ GA (x , x )f(x ) dl + GA (x , x ) φ(x )d2x (A.4) − n1 ∆ 1 2 1 1 ∇x1 ∆ 1 2 ∇x1 1 1 Z∂A ZA = ∂ GA (x , x )f(x ) dl , − n1 ∆ 1 2 1 1 Z∂A where we use integration by part twice.

117 Taking x2 to approach the boundary, we obtain the desired identity

A f(x2) = lim ∂n1 G∆(x1, x2)f(x1) dl. (A.5) − x2 ∂A → Z∂A

We demonstrate this result with the explicit example of the upper half plane, whose Green function is

1 1 GA (x , x ) = ln x x + ln x x¯ . (A.6) ∆ 1 2 −2π | 1 − 2| 2π | 1 − 2|

The normal derivative of the green function on the boundary – the x-axis – is

1 y A x x 2 ∂y1 G∆( 1, 2) = 2 2 . (A.7) − y1=0 π (x1 x2) + y2 −

The fact that ∞ 1 y2 dx2 2 2 = 1 (A.8) π (x1 x2) + y2 Z−∞ − and 1 y2 lim 2 2 = 0 if x1 = x2 (A.9) y2 0 π (x x ) + y 6 → 1 − 2 2 suggest that when x2 approach the x-axis,

1 y2 lim 2 2 = δ(x1 x2). (A.10) y2 0 π (x x ) + y − → 1 − 2 2

A.2 Equal Space Green Function for a Simply Connected Region

In this appendix, we calculate the derivative of the equal space Green function for a simply connected region

A Suppose the equal time Green function of Laplacian in region A is denoted as G∆(x1, x2), which satisﬁes

2 A A x G∆(x1, x2) = 0 G∆(x1, x2) = 0. (A.11) ∇ 1 ∂A

The equal space Green function we are seeking for is then

A A 2 2 G∆(τ1, τ2) = H(x, τ1; x1)G∆(x1, x2)H(x, τ2; x2) d x1 d x2. (A.12) A A Z ×

118 In the following, we will make use of the heat kernel property

2 ∂ H(x, τ; x0) = H(x, τ; x0) (A.13) τ ∇ when taking derivatives to the Green function.

To simplify the notation, we denote

A 2 φ(x1) = G∆(x1, x2)H(x, τ2; x2) d x2 (A.14) ZA to be the solution of the Poisson equation in region A with heat kernel as its source

2 x φ(x1) = H(x, τ2; x1) φ(x1) = 0, (A.15) ∇ 1 ∂A

then

A 2 G∆(τ1, τ2) = H(x, τ1; x1)φ(x1) d x1. (A.16) ZA

Now taking the τ1 derivative and integrating by part twice, we have

∂ GA (τ , τ ) = 2 H(x, τ ; x )φ(x )d2x = H(x, τ ; x ) φ(x )d2x − τ1 ∆ 1 2 − ∇x1 1 1 1 1 ∇x1 1 1 ∇x1 1 1 ZA ZA = H(x, τ ; x )( 2 )φ(x )d2x + H(x, τ ; x )n φ(x )dl (A.17) 1 1 −∇x1 1 1 1 1 1 · ∇x1 1 1 ZA Z∂A 2 = H(x, τ1; x1)H(x, τ2; x1)d x1 + H(x, τ1; x1)∂n1 φ(x1)dl1. ZA Z∂A

And then we take symmetrically a τ2 derivative

∂ ∂ GA (τ , τ ) = H(x, τ ; x )∂ H(x, τ ; x )d2x H(x, τ ; x )∂ ∂ φ(x ) dl (A.18) τ2 τ1 ∆ 1 2 − 1 1 τ2 2 1 1 − 1 1 n1 τ2 1 1 ZA Z∂A and analyze the resulting two terms.

Upon integration by part, the ﬁrst term becomes

H(x, τ ; x )( 2 )H(x, τ ; x ) d2x = H(x, τ ; x ) H(x, τ ; x ) d2x 1 1 −∇x1 2 1 1 ∇x1 1 1 ∇x1 2 1 1 ZA ZA (A.19) H(x, τ ; x )∂ H(x, τ ; x ) dl . − 1 1 n1 2 1 1 ZA

119 We turn to the τ2 derivative in the second term

∂ ∂ φ(x ) = ∂ GA (x , x )( 2 )H(x, τ ; x ) d2x n1 τ2 1 n1 ∆ 1 2 ∇x2 2 2 2 ZA A 2 = lim ∂n1 G∆(x1, x2)∂n2 H(x, τ2; x2) d x2 x ∂A 2→ ZA ∂ GA (x , x ) H(x, τ ; x ) d2x − ∇x2 n1 ∆ 1 2 ∇x2 2 2 2 ZA (A.20) A = ∂n1 H(x, τ2; x1) ∂n1 ∂n1 G∆(x1, x2)H(x, τ2; x2) dl − x1 A − ∂A ∈ Z 2 A x x x x 2 + lim ∂n1 ( x2 )G∆( 1, 2)H( , τ2; 2) d x2 x1 ∂A,x2 ∂A ∇ ∈ → ZA A = ∂n1 H(x, τ2; x1) ∂n1 ∂n1 G∆(x1, x2)H(x, τ2; x2) dl. − x1 A − ∂A ∈ Z

where I have used the theorem in App. A.1 in going from the second equality to the third.

Collecting all these results, we have

∂ ∂ GA (τ , τ ) = H(x, τ ; x ) H(x, τ ; x ) d2x τ2 τ1 ∆ 1 2 ∇x1 1 1 ∇x1 2 1 1 ZA (A.21) A + H(x, τ1; x1)∂n1 ∂n2 G∆(x1, x2)H(x, τ2; x2) dl1 dl2. ∂A ∂A Z ×

The result is symmetric about τ1 and τ2. It consists of a contribution purely from the bulk and boundary of region A, where the former is what would be there if the Dirichlet boundary condition were not imposed on the entanglement cut.

Therefore, compared with the free Green function, we have

no cut A A ∂τ2 ∂τ1 G∆(τ1, τ2) = H(x, τ1; x1)∂n1 ∂n2 G∆(x1, x2)H(x, τ2; x2) dl1 dl2. (A.22) Dirichlet − ∂A ∂A Z ×

A.3 Alternative Calculation for the Equal Space Green Function

on the Half Plane

In this section, we consider an alternative calculation of ∆Sn for upper half plane case. According to Eq. (2.61), the general equal space Green function with Dirichlet boundary condition on the entanglement cut for partition A and B is

2 2 A G∆(τ1, τ2) = d x1d x2 H(x, τ1; x1)G∆(x1, x2)H(x, τ2; x2) Dirichlet A A Z × (A.23) 2 2 B + d x1d x2 H(x, τ1; x1)G∆(x1, x2)H(x, τ2; x2). B B Z ×

120 The diﬀerence of the presence and absence of the cut will only come from the boundaries of A and B, which

is the x-axis. By applying the general formula (A.22) for both regions A and B, we have

A B ∪ ∞ ∞ ∂τ2 ∂τ1 G∆(τ1, τ2) = 2 dx2 dx1 H(x, τ1; x1)∂y1 ∂y2 G∆(x1, x2)H(x, τ2; x2) . Dirichlet − y1=0,y2 0 Z−∞ Z−∞ → (A.24)

We interpret (or regulate) this boundary integral as in App. A.5

A B 1 H(x, τ ; x ) H(x, τ ; x ) H(x, τ ; x ) H(x, τ ; x )] ∪ 1 1 − 1 2 2 1 − 2 2 ∂τ2 ∂τ1 G∆(τ1, τ2) = dx1dx2 2 . (A.25) Dirichlet π (x1 x2) Z −

The calculation is a little involved, so for clarity we provide the details in App. A.4, and the result is

A B 2 2 ∪ 1 1 1 y 1 y ∂τ2 ∂τ1 G∆(τ1, τ2) = 2 exp( ) exp( ). (A.26) Dirichlet 4π τ1 + τ2 √τ1 −4τ1 √τ2 −4τ2

The Green function can be obtained by integrating τ1 and τ2,

A B 2 2 ∪ 1 1 1 y 1 y G∆(τ1, τ2) = 2 dτ1dτ2 exp( ) exp( ) Dirichlet 4π τ1 + τ2 √τ1 −4τ1 √τ2 −4τ2 Z 1 √τ1 √τ2 y2 1 y2 = du dv exp( ) exp( ) π2 −4u2 u2 + v2 −4v2 Z0 Z0 1 2√τ1 2√τ2 y2 1 y2 = du dv exp( ) exp( ) (A.27) π2 −u2 u2 + v2 −v2 Z0 Z0 1 ∞ ∞ 2 2 1 2 2 = 2 du dv exp( u y ) 2 2 exp( v y ) π 1 1 − u + v − Z 2√τ1 Z 2√τ2 1 ∞ ∞ 1 = du dv exp( u2) exp( v2). π2 y y − u2 + v2 − Z 2√τ1 Z 2√τ2

The excess EE is proportional to the cross Green function, which is

A B ∪ G∆(τ1, τ2, ) = G∆(τ1, τ1) G∆(τ1, τ2) G∆(τ2, τ1) + G∆(τ2, τ2) × − − Dirichlet n y y o 1 2√τ2 2√τ2 1 = du dv exp( u2) exp( v2) π2 y y − u2 + v2 − Z 2√τ1 Z 2√τ1 y 2 τ 2 (A.28) 1 ∞ √ 2 2 = 2 dλ du exp( (1 + λ)u ) π 0 y − Z n Z 2√τ1 o √1+λy 2 τ 2 1 ∞ 1 √ 2 2 = 2 dλ du exp( u ) . π 0 1 + λ √1+λy − Z n Z 2√τ1 o

We calculate the contour integral in the curly braces as in Figure A.1. The path is √1+λy 0 √1+λy , 2√τ1 → → 2√τ2 whose two segments only changes the length of the complex number.

121 y 2√τ2

y 2√τ1

Figure A.1: We choose the integration path where only the length of the complex number changes.

√1+λy 0 r0 2√τ2 du exp( u2) = du √ i exp(iu2) + du √i exp( iu2) √1+λy − r0 − 0 − Z 2√τ1 Z Z r0 r0 = √2i du cos u2 sin u2du (A.29) − Z0 Z0 π 2 2 = √2i C( r ) S( r ) , 2 π 0 − π 0 r r r where C and S are the Fresnel cosine/sine integrals

z π z π C[z] = cos( x2)dx S[z] = sin( x2)dx (A.30) 2 2 Z0 Z0

√1+λy and the constant r0 = 2√2gt . This gives the cross Green function

2 1 ∞ 1 2 2 G (τ , τ , ) = dλ C( r ) S( r ) ∆ 1 2 × −π 1 + λ π 0 − π 0 Z0 r r 2 1 ∞ 1 λ λ = dλ C( y) S( y) −π 1 λ s4πgt − s4πgt Z (A.31) 1 ∞ 1 2 = dλ C(√λ) S(√λ) −π y2 λ − Z 4πgt 2 ∞ 1 2 = q dλ C(λ) S(λ) −π y2 λ − Z 4πgt and consequently the excess EE

2 2 α α ∞ 1 2 ∆Sn = G∆(τ1, τ2, ) = 4 q dλ C(λ) S(λ) . (A.32) − 4g × 8πg y2 λ − Z 2πt h i

122 A.4 Schwinger Parameter Calculation

This section is devoted to the analytic calculation of Eq. (A.25).

After doing the proper regularization, the integral in Eq. (A.25) is convergent, so it is safe to add a small positive constant in

A B 1 H(x, τ ; x ) H(x, τ ; x ) H(x, τ ; x ) H(x, τ ; x )] ∪ 1 1 − 1 2 2 1 − 2 2 ∂τ2 ∂τ1 G∆(τ1, τ2) = lim dx1dx2 2 Dirichlet π 0+ (x1 x2) + → Z − (A.33) to compute each term. The divergent pieces in the limit 0+ will automatically be canceled in the end. → The four integrals are independent of x

1 H(x, τ ; x )H(x, τ ; x ) 1 1 1 dx dx 1 i 2 j = exp ( + )y2 I , (A.34) π 1 2 (x x )2 + (4π)2τ τ − 4τ 4τ ij Z 1 − 2 1 2 1 2 where 1 exp( 1 x2 1 x2) I = dx dx − 4τ1 i − 4τ2 j (A.35) ij π 1 2 (x x )2 + Z 1 − 2 such that

A B ∪ 1 1 1 2 ∂τ2 ∂τ1 G∆(τ1, τ2) = 2 exp ( + )y lim I11 I12 I21 + I22 Dirichlet (4π) τ1τ2 − 4τ1 4τ2 0+ − − → (A.36) 1 τ1 + τ2 2 = 2 exp ( )y lim 2(I11 I12). (4π) τ1τ2 − 4τ1τ2 0+ − →

The integral I11 can be done directly

1 1 2 1 1 √ I11 = dx1 exp( ( + )x1) dx2 2 − 4τ1 4τ2 √ π (x1 x2) + Z Z − (A.37) π 4τ τ = 1 2 . τ + τ r r 1 2

I12 can be done using the Schwinger parameter

∞ 1 1 1 I = dλ dx dx exp x2 x2 λ(x x )2 λ 12 π 1 2 − 4τ 1 − 4τ 2 − 1 − 2 − Z0 Z 1 2 1 + λ ∞ 1 4τ1 = dλ e λ q − (A.38) 0 1 + λ ( 1 + 1 )λ + 1 Z 4τ1 4τ1 4τ2 16τ1τ2 q q 4τ1τ2 ∞ 1 λ = dλ e− . τ1 + τ2 0 λ + 1 r Z 4(τ1+τ2) q

123 We are able to do the following indeﬁnite integral

1 λ π λ dλ e− = e 0 erfc( (λ + λ )). (A.39) √ − 0 λ + λ0 r Z p At λ , erfc 0, so → ∞ → 4τ1τ2 π I = eλ0erfc( λ ), (A.40) 12 τ + τ 0 r 1 2 r p where 1 λ0 = . (A.41) 4(τ1 + τ2)

When taking the 0 limit, erfc(x) 1 2x → ∼ − √π

4τ τ π 2 I = 1 2 (1 λ ). (A.42) 12 τ + τ − √π 0 r 1 2 r p Therefore 4τ1τ2 √4τ1τ2 lim I11 I12 = 2 λ0 = (A.43) 0+ − τ1 + τ2 τ1 + τ2 → r p and the Green function becomes

A B 2 2 ∪ 1 1 1 y 1 y ∂τ2 ∂τ1 G∆(τ1, τ2) = 2 exp( ) exp( ). (A.44) Dirichlet 4π τ1 + τ2 √τ1 −4τ1 √τ2 −4τ2

A.5 Distributional Boundary Integral

In this section, we focus on the correct distributional interpretation of integral of the type (see exercise 6.8

of the book [233]) ∞ ∞ I = lim dx1 dx2 f(x1)∂y2 ∂y1 G∆(x1, x2) g(x2), (A.45) y2 0 y1=0 → Z−∞ Z−∞

where the kernel of the integral is

1 y x x 2 lim ∂y2 ∂y1 G∆( 1, 2) = lim ∂y2 2 2 y2 0 y1=0 π y2 0 (x1 x2) + y2 → → − (A.46) 1 1 2y2 2 = lim 2 2 2 2 2 . π y2 0 (x1 x2) + y2 − [(x1 x2) + y2] → n − − o

1 It is well approximated by (x x )2 when x1 x2 y2, but is singular in the other limit. The fact that 1− 2 | − | ≥

∞ 1 y2 dx1 2 2 = 1 (A.47) π (x1 x2) + y2 Z−∞ −

124 is independent of y2 suggests ∞ 1 y2 ∂y2 2 2 = 0. (A.48) π (x1 x2) + y2 Z−∞ −

Hence there is a highly localized distribution at x1 = x2 to compensate the (positive) divergent integral of

1 P ∞ dx 2 . 1 (x1 x2) −∞ − RWe notice that

1 (x x )2 y2 1 x x x x 1 2 2 1 2 ∂y2 ∂y1 G∆( 1, 2) = − 2 − 2 2 = ∂x2 − 2 2 (A.49) y1=0 π [(x1 x2) + y2] π (x1 x2) + y2 − −

and hence 1 ∞ ∞ x1 x2 I = lim dx1 dx2 f(x1)g(x2)∂x2 − 2 . (A.50) π 0+ (x1 x2) + → Z−∞ Z−∞ − The rewriting does not remove the singularity; what we do is to integrate the kernel in region x x > Λ, | 1 − 2| 1 where it can be approximated by (x x )2 and x1 x2 < Λ where g(x2) g(x1): 1− 2 | − | ∼

f(x1)g(x2) x1 x2 πI() = dx1dx2 2 + dx1 f(x1)g(x1) dx2 ∂x2 − 2 x x >Λ (x1 x2) + x x Λ (x1 x2) + Z| 2− 2| − Z Z| 2− 1|≤ − f(x1)g(x2) x1 x2 = dx1dx2 2 dx1 f(x1)g(x1) dx2 ∂x2 − 2 (A.51) x x >Λ (x1 x2) + − x x >Λ (x1 x2) + Z| 2− 2| − Z Z| 2− 1| − f(x1)g(x2) f(x1)g(x1) = dx1dx2 2 dx1dx2 2 . x x >Λ (x1 x2) + − x x >Λ (x1 x2) + Z| 2− 2| − Z| 2− 1| −

Taking 0 limit and symmetrimizing functions, we have →

1 f(x1) f(x2) g(x1) g(x2)] I = dx dx − − . (A.52) −2 1 2 π(x x )2 Z 1 − 2

Notice that the singularity at x1 = x2 has been regulated away.

125 Appendix B

Appendices for the Conformal Interface Problem

B.1 General Boundary State Amplitude

Sj(θ2)

Si(θ1)

Figure B.1: Partition function between the two boundary states of Si(θ1) and that of Sj(θ2)

In this appendix, we calculate the amplitude between general boundary states deﬁned in Eq. (3.34)

¯ ¯ Zab = 0 exp bRb(θ)b exp b†(I2 M)b exp b†Ra(θ)b† 0 , (B.1) h | − ⊗ | i n o n o n o where 4π2 M = diag(1, 2, ), I2 = diag(1, 1), β ··· (B.2)

Ri = Si(θ) I. ⊗ The graphical representation of the partition function is shown in Fig. B.1. Using the identity Eq. (B.57)

proven in App. B.6, we have 1 Zab = . (B.3) M det(1 Ra† e I2 R ) − − ⊗ b

From det(R R†) = 1, free energy becomes | a b |

M F = ln Z = ln det(R R† e−I2⊗ ) . (B.4) − | ab| | a b − |

There are two cases to be considered, and we only take out the leading order term in β.

126 • case 1: S (θ ) S (θ ), the free energy is 1 1 → 2 2

M F = ln det(R (θ )R†(θ ) e−I2⊗ ) | 1 1 2 2 − | M cos 2∆θI e− sin 2∆θI − − − = ln det   M sin 2∆θI cos 2∆θI e−  − −  (B.5)   2λ = ln[1 e− i ] − i X β ∞ 2x 1 = dx ln[1 e− ] = β. 4π2 − −48 Z0

• case 2: S (θ ) S (θ ), where i = 1 or 2, i 1 → i 2

M cos 2∆θI e− sin 2∆θI F = ln det −  M  sin 2∆θI cos 2∆θI e−  − −    λ 2λ (B.6) = ln[1 2 cos 2∆θe− i + e− i ] − i X β ∞ x 2x = dx ln[1 2 cos 2∆θe− + e− ], 4π2 − Z0

where ∆θ = θ θ . This integral is an even function of ∆θ and the ∆θ > 0 case reduces to the polylog 2 − 1 and Bernoulli polynomial β 2i ∆θ 2i ∆θ F = Li (e | |) Li (e− | |) 4π2 − 2 − 2 β h i = 2π2B ( x ) (B.7) 4π2 − 2 | | β β 1 = B ( x ) = ( x x2 ), − 2 2 | | 2 | | − − 6

∆θ where x = π .

B.2 Alternative Approach to DN λ Amplitude → In this appendix, we calculate the amplitude for the setup shown in Fig. B.2. In particular, the unfolded conﬁguration has D/N boundary conditions at y = L and conformal interface λ at y = 0. The general ± 2 solutions can be written as

1 A eikt cos ky + kL y < 0 1 2 f(k, y) =  . (B.8)  ikt 1  A2e sin ky kL y > 0 − 2  

127 y DN L N 2

0 λ λ

L - 2 D

Figure B.2: Partition function of Hamiltonian with DN and λ boundary conditions. We unfold the cylinder and the new stripe has N and D boundary conditions on the top and bottom plus a λ junction in the middle. The length L here is the height of the unfolded cylinder 2π.

As demonstrated in Sec. 3.2.1, if we denote f(k, y < 0) φ and f(k, y > 0) φ , the boundary condition ≡ 1 ≡ 2 at the junction becomes

∂ φ ∂ φ ∂ φ π x 1 = λ2 x 2 = tan2 θ x 2 , θ 0, , (B.9) ∂tφ1 ∂tφ2 ∂tφ2 ∈ 2 h i which implies 2π θ k = n , n Z. (B.10) L ± π ∈ 2π It is evident that the momentum k is shifted from an integer multiple of L due to the λ boundary condition in the middle.

The normalized eigenfunctions in Eq. (B.8) serves as an orthonormal basis in the mode expansion, we thus have 1 1 H = k a† a + , (B.11) 2 | | n n 2 n X∈Z where the momentum k is defined in Eq. (B.10), and the creation and annihilation operators are defined as usual 1 i an = k gφn + πn , √2 | | k g ! p | | (B.12) 1 p i an† = k gφn πn . √2 | | − k g ! p | | The Casimir energy is the vacuum energy brought up byp the finite size of the setup. From Eq. (B.11)

128 and using x θ in Eq. (B.10), we have ≡ π

1 π E = k = n + x + n x c 4 | | 2L | | | − | n n n ! X∈Z X∈Z X∈Z π = (n + x) + ( n x) + ( n + x) + (n x) 2L − − − − (B.13) n 0 n<0 n 0 n>0 ! X≥ X X≤ X π = 2 (n + x) + 2 (n x) . 2L  −  n 0 n>0 X≥ X   We use the Hurwitz zeta function ∞ 1 ζ (s, x) = (B.14) H (n + x)s n=0 X to regularize the sum π s s s Ec = (n + x)− + (n x)− ( x)− L − − − s= 1 n 0 n 0 − h X≥ X≥ i π = [ζ ( 1, x) + ζ ( 1, x) + x] (B.15) L H − H − − 1 1 = x2 + x , 2 − − 6 where in the last line we use L = 2π for the unfolded geometry.

Thus the free energy in the large β limit is

β F = βE = B (x), (B.16) c − 2 2 which agrees with the boundary state calculation in Sec. B.1.

B.3 Corrections to the Free Energy

In the course of deriving the free energy subject to various boundary conditions, we use conformal transformation to convert the spacetime diagram with slits to a cylinder diagram, where the boundary state calculation in App. B.1 (and ground state energy calculation in App. B.2) is applicable. However, the free energy is not invariant under the conformal transformation since the boundaries partially break the conformal symmetry. In this Appendix, we point out two corrections – one from the outer boundary regulator and the other from the inhomogeneous Schwartzian term to get the correct exponent of the ﬁdelity and

Loschmidt echo.

It is discussed in Cardy and Peschel’s work[80] that the boundary will contribute logarithmic term in the

129 free energy, c F = K(x)d2x + k ds ln L, (B.17) −6 g ZM Z∂M

where M is a 2d smooth manifold, K(x) is the Gaussian curvature, kg is the geodesic curvature of the boundary of the manifold and L is the system’s characteristic length.

The boundary term was not previous noticed in the literature, but is actually important even in the simplest example of the disk free energy. Consider an annulus on ﬂat space with inner radius r1 and outer radius r2. Its free energy is c r F (annulus) = ln 2 . (B.18) −6 r1

On the other hand the free energy of a disk of radius r2 is

c r F (disk) = ln 2 , (B.19) −6 a

where a is the short distance regulator. The disk free energy is completely contributed by its outer boundary

1 with other parts being conformal invariant. In fact, K = 0, kg = r for disk, and so

c r c r F (disk) = k ds ln 2 = ln 2 , (B.20) −6 g a −6 a Z∂M

where a is the short distance cut-oﬀ.

One can then interpret the annulus free energy as additive contributions from its outer and inner surfaces

c r c r c r F (annulus) = ln 2 + ln 1 = ln 2 . (B.21) −6 a 6 a −6 r1

An annulus becomes a disk when its inner radius is of order a, and we can see that the contribution from

c r1 the inner surface 6 ln a becomes negligible compared to the one from the outer surface. A similar outer surface logarithmic term also appears in the middle panel of Fig. 3.5. The conformal

map from the z plane to ξ plane bring the strip (with the small blue semi-circle) to the upper half plane

with the semi-circle around z = 1 extracted. This is in close analogy with the truncated corner calculation in Ref. [80]. In order to evaluate this diagram, we manually add the large blue semi-circle as IR cut-oﬀ, at the price of introducing an additional contribution c ln r2 of free energy which should not be there. − 6 a The same thing happened in Fig. 3.6 with a slightly diﬀerent mechanism. In the slit diagram (left panel

in Fig. 3.6), the regulators all have radii that are at the order of the short distance cut-oﬀ. They will have

negligible contributions to the free energy. However, in the new ξ plane, we implicitly switch to a new short

130 distance regulator such that only the blue semi-circle around 0 contributes negligibly. The outer surface

radius, despite being the image of a small semi-circle on the z plane, will contribute a c ln r2 term on the − 6 a ξ plane that should not be there.

c r2 Therefore in both cases we should compensate 6 ln a . Using the cylinder parameters in App. B.1, the ξ plane and z plane free energy are related through

c F = F + β (B.22) z ξ 6

for both the ﬁdelity and Loschmidt echo.

The annulus on the ξ plane is called the staircase geometry in Ref. [80] due to its evolution in angular

direction. The traditional radial quantization however has radial direction to be the time. One can show

that the Hamiltonian of the staircase and rectangle has a shift due to the Schwartzian[80] of the conformal

transform c H = H β. (B.23) ξ w − 24π

After the evolution for 2π (in the folding picture, the evolution is only π but there are two bosons), the

diﬀerence in the free energy is c F = F β. (B.24) ξ w − 12

c Gathering the two terms, we obtain the missing correction 12 β between the slit and cylinder diagram,

c F = F + β. (B.25) z w 12

B.4 Winding Modes of Compact Bosons

In this appendix, we address the issue of the winding modes of the compact bosons. In the main text, we

have exclusively worked with the oscillator modes of the free bosons. Here, we shall show that winding

modes for the compactiﬁed bosons will have no contribution to the ﬁdelity or Loschmidt echo in the leading

order. Therefore, our results are ready to be applied in the case where two compactiﬁed bosons of diﬀerent

radii are connected by a conformal interface[242].

Our derivation follows the general multi-component boson constraints in Ref. [75, 42]. A review of the

detailed parameterization of the states can be found in Ref. [72].

131 B.4.1 Mode Expansion of Compact Boson

Suppose the boson is compactiﬁed as φ = φ + 2πR, using the notation in Ref. [79], we have the following

mode expansion

n mR i an n n mR i a¯n n φ(z, z¯) =φ i + ln z + z− i lnz ¯ + z¯− , 0 − 4πgR 2 √4πg n − 4πgR − 2 √4πg n n=0 n=0 X6 X6 (B.26) where n, m are the momentum and winding modes quantum numbers.

In Ref. [79], the quantization is performed on an equal time slice. The boundary state we need here however lives on x = 0 – an equal-space slice. We therefore compact the theory in the time direction with period T , and identify the holomorphic and anti-holomorphic coordinates as

t x t + x z = exp(2πi − ), z¯ = exp(2πi ). (B.27) L L

This corresponds to exchange the x and t in Ref. [79].

We further identify n mR a = 4πg + , 0 4πgR 2 (B.28) p n mR a¯ = 4πg , 0 4πgR − 2 p to obtain the expression uniform for all the modes

1 n 1 i∂zφ = anz− − . (B.29) √4πg n X B.4.2 Gluing Condition for the Winding Modes

We recall that the gluing condition is written as

1 1 ∂+φ ∂ φ = S(θ) − . (B.30)  2  2 ∂ φ ∂+φ  −        Upon folding φ2 to the negative axis, its ∂ derivative becomes ∂ , so x − x

1 1 ∂+φ ∂ φ = S − . (B.31)  2  2 ∂+φ ∂ φ    −     

132 In terms of the holomorphic coordinates deﬁned in Eq. (B.27),

4πi 4πi ∂+ = z∂¯ z¯, ∂ = z∂z, (B.32) T − − T the S matrix establishes a relation between the modes

1 n 1 n a¯nz¯− anz− = S(θ) . (B.33)  2 n −  2 n n Z a¯nz¯− n Z anz− X∈   X∈       1 At the boundary x = 0,z ¯ = z− , we have

i 1 j (B.34) an + (Sij− )¯a n = 0. −

The solution of the n = 0 constraints is exactly the boundary state in Eq. (3.9). 6 We specialize to S = S1 to solve the n = 0 constraint. We introduce the compactiﬁcation lattice and its dual[75, 42] n1 n2 M = (m12πR1, m22πR2)>, M ∗ = ( , )>, (B.35) R1 R2 to rewrite the zero mode part as

i 1 j 1 1 a + S− a¯ = 0 = (M + M ∗) = S ( M + M ∗), (B.36) 0 ij 0 ⇒ g 1 − g which is basically the multi-component boson winding constraints given in [75, 42]. The solution gives the interface parameter λ n R m R λ = tan θ = 2 1 = 1 1 , (B.37) n1R2 −m2R2 and the conformal boundary state

in φ im φ¯ g e 1 0− 1 0 n , m n , m , S1 | 1 1i| 2 2i (B.38) XS1 where is the summation consistent with the constraint in Eq. (B.37). The g-factor can only be de- S1 terminedP by the Cardy condition[52]. Since it is not important for what follows, we shall not include the calculation here.

Since S = S2 is eﬀectively S1 on the dual boson, we can expect that it will end up in the same expression

133 as in Eq. (B.38), but with a diﬀerent constraint on the winding number

cot θ 0 1 − M =   M ∗. (B.39) g 1 0     B.4.3 Winding Mode Contribution to the Partition Function

We now calculate the winding mode part of the partition function as shown in Fig. B.1

πH 2π Z = S (θ ) e− S (θ ) ,H = (L + L¯ ). (B.40) h j 2 | | i 1 i β 0 0

For boundary states, we can simply replace L0(L¯0) with a0(¯a0).

For the amplitude between the same boundary states λ1 = λ2, we have the winding mode contribution as 4π n m R Z g g exp 2πg( 1 + 1 1 )2 , 0 Si Sj (B.41) ≤ − β 4πR1g 2 XSi n o where the equality is only taken when the two boundary states are identical.

In the limit β , Eq. (B.41) can be approximated by a simple two dimensional integral → ∞

x yR Z g g β dxdy exp 8π2g( + 1 )2 . (B.42) 0 Si Sj ≈ − 4πR1g 2 Z n o

The winding mode thus can contribute at most a ln β term to the free energy. Compared to the result in

Apps. B.1-B.2, we conclude that the winding mode contribution will not present in the leading order of the

large β limit.

B.5 Conformal Interface in Free Bosonic Lattice

In this appendix, we demonstrate how to realize the conformal interface in a lattice harmonic chain deﬁned

in Sec. 3.2.2, 1 1 H = π2 + (φ φ )2 2 i 2 i − i+1 i i=0 X X6 (B.43) 1 1 + Σ11 Σ12 φ0 + φ , φ , 2 0 1     Σ21 1 + Σ22 φ1         where the matrix Σ parameterizes the two-site interaction between site 0 and 1. We set up the plane wave scattering problem across the interface with the following ansatz (the use of (n 1) in φB simpliﬁes the − n

134 calculation) iωt inka iωt+inka A e − + A+e n 0 − ≤ φn =  , (B.44) iωt i(n 1)ka iωt+i(n 1)ka  B e − − + B+e − n 1 − ≥ where a is the lattice constant. The solutions on both semi-inﬁnite chains are gapless with the dispersion

ka relation ω = 2 sin 2 . The S matrix connecting them can be found by relating the incoming and outgoing

amplitudes

1 − ika ika A+ Σ11 + e Σ12 Σ11 + e− Σ12 A = − , (B.45)   −  ika  ika   B Σ21 Σ22 + e Σ21 Σ22 + e− B+  −               and the explicit expression is

ika ika 1 det Σ + Σ11e− + Σ22e + 1 2i sin kaΣ12 S = − . ika− 2ika   det Σ + e− trΣ + e− ika ika 2i sin kaΣ21 det Σ + Σ11e + Σ22e− + 1  −   (B.46) The reﬂection and transmission coeﬃcients associated with this interface contained in the S matrix and

both of them have to be k-independent to form a conformal interface[71]. A necessary condition is that S | 12| must be k-independent. If Σ12 = Σ21 = 0, we have

ika Σ11 + e S11 = ika , (B.47) −Σ11 + e−

which is not scale invariant. The only remaining possibility is

det Σ = 1, trΣ = 0, (B.48) −

which leads to a scale invariant S-matrix

1 S = ( 2i sin ka)Σ = eikaΣ. (B.49) 1 e 2ika − − − −

In this continuum limit where a 0, the matrix Σ can be parameterized as →

λ2 1 2λ −2 − 2 Σ = lim S = 1+λ 1+λ , (B.50) a 0  2λ 1 λ2  − → 1+− λ2 1+−λ2    

where λ R is the parameter for S1(θ), as introduced in Sec. 3.3.2. ∈

135 We use this two-site interaction to model the S1(θ) type conformal interface, as they give the same S matrix in the continuum limit. Therefore, the large t behavior of its Loschmidt echo should match with our

ﬁeld theoretic prediction.

B.6 A Determinant Identity for the Boundary State Amplitude

In this appendix, we provide more details for calculating the amplitude Zab in Sec. 3.3. We start to prove the following identity for a real symmetric matrix M

M b†Mb b†Rb¯† b†e− Rb¯† b†Mb e− e = e e− , (B.51)

where b and b¯ are vectors of bosonic operators. The matrix notation here should be understood as a bilinear

expression as explained below Eq. (3.31).

To prove Eq. (B.51), we ﬁrst consider the special case where R = I. We diagonalize M = OT ΛO and rotate the two sets of boson operators to the diagonal basis

T b†Mb = d†Λd d = Ob d¯† = O ¯b†, (B.52)

where we understand ¯b† as a column vector independent of b†. Thus the whole expression can be written as

b†Mb b†b¯† d†Λd d†d¯† λ d†d d†d¯† e− e = e− e = e− i i i e i i . (B.53) i Y

We recall for [X,Y ] = sY ,

eX eY = eexp(s)Y eX , (B.54)

which is a solvable case of the Baker-Campbell-Hausdorﬀ formula. Upon taking X = λ d†d , Y = d†d¯†, − i i i i i we have

[ λ d†d , d†d¯†] = λ d†d¯†, (B.55) − i i i i i − i i i

and so s = λ for each λ . This enables us to commute those exponentials − i i

λ M b†Mb b†b¯† e− i d†d¯† λ d†d b†e− b¯† b†Mb e− e = e i i e− i i i = e e− . (B.56) i Y

For the general case where R = I, we take d¯∗ = OT R¯b∗. This will not change the commutation relation of 6 d, and the role of ¯b is decorative in Eq. (B.55). Hence the rest of the proof follows the same way.

136 A direct consequence of Eq. (B.51) is the following

¯ ¯ bR∗ b b†Mb b†Rbb† 1 Zab = 0 e a e− e 0 = , (B.57) M h | | i det(1 Ra† e R ) − − b

where 0 is the vacuum for b and b¯. | i One can use the identity in Eq. (B.51) to reduce Zab to

M Z = 0 exp bR∗¯b exp b†e− R b¯† 0 , (B.58) ab h | a b | i n o n o

then a direct application of the MacMahon master theorem

1 0 exp b Xb exp b†Y b† 0 = (B.59) h | 1 2 1 2 | i det(1 XT Y ) n o n o −

proves Eq. (B.57).

B.7 Numerical Computation of Bipartite Fidelity and Loschmidt

Echo

In this appendix, we provide technical details about the numerical calculation of the bipartite ﬁdelity and

Loschmidt echo. Our strategy takes advantage of the symplectic structure of the bosonic Bogoliubov trans-

formation and explicitly construct the ”BCS” like ground state. With slight modiﬁcation[243], one can work

out its fermionic version and apply to the quadratic fermion models in Ref. [69, 74].

During the course of derivation in this and other appendices, we will repeatedly use the combinatorial

identity called the McMahon master theorem

1 1 1 0 exp b X b exp b†Y b† 0 = det− 2 (1 XY ), (B.60) h | 2 i ij j 2 i ij j | i −

for symmetric matrix X and Y and set of independent bosonic creation operators bi†. One can prove it for (simultaneously) diagonalizable matrices and then claim its legitimacy for its combinatorial nature.

137 B.7.1 Boson Bogoliubov transformation

We consider the following quadratic bosonic Hamiltonian

1 b A B∗ Hˆ = (b†, b)M ,M = − , (B.61) 2 −     b† B A∗    −      where b (b , ..., b )T is a vector of bosonic annihilation operators. The matrix M consists of a n n ≡ 1 n × Hermitian block A which plays the role of a single particle Hamiltonian in the fermionic case and symmetric block B of the pairing interaction.

We want to do a Bogoliubov transformation, which uses a 2n 2n matrix S to deﬁne a diagonal basis ×

(a, a†) (b, b†)S (B.62) ≡ of the Hamiltonian. The transformation is canonical, meaning that it preserves the commutation relation

0 I a b J = [ , a a ] = ST [ , b b ]S ≡     †   † I 0 a† b† (B.63) −            = ST JS,

where we have used the compact notation of the sort ([b, b†])ij = [bi, bj†] to denote the commutator matrix. The appearance of J makes the symplectic nature of the problem manifest and we ﬁnd S is in the symplectic group Sp(2n, C)[243, 244]. Furthermore, the requirement that a† is a complex conjugation of a leads to the block structure of S

u v∗ S =   . (B.64) v u∗     And the blocks are constrained by the symplectic property

u†u v†v = I, (B.65) − uT u vT v = 0. (B.66) −

With these conditions, the Hamiltonian in basis a becomes (the use of (b†, b) rather than (b†, b) can be −

138 appreciated in this step) a 1 1 H = (a†, a)(S>M(S>)− ) . (B.67) 2 −   a†     Quiet unusually, the diagonalization is performed by a symplectic group element.

To proceed, we introduce the real basis

b φ 1 1 i φ = C = , (B.68)     √2     b† π 1 i π      −            in which the Hamiltonian is

1 Re(A B∗) Im(A) + Im(B) φ Hˆ = φ π − − 2     Im(A) + Im(B) Re(A + B) π         (B.69) 1 φ = φ π . 2 M   π     It is not hard to check that is real and symmetric. M The general solution of the diagonalization problem is hard[245], however the positive deﬁnite (and M hence M) case can be solved by Williamson’s theorem[245, 246, 247, 248], which states the existence, uniqueness (up to reordering of eigenvalues) and explicit construction of the matrix Sp(2nR) such that S ∈

d T =   , (B.70) M S d S     where the diagonal matrix d are positive eigenvalues of iJ . After some algebra, we have M

d 1 T T 1 T 1 M = J(C− ) C J −   C C− . (B.71) S d S  −    One can show that,

T 1 S C C− ≡ S

is the required symplectic matrix in the complex basis.

We will not elaborate on Williamson’s theorem and its proof (see proofs in Ref. [246, 247, 248] and also

a recent application in the entanglement entropy context[249]). Instead we will show in App. B.7.5 that for

139 the problem of the harmonic chain we are interested in, the diagonalization can be easily done without using

the general recipe in the Williamson theorem.

B.7.2 Groundstate in b Basis

Suppose we have obtained the required matrix S, the ground state will be the vacuum of the annihilation

operators deﬁned in Eq. (B.62) and in the b basis it satisﬁes

(b u + b†v ) 0 a = 0. (B.72) i ij i ij | i

1 If the matrix u is invertible, then we can introduce a matrix T = vu− to rewrite Eq. (B.72) as

(b + b†T ) 0 a = 0. (B.73) i j ji | i

The constraint Eq. (B.66) on the blocks of u and v (followed by the symplectic constraint of S) implies that

T is a symmetric matrix. With the observation of

1 1 exp b†T b† b exp b†T b† = b + T b†, (B.74) −2 j jk k i 2 j jk k i ij j we solve the groundstate 1 1 0 a = det 4 (1 T †T ) exp b†T b† 0 b, (B.75) | i − −2 j jk k | i where the normalization is given by the McMahon master theorem Eq. (B.60). Applying constraint in

Eq. (B.65), it simpliﬁes to the top left corner of the symplectic matrix

1 1 det 4 (1 T †T ) = det(u) − 2 . (B.76) − | |

Eq. (B.75) takes a similar form as the superconducting ground state, with the pairing wavefunction Tij determined by the Bogoliubov transformation. In the next section, we will see that the normalization factor

gives the ﬁdelity and Loschmidt echo.

B.7.3 Boson ﬁdelity

Fidelity is deﬁned as the (squared) overlap of groundstates of two diﬀerent bosonic Hamiltonians.

We start with a quadratic bosonic Hamiltonian Hˆ0 in the b basis, as in Eq. (B.61). From the discussion in App. B.7.1, we are able to diagonalize it in the a basis for positive deﬁnite M. At t = 0, the Hamiltonian

140 becomes Hˆ1, which is still written in the b basis, but is diagonalized in a new basis c. The corresponding Bogoliubov transformations read

(b, b†)S0 = (a, a†), (b, b†)S1 = (c, c†), (B.77)

and so

1 1 − (a, a†) = (c, c†) S0− S1 . (B.78) One realizes that Eq. (B.78) is another Bogoliubov transformation and so the corresponding matrix has the

block structure

1 u1 v1∗ S1− S0 =   . (B.79) v1 u1∗     Thus 0 c is related to the 0 a in the same way as in Eq. (B.75). Their overlap is therefore given by the | i | i normalization factor 2 1 jk a 0 exp( 2 aj†T ak† ) 0 a 1 0 0 2 = h | − | i = . (B.80) a c | h | i | det(u1) det(u1) | | | | B.7.4 Boson Loschmidt echo

The Loschmidt echo is deﬁned as the (squared) overlap of the evolved state

iHˆ1t 0 e− 0 a, (B.81) | ia(t) ≡ | i

with 0 a the ground state of the Hamiltonian Hˆ before the quench. We introduce a dynamical basis | i 0

iHˆ1t iHˆ1t ai(t) = e− aie , (B.82)

which annihilate the evolved state at time t: a (t) 0 = 0. Upon using the diagonal basis Hˆ = E c†c , i | ia(t) 1 i i i i the Bogoliubov transformation at time t can be represented as a chain of symplectic transformationP

iHt iHt 1 iEt iEt 1 (a(t), a†(t)) = e− (a, a†)e = (a, a†)S0− S1diag(e , e− )S1− S0. (B.83)

It is evident that the evolved state 0 is related to the 0 a in the same way as in Eq. (B.75). The overlap, | ia(t) | i as we have seen in the ﬁdelity case, is the normalization factor of the ”BCS” ground state. It is related to

141 the top left block of the Bogoliubov transformation in Eq. (B.83),

2 iEt iEt 1 (t) = a 0 0 = det(u†e u v†e− v ) − . (B.84) L | h | ia(t)| | 1 1 − 1 1 |

B.7.5 Harmonic chain

In this subsection, we explicitly construct the matrix for the case of the harmonic chain introduced in S Sec. 3.2.2. In the basis deﬁned in Eq. (B.68), the Hamiltonian for 1D harmonic chain is

1 φ 1 φ Hˆ = φ π = φ π V , (B.85) 2 M   2     π I π             where is real symmetric matrix that can be diagonalized as = D2 T . The matrix depends on the V V O O V boundary condition, but positive deﬁniteness is the only requirement here.

The matrix that diagonalizes S M D T =   (B.86) M S D S     is given by the following real symplectic matrix

D1/2 O . (B.87) S ≡  1/2 D−  O    Thus the Hamiltonian is diagonalized as

D 1 M = S−   S, (B.88) D  −    where the Bogoliubov transformation takes the desired block form

O(D1/2 + D 1/2) O(D1/2 D 1/2) 1 − − S = C C− = − . (B.89) S  1/2 1/2 1/2 1/2  O(D D− ) O(D + D− )  −   

142 Appendix C

Appendices for the Operator Entanglement Entropy

C.1 Channel-State Duality

Here we view the opEE in the light of the channel-state duality originated from the quantum information community. We restrict to the unitary channel that is relevant to the opEE. A more detailed account and application can be found for example in Refs. [234, 235, 236, 237, 156].

For any linear operator U expanded in a basis i as | i ∈ H

U = U i j , ij| ih | (C.1) ij X we can always construct a corresponding state ψ in the enlarged Hilbert space | i H × H

ψ = i U j ∗ = i U i ∗. (C.2) | i | i ⊗ ij| i | i ⊗ | i ij i X X

Operationally, we just replace the bra j by a ket j ∗ which is the complex conjugation of the state h | | i j . This choice makes the state ψ basis independent, which can be easily veriﬁed by applying a unitary | i | i transformation V

ψ0 = i0 U i0 ∗ = Vi iV ∗ i U j ∗ | i | i ⊗ | i 0 i0j| i ⊗ | i i i ij X0 X0 (C.3) = δ i U j ∗ = ψ . ij| i ⊗ | i | i ij X So the unique state ψ dual to the unitary operator U contains all its information, and one can study this | i state instead to gain knowledge of the operator.

The dual state is deﬁned on two copies of the original system, and the unitary operator is acting only on one of them. Partitioning of the operator corresponds to an identical space partitions in these two copies of system, which is shown in Fig. C.1. The opEE is then identical to the state EE of the A, B partition for the dual state ψ . We use this picture to analytically compute the average opEE of a random unitary operator | i in App. C.2.

143 A C(=A)

B D(=B)

Figure C.1: Channel state duality point of view of opEE. The vertical lines correspond to the two copies of the original system and the bipartition of the system into A and B has to be performed equally in both copies.

C.2 Average opEE of Random Unitary Operator

In this appendix, we prove that the average opEE of random unitary operator (circular unitary ensemble) is equal to the Page value. Ref. [113] notices that the distribution of Schmidt eigenvalues of a random operator and a random state of doubled system are diﬀerent. However it is argued that in the large system limit, the ”reshuﬄed” matrix should asymptotically follow the same random matrix ensemble and hence will be consistent with numerically calculated Page value. We here present a direct mathematical calculation to prove this point.

We use the standard replica trick and average over the Haar measure [dU] of the unitary group U(N) to compute the EE

n S¯[U] = [dU] ∂nTr(ρ [U]) (C.4) − n=1 Z and further assume that the derivative and integral commute, so that we can compute the average ﬁrst

Tr(ρn) = [dU] Tr(ρn[U]). (C.5) Z

In a chosen basis, the matrix element can be written as UiAjB ,¯iA¯jB , where the combination of iA, jB exhausts the indices for a state (left line of Fig. C.1), and the same for ¯iA, ¯jB (right line of Fig. C.1). We need a partial transpose to obtain the expansion coeﬃcient in the operator basis

U ¯ ¯ U ¯ ¯ , (C.6) iAjB ,iAjB → iAiA,jB jB

where now iA and ¯iA are indexing the Ai basis etc. The density matrix for the operator is then

ρ[U]i ¯i ,i ¯i = Ui ¯i ,j ¯j U ∗ ¯ ¯ , (C.7) A A A0 A0 A A B B jB jB ,iA0 iA0 summing over repeated indices. The diagrammatic representation in Fig. C.2 can guide[156] us to write

144 n n n down the complicated index structure of Tr(ρ [U]). For a 2 2 matrix U ¯ ¯ , the upper two closed × iAjB ,iAjB

1 ¯1 ¯1 1 2 ¯2 ¯2 2 iA iA iA iA iA iA iA iA

U U † U U † 1 ¯1 2 ¯2 ¯2 2 ¯1 1 jB jB jB jB jB jB jB jB

Figure C.2: Diagrammatic representation of Tr(ρ2[U]).

lines on each block represent A region indices iA,¯iA and the lower two closed lines represent B region indices jB, ¯jB. The two ends of connecting lines are contracting indices. So for example, the diagram in Fig. C.2 can be translated to

2 1 1 1 1 2 2 2 2 Tr(ρ [U]) =Ui j ,¯i ¯j U † ¯i1 ¯j2 ,i1 j2 Ui j ,¯i ¯j U † ¯i2 ¯j1 ,i2 j1 A B A B A B A B A B A B A B A B (C.8) =Ui1 j1 ,¯i1 ¯j1 U ∗1 2 ¯1 ¯2 Ui2 j2 ,¯i2 ¯j2 U ∗2 1 ¯2 ¯1 , A B A B iAjB ,iAjB A B A B iAjB ,iAjB

where represents the complex conjugate of the indexed element. ∗ The same type of integral also appears in the discussion of Haar scrambling in Ref. [173] and [156], where

the n = 2 case is calculated by the Weingarten formula to obtain the R´enyi entropy. We here apply the

general Weingarten formula for the integration on the unitary group,

[dU]Ui ,j Ui ,j ...Ui ,j U ∗ U ∗ ...U ∗ 1 1 2 2 n n i10 ,j10 i20 ,j20 in0 ,jn0 Z (C.9) 1 = δ δ . . . δ δ δ . . . δ Wg(N, στ − ), i1iσ0 (1) i2iσ0 (2) iniσ0 (n) j1jτ0 (1) j2jτ0 (2) jnjτ0 (n) σ,τ S X∈ n

L where the sum is taken over all possible permutations in Sn, and N is the size of the matrix 2 . Wg is the Weingarten function (see detailed deﬁnition and the ﬁrst few examples in Ref. [230]), whose large N

limit[230] is given by

where the C are the cycle decomposition of σ, C are the number of elements in this cycle, Catalan is the i | i| i ith Catalan number, σ is its Cayley distance to the identity (minimal number of transpositions that makes | | it identity). Obviously, the dominant term in N limit is the one with σ = e → ∞

1 n 2 Wg(N, e) = + (N − − ). (C.11) N n O

145 Consequently, in the integration of the U only σ = τ are relevant, i.e. terms whose i index and j index share

the same permutation

1 [dU] δi i δi i . . . δi i δj j δj j . . . δj j . (C.12) · · · ∼ N n 1 σ0 (1) 2 σ0 (2) n σ0 (n) 1 σ0 (1) 2 σ0 (2) n σ0 (n) σ S Z X∈ n

The contractions of these delta functions can be converted to loop counting in planar diagrams. Let us illustrate the example of n = 2

2 [dU]Tr(ρ [U]) = [dU] Ui1 j1 ,¯i1 ¯j1 U ∗1 2 ¯1 ¯2 Ui2 j2 ,¯i2 ¯j2 U ∗2 1 ¯2 ¯1 , (C.13) A B A B iAjB ,iAjB A B A B iAjB ,iAjB Z Z

where the four indices may be represented as the lids in Fig. C.3. After doing the integration, the delta

iA,¯iA jB, ¯jB

Figure C.3: Eight indices in Tr(ρ2[U]), where contractions are performed for the 4 pairs. functions for each permutation element σ will close these diagrams. For example when σ = (12), there are

1 2 loops for A indices and 2 2 loops for B indices, and then the corresponding factor is 22`A+4`B . × ×

iA,¯iA jB, ¯jB σ = (12) σ = (12)

Figure C.4: Delta functions for each permutation element σ will close these diagrams. Each loop will contribute a (2`A )2 or (2`B ), where the square is for two copies of indices.

The loop counting is a combinatorial problem that can be formulated in terms of its generating function.

Let

χ(g) χ(τg) fn(x, y) = x y , (C.14) g S X∈ n where χ(g) is the number of cycles in the permutation and τ = (12 n). The average of the trace can be ··· expressed by this polynomial

1 1 n À `B À `B Tr(ρ [U]) = fn(4 , 4 ) = fn(4 , 4 ). (C.15) (22À+2`B )n 4nL

At this point, we can apply Page’s state result as a shortcut. For a random state, the component of the wavefunction is

ψ¯iA¯jB = U1,¯iA¯jB , (C.16)

146 where U is again taken from the Haar measure. To contrast, we write down the state version of the integral

for n = 2

2 [dU]Tr(ρ [ψ]) = [dU] U1,¯i1 ¯j1 U ∗¯1 ¯2 U1,¯i2 ¯j2 U ∗¯1 ¯1 . (C.17) A B 1,iAjB A B 1,iAjB Z Z The whole process using the Weingarten formula and its asymptotics can be similarly applied; the only

diﬀerence is that the state has no unbarred set of indices

1 Tr(ρn[ψ]) = f (2`A , 2`B ). (C.18) 2nL n

So the opEE will be the Page value of a state with length 2L and partition 2`A + 2`B = 2L, i.e.

2` 2` 1 S[U] = 2` ln 2 2 A− B − . (C.19) A −

We also manage to do a direct combinatorial computation for equal partition, where the top coeﬃcient of the generating function[238] an (cf. Ref. [239] for why the top power is n + 1 and the concept of genus)

f (x, x) = xχ(g)+χ(τg) = a xn+1 + (C.20) n n ··· g S X∈ n

determines the trace 1 L L an 1 n 2 2 Tr(ρ [ψ]) = nL fn(4 , 4 ) = (n 1)L + ( nL ). (C.21) 4 2 − O 2

By analytic continuation, the EE is

1 S[U] = L ln 2 ∂nan + ( ). (C.22) − n=1 O 2L

Through a series of bijections, one can show that an is the Catalan number (exercise 118 of Refs. [238], [240]) 2n! a = . (C.23) n n!(n + 1)!

Therefore 2n! 1 1 ∂n = ∂n = (C.24) n!(n + 1)! n=1 n(n + 1)B(n, n + 1) n=1 2

gives us the correct deﬁcit of the Page value

1 S = L ln 2 . (C.25) − 2

147 C.3 Lin Table Algorithm for Sz = 0 Sector

z We take the σz basis in the Hilbert space, each of which are eigenvectors of total i σi . Consider the subspace where the eigenvalue of the total Sz is zero. Each basis state then has an equalP number of spins ↑ and spins. In a 2N-site system, the dimension of the subspace is 2N . ↓ N For any state in this subspace, we can take advantage of the constraint to do a partial Schmidt decomposition

ij ψ = ψn n , i A n n , j B, (C.26) | i ↑ | ↑ i | − ↑ i n ij X↑ X ij where ψn is a block diagonal matrix, the number of up spins n in part A is the block index and ij are the ↑ ↑ N 2 row and column indices within the block. The dimension of each block is n , and the identity ↑ N N 2 2N = (C.27) n N n =0 ↑ X↑

z ij ensures that the coeﬃcients from the σ basis wavefunction to the block elements ψn is just a permutation. ↑ We use a 2N-bit binary number to represent the σz basis. In the example of N = 3, the σz basis

wavefunction elements are

000 001 010 011 100 101 110 111   000 ψ1     001 ψ2 ψ3 ψ4      010 ψ5 ψ6 ψ7        (C.28) 011 ψ8 ψ9 ψ10      100 ψ11 ψ12 ψ13      101 ψ14 ψ15 ψ16        110 ψ17 ψ18 ψ19      111 ψ20     

148 ij and the corresponding ψn matrix is ↑

111 (011 101 110) (001 010 100) 000   000 ψ1     001 ψ2 ψ3 ψ4      010 ψ5 ψ6 ψ7        . (C.29) 100 ψ11 ψ12 ψ13      011 ψ8 ψ9 ψ10      101 ψ14 ψ15 ψ16        110 ψ17 ψ18 ψ19      111 ψ20      ij If we store ψn in a row vector, then the index of ψ will be ↑

(1, 2, 3, 4, 5, 6, 7, 11, 12, 13, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20). (C.30)

The permutation element we are looking for in this n = 3 example is (8, 11), (9, 12), (10, 13). The algorithm needs to ﬁgure out the conversion table from the σz basis elements to the block elements and then do the

Schmidt decomposition for each block, which is much more eﬃcient than doing it in the full Hilbert space.

We note that this can be done eﬃciently using the method of Lin tables as pointed out by H. Lin in Ref. [241].

149 Appendix D

Appendices for the Random Tensor Network Problem

1 D.1 Combinatorial Calculation of Z3(t) at Order q2

In this section, we use a combinatorial technique to calculate the partition function Z3 and the entanglement

1 velocity v3 to order q2 . In doing this, we deﬁne and calculate a slightly more general function Ω3(t, q) which 1 takes the q2 correction into account. 2 2 2 2 According to the exact result in App. D.6, the two walks in Z have weight K q− e− q when they 3 ' are separate. When they meet, there are two types of interactions:

1. The leading order attractive interaction resulting from the 3 instead of 2 ways to split in Eq. (5.34).

3 2 2 2. A weak repulsive interaction when the two walks are overlapping. Such a step has weight q− e− q .

Taking this into account, we can write the partition function as

2t 2t 2 Z3(t) = q− e− q Ω3(t, q), (D.1)

where Ω3(t, q) is the following partition function for two walks

# splitting 3 1 # overlapping steps Ω (t, q) = e− q2 . (D.2) 3 2 conﬁgs of 2X walks

The entanglement velocity is 1 ln Ω3(t, q) v3 = 1 + lim . (D.3) 2 t q ln q − →∞ 2t ln q

Instead of counting Ω3(t, q), we consider the partition function where each step is assigned a fugacity √x

∞ t Ω3(√x, q) = Ω3(t, q)x 2 . (D.4) t=0 X

150 D.1.1 q = ∞ 1 If we neglect the weak repulsive interaction at q2 order, Ω3(t, q) becomes Ω3(t) deﬁned in Eq. (5.36). The corresponding partition function is

r 3 t Ω (√x, q = ) = √x , (D.5) 3 ∞ 2 t conﬁgs of X 2X walks

where r is the number of splittings.

It is simpler to consider the relative motion of the two walks. There are 3 possible displacements in a

single time step: 0, 2. A displacement of ∆ = 0 means that the two walks move in the same direction (left ± or right), while a displacement ∆ = 2 means moving in opposite directions. ± Let t be the number of time steps in which ∆ = 0 and t0 be the number of time steps in which ∆ = 2. 0 ± We can immediately perform the sum over t by noting that, in between one ∆ = 2 step and the next, 0 ± there can be an arbitrary number of ∆ = 0 steps. The sub-partition function for these steps is

∞ 1 (2√x)n = , (D.6) 1 2√x n=0 X − where the 2 represents the two choices of the center of mass going left or right. This leaves a partition

function for a single random walk (with steps of 2) representing the relative displacement, ±

3 r √x t0 Ω (√x, q = ) = 0 1. (D.7) 3 ∞ 2 1 2√x r Xt0 X − oneX walk Where the prime indicates that the walks can now only take ∆ = 2 steps, and where we sum over ± conﬁgurations with the speciﬁed t0, r.

3 To simplify, we assign 2 to the meeting event (when the single walk returns to the origin) and assume the two walks to meet at the end. This does not aﬀect the asymptotic behavior. The ﬁnal sum in the above

equation is the number of such single walks that return to the origin r times, which we denote Z(t0, r). This is [250] r2r 2n r Z(t0 = 2n, r) = − . (D.8) 2n r n r − − It has a generating function

∞ n r Z(t0 = 2n, r)y = [f(y)] , (D.9) n=r X

151 where

f(y) = 1 1 4y. (D.10) − − p Therefore 3 r x r Ω (√x, q = ) = f 3 ∞ 2 (1 2√x)2 r X − (D.11) 1 = . 3 x 1 2 f (1 2√x)2 − − We read oﬀ the smallest pole √x = 3√2 4 which determines the asymptotic behavior ∗ −

1 t 3 t Ω3(t) = 2 + . (D.12) ∼ √x √2 ∗

D.1.2 q large but ﬁnite

The analysis is the same except that the sub-partition function for a given string of consecutive ∆ = 0 steps is diﬀerent depending on whether the relative coordinate is zero or not. If it is, the sub-partition function is modiﬁed to ∞ 1 1 q2 n (2e− √x) = 1 . (D.13) 2 n=0 1 2e− q √x X −

The modiﬁcation to Eq. (D.7) is that there are r of these factors, and t0 r of the factors we had before. −

r t0 3 1 2√x √x 0 Ω3(√x, q = ) = − 1 1 ∞ 2 q2 1 2√x r 1 2e− √x! Xt0 X − − oneX walk 1 (D.14) = . 3 1 2√x x − 1 2 1 f (1 2√x)2 − 1 2e− q2 √x − − The pole satisﬁes 4 2 (1 + )x + (8 )√x 2 = 0. (D.15) q2 − q2 −

2 1 The smaller pole √x = 3√2 4 + (3 2√2) 2 determines the asymptotic behavior ∗ − − q

1 t 3 1 t Ω3(t, q) = 2 + . (D.16) ∼ √x √2 − 2q2 ∗

This gives the partition function Z (t) and velocity to order ( 1 ) 3 O q2 ln q

ln 2 + 3 √2 3√2 1 v = 1 + . (D.17) 3 − 2 ln q 4 q2 ln q

152 D.2 Slope-dependent Line Tension (v) E3 In this section, we derive the slope-dependent line tension (v) of the n = 3 bond state, which generalizes E3 the partition function Ω3(t) in App. D.1. To this end we must obtain the free energy of the walks as a function of their coarse-grained velocity.

Let x be the total displacement of the bound state, i.e. the mean displacement of the two walks, and let

Ω3(x, t) be the partition function with a ﬁxed displacement. We expect

2t 2 ln q (v)t Ω (vt, t)q− e− E3 (D.18) 3 ∼ in other words

(v) 2 ln q ln Ω (vt, t). (D.19) E3 ∼ − 3

Consider the generating function of Ω3(x, t)

x Ω˜ 3(φ, t) = Ω3(x, t)φ , (D.20) x X

1 where we assign weight φ for the mean displacement to go one step right and φ− for one step left.

We break up the sum according to the number t0 of time steps where the relative displacement is 0. The

1 steps with relative displacement 0 can change the mean displacement by φ± , while the steps with relative displacement 2 do not change the mean displacement. We therefore have ±

t t − 0 t t 2 t 1 t0 t 1 t0 9 Ω˜ (φ, t) = φ + φ− Z(t t ) φ + φ− 3 t − 0 ∼ t 2 t =0 0 t =0 0 X0 X0 (D.21) t 1 3 φ + φ− + . ∼ √2

If we regard Ω˜ 3(φ, t) as the partition function for a modiﬁed ensemble, the total displacement of the bound state, i.e. the mean displacement of the two walks, is

1 ∂ φ φ− ln Ω˜ 3(φ, t) = t − . (D.22) ∂ ln φ × φ + φ 1 + 3 − √2

Then the mean velocity is 1 φ φ− v(φ) = − , (D.23) φ + φ 1 + 3 − √2

153 which we can solve for the fugacity 3v + √8 + v2 φ = . (D.24) √8(1 v) − By saddle-point reasoning,

Ω˜ (φ, t) Ω (v(φ)t, t)φv(φ)t. (D.25) 3 ∼ 3

Therefore 1 3 1 3(v) =1 ln( + φ− + φ) v ln φ E − 2 ln q √2 − (D.26) 1 3v2 + √v2 + 8 3 3v + √8 + v2 =1 + v ln . − 2 ln q √2(1 v2) √2 − √8(1 v) ! − − D.3 Line Tension (v) Close to Lightcone E3 In this section we calculate (v) for v very close to the lightcone v = 1. Writing v = 1 α , with α of order E3 − q2 1 and q large, we obtain (1 α ) up to terms of order 1/q2 ln q: see Eqs. (5.47), (5.48) and (5.49) in the E3 − q2 main text. This allows a nontrivial check on the relation (v ) = v , and reveals an unbinding transition E3 B B for the two walks appearing in the partition function Z3 when the boundary conditions are modiﬁed so that their coarse-grained speed exceeds a critical value v 1 2/q2. This is consistent at this order with c ' − vc = vB, which we conjecture is true to all orders. To the order at which we are working we can neglect interactions between replicas. Then

ln Z(vt; t) 3(v) = lim − , (D.27) E t 2 ln q t →∞ ×

where Z(vt; t) is a partition function for two walks with the constraint that the displacement of their centre

of mass is vt. For deﬁniteness we can take their relative coordinate ∆ (the diﬀerence in the coordinates of

the two walks) to be zero at the initial and ﬁnal time.

From the exact triangle weights in Appendix. D.6, to relative order 1/q2, the weight is

2 2/q2 q− e− (D.28)

for a time step in which the walks are separate, and

2 3/q2 q− e− (D.29)

154 for a time step in which the walks are in a composite walk. Finally, for a time step in which the walks either split or merge we may take the weight to be

3 2 2/q2 q− e− . (D.30) r2 ×

Here we have shared the statistical weight 3/2 for each merge-split event (see Sec. 5.3.2) equally between the splitting event and the merging event. We neglect boundary terms, which are unimportant in the t → ∞ limit.

In order to ﬁx the velocity v, we introduce a ‘fugacity’ φ for leftward steps. Since v is close to one almost all steps are rightward, and the fugacity φ will be small. If Z(φ; t) is the partition function with fugacity

φ but with no constraint on the total displacement of the centre of mass, and if v(φ) = 1 α(φ)/q2 is the − average speed in the ensemble with ﬁxed φ, then

2 Z(φ; t) Z(v(φ); t)φα(φ)t/q . (D.31) ∼

In the present regime only an (1/q2) fraction of the time steps involve a walk taking a step to the left. We O can neglect conﬁgurations in which both walks take a step to the left in the same time step, since such an event occurs only once in every (1/q4) time steps. O The conﬁguration is then determined entirely by the relative displacement ∆ as a function of time, and

we can write Z(φ, t) in terms of a transfer matrix T∆,∆0 . This transfer matrix contains a factor of φ for each time step in which ∆ = ∆0, since in such a time step one of the walks takes a step to the left: 6

2t 2t/q2 t Z(φ; t) = q− e− T 0,0 (D.32) with (expanding in 1/q2)

...   1 φ      φ 1 φ 3/2     2  T =  φ 3/2 1 pq− φ 3/2  . (D.33)  −     p φ 3/2p 1 φ       p   φ 1       ...    

155 Let us deﬁne the (1) quantity O Φ = q2φ. (D.34)

For Φ > 1 the largest eigenvalue of the transfer matrix, determining the scaling of the partition function, is

3√1 + 8Φ2 1 λ = 1 + − , (D.35) 4q2

corresponding to the bound state ‘wavefunction’ ψ∆ with

2 3 ∆ 1 + √1 + 8Φ ψ0 = 1, ψ∆=0 = µ| |, and µ = . (D.36) 6 r2 4Φ

The bound state exists for µ < 1, i.e. for Φ > 1. At Φ = 1 the bound state disappears. For the range of

velocities where Φ < 1, when the walks are unbound, their typical separation is √t at large t. They are therefore eﬀectively independent and their free energy is twice that of a single walk, leading to (v) = (v). E3 E2 It is straightforward to check that Φ = 1 corresponds to α = 2: for Φ 1 the walks can be treated as c ≤ independent, and v is simply related to the weight φ = Φ/q2 for a left step by v = 1 2φ. − For the range of velocities where Φ > 1, Eqs. (D.31), (D.35) together with ln(T t) t ln λ give 00 ∼ ×

2 α 1 9 3√1 + 8Φ α 2 3(v) 1 2 + 2 + ln Φ . (D.37) E ' − q q ln q 8 − 8 4 !

We still need to relate Φ and v.

In the bound region, we note that v is equal to the probability that in a given time step the change in

∆ is zero. The sum over such conﬁgurations is obtained by replacing T with Tdiag for the given time step,

where Tdiag is the diagonal part of T . This leads to

1 ψ T ψ v = h | diag| i. (D.38) λ ψ ψ h | i

Using Eq. (D.36) gives

1 6Φ2(1 + √1 + 8Φ2) α2 9 v =1 , Φ2 = 2 + 4 + . 2 2 2 2 − q 1 + 8Φ + √1 + 8Φ 18 r α !

Together with Eq. (D.37) this gives Eqs. (5.48), (5.49) in the main text.

156 D.4 The Weingarten Function

In this section, we introduce the properties of the Weingarten function used in the main text.

We begin with the general formula for the average of the tensor product of a Haar random unitary[229,

230]

[dUd d]Ui1,j1 Ui2,j2 ...Uin,jn Ui∗ ,j Ui∗ ,j ...Ui∗ ,j × 10 10 20 20 n0 n0 Z (D.39) 1 = δ . . . δ Wg(d, στ − )δ . . . δ . i1iτ0 (1) iniτ0 (n) j1jσ0 (1) jnjσ0 (n) σ,δ S X∈ n In the main text, d = q2 and we pack formula compactly in the bracket notation (Eq. (5.18)), where the products of delta functions are identiﬁed as components of the permutation states σ and τ . | i | i The Weingarten function Wg(σ) Wg(q2, σ) is a function of the conjugacy class of the permutation. Its ≡ deﬁning property can be obtained from the left/right invariance of the Haar ensemble,

= . (D.40)

Translating this into algebra, we have

1 1 N τ1− σ1 1 1 Wg(στ1− )d −| |Wg(σ1τ − ) = Wg(στ − ). (D.41) τ σ X1 1

1 If we regard Wg(στ − ) as an invertable matrix with σ and τ as its row and column indices, then

1 1 N (τ − σb) Wg(σaτ − )d − = δσaσb . (D.42) τ X

1 1 N (σa− σb) Therefore Wg(σaσb− ) is the inverse of d − . This is the key to all the exact weights, see App. D.5. The Weingarten function can be expanded perturbatively, and the leading order term for each permuta-

tion is [230] 1 Moeb(σ) 1 Wg(σ) = + , (D.43) dN d σ O d σ +2 | | | | where the M¨obiusfunction for a permutation with cycle decomposition σ = c c c is deﬁned as 1 2 ··· k

k ci 1 Moeb(σ) = Catalan c 1( 1)| |− . (D.44) | i|− − i=1 Y

157 Some elementary examples are

Moeb(I) = 1, Moeb((12)) = 1, (D.45) − Moeb((12)(34)) = 1, Moeb((123)) = 2. (D.46)

For the convenience of the perturbative calculation, we deﬁne

wg(σ) = dN Wg(σ). (D.47)

1 1 Up to d2 ( q4 ) order, the only non-vanishing wg functions are

N 1 wg(I) = 1 + 2 + , d2 O d3 1 1 wg((12)) = + , −d O d3 (D.48) 2 1 wg((123)) = + , d2 O d3 1 1 wg((12)(34)) = + , d2 O d4 where the particular permutations inside, like (12), are representatives of their conjugacy classes. The last three relations come from the leading term expansion of Wg. The ﬁrst one can be worked out by subtracting

1 all the other d2 terms from the sum [251]

dN (d 1)! wg(σ) = − . (D.49) (d + N 1)! σ X −

D.5 Exact Weights with 1 Incoming Domain Wall ≤ In this section, we derive the exact wight of some down-pointing triangles by using the orthogonality relation in Eq. (D.42). We will denote the number of cycles in a permutation by χ(σ) = N σ . − | | First, according to the deﬁnition in Eq. (5.24)

1 1 2N 2 σ− τ J(σb, σb; σa) = Wg(σaτ − )q − | b |. (D.50) τ X

Comparing this with the orthogonality relation in Eq. (D.42) and setting q2 = d, we obtain

J(σb, σb; σa) = δσa,σb . (D.51)

158 Next, we consider the weight of a single domain wall

χ(τa)+χ(τa(12)) = Wg(τa)q . (D.52) τ Xa

We deﬁne

χ(τa)+χ(τa) Σ± = Wg(τa)q (D.53) χ(τ (12))=χ(τ ) 1 a X a ± then

+ 1 K = = qΣ + Σ−. (D.54) q

By taking σ = I and (12) in the variant of the orthogonality relation

χ(δσ) Wg(δ)d = δI,σ, (D.55) Xδ we have + Σ + Σ− = 1 (D.56) + 1 dΣ + Σ− = 0. d The solution is 2 + 1 d Σ = − , Σ− = . (D.57) d2 1 d2 1 − − Therefore q q3 q K = − + = . (D.58) q4 1 q4 1 1 + q2 − − Further, we consider a single domain wall that creates a pair of new (possibly composite) domain walls

χ(τa)+χ(τa(12)) K = = Wg(τaµ)q . (D.59) ∧ 1 τ µ µ− Xa

We deﬁne

χ(τa)+χ(τa) Σ± = Wg(τaµ)q , (D.60) χ(τ (12))=χ(τ ) 1 a X a ± then

+ 1 K = qΣ + Σ−. (D.61) ∧ q

159 On the other hand, from the orthogonality relation

χ(τa)+χ(τa) χ(τa(12))+χ(τa(12)) Wg(τaµ)q = Wg(τaµ)q = 0 (D.62) τ τ Xa Xa we have

+ + 1 Σ + Σ− = 0 dΣ + Σ− = 0 = Σ± = 0. (D.63) d ⇒

Hence

K = 0. (D.64) ∧

We conclude that q J(I, (12); σa) = δ ,σa + δ(12),σa . (D.65) q2 + 1 I D.6 Exact Weights for N = 3

3 This section presents the exact weights J(σb, σc; σa) for Z1 (the single replica of S3). There are 6 elements in the order 3 permutation group: I, (1, 2), (1, 3), (2, 3), (1, 2, 3), (1, 3, 2). The relevant Weingarten functions are (see for example [252]; d = q2)

q4 2 Wg(d, [1, 1, 1]) = − q2(q4 1)(q4 4) − − q2 Wg(d, [1, 2]) = − (D.66) q2(q4 1)(q4 4) − − 2 Wg(d, [3]) = , q2(q4 1)(q4 4) − − where the numbers inside the square brackets are the cycle sizes of the permutation. Then J(σb, σc; σa) becomes

160 additional non-trivial weights

(123) I q4 2q2 2 = − − (q2 + 1)(q4 4) − I (12) (23) q2(q2 1) = − (q2 + 1)(q4 4) (D.68) − I (123) (132) 2(q2 1) = − − . (q4 4)(q2 + 1) − I

D.7 Perturbative Calculation of the Triangle Weights

In this section, we present the perturbative calculation of the weight of a down-pointing triangle.

The weight of the down-pointing triangle is obtained by integrating out the τ spin. Formally

1 1 1 σ− τ τ − σc J(σb, σc; σa) = wg(τ − σa)q−| b |−| |. (D.69) τ S X∈ N

To represent this in diagrams, we put the τ spin in the center of the triangle and use dashed lines to connect

1 the τ spin and the three neighboring σs. The links between τ and σc or σb give an exact factor q for each 1 elementary domain wall, and the link between τ and σa gives wg(τ − σa).

First consider K (which we know exactly). The leading order diagram is the one where τ = σa, and we have 1 1 1 K = = + ( ) = + ( ). (D.70) O q2 q O q2

Now using the higher order series expansion of wg in Eq. (D.48), we can obtain a more accurate value of K

1 K = = + + + ( ) O q6

1 1 N 1 1 = wg(d, I) + wg(d, (12)) + 2 − wg(d, (13)) + ( ) q q q3 O q7 (D.71) N N 1 1 1 2 1 1 = 1 + 2 − + ( ) 4 2 2 2 7 q  q ! − q − q q  O q 1  1 1 1  = 1 + + ( ) . q − q2 q4 O q6

161 Here we see that the number of choices for the elementary domain wall on the vertical link in the last diagram cancels the N dependence in the expansion of wg(I) from the ﬁrst diagram, generating an N independent weight, which is consistent with the exact result in Eq. (D.58).

Now we consider two commutative incoming and outgoing domain walls (12) and (34), which is relevant

k to evaluating Z2 , = + 2 + +

1 1 1 = wg(d, I) + 2 wg(d, (12)) + wg(d, (12)(34)) q2 q2 q2 1 N 1 + wg(d, (12)) 2 + ( ) q4 2 − O q8 N N (D.72) 1 2 1 1 2 1 2 2 − = 2 1 + 4 2 + 4 2 2 + ( 6 ) q " q − q ! q − q q O q # 1 2 3 1 1 1 1 1 2 = 1 + + ( ) = 1 + + ( ) q2 − q2 q4 O q6 q2 − q2 q4 O q6 1 = 1 + ( ) . × × O q6

The calculation for the outgoing domain walls exiting in the opposite directions is similar. We thus obtain the factorization condition in Eq. (5.58).

The factorization fails if the incoming domain wall is a product of non-commutative transpositions. We take it to be (123), which is relevant to S3. There are now 3 ways to assign one elementary domain wall to the vertical link, and the weight wg(d, 123)) is 2/q4. Taking these into account, we have

= + 3 + +

1 1 1 = wg(d, I) + 3 wg(d, (12)) + wg(d, (123)) q2 q2 q2 1 n 3 1 + wg(d, (12)) + ( ) q4 2 − 2 O q8 n 1 2 1 2 = 2 1 + 4 + 2 3 + 4 (D.73) q q ! −q × q h 1 1 n 1 + 3 + ( ) q2 −q2 2 − O q6 i 1 1 1 1 2 1 1 = 1 + + ( ) 1 + ( ) q2 − q2 q4 O q6 − q2 O q6 1 1 = 1 + ( ) . × × − q2 O q6

162 We see that the factor of 1 1 + ( 1 ) gives rise to a repulsive interaction between the domain walls. − q2 O q6 Next we turn to correctionsh from addingi ‘bubbles’ to the domain wall conﬁgurations as in Fig. D.1.

Many such bubble conﬁgurations vanish due to the exact results in Eq. (D.51) and Eq. (D.64). The leading non-trivial diagrams corresponds to conﬁgurations (e) and (f) in Fig. D.1. Na¨ıve domain wall number

1 counting suggests a bubble is an order q4 correction to the diagram without the bubble. It is however at 1 most an order q6 correction if the bubble is created simply by adding a closed loop of a given domain wall type. For example, consider the bubble corrections to the leading diagram for K . Let the incoming domain ∧

(a) (c) (e)

(b) (d) (f)

Figure D.1: Possible bubble diagrams. Each line represents an elementary domain wall. (a) (b) (c) (d) have 0 weight: (a) and (b) vanish because the tip of the hexagon is J(I, I; (12)) = 0; (c) and (d) vanish because 1 1 on the top K = 0. (e) is an order 6 correction. (f) can be an order 6 correction if the dashed loop is an ∧ q q 1 elementary domain wall. It is an order q4 correction if it is a special hexagon as in Eq. (5.61).

wall be (12) and the outgoing ones be (12)(34) and (34). We can always choose a vertical link carrying (34)

to cancel the leading order diagram

1 1 = + + ( ) = ( ) (D.74) O q5 O q5

so that it is consistent with the exact result K = 0 in Eq. (D.64). The cancellation mechanism also exists ∧ for two commutative incoming domain walls in Eq. (5.59):

1 1 = + + ( ) = ( ). (D.75) O q6 O q6

When the newly generated domain wall pair annihilates in the time step immediately below, this gives an

1 order q6 correction in the bulk (Fig. D.1 (f)). In contrast, the special hexagons in Eq. (5.61) do not suﬀer 1 from the cancellation mechanism. As a result they are order q4 corrections in the bulk and lead to the k dominant pairwise attraction in Z2 .

163 D.8 Continuum Interaction Constants

Continuing from the discussion in Sec. 5.6.3, we use Z(k) to denote the partition function for k bosons

on a ring, or equivalently k walks on a torus. (Note that these BCs are not related to the entanglement

calculation.) To ﬁx λ we take L and t large enough that the continuum approximation is valid but small

enough that the interaction may be treated as as a perturbation: this is possible when λ 1. If ∆E is the change in the ground state energy of a pair of bosons when the small interaction is switched on

(2) (1) 2 t∆E then Z /[Z ] = e− . Since in the noninteracting problem the ground state wavefunction is spatially constant, λ λ ∆E = dx dx δ(x x ) = . (D.76) −L2 1 2 1 − 2 −L Z On the other hand, on the lattice

Z(2) 1 n (n 1) = exp b b − , (D.77) [Z(1)]2 2q4 2 * vertical + bondsX b

where the expectation value is taken for a pair of noninteracting walks on D. Expanding the exponential, and using translational invariance in both dimensions,

Z(2) tL 1 1 + P . (D.78) [Z(1)]2 ' 2 2q4 meet

Here tL/2 is the number of vertical bonds on the square lattice D, and Pmeet is the probability that a

2 given bond is visited by both walks. Using the independence of the walks, this is Pmeet = 1/L . Matching Eqs. (D.78), (D.76) gives λ = 1/(4q4), as stated in the main text.

164 References

[1] Pasquale Calabrese and John Cardy. Quantum Quenches in Extended Systems. Journal of Statistical Mechanics: Theory and Experiment, 2007(06):P06008–P06008, June 2007. arXiv: 0704.1880. [2] Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger. Many-body physics with ultracold gases. Reviews of Modern Physics, 80(3):885–964, July 2008.

[3] Toshiya Kinoshita, Trevor Wenger, and David S. Weiss. A quantum Newton’s cradle. Nature, 440(7086):900–903, April 2006. [4] Beln Paredes, Artur Widera, Valentin Murg, Olaf Mandel, Simon Flling, Ignacio Cirac, Gora V. Shlyapnikov, Theodor W. Hnsch, and Immanuel Bloch. Tonks-Girardeau gas of ultracold atoms in an optical lattice. Nature, 429(6989):277–281, May 2004.

[5] Aditi Mitra. Quantum quench dynamics. arXiv:1703.09740 [cond-mat], March 2017. arXiv: 1703.09740. [6] Rahul Nandkishore and David A. Huse. Many body localization and thermalization in quantum statistical mechanics. Annual Review of Condensed Matter Physics, 6(1):15–38, March 2015. arXiv: 1404.0686.

[7] Fabien Alet and Nicolas Laﬂorencie. Many-body localization: an introduction and selected topics. arXiv:1711.03145 [cond-mat], November 2017. arXiv: 1711.03145. [8] David J. Luitz, Nicolas Laﬂorencie, and Fabien Alet. Extended slow dynamical regime close to the many-body localization transition. Phys. Rev. B, 93(6):060201, February 2016.

[9] Song He, Tokiro Numasawa, Tadashi Takayanagi, and Kento Watanabe. Quantum Dimension as Entanglement Entropy in 2d CFTs. Physical Review D, 90(4), August 2014. arXiv: 1403.0702. [10] Adam Nahum, Sagar Vijay, and Jeongwan Haah. Operator Spreading in Random Unitary Circuits. arXiv:1705.08975 [cond-mat, physics:hep-th, physics:quant-ph], May 2017. arXiv: 1705.08975. [11] Curt von Keyserlingk, Tibor Rakovszky, Frank Pollmann, and Shivaji Sondhi. Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws. arXiv:1705.08910 [cond-mat, physics:quant-ph], May 2017. arXiv: 1705.08910. [12] Pasquale Calabrese and John Cardy. Entanglement Entropy and Quantum Field Theory. Journal of Statistical Mechanics: Theory and Experiment, 2004(06):P06002, June 2004. arXiv: hep-th/0405152.

[13] L. A. Rubel. Necessary and suﬃcient conditions for Carlsons theorem on entire functions. Transactions of the American Mathematical Society, 83(2):417–429, 1956. [14] Mark Srednicki. Entropy and area. Physical Review Letters, 71(5):666–669, August 1993. [15] M. B. Hastings. An area law for one-dimensional quantum systems. Journal of Statistical Mechanics: Theory and Experiment, 2007(08):P08024, 2007.

165 [16] Jens Eisert, Marcus Cramer, and Martin B. Plenio. Colloquium: Area laws for the entanglement entropy. Reviews of Modern Physics, 82(1):277, 2010. [17] Pasquale Calabrese and John Cardy. Entanglement entropy and conformal ﬁeld theory. Journal of Physics A: Mathematical and Theoretical, 42(50):504005, December 2009.

[18] Shinsei Ryu and Tadashi Takayanagi. Aspects of holographic entanglement entropy. Journal of High Energy Physics, 2006(08):045, 2006. [19] Fernando Pastawski, Beni Yoshida, Daniel Harlow, and John Preskill. Holographic quantum error- correcting codes: Toy models for the bulk/boundary correspondence. Journal of High Energy Physics, 2015(6), June 2015. arXiv: 1503.06237.

[20] Patrick Hayden, Sepehr Nezami, Xiao-Liang Qi, Nathaniel Thomas, Michael Walter, and Zhao Yang. Holographic duality from random tensor networks. arXiv preprint arXiv:1601.01694, 2016. [21] Pasquale Calabrese and John Cardy. Entanglement and correlation functions following a local quench: a conformal ﬁeld theory approach. Journal of Statistical Mechanics: Theory and Experi- ment, 2007(10):P10004–P10004, October 2007. arXiv: 0708.3750.

[22] Pasquale Calabrese and John Cardy. Evolution of Entanglement Entropy in One-Dimensional Systems. Journal of Statistical Mechanics: Theory and Experiment, 2005(04):P04010, April 2005. arXiv: cond- mat/0503393. [23] Pasquale Calabrese and John Cardy. Quantum quenches in 1+1 dimensional conformal ﬁeld theories. arXiv:1603.02889 [cond-mat, physics:hep-th], March 2016. arXiv: 1603.02889. [24] Hong Liu and S. Josephine Suh. Entanglement Tsunami: Universal Scaling in Holographic Thermal- ization. Physical Review Letters, 112(1), January 2014. arXiv: 1305.7244. [25] Mrk Mezei. On entanglement spreading from holography. arXiv:1612.00082 [cond-mat, physics:hep-th], November 2016. arXiv: 1612.00082.

[26] Mrk Mezei and Douglas Stanford. On entanglement spreading in chaotic systems. Journal of High Energy Physics, 2017(5), May 2017. arXiv: 1608.05101. [27] David Perez-Garcia, Frank Verstraete, J. Ignacio Cirac, and Michael M. Wolf. PEPS as unique ground states of local Hamiltonians. arXiv:0707.2260 [cond-mat, physics:math-ph, physics:quant-ph], July 2007. arXiv: 0707.2260. [28] Roman Orus. A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States. Annals of Physics, 349:117–158, October 2014. arXiv: 1306.2164. [29] Masahiro Nozaki, Tokiro Numasawa, and Tadashi Takayanagi. Quantum Entanglement of Local Op- erators in Conformal Field Theories. Physical Review Letters, 112(11), March 2014. arXiv: 1401.0539.

[30] Bin Chen, Wu-Zhong Guo, Song He, and Jie-qiang Wu. Entanglement Entropy for Descendent Local Operators in 2d CFTs. arXiv:1507.01157 [hep-th], July 2015. arXiv: 1507.01157. [31] Pawel Caputa and Alvaro Veliz-Osorio. Entanglement constant for conformal families. Physical Review D, 92(6), September 2015. arXiv: 1507.00582.

[32] Wu-Zhong Guo and Song He. R ’enyi entropy of locally excited states with thermal and boundary eﬀect in 2d CFTs. Journal of High\ Energy Physics, 2015(4), April 2015. arXiv: 1501.00757. [33] Pawel Caputa, Masahiro Nozaki, and Tadashi Takayanagi. Entanglement of Local Operators in large N CFTs. Progress of Theoretical and Experimental Physics, 2014(9):93B06–0, September 2014. arXiv: 1405.5946.

166 [34] Eddy Ardonne, Paul Fendley, and Eduardo Fradkin. Topological order and conformal quantum critical points. Annals of Physics, 310(2):493–551, 2004. [35] Eduardo Fradkin. Field Theories of Condensed Matter Physics. Cambridge University Press, February 2013. [36] Roderich Moessner and Kumar S. Raman. Quantum dimer models. In Introduction to Frustrated Magnetism, pages 437–479. Springer, 2011. [37] Eduardo Fradkin and Joel E. Moore. Entanglement entropy of 2d conformal quantum critical points: hearing the shape of a quantum drum. Physical Review Letters, 97(5), August 2006. arXiv: cond- mat/0605683. [38] Christopher L. Henley. Relaxation time for a dimer covering with height representation. Journal of Statistical Physics, 89(3-4):483–507, November 1997. [39] P.W. Kasteleyn. The statistics of dimers on a lattice. Physica, 27(12):1209–1225, December 1961. [40] Michael E. Fisher. Statistical Mechanics of Dimers on a Plane Lattice. Physical Review, 124(6):1664– 1672, December 1961. [41] B. Nienhuis. Two dimensional critical phenomena and the coulomb gas. Physics Phase Transitions and Critical Phenomena, 11, January 1987. [42] Masaki Oshikawa. Boundary Conformal Field Theory and Entanglement Entropy in Two-Dimensional Quantum Lifshitz Critical Point. arXiv:1007.3739 [cond-mat, physics:hep-th], July 2010. arXiv: 1007.3739. [43] Michael P. Zaletel, Jens H. Bardarson, and Joel E. Moore. Logarithmic Terms in Entanglement Entropies of 2d Quantum Critical Points and Shannon Entropies of Spin Chains. Physical Review Letters, 107(2):020402, July 2011. [44] Jean-Marie Stephan, Shunsuke Furukawa, Gregoire Misguich, and Vincent Pasquier. Shannon and entanglement entropies of one- and two-dimensional critical wave functions. Physical Review B, 80(18), November 2009. arXiv: 0906.1153. [45] Tianci Zhou, Xiao Chen, Thomas Faulkner, and Eduardo Fradkin. Entanglement Entropy and Mu- tual Information of Circular Entangling Surfaces in 2 + 1-dimensional Quantum Lifshitz Model. arXiv:1607.01771 [cond-mat], July 2016. arXiv: 1607.01771. [46] Tatsuma Nishioka, Shinsei Ryu, and Tadashi Takayanagi. Holographic entanglement entropy: an overview. Journal of Physics A: Mathematical and Theoretical, 42(50):504008, December 2009. [47] Masahiro Nozaki, Tokiro Numasawa, and Tadashi Takayanagi. Holographic local quenches and entanglement density. Journal of High Energy Physics, 2013(5):1–40, May 2013. [48] D. T. Son. Toward an AdS/cold atoms correspondence: A geometric realization of the Schr ”odinger symmetry. Physical Review D, 78(4):046003, August 2008. \ [49] Koushik Balasubramanian and John McGreevy. Gravity Duals for Nonrelativistic Conformal Field Theories. Physical Review Letters, 101(6):061601, August 2008. [50] Shamit Kachru, Xiao Liu, and Michael Mulligan. Gravity duals of Lifshitz-like ﬁxed points. Physical Review D, 78(10):106005, November 2008. [51] Petr Hoˇrava. Quantum gravity at a Lifshitz point. Physical Review D, 79(8):084008, April 2009. [52] J. Cardy. Boundary Conformal Field Theory. In Jean-Pierre Franoise, Gregory L. Naber, and Tsou She- ung Tsun, editors, Encyclopedia of Mathematical Physics, pages 333–340. Academic Press, Oxford, 2006. DOI: 10.1016/B0-12-512666-2/00398-9.

167 [53] Masaki Oshikawa and Ian Affleck. Boundary conformal field theory approach to the two-dimensional critical Ising model with a defect line. Nuclear Physics B, 495(3):533–582, June 1997. arXiv: cond- mat/9612187. [54] Ian Affleck. Boundary Condition Changing Operators in Conformal Field Theory and Condensed Matter Physics. Nuclear Physics B - Proceedings Supplements, 58:35–41, September 1997. arXiv: hep-th/9611064. [55] John L. Cardy. Conformal invariance and surface critical behavior. Nuclear Physics B, 240(4):514–532, November 1984. [56] Sebastian Eggert and Ian Affleck. Magnetic impurities in half-integer-spin Heisenberg antiferromag- netic chains. Physical Review B, 46(17):10866–10883, November 1992. [57] M. Fabrizio and Alexander O. Gogolin. Interacting one-dimensional electron gas with open boundaries. Physical Review B, 51(24):17827, 1995. [58] Ingo Peschel. Entanglement entropy with interface defects. Journal of Physics A: Mathematical and General, 38(20):4327–4335, May 2005. arXiv: cond-mat/0502034.

[59] Ferenc Igl´oi,Zsolt Szatm´ari,and Yu-Cheng Lin. Entanglement entropy with localized and extended interface defects. Physical Review B, 80(2):024405, July 2009. [60] Pasquale Calabrese, Mihail Mintchev, and Ettore Vicari. Entanglement Entropy of Quantum Wire Junctions. Journal of Physics A: Mathematical and Theoretical, 45(10):105206, March 2012. arXiv: 1110.5713. [61] Suraj Hegde, Vasudha Shivamoggi, Smitha Vishveshwara, and Diptiman Sen. Quench dynamics and parity blocking in Majorana wires. New Journal of Physics, 17(5):053036, 2015. [62] G. Francica, T. J. G. Apollaro, N. Lo Gullo, and F. Plastina. Local quench, Majorana zero modes, and disturbance propagation in the Ising chain. Physical Review B, 94(24):245103, December 2016.

[63] Carla Lupo and Marco Schir. Transient Loschmidt echo in quenched Ising chains. Physical Review B, 94(1):014310, July 2016. [64] Minchul Lee, Seungju Han, and Mahn-Soo Choi. Spatiotemporal evolution of topological order upon quantum quench across the critical point. New Journal of Physics, 18(6):063004, 2016.

[65] Ming-Chiang Chung, Yi-Hao Jhu, Pochung Chen, Chung-Yu Mou, and Xin Wan. A Memory of Majorana Modes through Quantum Quench. Scientiﬁc Reports, 6, July 2016. [66] P. D. Sacramento. Edge mode dynamics of quenched topological wires. Physical Review E, 93(6):062117, June 2016.

[67] Romain Vasseur, Christoph Karrasch, and Joel E. Moore. Expansion Potentials for Exact Far-from- Equilibrium Spreading of Particles and Energy. Physical Review Letters, 115(26):267201, December 2015. [68] Paolo P. Mazza, Jean-Marie Stphan, Elena Canovi, Vincenzo Alba, Michael Brockmann, and Masudul Haque. Overlap distributions for quantum quenches in the anisotropic Heisenberg chain. Journal of Statistical Mechanics: Theory and Experiment, 2016(1):013104, 2016. [69] R. Vasseur, J.P. Dahlhaus, and J.E. Moore. Universal Nonequilibrium Signatures of Majorana Zero Modes in Quench Dynamics. Physical Review X, 4(4):041007, October 2014. [70] C. Bachas, J. de Boer, R. Dijkgraaf, and H. Ooguri. Permeable conformal walls and holography. Journal of High Energy Physics, 2002(06):027–027, June 2002. arXiv: hep-th/0111210.

168 [71] Ingo Peschel and Viktor Eisler. Exact results for the entanglement across defects in critical chains. Journal of Physics A: Mathematical and Theoretical, 45(15):155301, April 2012. arXiv: 1201.4104. [72] Kazuhiro Sakai and Yuji Satoh. Entanglement through conformal interfaces. Journal of High Energy Physics, 2008(12):001–001, December 2008. arXiv: 0809.4548.

[73] Jean-Marie Stéphanand JérômeDubail. Logarithmic corrections to the free energy from sharp corners with angle 2π. Journal of Statistical Mechanics: Theory and Experiment, 2013(09):P09002, September 2013. [74] Jean-Marie Stéphanand JérômeDubail. Local quantum quenches in critical one-dimensional systems: entanglement, the Loschmidt echo, and light-cone effects. Journal of Statistical Mechanics: Theory and Experiment, 2011(08):P08019, 2011. [75] Ian Affleck, Masaki Oshikawa, and Hubert Saleur. Quantum brownian motion on a triangular lattice and c=2 boundary conformal field theory. Nuclear Physics B, 594(3):535–606, February 2001. arXiv: cond-mat/0009084. [76] Nobuyuki Ishibashi. The boundary and crosscap states in conformal field theories. Modern Physics Letters A, 04(03):251–264, February 1989. [77] Thomas Quella, Ingo Runkel, and Grard MT Watts. Reflection and transmission for conformal defects. Journal of High Energy Physics, 2007(04):095, 2007. [78] C. Bachas and I. Brunner. Fusion of conformal interfaces. Journal of High Energy Physics, 2008(02):085–085, February 2008. arXiv: 0712.0076. [79] Philippe Di Francesco, Pierre Mathieu, and David Snchal. Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer New York, New York, NY, 1997. [80] John L. Cardy and Ingo Peschel. Finite-size dependence of the free energy in two-dimensional critical systems. Nuclear Physics B, 300:377–392, 1988.

[81] Romain Vasseur, Kien Trinh, Stephan Haas, and Hubert Saleur. Crossover Physics in the Nonequi- librium Dynamics of Quenched Quantum Impurity Systems. Physical Review Letters, 110(24):240601, June 2013. [82] D. M. Kennes, V. Meden, and R. Vasseur. Universal quench dynamics of interacting quantum impurity systems. Physical Review B, 90(11):115101, September 2014. [83] James Ward Brown and Ruel V. Churchill. Complex variables and applications. Brown and Churchill series. McGraw-Hill Higher Education, Boston, 8th ed edition, 2009. OCLC: ocn176648981. [84] D. Schmeltzer and R. Berkovits. Zero mode bosonization of the Luttinger liquid: The eﬀect of a twisted boundary condition and a single impurity. Physics Letters A, 253(5):341–344, March 1999.

[85] Fabrizio Anfuso and Sebastian Eggert. Luttinger liquid in a ﬁnite one-dimensional wire with box-like boundary conditions. Physical Review B, 68(24):241301, December 2003. [86] Johannes Voit, Yupeng Wang, and Marco Grioni. Bounded Luttinger liquids as a universality class of quantum critical behavior. Physical Review B, 61(12):7930, 2000.

[87] Reinhold Egger and Hermann Grabert. Applying voltage sources to a Luttinger liquid with arbitrary transmission. Physical Review B, 58(16):10761–10768, October 1998. arXiv: cond-mat/9805268. [88] Thierry. Giamarchi and Oxford University Press. Quantum physics in one dimension. Oxford Science Publications; International Series of Monographs on Physics ; 121; International Series of Monographs on Physics 0950-5563 121.; Oxford Science Publications. Clarendon Press,, Oxford :, reprinted 2015. edition, 2015.

169 [89] F. Borgonovi, F. M. Izrailev, L. F. Santos, and V. G. Zelevinsky. Quantum chaos and thermalization in isolated systems of interacting particles. Physics Reports, 626:1–58, April 2016. [90] Luca D’Alessio, Yariv Kafri, Anatoli Polkovnikov, and Marcos Rigol. From Quantum Chaos and Eigen- state Thermalization to Statistical Mechanics and Thermodynamics. Advances in Physics, 65(3):239– 362, May 2016. arXiv: 1509.06411.

[91] David J. Luitz and Yevgeny Bar Lev. The Ergodic Side of the Many-Body Localization Transition. arXiv:1610.08993 [cond-mat], October 2016. arXiv: 1610.08993. [92] Gabriele De Chiara, Simone Montangero, Pasquale Calabrese, and Rosario Fazio. Entanglement entropy dynamics of Heisenberg chains. J. Stat. Mech., 2006(03):P03001, 2006.

[93] Marko Znidariˇc,TomaˇzProsen,ˇ and Peter Prelovˇsek.Many-body localization in the Heisenberg $XXZ$ magnet in a random ﬁeld. Phys. Rev. B, 77(6):064426, February 2008. [94] Jens H. Bardarson, Frank Pollmann, and Joel E. Moore. Unbounded Growth of Entanglement in Models of Many-Body Localization. Phys. Rev. Lett., 109(1):017202, July 2012.

[95] Maksym Serbyn, Z. Papić,and Dmitry A. Abanin. Universal Slow Growth of Entanglement in Inter- acting Strongly Disordered Systems. Physical Review Letters, 110(26):260601, June 2013. [96] Hyungwon Kim and David A. Huse. Ballistic spreading of entanglement in a diffusive nonintegrable system. Physical review letters, 111(12):127205, 2013. [97] Bela Bauer and Chetan Nayak. Area laws in a many-body localized state and its implications for topological order. J. Stat. Mech., 2013(09):P09005, 2013. [98] Jonas A. Kjäll,Jens H. Bardarson, and Frank Pollmann. Many-Body Localization in a Disordered Quantum Ising Chain. Phys. Rev. Lett., 113(10):107204, September 2014. [99] Ronen Vosk, David A. Huse, and Ehud Altman. Theory of the Many-Body Localization Transition in One-Dimensional Systems. Phys. Rev. X, 5(3):031032, September 2015. [100] Andrew C. Potter, Romain Vasseur, and S. A. Parameswaran. Universal Properties of Many-Body Delocalization Transitions. Phys. Rev. X, 5(3):031033, September 2015. [101] Xiao Chen, Xiongjie Yu, Gil Young Cho, Bryan K. Clark, and Eduardo Fradkin. Many-body localization transition in Rokhsar-Kivelson-type wave functions. Phys. Rev. B, 92(21):214204, December 2015. [102] David J. Luitz. Long tail distributions near the many body localization transition. Physical Review B, 93(13), April 2016. arXiv: 1601.04058. [103] Xiongjie Yu, David J. Luitz, and Bryan K. Clark. Bimodal entanglement entropy distribution in the many-body localization transition. Physical Review B, 94(18):184202, November 2016. [104] Arijeet Pal and David A. Huse. Many-body localization phase transition. Phys. Rev. B, 82(17):174411, November 2010. [105] David J. Luitz and Yevgeny Bar Lev. Anomalous thermalization in ergodic systems. Physical Review Letters, 117(17), October 2016. arXiv: 1607.01012.

[106] J. M. Deutsch. Quantum statistical mechanics in a closed system. Phys. Rev. A, 43(4):2046–2049, February 1991. [107] Mark Srednicki. Chaos and quantum thermalization. Phys. Rev. E, 50(2):888–901, August 1994. [108] Marcos Rigol, Vanja Dunjko, and Maxim Olshanii. Thermalization and its mechanism for generic isolated quantum systems. Nature, 452(7189):854–858, April 2008.

170 [109] Jayendra N. Bandyopadhyay and Arul Lakshminarayan. Entangling power of quantum chaotic evolutions via operator entanglement. arXiv:quant-ph/0504052, April 2005. arXiv: quant-ph/0504052. [110] TomaˇzProsen and Iztok Piˇzorn. Operator space entanglement entropy in a transverse Ising chain. Phys. Rev. A, 76(3):032316, September 2007.

[111] Iztok Piˇzornand TomaˇzProsen. Operator space entanglement entropy in $XY$ spin chains. Physical Review B, 79(18):184416, May 2009. [112] TomaˇzProsen. Chaos and complexity of quantum motion. Journal of Physics A: Mathematical and Theoretical, 40(28):7881, 2007. [113] Marcin Musz, Marek Ku´s,and Karol Zyczkowski.˙ Unitary quantum gates, perfect entanglers, and unistochastic maps. Physical Review A, 87(2):022111, February 2013. [114] Paolo Zanardi, Christof Zalka, and Lara Faoro. Entangling power of quantum evolutions. Physical Review A, 62(3):030301, August 2000. [115] Michael A. Nielsen, Christopher M. Dawson, Jennifer L. Dodd, Alexei Gilchrist, Duncan Mortimer, Tobias J. Osborne, Michael J. Bremner, Aram W. Harrow, and Andrew Hines. Quantum dynamics as a physical resource. Physical Review A, 67(5):052301, May 2003. [116] Karol Zyczkowski and Ingemar Bengtsson. On duality between quantum maps and quantum states. arXiv:quant-ph/0401119, January 2004. arXiv: quant-ph/0401119. [117] Ulrich Schollwoeck. The density-matrix renormalization group in the age of matrix product states. Annals of Physics, 326(1):96–192, January 2011. arXiv: 1008.3477. [118] David Pekker and Bryan K. Clark. Encoding the structure of many-body localization with matrix product operators. arXiv:1410.2224 [cond-mat], October 2014. arXiv: 1410.2224. [119] Frank Pollmann, Vedika Khemani, J. Ignacio Cirac, and S. L. Sondhi. Efficient variational diagonalization of fully many-body localized Hamiltonians. Phys. Rev. B, 94(4):041116, July 2016. [120] Xiaoguang Wang, Barry C. Sanders, and Dominic W. Berry. Entangling power and operator entanglement in qudit systems. Physical Review A, 67(4), April 2003. [121] Maksym Serbyn, Z. Papić,and Dmitry A. Abanin. Local Conservation Laws and the Structure of the Many-Body Localized States. Phys. Rev. Lett., 111(12):127201, September 2013. [122] David A. Huse, Rahul Nandkishore, and Vadim Oganesyan. Phenomenology of fully many-body- localized systems. Phys. Rev. B, 90(17):174202, November 2014. [123] Rahul Nandkishore and David A. Huse. Many-Body Localization and Thermalization in Quantum Statistical Mechanics. Annu. Rev. Condens. Matter Phys., 6(1):15–38, March 2015. [124] John Z. Imbrie. On Many-Body Localization for Quantum Spin Chains. J Stat Phys, 163(5):998–1048, June 2016. [125] John Z. Imbrie. Diagonalization and Many-Body Localization for a Disordered Quantum Spin Chain. Phys. Rev. Lett., 117(2):027201, July 2016. [126] J. Z. Imbrie, V. Ros, and A. Scardicchio. Review: Local Integrals of Motion in Many-Body Localized systems. arXiv:1609.08076 [cond-mat, physics:math-ph], September 2016. arXiv: 1609.08076. [127] Rajeev Singh, Jens H. Bardarson, and Frank Pollmann. Signatures of the many-body localization transition in the dynamics of entanglement and bipartite fluctuations. New J. Phys., 18(2):023046, 2016. [128] John Cardy. Quantum Revivals in Conformal Field Theories in Higher Dimensions. arXiv:1603.08267 [cond-mat, physics:hep-th], March 2016. arXiv: 1603.08267.

171 [129] Don N. Page. Average entropy of a subsystem. Physical Review Letters, 71(9):1291–1294, August 1993. [130] TomaˇzProsen. General relation between quantum ergodicity and ﬁdelity of quantum dynamics. Phys- ical Review E, 65(3), February 2002.

[131] Liangsheng Zhang, Hyungwon Kim, and David A. Huse. Thermalization of entanglement. arXiv:1501.01315 [cond-mat, physics:quant-ph], January 2015. arXiv: 1501.01315. [132] Liangsheng Zhang, Vedika Khemani, and David A. Huse. A Floquet Model for the Many-Body Local- ization Transition. arXiv:1609.00390 [cond-mat], September 2016. arXiv: 1609.00390. [133] M. Akila, D. Waltner, B. Gutkin, and T. Guhr. Particle-Time Duality in the Kicked Ising Chain I: The Dual Operator. Journal of Physics A: Mathematical and Theoretical, 49(37):375101, September 2016. arXiv: 1602.07130. [134] M. Akila, D. Waltner, B. Gutkin, and T. Guhr. Particle-Time Duality in the Kicked Ising Chain II: Applications to the Spectrum. arXiv:1602.07479 [cond-mat, physics:nlin, physics:quant-ph], February 2016. arXiv: 1602.07479.

[135] David J. Luitz, Nicolas Laﬂorencie, and Fabien Alet. Many-body localization edge in the random-ﬁeld Heisenberg chain. Physical Review B, 91(8), February 2015. arXiv: 1411.0660. [136] Yevgeny Bar Lev and David R. Reichman. Dynamics of many-body localization. Phys. Rev. B, 89(22):220201, June 2014.

[137] Yevgeny Bar Lev, Guy Cohen, and David R. Reichman. Absence of Diffusion in an Interacting System of Spinless Fermions on a One-Dimensional Disordered Lattice. Phys. Rev. Lett., 114(10):100601, March 2015. [138] Kartiek Agarwal, Sarang Gopalakrishnan, Michael Knap, Markus Müller,and Eugene Demler. Anoma- lous Diffusion and Griffiths Effects Near the Many-Body Localization Transition. Phys. Rev. Lett., 114(16):160401, April 2015. [139] Marko Znidariˇc,Antonelloˇ Scardicchio, and Vipin Kerala Varma. Diffusive and Subdiffusive Spin Transport in the Ergodic Phase of a Many-Body Localizable System. Phys. Rev. Lett., 117(4):040601, July 2016.

[140] Fritz Haake. Quantum Signatures of Chaos, volume 54 of Springer Series in Synergetics. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010. [141] David J. Luitz and Yevgeny Bar Lev. Information propagation in isolated quantum systems. arXiv:1702.03929 [cond-mat, physics:hep-th], February 2017. arXiv: 1702.03929. [142] Sunil K. Mishra, Arul Lakshminarayan, and V. Subrahmanyam. Protocol using kicked Ising dynamics for generating states with maximal multipartite entanglement. Physical Review A, 91(2):022318, February 2015. [143] Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. Quantum Entanglement Growth Under Random Unitary Dynamics. arXiv:1608.06950 [cond-mat, physics:hep-th, physics:quant-ph], August 2016. arXiv: 1608.06950.

[144] Juan M. Maldacena. Eternal Black Holes in AdS. Journal of High Energy Physics, 2003(04):021–021, April 2003. arXiv: hep-th/0106112. [145] Thomas Hartman and Juan Maldacena. Time evolution of entanglement entropy from black hole interiors. arXiv preprint arXiv:1303.1080, 2013.

172 [146] J. Dubail. Entanglement scaling of operators: a conformal ﬁeld theory approach, with a glimpse of sim- ulability of long-time dynamics in 1+1d. arXiv:1612.08630 [cond-mat, physics:hep-th, physics:quant- ph], December 2016. arXiv: 1612.08630. [147] Madan Lal Mehta. Random matrices, volume 142. Elsevier, 2004.

[148] Fritz Haake. Quantum signatures of chaos, volume 54. Springer Science & Business Media, 2013. [149] Terence Tao. Topics in Random Matrix Theory, volume 132 of Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island, March 2012. [150] Subir Sachdev and Jinwu Ye. Gapless spin-ﬂuid ground state in a random quantum Heisenberg magnet. Physical Review Letters, 70(21):3339–3342, May 1993.

[151] Juan Maldacena and Douglas Stanford. Comments on the Sachdev-Ye-Kitaev model. arXiv:1604.07818 [cond-mat, physics:hep-th], April 2016. arXiv: 1604.07818. [152] R Oliveira, OCO Dahlsten, and MB Plenio. Generic entanglement can be generated eﬃciently. Physical review letters, 98(13):130502, 2007.

[153] Marko Znidariˇc.Exactˇ convergence times for generation of random bipartite entanglement. Physical Review A, 78(3):032324, 2008. [154] Alioscia Hamma, Siddhartha Santra, and Paolo Zanardi. Quantum entanglement in random physical states. Physical review letters, 109(4):040502, 2012.

[155] Fernando GSL Brand˜ao,Aram W Harrow, and MichalHorodecki. Local random quantum circuits are approximate polynomial-designs. Communications in Mathematical Physics, 346(2):397–434, 2016. [156] Pavan Hosur, Xiao-Liang Qi, Daniel A. Roberts, and Beni Yoshida. Chaos in quantum channels. Journal of High Energy Physics, 2016(2):1–49, February 2016. [157] Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. Quantum Entanglement Growth under Random Unitary Dynamics. Physical Review X, 7(3):031016, July 2017. [158] Leonardo Banchi, Daniel Burgarth, and Michael J Kastoryano. Driven quantum dynamics: will it blend? Physical Review X, 7(4):041015, 2017. [159] Pasquale Calabrese and John Cardy. Evolution of entanglement entropy in one-dimensional systems. Journal of Statistical Mechanics: Theory and Experiment, 2005(04):P04010, 2005. [160] Hyungwon Kim and David A Huse. Ballistic spreading of entanglement in a diﬀusive nonintegrable system. Physical review letters, 111(12):127205, 2013. [161] Hong Liu and S Josephine Suh. Entanglement tsunami: universal scaling in holographic thermalization. Physical review letters, 112(1):011601, 2014.

[162] Adam M Kaufman, M Eric Tai, Alexander Lukin, Matthew Rispoli, Robert Schittko, Philipp M Preiss, and Markus Greiner. Quantum thermalization through entanglement in an isolated many-body system. Science, 353(6301):794–800, 2016. [163] Curtis T Asplund, Alice Bernamonti, Federico Galli, and Thomas Hartman. Entanglement scrambling in 2d conformal ﬁeld theory. Journal of High Energy Physics, 2015(9):110, 2015. [164] Horacio Casini, Hong Liu, and M´arkMezei. Spread of entanglement and causality. Journal of High Energy Physics, 2016(7):77, 2016. [165] Wen Wei Ho and Dmitry A Abanin. Entanglement dynamics in quantum many-body systems. Physical Review B, 95(9):094302, 2017.

173 [166] M´arkMezei and Douglas Stanford. On entanglement spreading in chaotic systems. Journal of High Energy Physics, 2017(5):65, 2017. [167] M´arkMezei. On entanglement spreading from holography. Journal of High Energy Physics, 2017(5):64, 2017.

[168] Yingfei Gu, Andrew Lucas, and Xiao-Liang Qi. Spread of entanglement in a Sachdev-Ye-Kitaev chain. arXiv:1708.00871 [cond-mat, physics:hep-th], August 2017. arXiv: 1708.00871. [169] Cheryne Jonay, David A. Huse, and Adam Nahum. Coarse-grained dynamics of operator and state entanglement. arXiv:1803.00089 [cond-mat, physics:hep-th, physics:nlin, physics:quant-ph], February 2018. arXiv: 1803.00089.

[170] M´ark Mezei. Membrane theory of entanglement dynamics from holography. arXiv preprint arXiv:1803.10244, 2018. [171] Elliott H Lieb and Derek W Robinson. The ﬁnite group velocity of quantum spin systems. In Statistical Mechanics, pages 425–431. Springer, 1972.

[172] A. Kitaev. Hidden correlations in the hawking radiation and thermal noise. In Talk given at the Fundamental Physics Prize Symposium, 2014. [173] Stephen H. Shenker and Douglas Stanford. Black holes and the butterfly effect. Journal of High Energy Physics, 2014(3):67, March 2014. [174] Stephen H. Shenker and Douglas Stanford. Multiple shocks. Journal of High Energy Physics, 2014(12):46, 2014. [175] Juan Maldacena, Stephen H. Shenker, and Douglas Stanford. A bound on chaos. arXiv:1503.01409 [cond-mat, physics:hep-th, physics:nlin, physics:quant-ph], March 2015. arXiv: 1503.01409. [176] Daniel A. Roberts and Douglas Stanford. Two-dimensional conformal field theory and the butterfly effect. Physical Review Letters, 115(13), September 2015. arXiv: 1412.5123. [177] Daniel A Roberts, Douglas Stanford, and Leonard Susskind. Localized shocks. Journal of High Energy Physics, 2015(3):51, 2015. [178] Daniel A. Roberts and Brian Swingle. Lieb-Robinson and the butterfly effect. arXiv preprint arXiv:1603.09298, 2016.

[179] Igor L Aleiner, Lara Faoro, and Lev B Ioffe. Microscopic model of quantum butterfly effect: out-of- time-order correlators and traveling combustion waves. Annals of Physics, 375:378–406, 2016. [180] Aavishkar A Patel and Subir Sachdev. Quantum chaos on a critical fermi surface. Proceedings of the National Academy of Sciences, 114(8):1844–1849, 2017.

[181] Debanjan Chowdhury and Brian Swingle. Onset of many-body chaos in the o (n) model. Physical Review D, 96(6):065005, 2017. [182] A Bohrdt, CB Mendl, M Endres, and M Knap. Scrambling and thermalization in a diﬀusive quantum many-body system. New Journal of Physics, 19(6):063001, 2017.

[183] Jun Li, Ruihua Fan, Hengyan Wang, Bingtian Ye, Bei Zeng, Hui Zhai, Xinhua Peng, and Jiangfeng Du. Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator. Physical Review X, 7(3):031011, 2017. [184] Martin G¨arttner,Justin G Bohnet, Arghavan Safavi-Naini, Michael L Wall, John J Bollinger, and Ana Maria Rey. Measuring out-of-time-order correlations and multiple quantum spectra in a trapped- ion quantum magnet. Nature Physics, 13(8):781, 2017.

174 [185] Tibor Rakovszky, Frank Pollmann, and C. W. von Keyserlingk. Diffusive hydrodynamics of out- of-time-ordered correlators with charge conservation. arXiv:1710.09827 [cond-mat, physics:hep-th, physics:quant-ph], October 2017. arXiv: 1710.09827. [186] Vedika Khemani, Ashvin Vishwanath, and D. A. Huse. Operator spreading and the emergence of dissipation in unitary dynamics with conservation laws. arXiv:1710.09835 [cond-mat, physics:hep-th, physics:quant-ph], October 2017. arXiv: 1710.09835. [187] Adam Nahum, Jonathan Ruhman, and David A. Huse. Dynamics of entanglement and transport in 1d systems with quenched randomness. arXiv:1705.10364 [cond-mat, physics:quant-ph], May 2017. arXiv: 1705.10364. [188] Amos Chan, Andrea De Luca, and J. T. Chalker. Solution of a minimal model for many-body quantum chaos. arXiv:1712.06836 [cond-mat, physics:hep-th, physics:quant-ph], December 2017. arXiv: 1712.06836. [189] Pavel Kos, Marko Ljubotina, and Tomaz Prosen. Many-body quantum chaos: The first analytic connection to random matrix theory. arXiv preprint arXiv:1712.02665, 2017. [190] Amos Chan, Andrea De Luca, and JT Chalker. Spectral statistics in spatially extended chaotic quantum many-body systems. arXiv preprint arXiv:1803.03841, 2018. [191] Patrick Hayden and John Preskill. Black holes as mirrors: quantum information in random subsystems. Journal of High Energy Physics, 2007(09):120, 2007. [192] Yasuhiro Sekino and Leonard Susskind. Fast scramblers. Journal of High Energy Physics, 2008(10):065, 2008. [193] Aram W Harrow and Richard A Low. Random quantum circuits are approximate 2-designs. Commu- nications in Mathematical Physics, 291(1):257–302, 2009. [194] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. Exact and approximate unitary 2-designs and their application to fidelity estimation. Physical Review A, 80(1):012304, 2009. [195] Nima Lashkari, Douglas Stanford, Matthew Hastings, Tobias Osborne, and Patrick Hayden. Towards the fast scrambling conjecture. Journal of High Energy Physics, 2013(4):22, 2013. [196] Stephen H Shenker and Douglas Stanford. Stringy effects in scrambling. Journal of High Energy Physics, 2015(5):132, 2015. [197] Brian Swingle. Entanglement renormalization and holography. Physical Review D, 86(6):065007, 2012. [198] Fernando Pastawski, Beni Yoshida, Daniel Harlow, and John Preskill. Holographic quantum error- correcting codes: Toy models for the bulk/boundary correspondence. Journal of High Energy Physics, 2015(6):149, 2015. [199] Patrick Hayden, Sepehr Nezami, Xiao-Liang Qi, Nathaniel Thomas, Michael Walter, and Zhao Yang. Holographic duality from random tensor networks. Journal of High Energy Physics, 2016(11):9, 2016. [200] Tianci Zhou and David J. Luitz. Operator entanglement entropy of the time evolution operator in chaotic systems. Physical Review B, 95(9), March 2017. arXiv: 1612.07327. [201] Xiao-Liang Qi and Zhao Yang. Space-time random tensor networks and holographic duality. arXiv:1801.05289 [hep-th], January 2018. arXiv: 1801.05289. [202] Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang. Dynamic Scaling of Growing Interfaces. Physical Review Letters, 56(9):889–892, March 1986. [203] David A. Huse, Christopher L. Henley, and Daniel S. Fisher. Huse, henley, and fisher respond. Phys. Rev. Lett., 55:2924–2924, Dec 1985.

175 [204] David A. Huse and Christopher L. Henley. Pinning and Roughening of Domain Walls in Ising Systems Due to Random Impurities. Physical Review Letters, 54(25):2708–2711, June 1985. [205] Mehran Kardar. Roughening by impurities at finite temperatures. Phys. Rev. Lett., 55:2923–2923, Dec 1985. [206] Mehran Kardar. Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities. Nuclear Physics B, 290:582–602, January 1987. [207] P. Calabrese, P. Le Doussal, and A. Rosso. Free-energy distribution of the directed polymer at high temperature. EPL (Europhysics Letters), 90(2):20002, 2010. [208] V. Dotsenko. Bethe ansatz derivation of the tracy-widom distribution for one-dimensional directed polymers. EPL (Europhysics Letters), 90(2):20003, 2010. [209] Tomohiro Sasamoto and Herbert Spohn. One-dimensional kardar-parisi-zhang equation: An exact solution and its universality. Phys. Rev. Lett., 104:230602, Jun 2010. [210] Tomohiro Sasamoto and Herbert Spohn. Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Physics B, 834(3):523 – 542, 2010. { } [211] Tomohiro Sasamoto and Herbert Spohn. The crossover regime for the weakly asymmetric simple exclusion process. Journal of Statistical Physics, 140(2):209–231, 2010. [212] Gideon Amir, Ivan Corwin, and Jeremy Quastel. Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions. Communications on Pure and Applied Mathematics, 64(4):466–537, 2011. [213] Pasquale Calabrese and Pierre Le Doussal. Exact solution for the kardar-parisi-zhang equation with flat initial conditions. Phys. Rev. Lett., 106:250603, Jun 2011. [214] Sylvain Prolhac and Herbert Spohn. Height distribution of the kardar-parisi-zhang equation with sharp-wedge initial condition: Numerical evaluations. Phys. Rev. E, 84:011119, Jul 2011. [215] Pierre Le Doussal and Pasquale Calabrese. The kpz equation with flat initial condition and the directed polymer with one free end. Journal of Statistical Mechanics: Theory and Experiment, 2012(06):P06001, 2012. [216] Takashi Imamura and Tomohiro Sasamoto. Exact solution for the stationary kardar-parisi-zhang equation. Phys. Rev. Lett., 108:190603, May 2012. [217] Takashi Imamura and Tomohiro Sasamoto. Stationary correlations for the 1d kpz equation. Journal of Statistical Physics, 150(5):908–939, 2013. [218] Thomas Kriecherbauer and Joachim Krug. A pedestrian’s view on interacting particle systems, kpz universality and random matrices. Journal of Physics A: Mathematical and Theoretical, 43(40):403001, 2010. [219] Ivan Corwin. The kardar-parisi-zhang equation and universality class. Random Matrices: Theory and Applications, 01(01):1130001, 2012. [220] Timothy Halpin-Healy and Kazumasa A. Takeuchi. A kpz cocktail-shaken, not stirred... Journal of Statistical Physics, 160(4):794–814, 2015. [221] Mehran Kardar. Directed paths in random media. arXiv preprint cond-mat/9411022, 1994. [222] Victor Dotsenko. Bethe ansatz derivation of the tracy-widom distribution for one-dimensional directed polymers. EPL (Europhysics Letters), 90(2):20003, 2010. [223] Pasquale Calabrese, Pierre Le Doussal, and Alberto Rosso. Free-energy distribution of the directed polymer at high temperature. EPL (Europhysics Letters), 90(2):20002, 2010.

176 [224] Pasquale Calabrese and Pierre Le Doussal. Exact Solution for the Kardar-Parisi-Zhang Equation with Flat Initial Conditions. Physical Review Letters, 106(25):250603, June 2011. [225] Paolo Zanardi. Entanglement of quantum evolutions. Phys. Rev. A, 63:040304, Mar 2001. [226] Alioscia Hamma, Siddhartha Santra, and Paolo Zanardi. Quantum entanglement in random physical states. Physical review letters, 109(4):040502, 2012. [227] Ernesto Medina, Mehran Kardar, Yonathan Shapir, and Xiang Rong Wang. Interference of Directed Paths in Disordered Systems. Physical Review Letters, 62(8):941–944, February 1989. [228] Celine Nadal, Satya N. Majumdar, and Massimo Vergassola. Statistical Distribution of Quantum Entanglement for a Random Bipartite State. Journal of Statistical Physics, 142(2):403–438, January 2011. [229] Don Weingarten. Asymptotic behavior of group integrals in the limit of infinite rank. Journal of Mathematical Physics, 19(5):999–1001, 1978. [230] BenoˆıtCollins and Piotr Sniady.´ Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group. Communications in Mathematical Physics, 264(3):773–795, March 2006. [231] Vedika Khemani, David A. Huse, and Adam Nahum. Velocity-dependent Lyapunov exponents in many-body quantum, semi-classical, and classical chaos. arXiv:1803.05902 [cond-mat, physics:hep-th, physics:nlin, physics:quant-ph], March 2018. arXiv: 1803.05902. [232] Albert-LaszlóBarabási. Fractal concepts in surface growth /. Press Syndicate of the University of Cambridge,, New York, NY, USA :, 1995. [233] Michael Stone and Paul Goldbart. Mathematics for Physics: A Guided Tour for Graduate Students. Cambridge University Press, Cambridge, UK ; New York, 1 edition edition, August 2009. [234] John de Pillis. Linear transformations which preserve hermitian and positive semidefinite operators. Pacific Journal of Mathematics, 23(1):129–137, October 1967. [235] Man-Duen Choi. Completely positive linear maps on complex matrices. Linear Algebra and its Appli- cations, 10(3):285–290, June 1975. [236] A. Jamiolkowski. Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics, 3(4):275–278, December 1972.

[237] Min Jiang, Shunlong Luo, and Shuangshuang Fu. Channel-state duality. Physical Review A, 87(2):022310, February 2013. [238] Richard P. Stanley. Catalan Numbers. Cambridge University Press, Cambridge, 2015. [239] Serge Dulucq and Rodica Simion. Combinatorial Statistics on Alternating Permutations. Journal of Algebraic Combinatorics, 8(2):169–191, 1998. [240] Rodica Simion. Noncrossing partitions. Discrete Mathematics, 217(1):367–409, April 2000. [241] H. Q. Lin. Exact diagonalization of quantum-spin models. Phys. Rev. B, 42(10):6561–6567, October 1990.

[242] Luiz H. Santos and Taylor L. Hughes. Parafermionic wires at the interface of chiral topological states. Phys. Rev. Lett., 118:136801, Mar 2017. [243] Jean-Paul Blaizot. Quantum theory of ﬁnite systems. MIT Press,, Cambridge, Mass. :, 1986. [244] William Fulton and Joe Harris. Representation Theory, volume 129 of Graduate Texts in Mathematics. Springer New York, New York, NY, 2004. DOI: 10.1007/978-1-4612-0979-9.

177 [245] V. I. Arnold. Mathematical methods of classical mechanics. Number 60 in Graduate Texts in Mathe- matics. Springer, New York, NY, 2. ed edition, 2010. OCLC: 837651184. [246] Ming-wen Xiao. Theory of transformation for the diagonalization of quadratic Hamiltonians. arXiv preprint arXiv:0908.0787, 2009.

[247] Stefano Pirandola, Alessio Seraﬁni, and Seth Lloyd. Correlation matrices of two-mode bosonic systems. APS, May 2009. [248] Maurice A. de Gosson. Symplectic Geometry and Quantum Mechanics. Springer Science & Business Media, August 2006. Google-Books-ID: q9SHRvay75IC. [249] Andrea Coser, Cristiano De Nobili, and Erik Tonni. A contour for the entanglement entropies in harmonic lattices. arXiv:1701.08427 [cond-mat, physics:hep-th, physics:quant-ph], January 2017. arXiv: 1701.08427. [250] Michael Z. Spivey. Enumerating Lattice Paths Touching or Crossing the Diagonal at a Given Number of Lattice Points. The Electronic Journal of Combinatorics, 19(3):P24, August 2012.

[251] Aram W. Harrow. The Church of the Symmetric Subspace. arXiv:1308.6595 [quant-ph], August 2013. arXiv: 1308.6595. [252] Yinzheng Gu. Moments of Random Matrices and Weingarten Functions. Thesis, Queens University, September 2013.

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