Wave Functions of Integrable Models

Home , Tensor network

UNIVERSITYOF CINCINNATI

Author: Committee Chair: Zhongtao Mei Prof. Carlos J. Bolech

A thesis submitted in fulﬁllment of the requirements for the degree of

Doctor of Philosophy

in the

McMicken College of Arts and Sciences Department of Physics

M.Sc., University of Cincinnati, Cincinnati, USA B.Sc., Huazhong University of Science and Technology, Wuhan, China

June 2018 Abstract

This thesis centers on Bethe wave functions for one-dimensional lattice and ﬁeld theories of quantum systems. The Bethe ansatz is a well established theoretical tool in the cross- cutting research ﬁeld of integrable systems. However, most of the work in the literature focuses on the eigenenergies, and very limited results are known about the eigenstates. We present here two pieces of work to extend and deepen our understanding about Bethe wave functions. In one work, we recast the Bethe states as exact matrix product states for the Heisenberg

1 XXZ spin- 2 chain and the Lieb-Liniger model with open boundary conditions, and find that the matrices do not depend on the spatial coordinate despite the open boundaries. Based on this result, we suggest generic ways of exploiting translational invariance both for finite size and in the thermodynamic limit. Our work makes the Bethe eigenstates more accessible and informs the choice of ansatz for tensor-network algorithms of both integrable and nonintegrable systems in one dimension. This achievement contributes in this way not only to the basic theory of integrable models but it will also influence the community that works using matrix product states and other tensor networks. In another work, we use the physical interpretation of rapidities in integrable models to calculate the asymptotic expansion velocity of interacting atomic gases. Which is accessible in sudden expansion experiments as those done routinely these days using optically- trapped cold atomic gases. Through our research, the calculations of the asymptotic forms of observables of integrable models in quantum quench problems become more clear and theoretically accessible.

ii iii Acknowledgments

At ﬁrst, I would like to thank my advisor Prof. Carlos Bolech. He introduced me to the area of integrable models, which I think is an interesting ﬁeld. Prof. Bolech also provided many useful comments for the writing of this thesis. I have learned much from him. I also want to thank Dr. Nayana Shah. The discussion with her broadens my knowledge of theoretical physics. I would like to thank my committee members Prof. Philip Argyres and Prof. Hans-Peter Wagner for their guidance in the past a few years. Some comments from Prof. Argyres are especially valuable. I am also grateful to Prof. Slava Serota for his help and encouragement during my years at Cincinnati. Next, I want to thank Prof. Michael Ma, Dr. Kuei Sun, and Dr. Sangwoo Chung for interesting discussion about physics and things beyond physics. I also want to thank my father and my mother. Without their constant support, I can not go so far. Last but not least, I want to thank my daughter and my wife. They are my best friends in the world, and they bring much happiness to me.

iv Contents

Abstract ii

Acknowledgments iv

List of Symbols vii

1 Introduction1 1.1 Outline...... 1 1.2 Bethe Ansatz and Integrability ...... 2 1.2.1 The Origin of Bethe Ansatz...... 2 1.2.2 Brief History of Bethe Ansatz...... 4 1.2.3 Basic Building Blocks of Algebraic Bethe Ansatz...... 4 1.2.4 A Note about Integrability and Exact Solvability...... 8 1.3 Matrix Product States...... 9 1.4 Cold Atoms and Quantum Quenches...... 11 1.5 Motivation of Our Research...... 13 1.5.1 Motivation for Our Research on Matrix Product States ...... 13 1.5.2 Motivation for Our Research on Quantum Quench in Cold Atoms . . 14

2 Derivation of MPSs for Integrable Models with OBCs 15 2.1 Background and Motivation...... 15 2.2 Derivation of MPSs for the Heisenberg Spin Chain with OBCs...... 18 1 2.2.1 ABA for the Heisenberg XXZ Spin- 2 Chain with OBCs...... 18 2.2.2 Comparison between ABA and CBA...... 21 1 2.2.3 Exact MPSs for the XXZ Spin- 2 Chain with OBCs...... 24 2.2.4 Discussion and Conclusions...... 28 2.2.5 Comments on the Key Idea of Our Research ...... 30 2.3 MPSs for the Lieb-Liniger Model with OBCs ...... 31 2.3.1 ABA for the Lieb-Liniger Model ...... 31 2.3.2 Exact MPSs for the Lieb-Liniger Model with OBCs...... 34

3 Expansion Velocity of 1D Hubbard Model 38 3.1 Background ...... 38 3.1.1 Introduction...... 38 3.1.2 Asymptotic Form of the Quasi-MDF at Half Filling for FHM ...... 41 3.2 Expansion Velocity of the 1D FHM...... 43 3.2.1 Eigenvalue Equations of the 1D FHM with OBCs ...... 43 3.2.2 BA Equations of the 1D FHM with OBCs...... 52

v 3.2.3 Expansion Velocity of the 1D FHM...... 55 3.3 Expansion Velocity of the 1D BHM...... 61 3.4 Discussion...... 63

4 Outlook 65 4.1 Summary...... 65 4.2 Possible Future Directions...... 65

Bibliography 67

A Proof of Eq. (2.10) for the XXZ Chain with OBCs 76 A.1 An Equation about the R-matrix of the XXZ Spin Chain...... 76 A.2 Proof of Eq. (2.10)...... 77

B Proof of the Integrability of the XXZ Chain with OBCs 79

C Recover the Hamiltonian of the XXZ Chain with OBCs 82

D Eigenenergy and BAEs of the XXZ Chain with OBCs 85 D.1 Commutation Relations among A, B, C, and D ...... 85 D.2 Eigenenergy and BAEs...... 91

E Expansion Velocity of the FHM 97

F Ground State Rapidities for the BHM 98

G List of Publications 101

vi List of Symbols

1ˆ identity matrix h¯ reduced Planck constant kB Boltzmann constant σi Pauli matrix |ψi wave function ⊗ tensor product δij Kronecker delta ∆ anisotropy parameter for the XXZ spin chain εijk Levi-Civita symbol ΞA(α) monodromy matrix for the Hubbard model |ωi reference ferromagnetic state for the XXZ spin chain ψ(x) boson annihilation operator (for field) ψ†(x) boson creation operator (for field) ai boson annihilation operator (for lattice) [A, B] commutator, defined as AB − BA ciσ fermion annihilation operator (for lattice) C set of complex numbers E energy H Hamiltonian L number of lattice sites ln x the natural logarithm of x log x the natural logarithm of x R set of real numbers Pij permutation matrix T thermodynamic temperature T0(λ) monodromy matrix for the Heisenberg model t(λ) transfer matrix Tr taking the trace tr0 trace over the auxiliary space V0 Z partition function Z set of integers

vii To my daughter Luchen, and my wife Jingjing.

viii Chapter 1

Introduction

1.1 Outline

For this thesis, the main focus is on the wave functions of integrable models. We do research on two topics related to Bethe wave functions. In one topic, we recast the Bethe wave

1 functions as exact matrix product states (MPSs) for Heisenberg XXZ spin- 2 chain and Lieb- Liniger model with open boundary conditions (OBCs), which was a difﬁcult problem defy- ing resolution for at least 11 years. The matrices do not depend on the spatial coordinate despite the open boundaries. Based on this result, we suggest generic ways of exploiting translational invariance both for ﬁnite size and in the thermodynamic limit. The results were published as Ref. [1]. In another topic, we invent a simple approximate method to calculate the asymptotic expansion velocity of interacting atomic gases. The results were published as Ref. [2]. In this chapter, we give the background and motivation of our research.

1 In Chapter2, we derive MPSs for the Heisenberg spin- 2 chain and Lieb-Liniger model with OBCs [1]. For each model, we ﬁrst give full details about the solution of the model by using algebraic Bethe ansatz (ABA). Then we recast the Bethe eigenstates as exact MPSs, which is our original research. This chapter not only presents our original research, but also it can be used as a detailed introduction to ABA. Chapter3 deals with the sudden expansion of cold atoms in one dimension [2]. Based on a physical interpretation of the rapidities that parameterize the Bethe wave function of integrable models, we invent a simple approximate method to calculate the asymptotic expansion velocity of cold atoms in one dimension. We deal with both Fermi-Hubbard model

1 Chapter 1. Introduction

(FHM) and Bose-Hubbard model (BHM). In Chapter4, we discuss the outlook of our work.

1.2 Bethe Ansatz and Integrability

1.2.1 The Origin of Bethe Ansatz

Theoretical physics is an exact science to help us to understand the universe. Newtonian mechanics can help us to understand the movements of a particle, a rigid body, a satellite, etc. J. C. Maxwell’s electromagnetic theory can be used to understand the nature of elec- tric and magnetic interactions, and underpins the workings of modern devices like the cell phone, the TV, etc. A. Einstein’s general relativity is important for us to understand the pre- cession of the perihelion of Mercury, and underpins the workings of modern technologies like the global positioning system (GPS), etc. To understand the behavior of a large number of particles, we need statistical physics. Statistical physics was developed by J. C. Maxwell (1831 − 1879), L. Boltzmann (1844 − 1906), J. W. Gibbs (1839 − 1903), etc. An interesting phenomena in nature is ferromagnetism. For a permanent magnet, there is magnetism if the temperature is below the Curie temperature. Above the Curie temperature, there is no magnetism even for permanent magnet. This example is one case of phase transition. In the 1930s, many physicists doubted whether statistical physics could describe phase transitions. The reason is the following: In statistical physics, the most important step is to calculate the partition function Z = ∑ e−βEi , (1.1) i where i is the index for the microstates of the system, β is given by 1 (k is the Boltzmann kBT B constant, T is the thermodynamic temperature), Ei is the total energy of the system in the respective microstate. e−βEi is a smooth function, the summation over a large number of e−βEi ’s will lead to a very smooth function. During a phase transition, some properties of

2 1.2. Bethe Ansatz and Integrability the system will change abruptly. How can a smooth function describe abrupt changes of some properties of a system? In 1944, L. Onsager (1903 − 1976) eliminated physicist’s doubts by giving an exact solution of two-dimensional (2D) Ising model [3], which is a simple statistical model of a magnet that orders at low temperatures. The Ising model consists a large number of spins, which are arranged in a lattice. At each site of the lattice, the spin can be in one of two states. Based on Onsager’s exact solution, we know that statistical physics can describe phase transition

N in the thermodynamical limit (N → ∞, S → ∞, S is ﬁnite, where N is the number of spins and S is the area of the lattice). Onsager’s achievement is a summit in the history of science, as it justiﬁes statistical physics. From this example, we know that exact solutions of statistical physics models are very important, they are cornerstones of many concepts and methods. Ising model is a clas- sical model, because each spin can be in only one of two states. A more realistic model is the Heisenberg model. The Heisenberg model is similar to the Ising model, the only difference is the following: For the Ising model, each spin can be in one of two states; for the Heisenberg model, each spin can point in any direction in the three-dimensional (3D) space, and the components of the spin operator satisfy the commutation relations

Si, Sj = ih¯ εijkSk, (1.2)

where Si is the ith component of the spin operator,h ¯ is the reduced Planck constant, εijk denotes the Levi-Civita symbol, and the Einstein summation convention1 is used. We see that Heisenberg model is more complex than Ising model. Of course, the solution of Heisenberg model is much more difﬁcult than the solution of Ising model. In 1931, H. Bethe (1906 − 2005) successfully solved the one-dimensional (1D) Heisenberg model [4]. People refer to his method as Bethe ansatz (BA).

1When an index appears twice in a term, one should do the summation over this index.

3 Chapter 1. Introduction

1.2.2 Brief History of Bethe Ansatz

After Bethe’s seminal work in 1931, there was not much work in the following few decades. In 1963, E. H. Lieb and W. Liniger solved a gas of 1D Bose particles interacting via a repulsive delta-function potential by BA [5,6]. In 1966, C. N. Yang and C. P. Yang proved the BA for the 1D Heisenberg model in the ground state [7]. In 1967, M. Gaudin and C. N.

1 Yang solved a gas of 1D spin- 2 fermions interacting via a repulsive delta-function potential by BA independently [8,9]. 2 In 1968, E. H. Lieb and F. Y. Wu solved the 1D Hubbard model

1 by using C. N. Yang’s results for spin- 2 fermions [11, 12]. In 1969, C. N. Yang and C. P. Yang did the thermodynamics for Lieb-Liniger model [13]. In the following a few years, M. Takahashi did the thermodynamics for many other models [14–17]. In the 1970s, R. J. Baxter and the Saint Petersburg school developed the algebraic methods for integrable models [18–23]. In the 1980s, Kondo model was solved by BA [24–27], the algebraic structure of Hubbard was discovered [28–30], general method to solve models with OBCs was developed [31]. From the 1990s on, there were a lot of new progresses, such as the separation of variable method [32], exact solution of spin-ladder model [33], ABA for the 1D Hubbard model [34– 38], calculations of the scalar products and the norm of Bethe eigenstates [39, 40], methods for solving integrable models with arbitrary boundary ﬁelds [41–44], etc. Overall, BA is a well established research direction.

1.2.3 Basic Building Blocks of Algebraic Bethe Ansatz

As we have shown, the theory of integrable models has been developed for many years. Currently, the most prevalent version of the theory is ABA. We point out the basic build-

1 ing blocks of ABA by considering the spin- 2 isotropic Heisenberg (XXX) quantum spin chain [45].

2A detailed proof of Yang’s results was published later in [10].

4 1.2. Bethe Ansatz and Integrability

1 FIGURE 1.1: The Heisenberg XXX spin- 2 chain. There is one spin at each site. Each spin only interacts with its nearest neighbors.

The Hamiltonian (see Fig. 1.1) is given by

N J ˆ H = ∑ ~σn ·~σn+1 − 1 , (1.3) 4 n=1

x y z x where J is the interaction constant, N is the number of sites,~σn = xˆσn + yˆσn + zˆσn (σn is the Pauli matrix σx at site n), 1ˆ is the identity matrix, and periodic boundary conditions (PBCs) are used (~σN+1 = ~σ1).

V2 V2

V3 V3

V1 V1

FIGURE 1.2: Graphical representation of the Yang-Baxter equation.

To solve this Hamiltonian, the most important step is to ﬁnd the suitable solution of the Yang-Baxter equation [9, 18]

0 0 0 0 R12 λ − λ R13 (λ) R23 λ = R23 λ R13 (λ) R12 λ − λ , (1.4) 1

2 where R12 is a 4 × 4 matrix deﬁned on the tensor product space V1 ⊗ V2 (V1 and V2 are C spaces). For a graphical representation of the Yang-Baxter equation, please see Fig. 1.2. For

1 Chapter 1. Introduction

Hamiltonian (1.3), the R-matrix is given by

R12 (λ) = λ1ˆ + iP12, (1.5)

where P12 is the permutation matrix, which is deﬁned by

1 P = 1ˆ +~σ ·~σ 12 2 1 2   1 0 0 0      0 0 1 0    =   . (1.6)    0 1 0 0    0 0 0 1 12

The permutation matrix has the following very useful properties:

P12X1 = X2P12, (1.7)

X1P12 = P12X2, (1.8)

P12P12 = 1,ˆ (1.9)

Pij = Pji, (1.10)

2 2 where X1 is any matrix deﬁned on C space V1, X2 is any matrix deﬁned on C space V2. One can prove the above equations easily (write X1 as X1 ⊗ 1ˆ2, ···). Using Eq. (1.7) and Eq. (1.8), we obtain

PijPik = PikPkj, (1.11)

PijPik = PjkPij, (1.12) where i, j, k are different from each other. We are now ready to prove that Eq. (1.5) is the solution of Eq. (1.4). For Eq. (1.4),

0 0 0 0 0 LHS = λλ λ − λ + iλ λ − λ P23 + iλ λ − λ P13

6 1.2. Bethe Ansatz and Integrability

0 0 0 − λ − λ P13P23 + iλλ P12 − λP12P23 − λ P12P13

−iP12P13P23, and

0 0 0 0 0 RHS = λλ λ − λ + iλλ P12 + iλ λ − λ P13

0 0 0 −λ P13P12 + iλ λ − λ P23 − λP23P12 − λ − λ P23P13

−iP23P13P12.

Then,

LHS − RHS = i (P23P13P12 − P12P13P23)

0 +λ (P13P12 − P12P13 − P23P13 + P13P23)

+λ (P23P12 − P12P23 + P23P13 − P13P23)

= i (P12P23P12 − P12P13P23)

= 0.

We have ﬁnished the proof. In quantum mechanics, there is an elegant algebraic method to solve the simple harmonic oscillator [46]. In that method, one deals with the boson annihilation operator a and creation operator a†. The most important equation in that method is a, a† = 1. Similarly, Yang-Baxter equation is the most important equation in the theory of integrable models. Another important quantity in ABA is the monodromy matrix [18]

T0 (λ) = L0N (λ) ··· L01 (λ) , (1.13)

where L0n (λ) is deﬁned as i L (λ) = R λ − . (1.14) 0n 0n 2

7 Chapter 1. Introduction

The monodromy matrix obeys the fundamental commutation relation

0 0 0 0 R000 λ − λ T0 (λ) T00 λ = T00 λ T0 (λ) R000 λ − λ , (1.15) which is a direct consequence of the Yang-Baxter equation. The transfer matrix t(λ) is deﬁned by tracing over the auxiliary space

t (λ) = tr0T0 (λ) . (1.16)

The transfer matrices with different spectral parameters commute with each other:

t (λ) , t λ0 = 0. (1.17)

We can prove this equation easily by using Eq. (1.15). From Eq. (1.17), one can obtain an inﬁnite number of commuting conserved charges,3 which guarantees the integrability of the Hamiltonian (1.3). The relationship of the transfer matrix and the Hamiltonian is given by

( ) J d H = i ln t(λ) − N . (1.18) 2 dλ i λ= 2

At this point, we are clear about the relationship between R-matrix, L-matrix, monodromy matrix, transfer matrix and the Hamiltonian. We also have some ideas about the meaning of integrability. To know the full story of ABA, please see [45], or [44], or Chap- ter2. Before we leave this section, we want to emphasize that the Yang-Baxter equation [9, 18] and monodromy matrix [18] are the most important building blocks of ABA.

1.2.4 A Note about Integrability and Exact Solvability

In principle, all integrable models can be solved exactly, although some models are easy to be solved and some are difﬁcult [44]. Not all exactly solvable models are integrable.

3The Hamiltonian is one of them.

8 1.3. Matrix Product States

The Rabi model [47] is a simple and important model in quantum optics. The exact solution of this model has been achieved a few year ago [48, 49]. Although the Rabi model is exactly solvable, it’s integrable only for a few special parameters [50].

1.3 Matrix Product States

In 1968, R. J. Baxter considered the problem of fitting dimers on to a plane rectangular lattice [51]. The thermodynamic limit of the properties of the system is given by a set of matrix equations, where the matrices are of infinite dimensionality. The equations cannot be solved directly, but if they are restricted to be of finite and small dimensionality, very good approximations to the thermodynamic properties of the system are obtained. The above example is one of the early examples that contributed to the concept of MPSs. Some other examples related to MPSs come from the work related to Haldane gap:4 there is a gap between the ground state and the excited states for the 1D Heisenberg antiferromagnet with integer, but not half-odd-integer, spin. Affleck, Kennedy, Lieb, and Tasaki (AKLT) considered a spin-1 chain in 1987, and exem- plified Haldane’s results [54, 55]. The exact ground state of the AKLT model can be written as a MPS. After AKLT’s seminal work, there are a lot of subsequent work to verify Haldane’s results by using matrix product approach [56–59]. Apart from the above work, we want to mention one more line of work about the exact solution of a 1D asymmetric exclusion model [60]. In this work, the authors derived quadratic algebras of the matrices by using a matrix product ansatz (MPA). This MPA has wide applicability because many quadratic algebras have explicitly known representations. All the above examples are exactly solvable models. However, exactly solvable models are very rare. Physicists need to develop a general numerical method to solve many-body problems. People achieved this for 1D lattice models in 1992 by inventing the density matrix renormalization group (DMRG) method [61, 62]. The DMRG method is very accurate,

4Proposed in 1983 by F. Duncan Haldane [52, 53].

9 Chapter 1. Introduction and can be used to a large number of situations. In 1995, it was discovered that the converged quantum states in the DMRG method could be simply represented by a MPS in the thermodynamic limit [63]. After that, the connection between MPS and DMRG was ﬁrmly established, and lead to a lot of advances [64, 65].

σ1 σ2 σ3 σ4 σ5 σ6 σ7

A1 A2 A3 A4 A5 A6 A7

FIGURE 1.3: Representation of a MPS with PBCs; the long line at the bottom indicates the trace operation.

So, MPS is a very useful concept in both the exactly solvable models and numerical methods. To use the MPS to do numerical calculations, one usually writes the ground state wave function (see Fig. 1.3) as

σ1 ··· σL |ψi = ∑ Tr A1 AL |σ1 ... σLi , (1.19) {σ}

5 σi where 1, ···, L denote the sites, |σii denotes the eigenstate at site i, Ai are n × n matrices (n is called the bond dimension), "Tr" means taking the trace, and PBCs are used. People use this |ψi as the variational wave function to calculate the ground state energy of the system. The state corresponding to the minimal energy is the approximate ground state. Generally speaking, the larger the bond dimension, the more accurate the result. The effectiveness of the numerical method is guaranteed by the area law [66–69], that is, the entanglement of the ground state wave function has nothing to do with the length of the chain. In recent years, MPSs have developed to the more general tensor networks [70]. At last, we want to mention continuous matrix product states (cMPSs) [71], which are a generalization of MPSs to the continuum limit. cMPSs are very effective in describing the low-energy states of quantum ﬁeld theories [71, 72]. We consider a 1D system of bosons on a ring of length L and associated ﬁeld operator ψ(x) with canonical commutation relation,

5 1 For Heisenberg spin- 2 chain, |σii can be |↑i or |↓i.

10 1.4. Cold Atoms and Quantum Quenches

ψ(x), ψ†(y) = δ(x − y) with 0 ≤ x, y ≤ L space coordinates. A cMPS is defined for PBCs as R L n dx[Q(x)⊗1ˆ+R(x)⊗ψ†(x)]o |χi = Traux Pe 0 |Ωi , (1.20) where Q(x), R(x) are position dependent matrices of dimension D × D that act on a D- dimensional auxiliary system (D is called the bond dimension), Pexp is the notation for the path-ordered exponential [73], Traux is the trace over the auxiliary space, and |Ωi is the vacuum state: ψ(x) |Ωi = 0. With the above defined cMPS, it’s easy to express the norm and expectation value of operators in terms of the matrices R and Q. Then one can express the expectation value of the Hamiltonian of the system in terms of R and Q. Next, one tries to find R and Q such that the expectation value of the Hamiltonian takes its minimum value, which is close to (but a little larger than) the ground state energy. In principle, the larger the bond dimension, the more accurate the results.

1.4 Cold Atoms and Quantum Quenches

Since the first realization of Bose-Einstein condensation in dilute vapors of alkali-metal atoms, a large number of research activities have been done [74–76], and a lot of work are ongoing. Compared to condensed matter systems, ultracold quantum gases have a lot of merits: easy to tune the dimensionality, easy to tune the interaction strengths, and possible to obtain clean realizations of standard many-body Hamiltonians such as FHM and BHM. These properties explain why ultracold atoms are so popular these days? Storing ultracold quantum gases in artificial periodic potentials of light is very convenient to simulate quantum many-body systems [77]. A periodic potential can be formed by overlapping two counter-propagating laser beams. The interference between the two laser beams forms an optical standing wave, which can trap the atoms. By interfering more laser beams, one can obtain 3D periodic potential (see Fig. 1.4). Three orthogonal optical standing waves correspond to a 3D simple cubic crystal (see Fig. 1.5), in which each trapping site acts as a tightly confining harmonic oscillator potential.

11 Chapter 1. Introduction

FIGURE 1.4: Two orthogonal optical standing waves. If one adds another optical standing wave in the direction that is perpendicular to the page, one obtains a 3D simple cubic lattice.

For the simple cubic lattice, if the laser intensities in two orthogonal directions (y and z) are made very large compared to the intensity in the remaining direction (x), then tunnelling predominantly happens along the x direction and the system is effectively reduced to a set of 1D lattices [78]. One research topic that is related to 1D ultracold atomic gases is quantum quench in integrable models [79]. This topic comes from the the study of isolated quantum systems out of equilibrium in systems of optically trapped ultracold atomic gases. Some fundamental questions are proposed: Do observables relax to time-independent values? What are the principles determining these values? How can one describe the relaxation towards stationary behavior?

12 1.5. Motivation of Our Research

FIGURE 1.5: A 3D simple cubic lattice created by three orthogonal optical standing waves.

A lot of work related to quantum quenches is ongoing. For a review of some recent progresses, please see [80].

1.5 Motivation of Our Research

1.5.1 Motivation for Our Research on Matrix Product States

In our work [1], we derive matrix product representations of the Bethe ansatz states of the

1 Heisenberg XXZ spin- 2 chain and Lieb-Liniger model with OBCs. The motivation for our work is twofold. Firstly, the theory of integrable models began in 1931 [4]. There is a lot of research activities until now. Most of the research work centered on the eigenenergy, but very limited

13 Chapter 1. Introduction information was known about the eigenstates. We wanted to know more about the Bethe eigenstates. Secondly, people postulates the form of MPSs for PBCs and OBCs differently in the DMRG algorithm. The MPSs for PBCs case are consistent with the exact MPSs derived from the ABA [81, 82]. What’s the situation for OBCs?

1.5.2 Motivation for Our Research on Quantum Quench in Cold Atoms

In our work [2], we invented a simple approximate method to calculate the asymptotic expansion velocity of interacting atomic gases, which is accessible in sudden expansion experiments with ultracold atoms. The motivation of this work is that we want to address in this particular context the fundamental questions of quantum quenches: Do observables relax to time-independent values? What are the principles determining these values?

14 Chapter 2

Derivation of MPSs for Integrable Models with OBCs

This chapter on the derivation of MPSs for integrable models with OBCs is based on the work published as Ref. [1].

2.1 Background and Motivation

1 In 2006, Alcaraz and Lazo solved the Heisenberg XXZ spin- 2 chain with OBCs [83]. They used a MPA, and gave ﬁnite-dimensional representation of the matrices. Also in 2006, Golinelli and Mallick derived a MPA for the asymmetric exclusion process (ASEP) on a 1D periodic lattice by using ABA [84]. In this MPA, the components of the eigenvectors of the ASEP Markov matrix were expressed as traces of products of non-commuting matrices. The matrices satisfy a quadratic algebra. Explicit ﬁnite-dimensional representations for the matrices were given. The authors stated in the paper that "the construction of a matrix product representation for the open ASEP remains an open problem". In 2010, Katsura and Maruyama derived a matrix product representation of the Bethe

1 ansatz state for the XXX and XXZ spin- 2 Heisenberg chains using the ABA [81]. Their method is similar to the one in [84], and their results are the same as the ones in [83]. Basi- cally speaking, the authors proved Alcaraz and Lazo’s results [83] by using ABA. For all the above work, PBCs were used. Similar work for OBCs was lacking.

15 Chapter 2. Derivation of MPSs for Integrable Models with OBCs

1 × 2 2 × 4 2 × 1 4 × 8 4 × 2

8 × 4

σ1 σ2 σ3 σ4 σL−3 σL−2 σL−1 σL A1 A2 A3 (A4 ··· AL−3 ) AL−2 AL−1 AL

FIGURE 2.1: Example of a typical MPS coefﬁcient generated by an itera- tive singular-value decomposition. In the generic case of all non-zero sin- σi 2 [L/2] gular values, the Ai matrix sizes telescope as (1 × d), (d × d ),..., (d × d[L/2]),..., (d2 × d), (d × 1); where at the center of the chain there will be one single square matrix if the number of sites, L, is odd or a pair of matrices of transposed dimensions if L is even [65] (the square brackets denote the integer-part of L/2). Here d is the dimension of the local Hilbert space for 1 each lattice site, —we have d = 2 for the case of the spin- 2 chains that we study and as depicted in the ﬁgure. The dashed matrix in the center rep- resents the product of all the central matrices that, in principle, telescope to very large sizes. In practical implementations these need to be truncated and are replaced by uniformly square matrices of size D × D, where D is called the bond dimension. This makes the practical OBC-MPS ansatz in the bulk of the chain look similar to that one used for PBCs discussed in the text.

16 2.1. Background and Motivation

The importance of boundary conditions should not be overlooked. It has been proved long ago (in 1973) that the bulk free energy of six-vertex model in the thermodynamic limit is the same for PBCs and OBCs [85]. However, Korepin and Zinn-Justin showed in 2000 that the bulk free energy of six-vertex model in the thermodynamic limit is different for PBCs and domain wall boundary conditions [86]. In our work [1], we derived MPSs for

1 the Heisenberg XXZ spin- 2 chain with OBCs. The motivation for our work comes from an interest in the properties of the DMRG algorithm as applied to spin chains. It was observed early on that the accuracy and speed of convergence of DMRG were much better in the case of OBCs as compared with PBCs. It was subsequently understood that the root cause of this is the structure of the MPSs generated by the algorithm. These have the form

σ1 σL ∑{σ} A1 ··· AL |σ1 ... σLi, where the matrices are in general rectangular and telescope both ways in size so that their product is ultimately a scalar quantity (for more details see Fig. 2.1). Without truncation of the matrix sizes, this can in principle represent any arbitrary quantum state. With truncation, it works well for OBCs, but it has the problem that the ﬁrst-few and last-few matrices are too restricted to fully capture the entanglement between the ﬁrst-few and last-few spins —which with PBCs is large since they actually become close neighbors by virtue of closing the chain on itself. The exact construction overcomes this problem by allowing exponentially large (in the number of sites) matrices for the central portion of the chain, but that is computationally impracticable. Thus, in the PBCs case, it is more natural

σ1 σL to postulate MPSs of the form ∑{σ} Tr A1 ··· AL |σ1 ... σLi, where all the matrices can now be taken to be square and equal in size. The effect of truncating the matrices is now distributed uniformly along the chain and is less detrimental. It was indeed shown that, in a variational MPS-based reformulation of DMRG, the algorithmic performance with either type of boundary conditions becomes comparable if the matching type of MPS is used [87].

1 Interestingly, the ABA construction of MPSs for the periodic Heisenberg XXZ spin- 2 chain naturally gives a traced-product of matrices in auxiliary space [81]. The study of what hap-

1 pens for the open Heisenberg XXZ spin- 2 chain is the main focus of our work [1]. 1 In this chapter, we shall construct MPSs for the Heisenberg XXZ spin- 2 chain with OBCs by using the ABA. For the MPSs of ditto system but with PBCs [81], the dimension of the

17 Chapter 2. Derivation of MPSs for Integrable Models with OBCs matrices is 2n, where n is the number of spin-ﬂips present in the state. For OBCs, we shall see that one cannot construct a MPS with the dimension of the matrices being 2n, but one can obtain a MPS with matrices of dimension 22n instead. The matrices of the MPS will not be given in closed form, but they can be obtained for any case of interest by using a set of recursion relations that will be derived. We shall also extend the same ideas to construct a MPS for a lattice-regularized version of the Lieb-Liniger model with OBCs. To obtain a closed-form cMPS similar to the one in [71] for the Lieb-Liniger model with OBCs, one has to solve explicitly the recursion relations of the matrices, which shall remain as an open problem. This chapter is outlined as follows. In section 2.2, we derive MPSs for the Heisenberg

1 spin- 2 chain with OBCs. In section 2.3, we derive MPSs for the Lieb-Liniger model with OBCs.

2.2 Derivation of MPSs for the Heisenberg Spin Chain with OBCs

1 2.2.1 ABA for the Heisenberg XXZ Spin- 2 Chain with OBCs

1 The Heisenberg XXZ spin- 2 chain with OBCs is described by the following Hamiltonian:

L−1 x x y y z z − − H = ∑ σi σi+1 + σi σi+1 + ∆(σi σi+1 1) ∆, (2.1) i=1

α where L denotes the total number of sites, σi (with α = x, y, z) are the Pauli matrices for the spin on the ith site, and ∆ is the anisotropy parameter (which is 1 for the XXX case). The eigenstates of this model can be constructed using the ABA [23, 44, 45]; we shall now brieﬂy outline this construction. The central object of the ABA is the quantum R-matrix which is the solution of the Yang-Baxter equation,

0 0 0 0 Rab(λ − λ )Rac(λ)Rbc(λ ) = Rbc(λ )Rac(λ)Rab(λ − λ ) . (2.2)

18 2.2. Derivation of MPSs for the Heisenberg Spin Chain with OBCs

1 For the Heisenberg XXZ spin- 2 chain, the R-matrix acting on the space Va ⊗ Vb has only six non-zero entries and is given by

  1 0 0 0    sinh η   0 sinh λ 0   sinh(λ+η) sinh(λ+η)  Rab(λ) =   , (2.3)  sinh η sinh λ   0 sinh(λ+η) sinh(λ+η) 0    0 0 0 1 ab where η is deﬁned by ∆ = cosh η. To prove that Eq. (2.3) is the solution of Eq. (2.2), we can write Eq. (2.3) as

ˆ x x y y z z Rab(λ) = f (λ)1ab + g(λ) σa σb + σa σb + h(λ)σa σb , (2.4) where

1 sinh(λ) f (λ) = 1 + , 2 sinh(λ + η) 1 sinh(η) g(λ) = · , 2 sinh(λ + η) 1 sinh(λ) h(λ) = 1 − . 2 sinh(λ + η)

Then we can use

σiσj = δij + iεijkσk (2.5) to prove that Eq. (2.3) is the solution of Eq. (2.2). Next, we introduce the quantum L-operator represented by a matrix acting on the tensor product of two 2D vector spaces V0 ⊗ Vi. The 2D auxiliary space is denoted by V0 (or also

Va or Vb when we need two of them below), and the physical Hilbert space at the ith site is 1 denoted by Vi. For the Heisenberg spin- 2 chain we have

η L (λ) = R (λ − ) . (2.6) 0i 0i 2

19 Chapter 2. Derivation of MPSs for Integrable Models with OBCs

The following intertwining relation can be shown as a direct consequence of the Yang-Baxter equation

Rab(λ − µ)Lai(λ)Lbi(µ) = Lbi(µ)Lai(λ)Rab(λ − µ) . (2.7)

Here, Rab acts on the space Va ⊗ Vb, and Lai acts on Va ⊗ Vi. Va and Vb are auxiliary spaces, and Vi is the physical space at site i. The construction so far did not involve the boundary conditions, but let us now focus on the OBCs case. Following Sklyanin, we can consider a looped monodromy matrix [31]

T0(λ) = L01(λ) ··· L0L(λ)L0L(λ) ··· L01(λ) . (2.8)

In the space V0, the monodromy matrix can be represented as a 2 × 2 matrix

  A(λ) B(λ)   T0(λ) =   , (2.9) C(λ) D(λ) 0 where the matrix elements A(λ), B(λ), C(λ) and D(λ) are themselves operators acting on the total Hilbert space of the chain V1 ⊗ V2 ⊗ · · · ⊗ VL. Using Eq. (2.2), we can obtain the following relation for the monodromy matrix

Rab(λ − µ)Ta(λ)Rab(λ + µ − η)Tb(µ) = Tb(µ)Rab(λ + µ − η)Ta(λ)Rab(λ − µ). (2.10)

The proof is given in AppendixA.

Taking the trace of the monodromy matrix over V0, we obtain a one-parameter family of transfer matrices acting on the total physical Hilbert space V1 ⊗ V2 ⊗ · · · ⊗ VL:

t(λ) = TrV0 T0(λ) = A(λ) + D(λ) . (2.11)

20 2.2. Derivation of MPSs for the Heisenberg Spin Chain with OBCs

One can show that [t(λ), t(µ)] = 0 (see AppendixB). Furthermore, the Hamiltonian can be recovered from the logarithmic derivative of t(λ)

∂ H = sinh η log t(λ) . (2.12) ∂λ λ=η/2

The derivation of the above equation is given in AppendixC. Since the Hamiltonian commutes with the transfer matrix, we can construct simultane- ous eigenstates of both H and t(λ). Such an eigenstate of t(λ) is given by

|λ1, λ2, ··· , λni = B(λ1)B(λ2) ··· B(λn) |ωi , (2.13)

where |ωi denotes the standard reference ferromagnetic state, i.e. |ωi = |↑i1 |↑i2 ···

|↑iL, and n denotes the number of down spins. We call the above state a Bethe state. Since one can show from Eq. (2.10) that the B(λi)’s commute with each other (see AppendixD), this state is invariant under permutations of λi’s. The latter have to be non-repeating and obey the OBC Bethe ansatz equations (BAEs), which are given by (see AppendixD)

n 2 η 2L η sinh λi + λj + η sinh λi − λj + η cosh λi − sinh λi + = 2 2 , ∏ sinh λ + λ − η sinh λ − λ − η 2 η 2L − η j=1(j6=i) i j i j cosh λi + 2 sinh λi 2 i = 1, 2, . . . n. (2.14)

The eigenenergy corresponding to |λ1, λ2, ··· , λni is given by (see AppendixD)

n 2 sinh2(η) E(λ1, λ2, ··· , λn) = ∑ η η − ∆ . (2.15) i=1 sinh(λi + 2 ) sinh(λi − 2 )

2.2.2 Comparison between ABA and CBA

1 In the last section, we solved the Heisenberg XXZ spin- 2 chain with OBCs by using ABA. This model can also be solved by using coordinate Bethe ansatz (CBA) [88]. We show that our results are exactly the same as the ones obtained by CBA. We pick up the results of CBA

21 Chapter 2. Derivation of MPSs for Integrable Models with OBCs from paper [89]. L−1 L−1 1 + − − + z z HC = ∑ Si Si+1 + Si Si+1 + ∆ ∑ Si Si+1, (2.16) 2 i=1 i=1 where the subscript "C" denotes coordinate Bethe ansatz, and

1 y S+ = σx + iσ , i 2 i i 1 y S− = σx − iσ , i 2 i i 1 Sz = σz. i 2 i

So, L−1 1 x x y y z z HC = ∑ σi σi+1 + σi σi+1 + ∆σi σi+1 . (2.17) 4 i=1 The eigenenergy is given by

M L − 1 EC = ∑ cos ki + − M ∆, (2.18) i=1 4 where M is the number of down spins. The BAEs are given by

M 2 i(kj−ki) ikj i(ki+kj) iki 1 − ∆eiki 1 + e − 2∆e 1 + e − 2∆e e2i(L+1)ki = , (2.19) −ik 2 ∏ i(k +k ) ik i(k −k ) (1 − ∆e i ) + e i j − e j + e j i − e−iki j=1(j6=i) 1 2∆ 1 2∆ and i = 1, 2, . . . , M. To compare our results from ABA (see section 2.2.1) with the above ones from CBA, we

1 change the "H" in Eq. (2.1) to "HA", and change the "E(λ1, λ2, ··· , λn)" in Eq. (2.15) to "EA". We should keep in mind that we use "n" to denote the number of down spins in the previous section, so "n" in the previous section is the same as "M" in Eq. (2.18) and Eq. (2.19). It’s easy to show that 1 1 H = H + ∆L. (2.20) C 4 A 4 1The subscript "A" denotes algebraic Bethe ansatz.

22 2.2. Derivation of MPSs for the Heisenberg Spin Chain with OBCs

Using2 sinh λ + η ikj j 2 e = η , (2.21) sinh λj − 2 we obtain

2 iki 2 η 2 η 1 − ∆e cosh λi + 2 sinh λi + 2 2 = 2 η 2 η , −iki (1 − ∆e ) cosh λi − 2 sinh λi − 2 sinh2L+2 λ + η 2i(L+1)ki i 2 e = 2L+2 η , sinh λi − 2 sinh η · sinh λ + λ + η i(kj−ki) ikj i j 1 + e − 2∆e = − η η , sinh λi + 2 sinh λj − 2 sinh η · sinh λ − λ + η i(ki+kj) iki i j 1 + e − 2∆e = η η , sinh λi − 2 sinh λj − 2 sinh η · sinh λ − λ − η i(ki+kj) ikj i j 1 + e − 2∆e = − η η , sinh λi − 2 sinh λj − 2 sinh η · sinh λ + λ − η i(kj−ki) −iki i j 1 + e − 2∆e = η η . sinh λi + 2 sinh λj − 2

Then, we can easily show that the BAEs obtained from CBA are the same as the ones obtained from ABA. Next, we consider the eigenenergy. Using

η sinh λi + 2 cos ki + i sin ki = η , sinh λi − 2 η sinh λi − 2 cos ki − i sin ki = η , sinh λi + 2 we obtain sinh2(η) cos ki = ∆ + η η . (2.22) 2 sinh λi + 2 sinh λi − 2 So M sinh2(η) (L − 1)∆ EC = ∑ η η + . (2.23) i=1 2 sinh λi + 2 sinh λi − 2 4

2Dr. Masudul Haque guessed that this equation might be able to build the connection between ABA and CBA. I’d like to thank Dr. C. J. Bolech for telling me Dr. M. Haque’s guess.

23 Chapter 2. Derivation of MPSs for Integrable Models with OBCs

Then, it’s easy to show that 1 1 E + ∆L = E , (2.24) 4 A 4 C which is consistent with Eq. (2.20). In summary, we have shown that CBA and ABA are consistent by using Eq. (2.21).

1 2.2.3 Exact MPSs for the XXZ Spin- 2 Chain with OBCs

In section 2.2.1, we outlined the construction of the eigenstates of the Hamiltonian by using the ABA. In this section, we derive MPS representations for those eigenstates. Using

B(λ) = 0h↑| T0(λ) |↓i0 , (2.25) we can write

|λ1, λ2, ··· , λni = nh↑| ··· 2h↑| 1h↑|

·Tn(λn) ··· T2(λ2)T1(λ1) · |↓in ··· |↓i2 |↓i1 · |ωi . (2.26)

To derive a MPS for this Bethe state, we need to reorganize the L-operators in that product of monodromy matrices. All of those belonging to the same physical Hilbert space should be brought together. Let us illustrate this by a simple example with L = 2, n = 2 and PBCs.

We have T2T1 = (L21L22) · (L11L12), of which the spectral parameters are omitted for brevity.

Then we can easily reorder them to obtain T2T1 = (L21L11) · (L22L12). This was a crucial step to derive a MPS for the Heisenberg spin chain with PBCs [81]. For the same example but with OBCs, we have T2T1 = (L21L22L22L21) · (L11L12L12L11) and it is clear that we cannot put the L-operators with the same physical-space subscript (say 1 or 2 in this example) together by means of a simple reordering. To overcome this difﬁculty, we rewrite Eq. (2.26) as

|λ , λ , ··· , λ i = [Tr (Q T (λ ))] ··· [Tr (Q T (λ ))][Tr (Q T (λ ))] |ωi , (2.27) 1 2 n Vn n n n V2 2 2 2 V1 1 1 1

24 2.2. Derivation of MPSs for the Heisenberg Spin Chain with OBCs where we introduced the deﬁnition

  0 0 |↓i · h↑|   Qi = i i =   (2.28) 1 0 i

Now we focus on the structure of an individual OBC looped monodromy matrix, Ti(λi), as given in Eq. (2.8). What we need to do is to change (Li1 ··· LiL) · (LiL ··· Li1) to something similar to (Li1 ··· LiL) · (Li1 ··· LiL). To achieve this, we denote

  EF ···   ≡ Li1(λi) LiL(λi) =   Mi, (2.29) GH i where E, F, G and H are matrices acting on the total Hilbert space V1 ⊗ V2 ⊗ · · · ⊗ VL. We also similarly denote   UV ···   ≡ LiL(λi) Li1(λi) =   Ni. (2.30) XY i

Next we introduce an additional auxiliary space Vi, and we want to ﬁnd a 4 × 4 matrix

Qii such that h t i [ T (λ )] = [ M N ] = M N i Q TrVi Qi i i TrVi Qi i i TrVi⊗Vi i i ii , (2.31)

ti where the superscript ti in Ni means the transpose of matrix N in the new auxiliary space only. One can easily ﬁnd that such a Qii is given by

  0 1 0 0      0 0 0 0  Q   ii =   , (2.32)    0 0 0 0    0 1 0 0 ii the subscript of the RHS matrix means that it is written in the space Vi ⊗ Vi. Using this, we have t t ( T (λ )) = [L (λ ) ··· L (λ )L i (λ ) ··· L i (λ ) ·Q ] TrVi Qi i i TrVi⊗Vi i1 i iL i i1 i iL i ii . (2.33)

25 Chapter 2. Derivation of MPSs for Integrable Models with OBCs

If we denote

tn t2 t1 Li(λ1, ··· , λn) = [Lni(λn)Lni(λn)] ··· [L2i(λ2)L2i(λ2)][L1i(λ1)L1i(λ1)], (2.34) then Eq.(2.26) becomes

|λ1, λ2, ··· , λni = Tr(V⊗V)⊗n {[L1(λ1, ··· , λn) |↑i1]

··· [LL(λ1, ··· , λn) |↑iL] · (Qnn ···Q22Q11)}, (2.35)

⊗n where (V ⊗ V) means the Hilbert space Vn ⊗ Vn ⊗ ... ⊗ V2 ⊗ V2 ⊗ V1 ⊗ V1.

It is convenient to introduce two matrices, Dn and Cn, deﬁned via

Li(λ1, ··· , λn) |↑ii = Dn(λ1, ··· , λn) |↑ii + Cn(λ1, ··· , λn) |↓ii , (2.36)

2n 2n and one should keep in mind that Dn and Cn are 2 × 2 matrices acting on the space

Vn ⊗ Vn ⊗ · · · ⊗ V2 ⊗ V2 ⊗ V1 ⊗ V1. Adopting the usual notations

η sinh(λ − 2 ) b(λ) ≡ η , (2.37) sinh(λ + 2 ) sinh(η) c(λ) ≡ η , (2.38) sinh(λ + 2 ) one can easily obtain

  1 0 0 0      0 b(λ ) 0 0   1  D1(λ1) =   , (2.39)    0 0 b(λ1) 0    2 2 c(λ1) 0 0 b(λ1) 11   0 0 c(λ1) 0      b(λ )c(λ ) 0 0 b(λ )c(λ )   1 1 1 1  C1(λ1) =   . (2.40)    0 0 0 0    0 0 c(λ1) 0 11

26 2.2. Derivation of MPSs for the Heisenberg Spin Chain with OBCs

For n > 1, recursion relations between Dn+1, Cn+1 and Dn, Cn can be found as follows. t ··· · (n+1) Starting from the recursion Li(λ1, , λn, λn+1) = L(n+1)i(λn+1) L(n+1)i(λn+1)

· Li(λ1, ··· , λn), we can introduce Dn and Cn, and derive expressions for Dn+1 and Cn+1. These are given by

  1 0 0 0      0 b 0 0   n+1  Dn+1 =   ⊗ Dn    0 0 bn+1 0    c2 0 0 b2 n+1 n+1 (n+1)(n+1)   0 cn+1 0 0      0 0 0 0    +   ⊗ Cn (2.41)    bn+1cn+1 0 0 bn+1cn+1    0 cn+1 0 0 (n+1)(n+1) and

  b2 0 0 c2  n+1 n+1     0 b 0 0   n+1  Cn+1 =   ⊗ Cn    0 0 bn+1 0    0 0 0 1 (n+1)(n+1)   0 0 cn+1 0      b c 0 0 b c   n+1 n+1 n+1 n+1  +   ⊗ Dn, (2.42)    0 0 0 0    0 0 cn+1 0 (n+1)(n+1)

where bn+1, cn+1 are shorthand notations for b(λn+1), and c(λn+1), respectively.

27 Chapter 2. Derivation of MPSs for Integrable Models with OBCs

Using the above deﬁnitions one can, after a little algebra, achieve the goal of rewriting the Bethe state |λ1, λ2, ··· , λni as a MPS involving a trace over all the auxiliary spaces:

h x1−1 x2−x1−1 |λ1, λ2, ··· , λni = ∑ Tr(V⊗V)⊗n (Dn) Cn (Dn) {x1,x2,···,xn} i xn−xn−1−1 L−xn ·Cn ··· (Dn) Cn (Dn) Qn · |x1, x2, ··· , xni , (2.43) where

Qn = Qnn ⊗ · · · ⊗ Q22 ⊗ Q11, (2.44) and |x1, x2, ··· , xni, with 1 ≤ x1 < x2 < ··· < xn ≤ L, denotes the conﬁguration with down spins at those (lattice) locations—remember that n is the total number of down spins. Remarkably, the MPSs obtained from ABA with OBCs have the same form as the ones commonly postulated for the PBCs case (ABA with PBCs was giving the same structure, but it was as expected in that case [81]). The correspondence is given by identifying the

σi=↑ σi=↓ matrices Ai = Dn and Ai = Cn, notice there is no dependence on the site index i, a kind of translational invariance. Of course, the appearance of the matrix Qn is an additional feature that alters the translational invariance of the ansatz.

2.2.4 Discussion and Conclusions

Let us consider the simplest non-trivial example to illustrate the way our MPS construction works. For the Hamiltonian in Eq. (2.1), we choose L = 3 and only one down spin. The dimension of the Hilbert space is thus 3. For such a small system, by direct calculation one can obtain the eigenenergies, Ei, and the corresponding (non-normalized) eigenstates, |φii, rather easily:

1. E1 = −3∆ with |φ1i = |↑i1 |↑i2 |↓i3 − |↓i1 |↑i2 |↑i3, √ 2 2. E2 = −4∆ + ∆ + 8 with √ 1 2 |φ2i = |↑i1 |↑i2 |↓i3 + 2 ( ∆ + 8 − ∆) |↑i1 |↓i2 |↑i3 + |↓i1 |↑i2 |↑i3, √ 2 3. E3 = −4∆ − ∆ + 8 with √ 1 2 |φ3i = |↑i1 |↑i2 |↓i3 − 2 ( ∆ + 8 + ∆) |↑i1 |↓i2 |↑i3 + |↓i1 |↑i2 |↑i3.

28 2.2. Derivation of MPSs for the Heisenberg Spin Chain with OBCs

Next, we solve the same eigenvalue problem by the Bethe ansatz and use the MPS formulas we derived above. To be speciﬁc, let us choose ∆ = 2, and then η ≈ 1.317. (1) (2) Solving Eq. (2.14), we obtain three possible solutions λ1 = iπ/2, or λ1 ≈ i0.3747, or (3) λ1 ≈ −0.831 + iπ/2. Then it is very easy to check that Eq. (2.15) exactly recovers the three energies found by exact diagonalization. Further, one can use Eq. (2.43) to get an expression

(i) 2 (i) (i) h 2 (i) for the corresponding eigenstates: |λ1 i = 2b (λ1 ) |↑i1 |↑i2 |↓i3 + b(λ1 ) 1 + b (λ1 ) + 2 (i) i h 2 (i) 2 (i) 2 (i) 4 (i) i c (λ1 ) |↑i1 |↓i2 |↑i3 + 1 + c (λ1 ) + b (λ1 )c (λ1 ) + b (λ1 ) |↓i1 |↑i2 |↑i3; which, as can be seen by simple evaluation, are in agreement (up to normalization) with the |φii found directly. Of course, these eigenstates need not be and are not translationally invariant. One would expect that removing the matrix Qn from the expression for the eigenstates will re- store that invariance (and be unphysical), but the effect is actually more dramatic and they all become null vectors (i.e., invalid eigenstates). Repeating this exercise but for PBCs, one finds translationally invariant eigenstates that are in agreement between exact diagonalization and the Bethe ansatz. Once again, removing the matrix Qn from the ABA expression for the eigenstates will turn them all into null vectors. This suggests, as we discuss below, that any practical MPS ansatz should incorporate as an element the correlate of Qn. Going beyond finite-system-size calculations and motivated by the infinite-system-size limit, one knows that the bulk physics does not depend on the boundary conditions, and one should be able to reach the same so-called infinite-MPS (iMPS) [90] construction. Focusing on the common global structure of our OBC and the PBC Bethe states after casting them into traced matrix-product forms, we propose the following unified practical MPS ansatz:

σ1 ··· σL Qr |MPSi = ∑ Tr A1 AL |σ1 ... σLi , (2.45) {σ}

σi σ where the Ai are D × D variational matrices. Notice that from the ABA we ﬁnd Ai = σ Aj ∀ i, j and independent of the size of the system. We thus have the usual site indepen- dence of the A-matrices in the spirit of iMPS but without needing to invoke translational

29 Chapter 2. Derivation of MPSs for Integrable Models with OBCs invariance. In contradistinction with the practice in the literature, ABA prompts the inclu- sion of an additional matrix into the ansatz that is also taken as a variational parameter (for a total of d + 1 variational matrices). We introduce the integer index r = 0, 1 which we call the rank index of the ansatz, and we consider it also a variational parameter. Rank zero is equivalent to not having Q and falls back into the standard iMPS structure. Rank one can be distinctively different and we posit it as the generic choice for Bethe-ansatz-integrable systems. In conclusion, we have successfully recast the Bethe eigenstates for Heisenberg XXZ

1 spin- 2 chain with OBCs to MPSs. We ﬁnd important similarities and differences with the various MPSs discussed in the tensor-networks literature for both open and periodic boundary conditions. In particular, our results clarify the conditions for the implementation of iMPS and prove that such an approach can address directly the thermodynamic limit in a uniﬁed way for both PBC and OBC. Our construction is rather generic and can be applied to other integrable models solvable via the standard ABA. Our work makes the Bethe eigenstates more accessible and thus informs the choice of ansatz for numerical solutions of both integrable and non-integrable systems in one dimension.

2.2.5 Comments on the Key Idea of Our Research

For our research, the most important equation is Eq. (2.31). The key idea used in this equation is doubling the auxiliary space. This idea appeared many times in the history of science before. In calculus, some kinds of definite integrals are difficult to evaluate directly, but the calculations are very easy if one use the theory of complex variables. Here one goes from 1D space to 2D space. In linear algebra, it’s difficult to calculate some kinds of n × n determinants. If one changes the n × n determinants to (n + 1) × (n + 1) determinants first, then it will be easy to finish the calculations. In the early days of quantum mechanics, people used 2 × 2 matrices to describe electron. P. A. M. Dirac (1902 − 1984) wanted to have an equation of electron that had first order

30 2.3. MPSs for the Lieb-Liniger Model with OBCs derivatives of space and time, and he succeeded by using 4 × 4 matrices [91]. To solve spin chain by ABA, one adds an auxiliary spin to the original spin chain (see Eq. (2.8)). We believe that similar ideas will continue to appear in many different situations in the future.

2.3 MPSs for the Lieb-Liniger Model with OBCs

1 We have derived the MPSs for Heisenberg XXZ spin- 2 chain with OBCs. Our method is very general, it can be used for any non-nested integrable model. We will show this for the Lieb-Liniger model.

2.3.1 ABA for the Lieb-Liniger Model

The Hamiltonian of the Lieb-Liniger model is given by

Z L † † † H = [∂xψ (x)∂xψ(x) + κψ (x)ψ (x)ψ(x)ψ(x)]dx, (2.46) 0 where κ > 0 and we have ﬁxedh ¯ = 1 = 2m. The bosonic ﬁeld operators satisfy the

† canonical commutation relation ψ(x), ψ (y) = δ(x − y). For the n-particle state |Ψ0i =

R † † dx1 ··· dxnΨ0(x1,... xn)ψ (x1) ··· ψ (xn) |0i, one can derive the Schrödinger equation from the above Hamiltonian [5]:

" n # 2 − ∂ + 2κ δ x − x 0 Ψ = EΨ , (2.47) ∑ xj ∑ j j 0 0 j=1 1≤j

where Ψ0 = Ψ0(x1,..., xn). We shall now introduce a lattice-regularized version of this model [92], where the spatial position x ∈ R in the continuum model is replaced by the site i ∈ Z with the lattice spacing a. Let Vi be a physical Hilbert space at the ith site spanned √ † m † by |mii = (1/ m!)(ψi ) |0ii with m ≥ 0. Here, ψi and ψi are the bosonic creation and † annihilation operators on V , respectively. They satisfy [ψ , ψ 0 ] = δ 0 , and [ψ , ψ 0 ] = 0. In the i i i ii i i

31 Chapter 2. Derivation of MPSs for Integrable Models with OBCs

√ continuum limit (a → 0), ψi → ψ(xi) a. Note that ψi and κa are dimensionless. There are N sites in total, and L = Na. The L-operator at the ith site is given by

 √  1 − iλa + κa ψ†ψ −i κaψ†ρ  2 2 i i i i  L0i(λ) = √ , (2.48)  iλa κa †  i κaρiψi 1 + 2 + 2 ψi ψi 0

q κa † where ρi = 1 + 4 ψi ψi. The matrix elements of L0i and ρi are operators acting on the space

Vi and L0i is represented as a 2 × 2 matrix in the auxiliary space V0. This L-operator satisﬁes the following relation:

Rab(λ − µ)Lai(λ)Lbi(µ) = Lbi(µ)Lai(λ)Rab(λ − µ) , (2.49) with the R-matrix given by3

  1 0 0 0      0 λ − iκ 0   λ−iκ λ−iκ  Rab(λ) =   . (2.50)  − iκ λ   0 λ−iκ λ−iκ 0    0 0 0 1 ab

To construct the looped monodromy matrix of the Lieb-Liniger model with OBCs, we need to know the inverse of the L-operator. One can write Lai(λ) as

  A(λ) B(λ)   Lai(λ) =   . (2.51) C(λ) D(λ) a

Choosing λ − µ = iκ in Eq. (2.49), one obtains the following equations:

iκ iκ iκ iκ A(µ − )D(µ + ) − B(µ − )C(µ + ) = det L (µ), 2 2 2 2 q 0i iκ iκ iκ iκ B(µ − )A(µ + ) − A(µ − )B(µ + ) = 0, 2 2 2 2 3 I’d like to thank Dr. Hosho Katsura for his comments about R12 and Rˇ 12 = P12R12 at this point.

32 2.3. MPSs for the Lieb-Liniger Model with OBCs

iκ iκ iκ iκ C(µ − )D(µ + ) − D(µ − )C(µ + ) = 0, 2 2 2 2 iκ iκ iκ iκ D(µ − )A(µ + ) − C(µ − )B(µ + ) = det L (µ), 2 2 2 2 q 0i

where detq L0i(µ) is known as the quantum determinant [23] and is given by a2 iκ 2i iκ 2i detq L0i(µ) = 4 (µ − 2 + a )(µ + 2 − a ). Using these four relations, we obtain

iκ y iκ y L (µ − )σ Lt0 (µ + )σ = det L (µ) · 1ˆ , (2.52) 0i 2 0 0i 2 0 q 0i 0

where 1ˆ0 is the identity matrix in the auxiliary space V0. Then, it follows immediately that

4 y y L−1(µ) = σ Lt0 (µ + iκ)σ . (2.53) 0i 2 2i 2i 0 0i 0 a (µ + a )(µ + iκ − a )

Then, the looped monodromy matrix of the Lieb-Liniger model with OBCs is given by (after shifting the parameter and omitting a constant factor)

iκ iκ T (λ) = L (λ + ) ··· L (λ + ) 0 01 2 0N 2 h y iκ yi h y iκ yi · σ Lt0 (−λ + )σ ··· σ Lt0 (−λ + )σ . (2.54) 0 0N 2 0 0 01 2 0

The loop monodromy matrix satisﬁes the fundamental commutation relation:

0 0 0 0 0 0 R00 (λ − µ)T0(λ)R00 (λ + µ + iκ)T0 (µ) = T0 (µ)R00 (λ + µ + iκ)T0(λ)R00 (λ − µ). (2.55)

In the auxiliary space V0, the matrix can be written in block form as

  A(λ) B(λ)   T0(λ) =   (2.56) C(λ) D(λ) 0

and the transfer matrix is t(λ) = trV0 T0(λ) = A(λ) + D(λ). One can easily verify that t(λ)t(µ) = t(µ)t(λ) (see AppendixB). A Bethe eigenstate of the Hamiltonian is then given by (N) |λ1, λ2, ··· , λni = B(λn) ··· B(λ2)B(λ1) |0i , (2.57)

33 Chapter 2. Derivation of MPSs for Integrable Models with OBCs

(N) where |0i = |0i1 |0i2 ··· |0iN is the global vacuum state and the |0ii are the local vacuum states at sites labeled by i. From the fundamental commutation relation, we know that B(λ)B(µ) = B(µ)B(λ). Using the general procedure of the quantum inverse scattering method for OBCs [31], one gets the following Bethe ansatz equations:

n (λi + λj + iκ)(λi − λj + iκ) ei2λi L = , ∏ (λi + λj − iκ)(λi − λj − iκ) j=1(j6=i) i = 1, 2, ..., n, (2.58) which are consistent with the results in [93].

2.3.2 Exact MPSs for the Lieb-Liniger Model with OBCs

We move on to derive a MPS for the lattice version of the Lieb-Liniger model with OBCs.

Using B(λ) = 0h↑| T0(λ) |↓i0, one obtains

| ··· i = { [ T ( )]} ··· [ T ( )] λ1, λ2, , λn TrVn Qn n λn TrV2 Q2 2 λ2

Tr [Q T (λ )] · |0i(N) , (2.59) V1 1 1 1

iκ where the matrix Qi is the same as the one in Eq. (2.28). Deﬁning Ani(λn) = Lni(λn + 2 ) y tn − iκ y { · ··· · and Bni(λn) = σn Lni( λn + 2 )σn, then TrVn [QnTn(λn)] = TrVn Qn [An1(λn) AnN(λn)]

[BnN(λn) ··· Bn1(λn)]}. Using now the same method as in Section 2.2.3, we obtain

{ ··· · tn ··· tn ·Q } TrVn [QnTn(λn)] = TrVn⊗Vn [An1(λn) AnN(λn)] [Bn1(λn) BnN(λn)] nn , (2.60)

tn y iκ y where Qnn is the same as in Eq. (2.32), and Bni(λn) = σn Lni(−λn + 2 )σn. With this, one can easily obtain

h |λ1, λ2, ··· , λni = Tr(V⊗V)⊗n L1(λ1, ··· , λn) |0i1 i ··· LN(λ1, ··· , λn) |0iN · (Qnn ···Q22Q11) , (2.61)

34 2.3. MPSs for the Lieb-Liniger Model with OBCs

⊗n where (V ⊗ V) means Vn ⊗ Vn ⊗ · · · ⊗ V2 ⊗ V2 ⊗ V1 ⊗ V1 as deﬁned before, and here

tn t2 t1 Li(λ1, ··· , λn) = Ani(λn)Bni(λn) ··· A2i(λ2)B2i(λ2)A1i(λ1)B1i(λ1). Let us introduce the notations

r mκa δ = i (m + 1) · κa · (1 + ), (2.62) m 4 iκ i(λn + )a mκa γ = 1 − 2 + , (2.63) n,m 2 2 iκ i(λn + )a mκa β = 1 + 2 + , (2.64) n,m 2 2 and also deﬁne

Cn,m(λ1, λ2, ··· , λn) = ihm| Li(λ1, λ2, ··· , λn) |0ii . (2.65)

For a given n, it is obvious that Cn,m 6= 0 only when m = 0, 1, 2, ··· , 2n. Going forward, one

2n 2n ⊗n should also keep in mind that Cn,m are 2 × 2 matrices acting on the space (V ⊗ V) . We can easily obtain, for n = 1,

  ∗ γ1,0β 0 0 0  1,0     0 γ γ∗ 0 0   1,0 1,0  C1,0(λ1) =   ,  ∗   0 0 β1,0β1,0 0    2 ∗ (δ0) 0 0 β1,0γ 1,0 11   ∗ 0 0 −δ0β 0  1,0     δ γ 0 0 −δ γ∗   0 1,1 0 1,0  C1,1(λ1) =   ,    0 0 0 0    0 0 δ0β1,1 0 11   0 0 0 0      0 0 −δ δ 0   0 1  C1,2(λ1) =   .    0 0 0 0    0 0 0 0 11

35 Chapter 2. Derivation of MPSs for Integrable Models with OBCs

(Here we have used the fact that all solutions of the Bethe-ansatz equations for the repulsive Lieb-Liniger model with OBCs are real.) Using

t h | · (n+1) ··· | i Cn+1,m = i m A(n+1)i(λn+1) B(n+1)i(λn+1)Li(λ1, , λn) 0 i , (2.66)

we can easily obtain with a little bit of algebra the recursion relation between Cn+1,m and

0 Cn,m , which is given by

  0 0 0 0      0 0 0 0    Cn+1,m =   ⊗ Cn,m+2    0 −δmδm+1 0 0    0 0 0 0 (n+1)(n+1)   0 0 0 0      0 0 −δ δ 0   m−1 m−2  +   ⊗ Cn,m−2    0 0 0 0    0 0 0 0 (n+1)(n+1)   0 −δmγn+1,m 0 0      0 0 0 0    +   ⊗ Cn,m+1  ∗ −   δmβn+1,m+1 0 0 δmβn+1,m    ∗ 0 δmγ 0 0 n+1,m+1 (n+1)(n+1)   ∗ 0 0 −δm−1β − 0  n+1,m 1     δ γ 0 0 −δ γ∗   m−1 n+1,m m−1 n+1,m−1  +   ⊗ Cn,m−1    0 0 0 0    0 0 δm−1βn+1,m 0 (n+1)(n+1)   ∗ 2 γn+1,mβ 0 0 (δm−1)  n+1,m     0 γ γ∗ 0 0   n+1,m n+1,m  +   ⊗ Cn,m.  ∗   0 0 βn+1,mβn+1,m 0    2 ∗ (δm) 0 0 βn+1,mγ n+1,m (n+1)(n+1)

36 2.3. MPSs for the Lieb-Liniger Model with OBCs

If N is large enough, it becomes a good approximation that there be at most one particle at each discretized lattice site. We then have

Li(λ1, ··· , λn) |0ii = Cn,0 |0ii + Cn,1 |1ii (2.67) and we obtain a MPS of the discretized Lieb-Liniger model with OBCs given as

h i1−1 i2−i1−1 |λ1, λ2, ··· , λni = ∑ Tr(V⊗V)⊗n (Cn,0) Cn,1 (Cn,0) Cn,1 ··· {i1,i2,···,in} i in−in−1−1 N−in (Cn,0) Cn,1 (Cn,0) Qn |x1, x2, ··· , xni , (2.68)

where 1 ≤ i1 < i2 < ··· < in ≤ N, xm is related to im by xm = ima, |x1, x2, ··· , xni denotes a configuration with particles at (discrete) positions (x1, x2, ··· , xn), the subscript n indicates the total number of particles, and Qn is again given by Eq. (2.44). It is clear that we cannot derive a cMPS similar to the ones in the literature [71, 82], because we cannot find a basis in which the matrices have a simple form. In [82], the authors derived a cMPS for the Lieb-Liniger model with PBCs by finding a basis in which the matrices are considerably simpler. Achieving the same for OBCs is an interesting and challenging open problem.

37 Chapter 3

Expansion Velocity of 1D Hubbard Model

This chapter on the prediction of the expansion velocity of ultracold 1D quantum gases for integrable models is based on the work published as Ref. [2].

3.1 Background

3.1.1 Introduction

As is mentioned in section 1.4, the study of isolated quantum systems out of equilibrium in systems of optically trapped ultracold atomic gases is an active research field these days. Quantum quench in integrable models is one direction in this field. Two fundamental questions of quantum quenches are the following : Do observables relax to time-independent values? What are the principles determining these values? Our work [2] is an effort to try to answer the above two questions. We consider some interacting atoms which move in a 1D optical lattice, and the atoms are also confined in a box trap. If we remove the box trap suddenly, we show that the asymptotic expansion velocity (for a definition of this velocity, please see the subsequent paragraphs) becomes time independent very rapidly and it can be calculated approximately.

38 3.1. Background

We consider hard-core bosons (HCBs) ﬁrst, because it’s the simplest, completely understood example. The Hamiltonian of the HCBs model is given by

L−1 † H = −J ∑ (ai+1ai + H.c.), (3.1) i=1

† 2 where ai is a boson annihilation operator, and (ai ) = 0. The density operator is given by † ni = ai ai. For HCBs, the physical quasimomentum distribution function (quasi-MDF) nk † (deﬁned as the Fourier transform of one-particle correlations hai aji) after a long expansion f time becomes identical to the conserved set nk of the spinless, noninteracting fermions that

HCBs can be mapped to [94]. Density proﬁles hni(t)i undergo a ballistic expansion for HCBs in a 1D lattice, which was observed in experiments [95]. The ballistic expansion manifests itself in a linear increase R(t) = vrt of the radius deﬁned as

2 1 2 R (t) = ∑(i − i0) hni(t)i , (3.2) N i where N denotes the number of particles and i0 = (L + 1)/2. The expansion velocity vr of f HCBs is related to the conserved nk and thus also to the asymptotic form of the physical f nk(t = ∞) = nk via [95–97] 2 1 2 f vr = ∑(vk) nk , (3.3) N k where vk = dek/dk are the group velocities of noninteracting particles with a tight-binding dispersion ek = −2J cos(k) (the lattice spacing is set to unity). In our work [2], we generalize these observations to Bethe-ansatz (BA) integrable lattice models with repulsive interactions that do not map onto noninteracting particles. Following Sutherland [98], for such systems, quasimomenta are replaced by so-called rapidities, which have the interpretation that they become physical momenta in the asymptotic regime of an expansion. This happens once particles have spatially rearranged themselves according to increasing momenta and thus stop crossing each other as they continue to expand. If we then deﬁne a distribution nκ of rapidities κ deﬁned by the initial condition, then our hypothesis is that the asymptotic physical momentum distribution function nk(t → ∞) becomes equal

39 Chapter 3. Expansion Velocity of 1D Hubbard Model

to the conserved nκ (assuming, for simplicity, real κ),

nk(t → ∞) = nκ . (3.4)

As a consequence, since in the asymptotic regime the expansion is expected to be ballistic because diluteness suppresses any scattering, we expect that the asymptotic expansion velocity can be written as1

2 1 2 vr (t = ∞) = ∑(vκ) nκ. (3.5) N κ

The main result of our work is that Eq. (3.5) indeed holds for the FHM with repulsive interactions and initial densities n0 = N/L0 smaller than or equal to 1, expanding from the correlated ground state within a box of size L0. Our results are based on a comparison of a BA calculation of nκ with numerical results for the density proﬁles obtained from time- dependent density matrix renormalization group (tDMRG) calculations. This implies that a measurement of density proﬁles, accessible in quantum-gas experiments, gives access to the very abstract concept of rapidities of an integrable quantum model, which are very important for actually carrying out calculations, but which are usually hidden. Of course, vr is just a single number and contains only partial information about the full rapidity distribution. A possible obstacle could be that the times needed to reach the asymptotic regime are not accessible to either experiments or tDMRG. This is true for the quasi-MDF, which, using tDMRG, we are only able to obtain for N = 2, 4 particles in the long-time limit. The expansion, however, turns out to be ballistic to a good approximation (i.e., R ∝ t) for the

FHM under the aforementioned conditions. Such a behavior implies that vr becomes time independent very rapidly and it can thus be extracted already from short-time dynamics, long before nk has converged to its asymptotic regime. Thus, experiments do not need to reach the asymptotic regime.

An example, in which nk nevertheless becomes stationary very fast, is the spin-imbalanced FHM with attractive interactions [99], where a quantum-distillation process ensures a fast separation of pairs and excess fermions. In that case, the generalization of Eq. (3.4) to both

1 Because of Eq. (3.4), we will not distinguish between nk and nκ in the following part of this chapter.

40 3.1. Background real and complex rapidities (the latter present because of the bound states in the ground state of the attractive FHM) seems to hold, based on a comparison of tDMRG and BA calculations. Interestingly, we will show here that even for the nonintegrable BHM, one can exploit a BA approach along the lines of [100] to deﬁne nκ, which via Eq. (3.5) leads to a good agreement with tDMRG data from [96].

3.1.2 Asymptotic Form of the Quasi-MDF at Half Filling for FHM

FIGURE 3.1: Sudden expansion in the FHM. (a1)-(a3) Density distribution hni(t)i and (b1)-(b3) renormalized quasi-MDF nk(t) at (a1), (b1) t = 0, (a2), (b2) tJ = 2, and (a3), (b3) tJ = 20 (U = 8J, N = 4, L0 = N, we seth ¯ = 1). (b4) Results for the BHM and the same model parameters. Solid lines are tDMRG results, dashed lines in (b3) and (b4) show the corresponding Fermi- Dirac function, Eq. (3.8). In the ﬁgures, all quantities are expressed in dimensionless units.

We deal with the Fermi-Hubbard model (FHM)

L−1 L † H = −J ∑ ∑ (ci+1σciσ + H.c.) + U ∑ ni↑ni↓ , (3.6) i=1 σ∈{↓,↑} i=1

† where L is the number of lattice sites, ciσ is a fermion annihilation operator, and niσ = ciσciσ. We begin by describing the overall time evolution of densities and the quasi-MDF, and for

41 Chapter 3. Expansion Velocity of 1D Hubbard Model the latter, we propose a simple expression for its asymptotic form for the example of initial states with half ﬁlling. Fig. 3.1 shows typical results for the FHM at U/J = 8 obtained with

2 tDMRG for the density proﬁles hni(t)i and the quasi-MDF nk(t). We calculated the observables in the initial state [Figs. 3.1(a1) and 3.1(b1)], in the transient regime of the expansion

[Figs. 3.1(a2) and 3.1(b2)] and in the asymptotic regime, where nk approaches its stationary form [Figs. 3.1(a3) and 3.1(b3)]. In the long-time limit, the quasi-MDF of a gas that has expanded from a Mott insulator approaches a particle-hole symmetric form, both for fermions and bosons [96]. For the BHM, this can be viewed as a generalized dynamical fermionization, similar to integrable bosonic models. The density profile at the longest times reached in the simulations is practically flat, except for the propagating wavefronts. Therefore, the gas can be well approximated by assuming both diluteness and a homogeneous density. We find that the final quasi-MDF approaches a simple Fermi-Dirac distribution

1 fk = , (3.7) eβ(ek−µ) + 1 where temperature T = 1/β and chemical potential µ are determined to match the energy E, which is conserved during the expansion, and the particle number N of the strongly correlated system. This effective noninteracting gas, containing the same number of particles, can be viewed as having originated from the same box. For large U/J, the total energy corresponds to relatively high temperatures of the free fermions and, in addition, µ = 0 at half ﬁlling. This simpliﬁes the expression of Eq. (3.7) since the only parameter that determines the quasi-MDF of the free particles fk is the energy

3 density ε = |Etot|/N. Expressing the quasi-MDF up to O(ε ), we get

N 1 3 f = 1 + ε cos k − ε3 cos2 k − cos k . (3.8) k L 3 4

We use Eq. (3.8) to compare to the tDMRG result at N = 4 for the FHM (ε = −0.279J) and

2My work [2] is ﬁnished with the collaboration with Dr. L. Vidmar, Dr. F. Heidrich-Meisner, and Dr. C. J. Bolech. The tDMRG results are obtained by Dr. L. Vidmar and Dr. F. Heidrich-Meisner.

42 3.2. Expansion Velocity of the 1D FHM the BHM (ε = −0.364J) in Figs. 3.1(b3) and 3.1(b4), respectively. The numerical data away from k = ±π/2 agree very well with the free-fermion reference system. From the above discussion, we see that atoms become non-interacting particles in the asymptotic regime. So, we gain some conﬁdence that Eq. (3.5) is reasonable. In the next section, we use Eq. (3.5) to calculate the expansion velocity of a 1D FHM.

3.2 Expansion Velocity of the 1D FHM

In this section, we calculate the expansion velocity of the 1D FHM. Firstly, we solve the 1D FHM with OBCs by BA. Secondly, we use Eq. (3.5) to ﬁnish the calculation.

3.2.1 Eigenvalue Equations of the 1D FHM with OBCs

The Hamiltonian of the 1D FHM with OBCs is given by Eq. (3.6):

L−1 L † H = −J ∑ ∑ (ci+1σciσ + H.c.) + U ∑ ni↑ni↓ . (3.9) i=1 σ∈{↓,↑} i=1

We solve this model by BA. The reader can also refer to [26, 38]. The number operator is given by L N = ∑ ni↑ + ni↓ , (3.10) i=1 and its a conserved quantity. So, we can study the Hamiltonian for a ﬁxed number of electrons. For the Hilbert space HN of N particles, we have

ciσ |0i = 0, (3.11)

where |0i is the vacuum state. The states that span HN are given by

N |Fi = F (n ··· n ) c† |0i . (3.12) ∑ ∑ σ1···σN 1 N ∏ niσi σ1···σN n1···nN i=1

Then, the Schrödinger equation H |Fi = E |Fi (3.13)

43 Chapter 3. Expansion Velocity of 1D Hubbard Model becomes hF = EF, (3.14) where N − h = J ∑ ∆j + U ∑ δnjnl , (3.15) j=1 j

∆jFσ1···σN n1 ··· nj ··· nN = Fσ1···σN n1 ··· nj + 1 ··· nN + Fσ1···σN n1 ··· nj − 1 ··· nN . (3.16) When N = 1, we have

h = −t∆, (3.17) and the solution is given by

ikn Fσ(n) = Aσe , E = −2J cos k.

When N = 2, we have

h = −J (∆1 + ∆2) + Uδn1n2 . (3.18)

If n1 6= n2,

h = −J∆1 − J∆2 , (3.19) and the wave function is given as

ik1n1+ik2n2 Fσ1σ2 (n1, n2) = Ae [Aσ1σ2 θ (n1 − n2) + Bσ1σ2 θ (n2 − n1)]

ik1n1+ik2n2 = e [Aσ1σ2 θ (n1 − n2) + Bσ1σ2 θ (n2 − n1)]

ik1n2+ik2n1 −e [Aσ2σ1 θ (n2 − n1) + Bσ2σ1 θ (n1 − n2)] , (3.20)

44 3.2. Expansion Velocity of the 1D FHM where A is the antisymmetrizer, and θ(x) is deﬁned as

  1, x < 0; θ(x) =  0, x > 0.

For n1 < n2 , we have

ik1n1+ik2n2 ik1n2+ik2n1 Fσ1σ2 (n1, n2) = Aσ1σ2 e − Bσ2σ1 e . (3.21)

From hF = EF, we obtain

E = −2J (cos k1 + cos k2) . (3.22)

For n1 > n2 , we have

ik1n1+ik2n2 ik1n2+ik2n1 Fσ1σ2 (n1, n2) = Bσ1σ2 e − Aσ2σ1 e . (3.23)

From hF = EF, we again obtain Eq. (3.22). Deﬁne the S-matrix as follows :

B = S1,2 A. (3.24)

The matrix A has the following form :

  A↑↑      A   ↑↓  A =   . (3.25)    A↓↑    A↓↓

S1,2 is a 4 × 4 matrix acting on the space V1 ⊗ V2. To determine the S-matrix, we use two conditions : (1) Uniqueness. That is, F(n, n) is independent of the region. We obtain

A − (S A) = (S A) − A . (3.26) σ1σ2 1,2 σ2σ1 1,2 σ1σ2 σ2σ1

45 Chapter 3. Expansion Velocity of 1D Hubbard Model

To simplify this equation, we need to use the permutation matrix. Using

  1 0 0 0      0 0 1 0  1 ˆ   P1,2 = 1 +~σ1 ·~σ2 =   , (3.27) 2    0 1 0 0    0 0 0 1 we have

      1 0 0 0 A↑↑ A↑↑              0 0 1 0   A   A     ↑↓   ↓↑  P1,2 A =     =   . (3.28)        0 1 0 0   A↓↑   A↑↓        0 0 0 1 A↓↓ A↓↓

That is,

(P1,2 A)↑↑ = A↑↑ ,

(P1,2 A)↑↓ = A↓↑ ,

(P1,2 A)↓↑ = A↑↓ ,

(P1,2 A)↓↓ = A↓↓ .

So, (P A) = A . (3.29) 1,2 σ1σ2 σ2σ1

Then, we obtain

1ˆ − P1,2S1,2 = S1,2 − P1,2 . (3.30)

(2) The Schrödinger equation on the boundary. For n1 = n2 = n,

h = −J (∆1 + ∆2) + U, (3.31)

46 3.2. Expansion Velocity of the 1D FHM and

ik1n+ik2n Fσ1σ2 (n, n) = (Aσ1σ2 − Bσ2σ1 ) e . (3.32)

From hF = EF, we obtain

ik1 −ik2 −ik1 ik2 −J e + e S1,2 + J e + e P1,2 −ik1 ik2 ik1 −ik2 −J e + e 1ˆ + J e + e P1,2S1,2

+U1ˆ − UP1,2S1,2

= −2J (cos k1 + cos k2) 1ˆ + 2J (cos k1 + cos k2) P1,2S1,2 . (3.33)

Observe the above equation, we propose the following solution:

S1,2 = a1ˆ + bP1,2 . (3.34)

From Eq. (3.33) and Eq. (3.34), we obtain

a + b = 1 , (3.35)

(sin k1 − sin k2) a = U , (3.36) (sin k1 − sin k2) + i 2J U i 2J b = U . (3.37) (sin k1 − sin k2) + i 2J

From Eq. (3.30) and Eq. (3.34), we just obtain Eq. (3.35). Denote

U c = , (3.38) 2J we have the S-matrix (sin k1 − sin k2) 1ˆ + icP1,2 S1,2 = . (3.39) (sin k1 − sin k2) + ic

For N particles, we begin by dividing the conﬁguration space into N! regions according to the ordering of the particles. If we have n7 < n1 < n20 < ···, then we label the conﬁgu- ration as Q = (Q1 = 7, Q2 = 1, Q3 = 20, ···) ∈ SN (SN is the permutation group). The wave

47 Chapter 3. Expansion Velocity of 1D Hubbard Model functions and eigenenergy of the N particles are given by the Bethe ansatz :

··· A i ∑j kjnj Fσ1···σN (n1 nN) = e ∑ Aσ1···σN (Q)θ (nQ) , (3.40) Q N E = −2J ∑ cos kj . (3.41) j=1

In Eq. (3.40) there are N! different Qs, θ (nQ) is equal to 1 if the particles are ordered according to Q and vanishes otherwise, and Aσ1···σN (Q) is the coefﬁcient of the wave functions in region Q. Once two particles exchange their position, there is interaction, and the co- efﬁcients are related by the S-matrix. If Q = (Q1, Q2, ··· , Ql = i, Ql+1 = j, ··· , QN), and 0 0 Q = (Q1, Q2, ··· , Ql+1 = j, Ql = i, ··· , QN) = PijQ, then A (Q ) = Sij A(Q). Similarly, 0 −1 A(Q) = Sji A (Q ). Note that Sij = Sji . When two regions are not adjacent, we can always connect them by successive permutations. For N = 3, we denote the ordering (n2 < n1 < n3) as (213). From (123) to (321), we have different paths: (A)

A(123) → S12 A(123) = A(213) → S13 A(213) = A(231) → S23 A(231) = A(321). (3.42)

So, A(321) = S23S13S12 A(123). (B)

A(123) → S23 A(123) = A(132) → S13 A(132) = A(312) → S12 A(312) = A(321). (3.43)

So, A(321) = S12S13S23 A(123). To make sure that the results are consistent, we should have the Yang-Baxter equation:

S12S13S23 = S23S13S12. (3.44)

One can easily prove that the above equation is correct by using Eq. (3.39). The Yang-Baxter equation can guarantee consistency for the case of any number of particles.

48 3.2. Expansion Velocity of the 1D FHM

For N electrons, the coefﬁcient of plane waves for the conﬁguration

1 < 2 < ··· < j − 1 < j < j + 1 < ··· < N − 1 < N (3.45) is denoted by A(Q). After the electron (j) exchanges positions with all the electrons at right, we have the conﬁguration

1 < 2 < ··· < j − 1 < j + 1 < j + 2 < ··· < N − 1 < N < (j+), (3.46) and the coefﬁcient of plane waves is given by

Sj,NSj,N−1 ··· Sj,j+2Sj,j+1 A(Q). (3.47) | {z } (+kj)

For the above configurations, the wave function of electron j is given by eikj L. When the electron reaches site L, it goes back. Then the wave function becomes e−ikj L, and the coefficient is multiplied by a number Re. The configuration becomes

1 < 2 < ··· < j − 1 < j + 1 < j + 2 < ··· < N − 1 < N < (j−), (3.48) and the coefﬁcient of plane waves becomes

RSe j,N ··· Sj,j+1 A(Q). (3.49) | {z } (+kj)

Then the electron will move to the left:

(j−) < 1 < 2 < ··· < j − 1 < j + 1 < j + 2 < ··· < N − 1 < N. (3.50)

The coefﬁcient of plane waves is

S1,jS2,j ··· Sj−1,jSj+1,jSj+2,j ··· SN−1,jSN,j RSe j,N ··· Sj,j+1 A(Q). (3.51) | {z } | {z } (−kj) (+kj)

49 Chapter 3. Expansion Velocity of 1D Hubbard Model

Next, we have:

(j+) < 1 < 2 < ··· < j − 1 < j + 1 < j + 2 < ··· < N − 1 < N, (3.52) and

RS1,j ··· Sj−1,j Sj+1,j ··· SN,j RSe j,N ··· Sj,j+1 A(Q), (3.53) | {z } | {z } | {z } (−kj) (−kj) (+kj) where R is a number. Finally, we have:

1 < 2 < ··· < j − 1 < (j+) < j + 1 < j + 2 < ··· < N − 1 < N, (3.54) and

Sj,j−1 ··· Sj,1 RS1,j ··· Sj−1,j Sj+1,j ··· SN,j RSe j,N ··· Sj,j+1 A(Q) = A(Q). (3.55) | {z } | {z } | {z } | {z } (+kj) (−kj) (−kj) (+kj)

Next, we need to calculate Re and R. For conﬁguration given by Eq. (3.46), the Hamilto- nian for electron j is given by

h = −J∆j, (3.56) and the wave function is given by

− F(n) = Aeikjn + RAee ikjn. (3.57)

From hF = EF, we obtain

E = −2J cos kj. (3.58)

When n = L, we obtain from hF = EF that

−J · F(L − 1) = E · F(L). (3.59)

50 3.2. Expansion Velocity of the 1D FHM

From the above equation, we obtain

Re = −e2ikj(L+1). (3.60)

For conﬁguration given by Eq. (3.50), the Hamiltonian for electron j is again given by Eq. (3.56), and the wave function is

F(n) = Ae−ikjn + RAeikjn. (3.61)

From hF = EF, we obtain Eq. (3.58) again. For n = 1, we have

−J · F(2) = E · F(1), (3.62) from which we obtain R = −1. (3.63)

Denote β1ˆ + icP S (β) = 1,2 1,2 . (3.64) 1,2 β + ic

We have

S1,2(β) = S2,1(β). (3.65)

Then, Sj,m = Sj,m αj − αm , Sm,j = Sm,j αm + αj , (3.66) |{z} |{z} (+kj) (−kj) where αj = sin kj. Denote

Zj = Sj,j−1 αj − αj−1 ··· Sj,1 αj − α1 · S1,j α1 + αj ··· Sj−1,j αj−1 + αj · Sj+1,j αj+1 + αj ··· SN,j αN + αj · Sj,N αj − αN ··· Sj,j+1 αj − αj+1 , (3.67)

51 Chapter 3. Expansion Velocity of 1D Hubbard Model the eigenvalue equations (Eq. (3.55)) become

−2ikj(L+1) Zj · A(Q) = e · A(Q). (3.68)

3.2.2 BA Equations of the 1D FHM with OBCs

We derive the BA equations of the 1D FHM with OBCs in this section.3 The monodromy matrix of the 1D FHM with OBCs is given by

ΞA(α) = [S1A (α1 − α) ··· SNA (αN − α)] · [SNA (−αN − α) ··· S1A (−α1 − α)] , (3.69) where the S-matrix satisﬁes the continuous version of Yang-Baxter equation:

Skj(α − β)Ski(α)Sji(β) = Sji(β)Ski(α)Skj(α − β). (3.70)

The transfer matrix is given by

T(α) = trAΞA(α). (3.71)

Using SjA(α)SjA(β) = SjA(β)SjA(α) and Eq. (3.70), we obtain

2αj − 2ic T αj = Sj+1,j αj+1 − αj ··· SN,j αN − αj SN,j −αN − αj 2αj − ic ··· Sj+1,j −αj+1 − αj Sj−1,j −αj−1 − αj ··· S1,j −α1 − αj S1,j α1 − αj ··· Sj−1,j αj−1 − αj

≡ Tj. (3.72)

We can easily verify that

SjA(α)SjA(−α) = 1.ˆ (3.73)

3Dr. C. J. Bolech’s notes for Fermi-Hubbard model with OBCs are very helpful for this section. I’d like to thank Dr. Bolech for sharing his notes with me.

52 3.2. Expansion Velocity of the 1D FHM

So,

−1 SjA (α) = SjA(−α). (3.74)

Then, we obtain

2αj − 2ic −1 Zj = Tj , (3.75) 2αj − ic and the eigenvalue equations (Eq. (3.68)) become

2α − 2ic j 2ikj(L+1) Tj · A(Q) = e · A(Q). (3.76) 2αj − ic

Using Yang-Baxter equation, we can prove that Tj, Tk = 0, which guarantees the integrability of the FHM. In fact, we can prove the more general equation [T(α), T(β)] = 0 (see AppendixB). We can also prove that the monodromy matrix satisﬁes the fundamental commutation relation :

RAB(β − α)ΞA(α)RAB(−α − β)ΞB(β) = ΞB(β)RAB(−α − β)ΞA(α)RAB(β − α), (3.77)

where Rjk(α) = Sjk(α).

In the auxiliary space VA, we have

  A(α) B(α)   ΞA(α) =   . (3.78) C(α) D(α) A

So, T(α) = A(α) + D(α). (3.79)

The eigenvalue equation is given by

2α − 2ic T(α) · A(Q) = e2ik(L+1) · A(Q). (3.80) 2α − ic

53 Chapter 3. Expansion Velocity of 1D Hubbard Model

From Eq. (3.77), we can obtain the commutation relations among A, B, C, and D (see AppendixD). Denote ic De(α) = D(α) − A(α). (3.81) −2α + ic

We have 2α − 2ic 2α − ic T(α) = A(α) + De(α) . (3.82) 2α − ic 2α − 2ic

Next, denote

A(α) = A(α), (3.83) 2α − ic D(α) = De(α), (3.84) 2α − 2ic the eigenvalue equation becomes

A(α) + D(α) A(Q) = e2ik(L+1) A(Q). (3.85)

Using Eq. (3.77), we have

(α + β)(α − β + ic) A(α)B(β) = B(β)A(α) (α − β)(α + β − ic) 2icβ + B(α)A(β) (−2β + ic)(α − β) 2ic(β − ic) + B(α)D(β), (3.86) (2β − ic)(α + β − ic) and

(β − α + ic)(−α − β + 2ic) D(α)B(β) = B(β)D(α) (β + α − ic)(α − β) 2ic(−β + ic) + B(α)D(β) (α − β)(−2β + ic) (−2iβc) + B(α)A(β). (3.87) (−β − α + ic)(−2β + ic)

54 3.2. Expansion Velocity of the 1D FHM

We construct the ferromagnetic vacuum state as follows :

|ωi = |↑i1 |↑i2 ··· |↑iN . (3.88)

From ΞA(α) |ωi, we can calculate A(α) |ωi and D(α) |ωi. Next, we construct more eigenstates :

|β1, ··· , βMi = B (β1) ··· B (βM) |ωi . (3.89)

From T αj |β1, ··· , βMi = (const.) |β1, ··· , βMi , (3.90) we obtain the BA equations :

M c c α + Λn + i α − Λn + i j 2 j 2 2ikj(L+1) c c = e , (3.91) ∏ αj + Λn − i αj − Λn − i n=1 2 2

M N c c (Λn + Λm + ic)(Λn − Λm + ic) Λn + αj + i 2 Λn − αj + i 2 = c c , ∏ (Λn + Λm − ic)(Λn − Λm − ic) ∏ Λn + αj − i Λn − αj − i m=1(m6=n) j=1 2 2 (3.92)

c where Λn = βn − i 2 , j = 1, 2, ··· , N, and n = 1, 2, ··· , M. We should remember that αj = sin kj.

3.2.3 Expansion Velocity of the 1D FHM

We turn now to determine the asymptotic expansion velocity of atoms from Eq. (3.5). Gen- erally speaking, we are attempting to predict the asymptotic form of observables from integrability in a specific quantum quench problem. We first need to determine the distribution nκ, which is conserved for t > 0. The main technical complication is that the wave function needs to be expanded in the postquench eigenstates (after quenching the trapping potential to zero) of the integrable homogeneous FHM in an infinitely large lattice. This problem is notoriously difficult, and we therefore compute the discrete set of prequench rapidities with respect to the initial box (i.e., for t < 0, what we denote as ”rawBA”) and then use

55 Chapter 3. Expansion Velocity of 1D Hubbard Model

single-particle projection techniques to approximate nκ for t > 0 (”projBA”). The calculation of the discrete set of rapidities for particles in a box is well deﬁned and can be done exactly, albeit numerically. Next, as the trapping potential is suddenly turned off, the initial distri-

− + bution nκ(t = 0 ) is projected into a modiﬁed one nκ(t = 0 ) that is consistent with the new size and boundary conditions of the system. Thus, in principle, one needs to compute the overlaps between the initial state of the system and a complete basis of Bethe states for the system without trap. Each of these Bethe states is in one-to-one correspondence with a rapidity distribution and the overlaps give the probability amplitudes for combining those

+ into the resultant nκ ≡ nκ(t = 0 ). If one can identify the largest overlaps, those will give the dominant contributions, while small overlaps will give small corrections that can be left out in an approximate calculation. On the one hand, for the repulsive models we focus on in here, the initial ground states are characterized by having only real-valued charge (and, for the FHM, spin) rapidities. On the other hand, both real and complex rapidities (strings) are present in the full spectrum of these models, but the overlaps in the case of the latter are comparatively smaller and can be ignored in a ﬁrst approximation. Complex rapidities are associated with different types of bound states and will most often expand more slowly, so by neglecting their contribu- tion we overestimate the expansion velocity of the cloud. For an initial density approaching the value of one particle per lattice site, the system gets closer to a Mott-insulating state for which double occupancy is relatively suppressed (though still non-zero) and thus the projection onto bound states is also expected to be relatively suppressed. The approximation of ignoring the bound-state contributions should thus become better the closer the trapped system is to n0 = 1; consistent with our numerical comparisons to be shown below. To further simplify the calculation, we consider the discrete set of initial rapidities treated individually as one-particle distributions and then combine the resulting post-quench distributions to get the ﬁnal result. Finally, different moments of the distribution nκ are combined according to

Eq. (3.5) to yield the asymptotic vr. Now we give the details below.

56 3.2. Expansion Velocity of the 1D FHM

− At the time t = 0 , the atoms move among the sites 1, 2, ···, L0. The wave function of one particle is given by L0 ik1n −ik1n † |F0i = A ∑ e − e cn↑ |0i , (3.93) n=1 where s sin k A = 1 . (3.94) 2 {L0 sin k1 − cos [k1 (L0 + 1)] sin (k1L0)}

At the time t = 0+, the atoms move in a very large space, which is chosen to contain sites −L, −(L − 1), ···, L (L is very large). The one-particle wave function is given by

E L k ikn −2ik(L+1) −ikn † F = Bk ∑ e − e e cn↑ |0i , (3.95) n=−L

π where k = 2L+2 l (l = 1, 2, ··· , (2L + 1)). Using L D E k † k 1 = ∑ F cl↑cl↑ F , (3.96) l=−L we obtain s sin k B = . (3.97) k 2 {(2L + 1) sin k − cos [2k(L + 1)] sin [k(2L + 1)]}

k0 k We can easily prove that hF |F i = δk0k. Next,

E k |F0i = ∑ Ck F . (3.98) k

k So, Ck = hF |F0i. Finally, we obtain the one-particle MDF:

sin k n = |C |2 = k,k1 k 2 {(2L + 1) sin k − cos [2k(L + 1)] sin [k(2L + 1)]} sin k k − k k + k · 1 · csc2 1 · csc2 1 2 {L0 sin k1 − cos [k1 (L0 + 1)] sin (k1L0)} 2 2

·{ cos(kL) sin k sin k1 + cos k sin(kL) sin k1 + sin [k (2 + L0 + L)] sin (k1L0) 2 − sin [k (1 + L0 + L)] sin [(1 + L0) k1] } . (3.99)

57 Chapter 3. Expansion Velocity of 1D Hubbard Model

TABLE 3.1: Expansion velocity of the FHM for U/J = 2 and L0 = 20.

n0 vr/J n0 vr/J 0.1 0.463939 0.6 1.29041 0.2 0.734731 0.7 1.37596 0.3 0.892473 0.8 1.43157 0.4 1.04686 0.9 1.45392 0.5 1.18334 1.0 1.42891

Then, the MDF of the system is given by

N n = n k ∑ k,kj , (3.100) j=1 where kjs are rapidities for particles at the ground state in the initial box. From Eq. (3.5), we have s vr 4 2 = ∑ nk · sin (k). (3.101) J N k

4 For U/J = 2, L0 = 20, and N = 4 (n0 = N/L0 = 0.2), we have:

k1 0.125481

k2 0.250934

k3 0.376816

k4 0.503889

Using Eq. (3.101), we can easily obtain the expansion velocity as vr/J = 0.734731 (see Ap- pendixE). For the expansion velocities corresponding to other initial densities, please see Table 3.1.

For U/J = 8, L0 = 20, and N = 4, we have

k1 0.141749

k2 0.283646

k3 0.42584

k4 0.568477

From Eq. (3.101), we obtain vr/J = 0.761587. The expansion velocities corresponding to other initial densities are given in Table 3.2.

4The rapidities are calculated by Dr. C. J. Bolech. See [101].

58 3.2. Expansion Velocity of the 1D FHM

TABLE 3.2: Expansion velocity of the FHM for U/J = 8 and L0 = 20.

n0 vr/J n0 vr/J 0.1 0.453511 0.6 1.44616 0.2 0.761587 0.7 1.50585 0.3 1.01686 0.8 1.5199 0.4 1.20032 0.9 1.49012 0.5 1.34113 1.0 1.43186

Figure 3.2(a) shows the results for the FHM for two different values of the interaction

5 U/J = 2, 8 and L0 = 20. The agreement between the tDMRG and BA calculations is generally very good. Both approaches consistently show that the maximum values of vr/J for strong interactions are found at initial in-trap densities 0.5 < n0 < 1 (at n0 = 1 one √ has vr/J = 2 regardless of the interaction strength [97]). For guidance, we also show the exact results for vr for the noninteracting and the U/J → ∞ limits, computed by taking

L0 → ∞ [97], s sin (kF) cos (kF) vr/J = 2 1 − . (3.102) kF

Here, the Fermi momentum kF is set by the initial density n0. For a noninteracting two- component gas, kF = n0π/2, while for the single-component gas that describes the charge dynamics in the limit of inﬁnite on-site repulsion, kF = n0π. In Fig. 3.2(b), we compare the BA results obtained with or without the rapidity projection techniques, with the tDMRG results. Without the rapidity projection techniques, the asymptotic expansion velocity is given by6

v u N u 4 2 vr/J = t ∑ sin (kj). (3.103) N j=1

Remarkably, the ﬁnite-size scaling with respect to the initial box size L0 shows that all data sets approach the same value as L0 increases. Thus, the approximations used in the BA- based approach become increasingly unimportant. That the asymptotic and the scaling limits of the expansion velocities coincide is one of the main nonintuitive ﬁndings of our

5The results of tDMRG are provided by Dr. L. Vidmar and Dr. F. Heidrich-Meisner. 6See Eq. (3.5).

59 Chapter 3. Expansion Velocity of 1D Hubbard Model

2.0 (a) U/J = 8 1.5 U/J = 2 J /

r 1.0 L = 20 v 0 Filled symbols: projBA 0.5 Open symbols: tDMRG 0 0 0.2 0.4 0.6 0.8 1 n 0 1.45 (b) U/J n = 8, 0 = 0.5 1.40 projBA

J rawBA / r v 1.35 tDMRG 1.30 0 0.05 0.1 0.15 0.2 0.25 L 1/ 0

FIGURE 3.2: Expansion velocity vr for the FHM. Comparison between BA (ﬁlled symbols) and tDMRG (open symbols). The BA results are obtained with (projBA) or without (rawBA) the rapidity projection techniques. (a) vr versus initial density n0 [L0 = 20; U/J = 2, 8]. The dot-dashed and the dashed lines are the exact expressions for U = 0 and U/J = ∞, respectively [see Eq. (3.102)]. (b) vr versus the initial box size L0 [U/J = 8; n0 = 0.5]. Lines are ﬁts to the rawBA and the tDMRG data in the range 1/L0 < 0.1.

60 3.3. Expansion Velocity of the 1D BHM work.

3.3 Expansion Velocity of the 1D BHM

In this section, we use similar method as in last section to calculate the asymptotic expansion velocity of the BHM. Although the BHM is nonintegrable, we still get good results for the case of large repulsive on-site interactions. The Hamiltonian of the BHM is

L−1 L † U H = −J ∑ (ai+1ai + H.c.) + ∑ ni(ni − 1), (3.104) i=1 2 i=1

† where ai is a boson annihilation operator and ni = ai ai. Similar to the case of FHM, we can obtain the S-matrix of the BHM:

(sin k1 − sin k2) − ic S1,2 = 1,ˆ (3.105) (sin k1 − sin k2) + ic

U where c = 2J . From the process of obtaining the S-matrix, we know that the BHM is nonintegrable. To derive the S-matrix, we should use the boundary condition that two particles are occupying one site. For FHM, at most two particles can occupy one site, there is no problem here. For BHM, an arbitrary number of particles can occupy one site, there is a problem here. The nonintegrability of the BHM does not mean that the technique of BA is useless for this model. If the number of sites is larger than the number of particles, and/or the repulsive interaction U is sufﬁciently strong, it is a good approximation that at most two particles can occupy one site at the same time. With these considerations, we can use the machinery of the BA to solve the BHM with OBCs approximately. For N particles, the eigenenergy is given by

N E = −2J ∑ cos kj, (3.106) j=1

61 Chapter 3. Expansion Velocity of 1D Hubbard Model

TABLE 3.3: Expansion velocity of the BHM for U/J = 8 and L0 = 10.

n0 vr/J n0 vr/J 0.1 0.537243 0.6 1.4341 0.2 0.815025 0.7 1.48736 0.3 1.05526 0.8 1.50611 0.4 1.23483 0.9 1.4888 0.5 1.3528 1.0 1.44372

where k1, ···, kN are determined by the following BA equations:

N sin kj − sin km − ic sin kj + sin km − ic −2ik (L+1) = e j . (3.107) ∏ sin kj − sin km + ic sin kj + sin km + ic m=1(m6=j)

For BHM, the one-particle wave function has the same form as the FHM case. As a result, the one-particle MDF is the same as the one for the FHM. Then, we know that the MDF of the system is again given by Eq. (3.100). Next, we need to calculate the ground state rapidities corresponding to the initial box. The details are given in AppendixF. It’s easy to calculate the asymptotic expansion velocity of the BHM now. The results are presented in Table 3.3.

Figure 3.3 shows the results for the BHM with U/J = 8 and L0 = 10. The approximate BA equations for the BHM are reliable in the dilute limit, as they tend to the corresponding equations for the Lieb-Liniger model, where integrability is restored. In this limit (see the dot-dashed line in Figure 3.3), it follows from Eq. (3.5) and the standard BA expression for p the energy of the system that vr/J = 2 E/(JN), where E is the energy of the system (as calculated in its prequench state and measured with respect to the bottom of the tight-binding band); in precise agreement with the exact result for the Lieb-Liniger model [102]. The reason for the recovery of the exact result is two-fold: (i) there are no bound states (and thus no complex rapidities) in the continuum limit as that is a lattice effect, and the single-rapidity approximation becomes more accurate in the dilute limit. In addition, (ii) E is constant after the quench. These considerations also apply to the dilute limit of the FHM, so the same relation is expected to hold for a Gaudin-Yang gas. In all cases studied here, because of the

62 3.4. Discussion

2.0 U/J = 8 1.5 J /

r 1.0 L = 10 v 0 Filled symbols: projBA 0.5 Open symbols: tDMRG 0 0 0.2 0.4 0.6 0.8 1 n 0

FIGURE 3.3: Expansion velocity vr for the BHM. Comparison of data for vr versus n0 between BA (ﬁlled symbols) using rapidity projection (projBA), and tDMRG (open symbols) [U/J = 8, L0 = 10]. The dashed line is the exact expression for U/J = ∞, see Eq. (3.102). The dot-dashed line is the result in p the dilute limit vr/J = 2 E/(JN). repulsive interactions, the asymptotic free Hamiltonian is fermionic [96]; even for bosons, in the Lieb-Liniger case, the underlying time dependence is captured by knowing that of an antisymmetric free-fermion-like wavefunction characterized by the values of the rapidities.

3.4 Discussion

We showed how sudden expansion experiments, already at short times, give access to information about integrals of motion that are usually hidden in the structure of the wave function. While we have substantiated the validity of the approximations in the BA calculation by a comparison to tDMRG for few particles, one can push the BA calculation of asymptotic quantities to much larger initial system sizes or particle numbers. We speculate that systems close to an integrable point are constrained at short times by the integrals of motion of that point and reach asymptotic expansion states that reﬂect them since the gas becomes increasingly more dilute as it expands [96]. Therefore, perturbations from integrability have no time to generate deviations, similar to but more robustly so than within the prethermalization scenarios realizable on the same type of systems under different

63 Chapter 3. Expansion Velocity of 1D Hubbard Model experimental conditions. This conclusion is corroborated by our results for the nonintegrable BHM. The identiﬁcation and study of asymptotic expansion states seems to be a very fertile ground to explore the physics of nonequilibrium systems as they constitute a very special case of asymptotic states of effectively noninteracting systems that nevertheless contain much of the information about the character and correlations on the parent equilibrium states of the actual interacting systems.

64 Chapter 4

Outlook

4.1 Summary

At this point, we have successfully ﬁnished the research we wanted to do (see section 1.5). In paper [1], we successfully recast the Bethe states as exact MPSs for the Heisenberg

1 XXZ spin- 2 chain and the Lieb-Liniger model with OBCs. This is an exact analytical piece of work in mathematical physics. It extends and deepens our knowledge about Bethe wave functions, and it may have important inﬂuence on the community that works using MPSs and other tensor networks. In paper [2], we approximately calculate the asymptotic expansion velocity of cold atoms, which are described by integrable models. We use the one-particle Bethe wave functions and some plausible approximations to ﬁnish the calculation. Our work builds a connection between an abstract concept of mathematical physics (rapidities for integrable models) and an experimentally accessible quantity (asymptotic expansion velocity). Overall, our research greatly extends and deepens our understanding about Bethe wave functions.

4.2 Possible Future Directions

At ﬁrst, we want to point out one open question. In paper [1], we obtain matrix product

1 representations of Bethe wave functions for the Heisenberg XXZ spin- 2 chain and the Lieb- Liniger model with OBCs. The matrices can be obtained by using a set of recursion relations.

65 Chapter 4. Outlook

One open question is whether one can solve the recursion relations. If one can do this, then we may have very simple expressions for the matrices. Further more, one may be able to write the Bethe states of the Lieb-Liniger model with OBCs as cMPSs, which will greatly attract the attention of the community that works using tensor networks. There is another interesting question. Until now, people only recasts the Bethe states of non-nested integrable models as MPSs. Whether one can do similar things for nested integrable models1 is not clear right now. Writing the Bethe states of nested integrable models as MPSs is a valuable and highly interesting research problem. Besides the above open questions, we want to point out another research direction that is closely related to our research. The nonequilibrium behavior of open quantum systems is an active research field these days [103–105]. In [104], the authors solved a 1D Heisenberg XXZ spin chain driven by special boundary Lindblad operators [103] by using a matrix product ansatz. The form of the matrix product is very similar to the monodromy matrix in ABA. The relationship between the matrix product ansatz in [104] and ABA was clarified later [105]. Until now, one can only solve integrable quantum chains with very limited kinds of boundary Lindblad operators. An interesting open problem is how to solve integrable quantum chains with more general boundary Lindblad operators. We want to end this thesis by mentioning an interesting question about integrable models. As is well known, the solution of the Heisenberg XYZ spin chain by BA is much more difficult than the solutions of the Heisenberg XXX and XXZ spin chains by the same method. The reason is that the XYZ spin chain breaks the U(1) symmetry, which is crucial to construct a vacuum state. Recently, some authors invented a method to solve the XYZ, XXZ, and XXX spin chains easily [106]. Finally, we have a method to solve the XYZ spin chain as easy as the XXX spin chain. The drawback of this new method is that it can only give information about the eigenvalues, one can not construct the corresponding eigenstates. How to overcome this drawback is a highly interesting open question.

1For non-nested integrable models, people use BA once. For nested integrable models, people use BA more than once. For example, Heisenberg spin chain is a non-nested system, while Hubbard chain is a nested system (one should use BA two times to solve it).

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75 Appendix A

Proof of Eq. (2.10) for the XXZ Chain with OBCs

A.1 An Equation about the R-matrix of the XXZ Spin Chain

1 For the R-matrix of the Heisenberg XXZ spin- 2 chain (see Eq. (2.3)), we have the following equation:

Rab(λ)Rab(µ) = Rab(µ)Rab(λ). (A.1)

To prove the above equation, we just need to use

ˆ z z Rab(λ) = f (λ)1ab + g(λ)Pab + h(λ)σa σb , (A.2) where

1 sinh λ sinh η f (λ) = 1 + − , 2 sinh(λ + η) sinh(λ + η) sinh η g(λ) = , sinh(λ + η) 1 sinh λ sinh η h(λ) = 1 − − , 2 sinh(λ + η) sinh(λ + η) 1 P = 1ˆ +~σ ·~σ . ab 2 ab a b

Use

z z z z Pabσa σb = σa σb Pab, (A.3)

76 A.2. Proof of Eq. (2.10) we can easily prove Eq. (A.1).

A.2 Proof of Eq. (2.10)

We prove the case L = 2, the general case is similar. We use the Yang-Baxter equation (see Eq. (2.2)), the relationship between L-matrix and R-matrix(see Eq. (2.6)), and Eq. (A.1) to ﬁnish the proof. The monodromy matrix is given by

T0(λ) = L01(λ)L02(λ)L02(λ)L01(λ). (A.4)

For Eq. (2.10),

LHS = Rab(λ − µ)La1(λ)La2(λ)La2(λ)La1(λ)

Rab(λ + µ − η)Lb1(µ)Lb2(µ)Lb2(µ)Lb1(µ) η η η = R (λ − µ)R (λ − )R (λ − )R (λ − ) ab a1 2 a2 2 a2 2 n η η o R (λ − )R (λ + µ − η)R (µ − ) a1 2 ab b1 2 η η η R (µ − )R (µ − )R (µ − ) b2 2 b2 2 b1 2 η h η η i h η i = R (λ − µ)R (λ − ) R (λ − )R (λ − ) R (µ − ) ab a1 2 a2 2 a2 2 b1 2 h η i h η η i η R (λ + µ − η) R (λ − ) R (µ − )R (µ − ) R (µ − ) ab a1 2 b2 2 b2 2 b1 2 η η η = R (λ − µ)R (λ − )R (µ − )R (λ − ) ab a1 2 b1 2 a2 2 n η η o R (λ − )R (λ + µ − η)R (µ − ) a2 2 ab b2 2 η η η R (µ − )R (λ − )R (µ − ) b2 2 a1 2 b1 2 h η η i h η η i = {R (λ − µ)} R (λ − )R (µ − ) R (λ − )R (µ − ) ab a1 2 b1 2 a2 2 b2 2 η η η η R (λ + µ − η)R (λ − )R (µ − )R (λ − )R (µ − ) ab a2 2 b2 2 a1 2 b1 2 η h η η i η = R (µ − ) R (λ − )R (µ − ) R (λ − ) b1 2 a1 2 b2 2 a2 2 η η [R (λ − µ)R (λ + µ − η)] R (λ − )R (µ − ) ab ab a2 2 b2 2 η η R (λ − )R (µ − ) a1 2 b1 2 η η η η = R (µ − )R (µ − )R (λ − )R (λ − )R (λ + µ − η) b1 2 b2 2 a1 2 a2 2 ab

77 Appendix A. Proof of Eq. (2.10) for the XXZ Chain with OBCs

h η η i h η η i {R (λ − µ)} R (λ − )R (µ − ) R (λ − )R (µ − ) ab a2 2 b2 2 a1 2 b1 2 η n η η o = L (µ)L (µ)R (λ − ) R (λ − )R (λ + µ − η)R (µ − ) b1 b2 a1 2 a2 2 ab b2 2 h η η i η R (λ − )R (µ − ) R (λ − )R (λ − µ) a2 2 b1 2 a1 2 ab h η η i = L (µ)L (µ) R (λ − )R (µ − ) R (λ + µ − η) b1 b2 a1 2 b2 2 ab h η η i η η R (λ − )R (µ − ) R (λ − )R (λ − )R (λ − µ) a2 2 b1 2 a2 2 a1 2 ab η = L (µ)L (µ)R (µ − ) b1 b2 b2 2 n η η o R (λ − )R (λ + µ − η)R (µ − ) a1 2 ab b1 2 η R (λ − )L (λ)L (λ)R (λ − µ) a2 2 a2 a1 ab η = L (µ)L (µ)L (µ)R (µ − )R (λ + µ − η) b1 b2 b2 b1 2 ab η R (λ − )L (λ)L (λ)L (λ)R (λ − µ) a1 2 a2 a2 a1 ab

= [Lb1(µ)Lb2(µ)Lb2(µ)Lb1(µ)] Rab(λ + µ − η)

[La1(λ)La2(λ)La2(λ)La1(λ)] Rab(λ − µ)

= Tb(µ)Rab(λ + µ − η)Ta(λ)Rab(λ − µ)

= RHS. (A.5)

We have ﬁnished the proof.

78 Appendix B

Proof of the Integrability of the XXZ Chain with OBCs

1 For the Heisenberg XXZ spin- 2 chain with OBCs, we prove [t(λ), t(µ)] = 0, which guarantees the integrability of the model. The ideas of the proof are provided in Ref. [31], but we give all details here, which will be useful for beginners of integrable models. We have

t(λ) = tr0T0(λ),

R000 (λ − µ)T0(λ)R000 (λ + µ − η)T00 (µ) = T00 (µ)R000 (λ + µ − η)T0(λ)R000 (λ − µ).

t(λ)t(µ) = [tr0T0(λ)] · [tr00 T00 (µ)] h i t0 = tr0T0 (λ) · [tr00 T00 (µ)] h i t0 = tr000 T0 (λ)T00 (µ) , (B.1)

t0 where the superscript t0 in T0 (λ) means the transpose of matrix T0(λ) in the auxiliary space

V0 only. Using

1 sinh λ R (λ) = 1 + 1ˆ 12 2 sinh(λ + η) 12

79 Appendix B. Proof of the Integrability of the XXZ Chain with OBCs

sinh η + σxσx + σyσy 2 sinh(λ + η) 1 2 1 2 1 sinh λ + 1 − σzσz, (B.2) 2 sinh(λ + η) 1 2 we obtain

sinh(λ) · sinh(λ + 2η) Rt1 (−λ − 2η)Rt1 (λ) = , (B.3) 12 12 sinh2(λ + η) sinh(λ) · sinh(λ + 2η) Rt1 (λ)Rt1 (−λ − 2η) = , (B.4) 12 12 sinh2(λ + η)

t1 t2 R12(λ) = R12(λ), (B.5)

R12(λ)R12(−λ) = 1.ˆ (B.6)

So,

· − t0 t0 sinh(λ + µ + η) sinh(λ + µ η) R 0 (−λ − µ − η)R 0 (λ + µ − η) = . (B.7) 00 00 sinh2(λ + µ)

Then

sinh2(λ + µ) t(λ)t(µ) = sinh(λ + µ + η) · sinh(λ + µ − η) h t t t i 0 0 0 0 0 tr00 R000 (−λ − µ − η)R000 (λ + µ − η)T0 (λ)T0 (µ) sinh2(λ + µ) = sinh(λ + µ + η) · sinh(λ + µ − η) n o t t0 0 0 0 0 tr00 R000 (−λ − µ − η) [T0(λ)R00 (λ + µ − η)T0 (µ)] sinh2(λ + µ) = sinh(λ + µ + η) · sinh(λ + µ − η)

tr000 {R000 (−λ − µ − η)T0(λ)R000 (λ + µ − η)T00 (µ)} sinh2(λ + µ) = tr 0 {R 0 (−λ − µ − η) sinh(λ + µ + η) · sinh(λ + µ − η) 00 00

R000 (−λ + µ)R000 (λ − µ)T0(λ)R000 (λ + µ − η)T00 (µ)} sinh2(λ + µ) = tr 0 {R 0 (−λ − µ − η) sinh(λ + µ + η) · sinh(λ + µ − η) 00 00

R000 (−λ + µ)T00 (µ)R000 (λ + µ − η)T0(λ)R000 (λ − µ)} sinh2(λ + µ) = tr 0 {R 0 (λ − µ) sinh(λ + µ + η) · sinh(λ + µ − η) 00 00

80 Appendix B. Proof of the Integrability of the XXZ Chain with OBCs

R000 (−λ − µ − η)R000 (−λ + µ)T00 (µ)R000 (λ + µ − η)T0(λ)} sinh2(λ + µ) = sinh(λ + µ + η) · sinh(λ + µ − η)

tr000 {R000 (−λ − µ − η)T00 (µ)R000 (λ + µ − η)T0(λ)} sinh2(λ + µ) = sinh(λ + µ + η) · sinh(λ + µ − η) t t h t i 0 h t t i 0 0 0 0 0 0 tr00 R000 (−λ − µ − η)T0 (µ) T0 (λ)R000 (λ + µ − η) sinh2(λ + µ) = sinh(λ + µ + η) · sinh(λ + µ − η) n t t t o 0 0 0 0 0 tr00 R000 (−λ − µ − η)T0 (µ)T0 (λ)R000 (λ + µ − η) sinh2(λ + µ) = sinh(λ + µ + η) · sinh(λ + µ − η) n t t t o 0 0 0 0 0 tr00 R000 (λ + µ − η)R000 (−λ − µ − η)T0 (µ)T0 (λ) n o t0 = tr000 T00 (µ)T0 (λ) n o t0 = {tr00 T00 (µ)} tr0T0 (λ)

= {tr00 T00 (µ)}{tr0T0(λ)}

= t(µ)t(λ). (B.8)

So, t(λ)t(µ) = t(µ)t(λ). (B.9)

81 Appendix C

Recover the Hamiltonian of the XXZ Chain with OBCs

1 For the Heisenberg XXZ spin- 2 chain with OBCs, the Hamiltonian is given by

L−1 x x y y z z − − HXXZ = ∑ σi σi+1 + σi σi+1 + ∆(σi σi+1 1) ∆. (C.1) i=1

Using 1 P = (1 +~σ ·~σ ) , (C.2) 12 2 1 2 we can rewrite the Hamiltonian as

L−1 z z HXXZ = ∑ 2Pi,i+1 + (∆ − 1)σi σi+1 − (∆ + 1) − ∆. (C.3) i=1

The z-component of the total spin is

L 1 z Sz = ∑ σi . (C.4) 2 i=1

We have

[HXXZ, Sz] = 0. (C.5)

Next, we have η L ( ) = P , (C.6) 0n 2 0n

82 Appendix C. Recover the Hamiltonian of the XXZ Chain with OBCs and

η M = L0 ij ij 2 1 + cosh η cosh η 1 − cosh η = 1ˆ − P − σzσz, (C.7) 2 sinh η ij sinh η ij 2 sinh η i j and

Mij = Mji, (C.8)

M0iP0j = P0j Mij, (C.9)

P0i M0j = MijP0i, (C.10) where i 6= j. We also have

z z z z P12σ1 σ2 = σ1 σ2 P12

z z = 1 + σ1 σ2 − P12,

σiσj = δij + iεijkσk,

P12 M12 = M12P12,

z z 2 sinh η · P12 M12 = 2P12 + (∆ − 1)σ1 σ2 − (∆ + 1), ∆ tr (P M ) = − . 0 01 01 sinh η

Denote

Γ12 = P12 M12, (C.11) we have

Γ12 = Γ21, ∆ tr (Γ ) = − , 0 01 sinh η 1 Γ = [2P + (∆ − 1)σzσz − (∆ + 1)] . 12 2 sinh η 12 1 2

83 Appendix C. Recover the Hamiltonian of the XXZ Chain with OBCs

For i 6= j, i 6= k, j 6= k, we have

PijΓik = ΓjkPij. (C.12)

Now, we can rewrite the Hamiltonian as

L−1 HXXZ = 2 sinh η · ∑ Γi,i+1 − ∆. (C.13) i=1

We have

T0(λ) = L01(λ) ··· L0L(λ)L0L(λ) ··· L01(λ),

t(λ) = tr0T0(λ), η t( ) = 2. 2

So, 0 η ∆ t ( ) = 2 − + 2Γ + ··· + 2Γ − + ··· + 2Γ − . (C.14) 2 sinh η 12 i 1,i L 1,L

Next, we have

∂ h η i−1 η sinh η · log t(λ) = sinh η · t · t0 ∂λ η 2 2 λ= 2 L−1 = 2 sinh η · ∑ Γi,i+1 − ∆. i=1

We have ﬁnished the proof that

∂ HXXZ = sinh η · log t(λ) (C.15) ∂λ η λ= 2 is valid for any L.

84 Appendix D

Eigenenergy and BAEs of the XXZ Chain with OBCs

D.1 Commutation Relations among A, B, C, and D

1 For the Heisenberg XXZ spin- 2 chain with OBCs, the monodromy matrix is given by

T0(λ) = L01(λ) ··· L0L(λ)L0L(λ) ··· L01(λ). (D.1)

In the auxiliary space V0, we have

  A(λ) B(λ)   T0(λ) =   , (D.2) C(λ) D(λ) 0 so

t(λ) = tr0T0(λ)

= A(λ) + D(λ). (D.3)

85 Appendix D. Eigenenergy and BAEs of the XXZ Chain with OBCs

We have   1 0 0 0      0 a(λ) b(λ) 0    R000 (λ) =   , (D.4)    0 b(λ) a(λ) 0    0 0 0 1 000 where

sinh λ a(λ) = , (D.5) sinh(λ + η) sinh η b(λ) = . (D.6) sinh(λ + η)

The fundamental commutation relation is

R000 (λ − µ)T0(λ)R000 (λ + µ − η)T00 (µ) = T00 (µ)R000 (λ + µ − η)T0(λ)R000 (λ − µ). (D.7)

We have   A(λ) 0 B(λ) 0      0 A(λ) 0 B(λ)  ˆ   T0(λ) ⊗ 100 =   , (D.8)    C(λ) 0 D(λ) 0    0 C(λ) 0 D(λ) 000 and   A(µ) B(µ) 0 0      C(µ) D(µ) 0 0  ˆ   10 ⊗ T00 (µ) =   . (D.9)    0 0 A(µ) B(µ)    0 0 C(µ) D(µ) 000 For Eq. (D.7), we have

(LHS)11 = A(λ)A(µ) + b(λ + µ − η)B(λ)C(µ),

(LHS)12 = A(λ)B(µ) + b(λ + µ − η)B(λ)D(µ),

(LHS)13 = a(λ + µ − η)B(λ)A(µ),

86 D.1. Commutation Relations among A, B, C, and D

(LHS)14 = a(λ + µ − η)B(λ)B(µ),

(LHS)21 = b(λ − µ)C(λ)A(µ) + a(λ − µ)a(λ + µ − η)A(λ)C(µ)

+b(λ − µ)b(λ + µ − η)D(λ)C(µ),

(LHS)22 = b(λ − µ)C(λ)B(µ) + a(λ − µ)a(λ + µ − η)A(λ)D(µ)

+b(λ − µ)b(λ + µ − η)D(λ)D(µ),

(LHS)23 = a(λ − µ)B(λ)C(µ) + a(λ − µ)b(λ + µ − η)A(λ)A(µ)

+b(λ − µ)a(λ + µ − η)D(λ)A(µ),

(LHS)24 = a(λ − µ)B(λ)D(µ) + a(λ − µ)b(λ + µ − η)A(λ)B(µ)

+b(λ − µ)a(λ + µ − η)D(λ)B(µ),

(LHS)31 = a(λ − µ)C(λ)A(µ) + b(λ − µ)a(λ + µ − η)A(λ)C(µ)

+a(λ − µ)b(λ + µ − η)D(λ)C(µ),

(LHS)32 = a(λ − µ)C(λ)B(µ) + b(λ − µ)a(λ + µ − η)A(λ)D(µ)

+a(λ − µ)b(λ + µ − η)D(λ)D(µ),

(LHS)33 = b(λ − µ)B(λ)C(µ) + b(λ − µ)b(λ + µ − η)A(λ)A(µ)

+a(λ − µ)a(λ + µ − η)D(λ)A(µ),

(LHS)34 = b(λ − µ)B(λ)D(µ) + b(λ − µ)b(λ + µ − η)A(λ)B(µ)

+a(λ − µ)a(λ + µ − η)D(λ)B(µ),

(LHS)41 = a(λ + µ − η)C(λ)C(µ),

(LHS)42 = a(λ + µ − η)C(λ)D(µ),

(LHS)43 = D(λ)C(µ) + b(λ + µ − η)C(λ)A(µ),

(LHS)44 = D(λ)D(µ) + b(λ + µ − η)C(λ)B(µ),

87 Appendix D. Eigenenergy and BAEs of the XXZ Chain with OBCs and

(RHS)11 = A(µ)A(λ) + b(λ + µ − η)B(µ)C(λ),

(RHS)12 = b(λ − µ)A(µ)B(λ) + a(λ − µ)a(λ + µ − η)B(µ)A(λ)

+b(λ − µ)b(λ + µ − η)B(µ)D(λ),

(RHS)13 = a(λ − µ)A(µ)B(λ) + b(λ − µ)a(λ + µ − η)B(µ)A(λ)

+a(λ − µ)b(λ + µ − η)B(µ)D(λ),

(RHS)14 = a(λ + µ − η)B(µ)B(λ),

(RHS)21 = C(µ)A(λ) + b(λ + µ − η)D(µ)C(λ),

(RHS)22 = b(λ − µ)C(µ)B(λ) + a(λ − µ)a(λ + µ − η)D(µ)A(λ)

+b(λ − µ)b(λ + µ − η)D(µ)D(λ),

(RHS)23 = a(λ − µ)C(µ)B(λ) + b(λ − µ)a(λ + µ − η)D(µ)A(λ)

+a(λ − µ)b(λ + µ − η)D(µ)D(λ),

(RHS)24 = a(λ + µ − η)D(µ)B(λ),

(RHS)31 = a(λ + µ − η)A(µ)C(λ),

(RHS)32 = a(λ − µ)B(µ)C(λ) + a(λ − µ)b(λ + µ − η)A(µ)A(λ)

+b(λ − µ)a(λ + µ − η)A(µ)D(λ),

(RHS)33 = b(λ − µ)B(µ)C(λ) + b(λ − µ)b(λ + µ − η)A(µ)A(λ)

+a(λ − µ)a(λ + µ − η)A(µ)D(λ),

(RHS)34 = b(λ + µ − η)A(µ)B(λ) + B(µ)D(λ),

(RHS)41 = a(λ + µ − η)C(µ)C(λ),

(RHS)42 = a(λ − µ)D(µ)C(λ) + a(λ − µ)b(λ + µ − η)C(µ)A(λ)

+b(λ − µ)a(λ + µ − η)C(µ)D(λ),

88 D.1. Commutation Relations among A, B, C, and D

(RHS)43 = b(λ − µ)D(µ)C(λ) + b(λ − µ)b(λ + µ − η)C(µ)A(λ)

+a(λ − µ)a(λ + µ − η)C(µ)D(λ),

(RHS)44 = b(λ + µ − η)C(µ)B(λ) + D(µ)D(λ).

Using (LHS)14 = (RHS)14, we obtain

B(λ)B(µ) = B(µ)B(λ). (D.10)

Using (LHS)13 = (RHS)13, we obtain

a(µ + λ − η) A(λ)B(µ) = B(µ)A(λ) a(µ − λ) b(µ − λ)a(µ + λ − η) − B(λ)A(µ) − b(µ + λ − η)B(λ)D(µ). (D.11) a(µ − λ)

Using (LHS)34 = (RHS)34, we obtain

1 D(λ)B(µ) = B(µ)D(λ) a(λ − µ)a(λ + µ − η) b(λ − µ) − B(λ)D(µ) a(λ − µ)a(λ + µ − η) b(λ + µ − η) + A(µ)B(λ) a(λ − µ)a(λ + µ − η) b(λ − µ)b(λ + µ − η) − A(λ)B(µ). (D.12) a(λ − µ)a(λ + µ − η)

Combine Eq. (D.11) and Eq. (D.12), we have

1 − [b(λ + µ − η)]2 D(λ)B(µ) = B(µ)D(λ) a(λ − µ)a(λ + µ − η) h i b(λ − µ) (b(λ + µ − η))2 − 1 + B(λ)D(µ) a(λ − µ)a(λ + µ − η) b(λ + µ − η) 1 b(λ − µ)b(µ − λ) + + B(λ)A(µ) a(λ − µ) a(λ − µ) a(µ − λ) b(λ − µ)b(λ + µ − η) 1 1 − + B(µ)A(λ). (D.13) a(λ − µ) a(µ − λ) a(λ − µ)

89 Appendix D. Eigenenergy and BAEs of the XXZ Chain with OBCs

Compare Eq. (D.11) and Eq. (D.13), we want to kill the term "(···)B(µ)A(λ)" in Eq. (D.13):

D(λ) ↔ De(λ),

De(λ) = f (λ)D(λ) + g(λ)A(λ), 1 g(λ) D(λ) = De(λ) − A(λ). f (λ) f (λ)

We choose

f (λ) = 1, (D.14) sinh η g(λ) = − . (D.15) sinh(2λ)

Finally, we obtain

sinh η De(λ) = D(λ) − A(λ), (D.16) sinh(2λ) sinh(2λ) + sinh η t(λ) = A(λ) + De(λ), (D.17) sinh(2λ) and

sinh(µ + λ − η) sinh(µ − λ + η) A(λ)B(µ) = B(µ)A(λ) sinh(µ + λ) sinh(µ − λ) sinh(η) sinh(2µ − η) − B(λ)A(µ) sinh(2µ) sinh(µ − λ) sinh(η) − B(λ)De(µ), (D.18) sinh(µ + λ) and

sinh(λ + µ + η) sinh(λ − µ + η) De(λ)B(µ) = B(µ)De(λ) sinh(λ + µ) sinh(λ − µ) sinh(η) sinh(2λ + η) − B(λ)De(µ) sinh(2λ) sinh(λ − µ) sinh(η) sinh(2λ + η) sinh(2µ − η) + B(λ)A(µ). (D.19) sinh(λ + µ) sinh(2λ) sinh(2µ)

90 D.2. Eigenenergy and BAEs

D.2 Eigenenergy and BAEs

In the auxiliary space V0, we have

  ↑ ↓ − σi + a(λ)σi b(λ)σi R0i(λ) =   , (D.20)  + ↓ ↑  b(λ)σi σi + a(λ)σi 0 where

1 y σ+ = σx + iσ , i 2 i i 1 y σ− = σx − iσ , i 2 i i ↑ 1 σ = (1 + σz) , i 2 i ↓ 1 σ = (1 − σz) . (D.21) i 2 i

Next,

T0(λ) |ωi = L01(λ) ··· L0L(λ)L0L(λ) ··· L01(λ) |↑i1 ··· |↑iL , (D.22) and denote

η η sinh λ − 2 a = a λ − = η , 2 sinh λ + 2 η sinh (η) b = b λ − = η , 2 sinh λ + 2 we have   |↑i 2ab |↓i  1 1  L01(λ)L01(λ) |↑i1 = . (D.23)  2 2  0 a + b |↑i1 0 For L = 2,   |ωi (∗) |↑i |↓i + (∗) |↓i |↑i  1 2 1 2  T0(λ) |ωi =   . (D.24) 0 b2 + a2 a2 + b2 |ωi 0

91 Appendix D. Eigenenergy and BAEs of the XXZ Chain with OBCs

For L = 3,

  |ωi (···)  12  T0(λ) |ωi =   , (D.25) 0 b2 + a2 b2 + a2 a2 + b2 |ωi 0 where

(···)12 = (∗) |↑i1 |↑i2 |↓i3 + (∗) |↑i1 |↓i2 |↑i3 + (∗) |↓i1 |↑i2 |↑i3 . (D.26)

By induction, for arbitrary L,

  |ωi ∗   T0(λ) |ωi =   , (D.27) 0 ∆L(λ) |ωi 0 where

2 2 ∆L(λ) = b + a ∆L−1(λ), L = 2, 3, 4, ··· ;

2 2 ∆1(λ) = b + a .

We can easily solve the above recursion relation:

L 1 − a2 ∆ (λ) = b2 + a2L. (D.28) L 1 − a2

In the auxiliary space, we also have

A(λ) |ωi = |ωi , (D.30)

D(λ) |ωi = ∆L(λ) |ωi , (D.31)

C(λ) |ωi = 0, (D.32)

92 D.2. Eigenenergy and BAEs and B(λ) |ωi ﬂips one spin. We propose the common eigenstate of the transfer matrix and the Hamiltonian as

|λ1, λ2, ··· , λni = B(λ1)B(λ2) ··· B(λn) |ωi , (D.33) where n denotes the number of down spins. We have

sinh(2λ) + sinh η t(λ) |λ , λ , ··· , λ i = A(λ)B(λ ) ··· B(λ ) |ωi 1 2 n sinh(2λ) 1 n

+De(λ)B(λ1) ··· B(λn) |ωi . (D.34)

We know that

A(λ)B(λ1) ··· B(λn)

= α0B(λ1) ··· B(λn)A(λ)

+α1B(λ)B(λ2) ··· B(λn)A(λ1)

+ ···

+αiB(λ)B(λ1) ··· B(λi−1)B(λi+1) ··· B(λn)A(λi)

+ ···

+αnB(λ)B(λ1) ··· B(λn−1)A(λn)

+β1B(λ)B(λ2) ··· B(λn)De(λ1)

+ ···

+βiB(λ)B(λ1) ··· B(λi−1)B(λi+1) ··· B(λn)De(λi)

+ ···

+βnB(λ)B(λ1) ··· B(λn−1)De(λn). (D.35)

It’s easy to obtain

n sinh (λ + λ − η) sinh (λ − λ + η) α = i i . (D.36) 0 ∏ sinh (λ + λ) sinh (λ − λ) i=1 i i

93 Appendix D. Eigenenergy and BAEs of the XXZ Chain with OBCs

Using

A(λ)B(λ1) ··· B(λn)

= A(λ)B(λi)B(λ1) ··· B(λi−1)B(λi+1) ··· B(λn)

sinh (λi + λ − η) sinh (λi − λ + η) = { B(λi)A(λ) sinh (λi + λ) sinh (λi − λ) sinh (η) sinh (2λi − η) − B(λ)A(λi) sinh (2λi) sinh (λi − λ) sinh (η) − B(λ)De(λi)}B(λ1) ··· B(λi−1)B(λi+1) ··· B(λn), sinh (λi + λ) we can easily obtain

n " # sinh (η) sinh (2λi − η) sinh λj + λi − η sinh λj − λi + η αi = − , (D.37) sinh (2λi) sinh (λi − λ) ∏ sinh λj + λi sinh λj − λi j=1(j6=i) and

n " # sinh (η) sinh λi + λj + η sinh λi − λj + η βi = − . (D.38) sinh (λi + λ) ∏ sinh λi + λj sinh λi − λj j=1(j6=i)

Similarly,

De(λ)B(λ1) ··· B(λn)

= γ0B(λ1) ··· B(λn)De(λ)

+γ1B(λ)B(λ2) ··· B(λn)De(λ1)

+ ···

+γiB(λ)B(λ1) ··· B(λi−1)B(λi+1) ··· B(λn)De(λi)

+ ···

+γnB(λ)B(λ1) ··· B(λn−1)De(λn)

+δ1B(λ)B(λ2) ··· B(λn)A(λ1)

+ ···

94 D.2. Eigenenergy and BAEs

+δiB(λ)B(λ1) ··· B(λi−1)B(λi+1) ··· B(λn)A(λi)

+ ···

+δnB(λ)B(λ1) ··· B(λn−1)A(λn), where n sinh (λ + λ + η) sinh (λ − λ + η) γ = i i , (D.39) 0 ∏ sinh (λ + λ ) sinh (λ − λ ) i=1 i i and

n " # sinh (η) sinh (2λ + η) sinh λi + λj + η sinh λi − λj + η γi = − , (D.40) sinh (2λ) sinh (λ − λi) ∏ sinh λi + λj sinh λi − λj j=1(j6=i) and

sinh (η) sinh(2λ + η) sinh (2λi − η) δi = sinh (λ + λi) sinh (2λ) sinh (2λi) n " # sinh λj + λi − η sinh λj − λi + η × . (D.41) ∏ sinh λj + λi sinh λj − λi j=1(j6=i)

We have

t(λ) |λ1, λ2, ··· , λni = (constant) |λ1, λ2, ··· , λni + (unwanted terms). (D.42)

From (unwanted terms) = 0, we obtain the BAEs:

n 2 η 2L η sinh λi + λj + η sinh λi − λj + η cosh λi − sinh λi + = 2 2 , (D.43) ∏ sinh λ + λ − η sinh λ − λ − η 2 η 2L − η j=1(j6=i) i j i j cosh λi + 2 sinh λi 2 and i = 1, 2, . . . n. Under these conditions, we have

t(λ) |λ1, λ2, ··· , λni = τ(λ) |λ1, λ2, ··· , λni , (D.44)

95 Appendix D. Eigenenergy and BAEs of the XXZ Chain with OBCs where

n sinh η sinh (λ + λ − η) sinh (λ − λ − η) τ(λ) = 1 + i i + sinh(2λ) ∏ sinh (λ + λ ) sinh (λ − λ ) i=1 i i " η #2L n sinh η sinh λ − sinh (λ + λ + η) sinh (λ − λ + η) 1 − 2 i i . sinh(2λ) + η ∏ sinh (λ + λ ) sinh (λ − λ ) sinh λ 2 i=1 i i (D.45)

From Eq. (C.15), we have

1 0 EXXZ = sinh η · τ (λ) | η . (D.46) η λ= 2 τ 2

η We can easily obtain τ 2 = 2. Using these results, we obtain

n 2 sinh2(η) EXXZ(λ1, λ2, ··· , λn) = ∑ η η − ∆ . (D.47) i=1 sinh(λi + 2 ) sinh(λi − 2 )

96 Appendix E

Expansion Velocity of the FHM

Below is a Mathematica code that is used to calculate the expansion velocity of the FHM:

In[31]:= NN= 4;(* Number of particles*) DD= 20;(* Initial number of sites*) L= 1000;(* Final number of sites is2L+1*) kk[1] = 0.125481; kk[2] = 0.250934; kk[3] = 0.376816; kk[4] = 0.503889; Sin[k] f[k_, q_]:= × 2.0× 2×L+1× Sin[k]- Cos2×k×L+1× Sink×2×L+1 Sin[q] k-q 2 k+q 2 × Csc  Csc  2.0×DD× Sin[q]- Cosq×DD+1× Sin[q×DD] 2 2 Cos[k L] Sin[k] Sin[q] + Cos[k] Sin[k L] Sin[q] + Sink2+DD+L Sin[DD q]- Sink1+DD+L Sin1+DDq 2 ;(*one-particle MDF*) NN h[k_]:=  f[k, kk[j]];(*MDF*) j=1 π Sumh ×m,{m, 1, 2×L+1} 2×L+2

Out[36]= 4.

Expand velocity:

4 π π 2 In[37]:= × Sumh ×m× Sin ×m ,{m, 1, 2×L+1} NN 2×L+2 2×L+2

Out[37]= 0.734731

97 Appendix F

Ground State Rapidities for the BHM

For the BHM (see Eq. (3.104)), the BAEs are given by

N sin kj − sin km − ic sin kj + sin km − ic −2ik (L +1) = e j 0 . (F.1) ∏ sin kj − sin km + ic sin kj + sin km + ic m=1(m6=j)

Do "ln(···)" on both sides, we obtain:

N " 2 # 1 sin kj π k = − arctan c + n , (F.2) j L + 1 ∑ − 1 2 − 2 L + 1 j 0 m=1(m6=j) 1 c2 sin kj sin km 0 where j = 1, ··· , N, and nj = 1, ··· , N. We can rewrite the above equation as

N " 2 # 1 sin kj π k = − arctan c · 1 − δ + n . (F.3) j L + 1 ∑ − 1 2 − 2 j,m L + 1 j 0 m=1 1 c2 sin kj sin km 0

We solve the above equations by an iteration procedure. If U = 0, then c = 0, so kj = π n . We illustrate the iteration procedure for L = 10, U/J = 8 and N = 3. In this case, L0+1 j 0 1 ··· 2 20 1 100 n0 = 0.3 and c = 4. We iterate 40 times, so c = 10 i (i = 1, , 40), c = i and c2 = i2 .A Mathematica code is provided in the following page. When c reaches 4, we see that the results are not convergent. So, we ﬁx c = 4 and iterate a few more times. After that, we obtain the convergent results.

98 Appendix F. Ground State Rapidities for the BHM

Calculation of the ground state rapidities (Step I)

π kk[1] = ; 11.0 π kk[2] = × 2; 11.0 π kk[3] = × 3; 11.0 20 1.0 × Sin[x] f[x_, y_]:= - × ArcTan i ; 11 1- 100 ×Sin[x]^2- Sin[y]^2 i2 π 3 Dott[1] = +  f[kk[1], kk[j]] ×1- KroneckerDelta[1, j], 11.0 j=1 π 3 tt[2] = ×2+  f[kk[2], kk[j]] ×1- KroneckerDelta[2, j], 11.0 j=1 π 3 tt[3] = ×3+  f[kk[3], kk[j]] ×1- KroneckerDelta[3, j]; 11.0 j=1 Print[{i, tt[1], tt[2], tt[3]}]; {kk[1] = tt[1], kk[2] = tt[2], kk[3] = tt[3]},{i, 1, 40}

{1, 0.252926, 0.583146, 0.930993}

{2, 0.23992, 0.599121, 0.979879}

{3, 0.232875, 0.613841, 1.02031}

{4, 0.229549, 0.628082, 1.05362}

{5, 0.228265, 0.641171, 1.08208}

Omit------Omit

{36, 0.260414, 0.522258, 0.787068}

{37, 0.261002, 0.523418, 0.788755}

{38, 0.261564, 0.524526, 0.790364}

{39, 0.262102, 0.525584, 0.7919}

{40, 0.262616, 0.526596, 0.793368}

99 Appendix F. Ground State Rapidities for the BHM

Calculation of the ground state rapidities (Step II)

kk[1] = 0.262616; kk[2] = 0.526596; kk[3] = 0.793368; 2 1.0 × Sin[x] f[x_, y_]:= - × ArcTan 4 ; 11 1- 1 ×Sin[x]^2- Sin[y]^2 42 π 3 Dott[1] = +  f[kk[1], kk[j]] ×1- KroneckerDelta[1, j], 11.0 j=1 π 3 tt[2] = ×2+  f[kk[2], kk[j]] ×1- KroneckerDelta[2, j], 11.0 j=1 π 3 tt[3] = ×3+  f[kk[3], kk[j]] ×1- KroneckerDelta[3, j]; 11.0 j=1 Print[{tt[1], tt[2], tt[3]}];{kk[1] = tt[1], kk[2] = tt[2], kk[3] = tt[3]},{10}

{0.262574, 0.526522, 0.79328}

{0.262577, 0.526527, 0.793285}

100 Appendix G

List of Publications

1. Zhongtao Mei and C. J. Bolech, “Derivation of matrix product states for the Heisen- berg spin chain with open boundary conditions”, Phys. Rev. E 95, 032127 (2017).

2. Zhongtao Mei, L. Vidmar, F. Heidrich-Meisner, and C. J. Bolech, “Unveiling hidden structure of many-body wave functions of integrable systems via sudden- expansion experiments”, Phys. Rev. A 93, 021607(R) (2016).

101