Periodic Table of Ordinary and Supersymmetric Sachdev-Ye-Kitaev Models

Mississippi State University Scholars Junction

Theses and Dissertations Theses and Dissertations

8-7-2020

Periodic table of ordinary and supersymmetric Sachdev-Ye-Kitaev models

Fadi Sun

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Fadi Sun

A Dissertation Submitted to the Faculty of Mississippi State University in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy in Engineering/Applied Physics in the Department of Physics and Astronomy

Mississippi State, Mississippi

August 2020 Copyright by

Fadi Sun

2020 Periodic table of ordinary and supersymmetric Sachdev-Ye-Kitaev models

Fadi Sun

Approved:

Jinwu Ye (Major Professor)

Yaroslav Koshka (Committee Member)

R. Torsten Clay (Committee Member)

Steven R. Gwaltney (Committee Member)

Hendirk F. Arnoldus (Committee Member/Graduate Coordinator)

Jason M. Keith Interim Dean Bagley College of Engineering Name: Fadi Sun

Date of Degree: August 7, 2020

Institution: Mississippi State University

Major Field: Engineering/Applied Physics

Major Professor: Jinwu Ye

Title of Study: Periodic table of ordinary and supersymmetric Sachdev-Ye-Kitaev models Pages of Study: 44

Candidate for Degree of Doctor of Philosophy

This dissertation is devoted to investigation of quantum chaos in the Sachdev-Ye-Kitaev

(SYK) and supersymmetric SYK models. First, a uniﬁed minimal scheme is developed to classify quantum chaos in the SYK and supersymmetric SYK models and also work out the structure of the energy levels in one periodic table. The SYK with even q-body or supersymmetric SYK with odd q-body interaction, with N even or odd number of sites, are put on an equal footing in the minimal Hilbert space; N (mod 8), q (mod 4) double

Bott periodicity, and a reflection relation are identified. Then, exact diagonalizations are performed to study both the bulk energy level statistics and hard-edge behaviors. Excellent agreements between the exact diagonalization results and the symmetry classifications are demonstrated. This compact and systematic method can be transformed to map out more complicated periodic tables of SYK models with more degrees of freedom, tensor models, or symmetry protected topological phases. Key words: Sachdev-Ye-Kitaev models, Quantum Chaos, Random Systems, AdS-CFT Correspondence ACKNOWLEDGEMENTS

I thank my committee for their comments on this dissertation, and I thank Jinwu Ye for directing this research.

ii TABLE OF CONTENTS

ACKNOWLEDGEMENTS ...... ii

LIST OF TABLES ...... v

LIST OF FIGURES ...... vi

LIST OF SYMBOLS, ABBREVIATIONS, AND NOMENCLATURE ...... vii

CHAPTER

1. INTRODUCTION ...... 1

1.1 Origin and Motivation ...... 1 1.2 Family of SYK models ...... 5 1.3 Organization of the dissertation ...... 7

2. RANDOM MATRIX THEORY AND QUANTUM CHAOS ...... 9

2.1 Ten symmetry classes in random matrix theory ...... 9 2.2 Energy level statistics in the bulk and edge ...... 11 2.3 Wigner surmises of energy level spacing and two universal ratios . 13 2.3.1 Wigner surmise of energy level spacing ...... 13 2.3.2 The ratio r and universality in the bulk ...... 14 2.3.3 The ratio A and universality in the edge ...... 15

3. A UNIFIED MINIMAL SCHEME TO CLASSIFY ORDINARY AND SUPERSYMMETRIC SYK MODELS ...... 19

3.1 Representation and relevant operators ...... 19 3.2 The Bott periodicity and reﬂection relation ...... 21 3.3 Application to SYK models ...... 24 3.3.1 SYK model with q (mod 4) = 0 ...... 24 3.3.2 SYK model with q (mod 4) = 2 ...... 25 3.3.3 SYK supercharge with q (mod 4) = 1 ...... 26 3.3.4 SYK supercharge with q (mod 4) = 3 ...... 27

iii 3.4 The Periodic Table of SYK Models ...... 29

4. NUMERICAL SIMULATIONS ...... 30

4.1 Algorithm for Simulation of SYK Models ...... 30 4.2 Numerical Results ...... 31

5. CONCLUSIONS ...... 36

REFERENCES ...... 38

APPENDIX

A. SOME USEFUL OPERATOR ALGEBRA IDENTITIES ...... 43

iv LIST OF TABLES

2.1 The ten-fold way of RMT classes and theirs operator algebras...... 11

2.2 RMT index β and the nearest-neighbor ratio r˜ ...... 15 h i 2.3 RMT indices (α, β) and the corresponding moment ratio A for the seven classes with a mirror symmetry (but no zero mode)...... 18

3.1 The periodic table of SYK for even q > 2 and supersymmetric SYK for odd q > 1...... 29

v LIST OF FIGURES

4.1 Many-body energy level statistics of q = 6 SYK model for N = 17, 19, 21, 23. (a) Distributions of consecutive level spacing ratio r in q = 6 SYK model are presented to show level statistics in the bulk. (b) Distributions of the smallest 3 energy level in q = 6 SYK model are presented to show level statistics in the hard edge...... 33

4.2 Energy level statistics of q (mod 4) = 1 SYK supercharge. The same as ﬁgure 4.1 but for q = 5 SYK supercharge. The spectral density from 1 sample is plotted in (b) to show the spectral mirror symmetry is absent for a

single realization of Ci1,··· ,iq ...... 34

4.3 The same as ﬁgure 4.2 but for q = 3 SYK supercharge. Notice the RMT classes appear in a reversed order rather than q (mod 4) = 1 case. . . . . 35

vi LIST OF SYMBOLS, ABBREVIATIONS, AND NOMENCLATURE

SY model Sachdev-Ye model

SYK model Sachdev-Ye-Kitaev model

OTOC out-of-time-order correlator

ELS energy level statistics

RMT random matrix theory

GOE Gaussian orthogonal ensemble

GUE Gaussian unitary ensemble

GSE Gaussian symplectic ensemble

integer ﬂoor: x means the greatest integer less than or equal to x b·c b c integer ceiling: x means the least integer greater than or equal to x d·e d e

vii CREDIT

This dissertation is an expanded and modiﬁed version of a paper in the Physics Review

Letter 124, 244101 (2020).

viii CHAPTER 1

INTRODUCTION

1.1 Origin and Motivation

The origin of Sachdev-Ye-Kitaev (SYK) model may date back to 1993 when Subir

Sachdev and Jinwu Ye introduce a model to describe a quantum Heisenberg magnet with

Gaussian random all-to-all exchange interactions [1, 2]. In the literature, the model is often

referred to as Sachdev-Ye (SY) model, and its Hamiltonian takes the form

N 1 X H = J S S , (1.1) SY √ ij i j NM i>j ·

where S is the spin operator of the group SU(M), and exchange constants Jij are random numbers drawn from a Gaussian distribution. The SY model becomes solvable at N → ∞ ﬁrst and then M limit, and it admits a gapless quantum spin liquid ground phase, → ∞ but may experience a transition to a spin-glass phase at low temperature. The possible

holographic duality of SY model was ﬁrst discussed in [3], where Sachdev showed a close

correspondence between the physical properties of holographic metals near charged black

holes in anti-de Sitter (AdS) space, and the fractionalized Fermi liquid phase of the lattice

Anderson model. Recently, the SY model is revisited by replacing SU(M) group with

SO(M) group in [4], and a deconﬁned critical point is found in the SY model with a

critical hole doping [5].

1 In a series of talks in 2015, Alexei Kitaev proposed a variant of SY model by replacing the SU(M) spins with N Majorana fermions as well as the two indices Jij with four indices

Jijkl, which now known as the SYK model [6]. The Hamiltonian of the SYK model is given by

N X H = Jijklχiχjχkχl , (1.2) SYK4 − i>j>k>l where the χ are Majorana fermions and the coupling Jijkl is totally antisymmetric and drawn from a Gaussian distribution. The SYK model can be solved in the single N → ∞ limit, in contrast to the double N and M limit in the SY model, which leads → ∞ → ∞ to a considerable simpliﬁcation of the original model in both analytical and numerical. It is also showed that the SYK model successfully avoids the spin-glass phase in the SY model

[7]. Despite the apparent simplicity of the SYK model, it possesses many remarkable properties, as explained in the following, including exact solvable in large N limit, an emergent conformal symmetry in the infrared limit, quantum chaotic at both early time and late time, and so on. It is the ﬁrst time to have these properties all in one model; thus it becomes a subject undergoing intense study in both high energy and condensed matter physics.

In the large N limit, the leading Feynman diagrams for the two-point function and the four-point function are melonic diagrams and ladder diagrams, respectively, thus exact Schwinger-Dyson equations can be derived from summing all corresponding diagrams

[8]. And the next leading order corrections can be calculated in a systematic way. The

Schwinger-Dyson equations can be solved numerically by a weighted iterative method or

2 solved analytically by perturbation method [8]. The solvability makes an in-depth study of

its notable properties possible.

In the infrared limit or, equivalently, the low energy limit, the Schwinger-Dyson equa-

tion of two-point function can be simpliﬁed and solved analytically exactly. It is found

1 the equation is invariance under the group Diff(S )∼=Conf(R) of reparameterization, but the solution is only invariance under the group SL(2,R) which is a subgroup of Diff(S1).

So there is a spontaneously breaks from Diff(S1) symmetry to PSL(2,R) symmetry, which

leads to Goldstone modes live in Diff(S1)/PSL(2,R). Such an emergent conformal sym-

metry in the infrared suggests it corresponding to a 1-dimensional conformal ﬁeld theory

(CFT1). When going beyond the infrared limit, the reparameterization symmetry becomes

approximate thus lift the Goldstone modes to pseudo-Goldstone modes, which sometimes

referred to as soft modes. The dynamics of soft modes are governed by an effective ﬁeld

theory called the Schwarzian action. It is well known that the Jackiw-Teitelboim gravity

theory is a model of 2-dimensional (2D) dilaton gravity, which is holographically dual to

Schwarzian theory. The connection between the SYK model and 2D dilaton gravity pro-

vides a way to identify the soft modes in SYK model with t’Hooft’s gravitational modes in

2D dilaton gravity theory [9, 10, 11, 12].

The quantum chaos may be diagnosed by the out-of-time-order correlators (OTOCs).

If the system is quantum chaotic, the OTOC will show exponential an growth behavior

in the early-time and then saturated in the late-time, and the exponential growth rate in

the early-time deﬁnes a Lyapunov exponent λL. For the SYK model, the Lyapunov ex-

ponent reaches its maximum possible value, Maldacena-Shenker-Stanford bound [8, 13],

3 λL = (kB/~)(2π/β) = πkBT/~ at low temperature; and becomes λL = J/~ at high temperature. In this sense, the SYK model is maximally chaotic. It may be easy to construct a concrete model that is quantum chaotic, but it is rather challenging to ﬁnd a model which is maximal quantum chaotic. Hence maximally chaos property becomes a salient feature of the SYK model. On the other hand, black holes are the fastest scramblers in nature [14], so the maximal quantum chaos behavior might also suggest an underlying duality to a black hole.

The quantum chaos may also be characterized by the energy level statistics (ELS) or spectral form factor. Since the OTOC saturated in the late-time, a complimented way is to use the random matrix theory (RMT) to characterize the ELS to deﬁne quantum chaos.

The RMT predicts the energy levels of quantum chaos system will have long range spectral rigidity and short range level repulsion. These spectral properties have been veriﬁed via exact diagonalization of SYK model, and the ELS indeed satisfy certain random matrix ensembles [15, 16, 17, 18, 19, 20, 21, 22, 23]. These facts justify the SYK model is also quantum chaotic in the late time. On the other hand, the quantum black holes are bound quantum chaotic systems, of which the energy levels should look like the eigenvalues of a very large random matrix. The RMT has been employed to study both black holes and

SYK models, which also suggest a close relationship between the two.

In summary, the SYK model is originally deﬁned in a condensed matter system with well-deﬁned infrared and ultraviolet properties, but the holography duality between the

SYK model and black holes suggests that the SYK model is an example of NAdS2/NCFT1,

4 where N stands for nearly. The SYK model and black holes are both maximal chaotic, and

the SYK model becomes a tool to investigate black holes and quantum chaos.

1.2 Family of SYK models

The SYK model deﬁned in Eq.(1.2) belongs to a larger family of SYKq model with q-fermion all-to-all interactions, where q is a positive even integer [6, 24, 8]. The Hamil- tonian of SYKq model can be expressed as

N q/2 X H q = i Ji ···i χi χi χi (1.3) SYK 1 q 1 2 ··· q i1<···

where χ are Majorana fermions satisfying condition χ = χ† as well as anticommutation

relation χi, χj = δij, the coupling Ji ···i is real antisymmetric and satisfy the Gaussian { } 1 q 2 2 q−1 distributions with mean Ji ···i = 0 and variance J = J (q 1)!/N . The h 1 q i h i1···iq i SYK − q/2 overall factor i factor ensures the Hermitian of HSYKq.

The original motivation to introduce the SYKq family is to solve the Schwinger-Dyson equation analytically in the large q limit or double scaling limit, which means N → ∞ but 2q2/N ﬁxed. It is also found that these SYKq models with q 4 share qualitatively ≥ similar properties with SYK4 as discussed before. The SYK2 model is special because it can be reduced to a free random-mass matrix model [25], which will show single-body chaos instead of many-body chaos.

Furthermore, the above SYKq model can be generalized to the case with odd q. Notice that Hq is a bosonic operator when q is even, while Hq is a fermionic operator when q is

5 odd, which should interpreted as a supercharge Qq. As in supersymmetric theories, the

Hamiltonian is the square of the supercharge

2 (q−1)/2 X H = Q ,Qq = i Ci ···i χi χi χi (1.4) SUSY-SYK q 1 q 1 2 ··· q i1<···

mitian of supercharge Qq. Since the theory has only one supercharge, it is called = 1 N supersymmetric SYK model [26, 20, 21] in order to distinguish from the = 2 super- N symmetric SYK model, which is a supersymmetric generalizations of complex SYK. In

the dissertation, unless otherwise speciﬁed, supersymmetric SYK model means the = 1 N supersymmetric SYK model. The supersymmetric SYK model is qualitatively similar to

the original SYK model. For example, it is also maximal chaotic, and its effective theory

is described by a supersymmetric generalizations of the Schwarzian action.

The supersymmetry in the = 1 supersymmetric SYK models is unbroken in the N N limit, but broken nonperturbatively in the 1/N expansion [26]. The borken → ∞ supersymmetry means the ground state energy in nonzero. The entire SYKq family have no

zero mode, which is an important fact we will use in the later discussions. This dissertation

will focus on the SYKq family, a.k.a. SYK models and supersymmetric SYK models, and

only brieﬂy review other variants of SYK model.

There is another important version of SYK model with complex fermions [27, 17, 28,

29], named as complex SYK model,

X X H = J c† c† c† c c µ c†c (1.5) cSYK i1i2···iq/2;iq/2+1···iq i1 i2 iq/2 iq/2+1 iq i i ··· ··· − i 6 † where c is complex fermion satisfying anticommutation relation c , cj = δij and ci, cj = { i } { } c†, c† = 0, µ denotes the chemical potential, and the random coupling subject to the con- { i j} straint J ∗ = J . Although Majorana fermions are replaced i1i2···iq/2;iq/2+1···iq iq···iq/2+1;iq/2···i2i1

with complex fermions, the complex SYK shares similar properties with Majorana SYK

discussed above. It is believed that the Majorana SYK model is dual to a neutral black hole

(Schwarzschild black hole), and the complex SYK model is due to a charged black hole

(Reissner-Nordstrom black hole). In fact, complex SYK model is the simplest SYK with

a global symmetry, U(1) symmetry, while the SYK model with the other global symmetry

also has been constructed [30, 31].

To end this section, it is worth pointing out the quenched disorder (random couplings)

in the SYK model is not essential. For example, the Gurau-Witten model is a type of col-

ored tensor model without quenched disorder which ﬁrst proposed by Witten [32, 33], and

it modiﬁed to Klebanov-Tarnopolsky models by simplifying colored to uncolored tensors

[34]. To the leading order in 1/N expansion, these tensor models have the same type of

Feynman diagrams and thus share similar properties as SYK models [35].

1.3 Organization of the dissertation

The focus of this dissertation is unveiling the many-body quantum chaos of the SYK

model and supersymmetric SYK model by studying their ELS property. In chapter 2, we

ﬁrst review some known results on ten-fold way random matrix classiﬁcations, bulk ELS

characterized by the ratio of nearest neighbor spacing. Then, we derive Wigner surmises

for the distribution function of the smallest eigenvalue for the four BdG classes and also

7 propose a new moment ratio A to determine the edge exponent α effectively. In chapter

3, we develop a minimal unified scheme to classify quantum chaos in the SYK and supersymmetric SYK models, which not only reproduces the previously known results in a much more efficient and compact waybut also leads to new findings on q (mod 4) = 1, 2, 3 with odd N, thus complete the whole periodic table for the SYK models. In chapter 4, we present numerical simulations of the SYK and supersymmetric SYK models for various

N and q. The ELS from numerical data are in excellent agreement with predictions from random matrix classiﬁcations. In chapter 5, we make an analogy between the periodic table of SYK and supersymmetric SYK and the periodic table of Topological insulators and superconductorsand address an open problem, which is left for further work. In appendix

A, we provide some useful operator algebra identities.

8 CHAPTER 2

RANDOM MATRIX THEORY AND QUANTUM CHAOS

2.1 Ten symmetry classes in random matrix theory

In quantum mechanics, the Hamiltonian can be represented by a matrix acting on a

Hilbert space of states. However, when there are a large number of degrees of freedom or complexity of interactions, i.e., large Z nuclear atoms, it becomes impossible to obtain all the energy levels and eigenfunctions. In the pioneering works [36, 37], Wigner and Dyson proposed a statistic way to describe the energy levels by replacing the Hamiltonian by a random matrix taken from a certain ensemble. As a consequence, the energy level statics

(ELS) are determined by the certain ensemble. In general, they found only three possible cases: The system is invariant under time-reversal and 2 = +1, and the ELS satisfy • Gaussian orthogonal ensemble (GOE); T T The system is invariant under time-reversal and 2 = 1, and the ELS satisfy • Gaussian symplectic ensemble (GSE); T T − The system is not time-reversal invariant, and the ELS satisfy Gaussian unitary en- • semble (GUE).

In fact, the key in Wigner-Dyson’s way to classify random matrix ensembles is the anti-unitary symmetry operator, which may not have to be the time-reversal symmetry operator. Thus the classiﬁcation can be restated as following: consider an anti-unitary operator T+, i.e. time-reversal symmetry operator, which commutes with the Hamiltonian 9 2 T+H = HT+, In the absence of T+, the symmetry class is A (GUE); while T+ = +1, it is

AI (GOE) and T 2 = 1, it is AII (GSE). These three symmetry classes are also known as + − three Wigner-Dyson classes.

Later, seven additional symmetry classes were found by considering another antiunitary operator T− and a unitary operator Λ, both of which anticommute with the Hamiltonian

T−H = HT− and ΛH = HΛ. Note that Λ anticommutes with the Hamiltonian, thus − − will not lead to a conserved quantity, and the existence of T− or Λ suggests a mirror sym-

metry in the spectra. If both antiunitary operators T+ and T− exist, then unitary operator

Λ = T+T− must exist, but the converse is not valid. It is easy to count there are only ten possible cases, and the classiﬁcation of these ten symmetry classes 1 can state as [38]:

2 T− and Λ do not exist: If T+ exists and T+ = +1, the Hamiltonian belongs to class • 2 AI (GOE). If T+ exists and T = 1, the Hamiltonian belongs to class AII (GSE). + − If T+ also does not exist, the Hamiltonian belongs to class AI (GUE). In fact, this set of classes is reduced to Wigner-Dyson’s threefold way discussed before.

2 2 2 2 T+ and T− exist and T+ = T−, or only Λ exists: If T+ = T− = +1, the Hamiltonian • 2 2 belongs to class BDI (chGOE). If T+ = T− = 1, the Hamiltonian belongs to class CII (chGSE). If only Λ exists, the Hamiltonian− belongs to class AIII (chGUE). As is written in the parentheses2, these are the three chiral ensembles.

2 2 2 2 T+ and T− exist and T+ = T−, or only T− exists: If T+ = T− = +1, the Hamilto- • 6 2 2 − nian belongs to class CI (BdG). If T+ = T− = 1, the Hamil- tonian belongs to − − 2 class DIII (BdG). If T+ does not exist but T− exists, with T− = +1, the Hamiltonian 2 belongs to class D (BdG). If T+ does not exist but T− exists, with T− = 1, the Hamiltonian belongs to class C (BdG). As is written in the parentheses, these− are the four Bogoliubov–de Gennes (BdG) ensembles.

Therefore, all the ten symmetry classes [39, 40, 41] can be completely determined by

2 2 2 squared values of T+, T− and Λ . Dropping the T− and Λ operators recovers the original 1The RMT classes are often named by its Cartan label of corresponding symmetric spaces. In the dissertation, we use their Cartan names followed with GOE-GUE-GSE, etc., in parentheses, only use the Cartan names in the tables and ﬁgures. 2The “ch” is short for “chiral”. 10 three Wigner-Dyson classes. The symmetry classiﬁcation scheme also applies to topological insulators and superconductors [42, 43, 44] in the context of topological equivalent classes. We also summarize the ten-fold way of RMT in Table 2.1, which was heavily consulted in the dissertation.

It is worth pointing out that, in the presence of conserved quantities [38], one needs to identify all the unitary and antiunitary operators which preserve all the compatible conserved quantities. Then the classiﬁcation needs to be applied to each block of the Hamil- tonian.

Table 2.1: The ten-fold way of RMT classes and theirs operator algebras.

RMT Class A AI AII AIII BDI CII C CI D DIII T 2 — +1 1 — +1 1 — +1 — 1 + − − − T 2 ——— — +1 1 1 1 +1 +1 − − − − Λ2 ——— 1 1 1 — 1 — 1

2.2 Energy level statistics in the bulk and edge

Different random matrix ensembles will result in different kinds of ELS. For three

Wigner-Dyson classes, the joint probability density function for the eigenvalues of the random matrix takes the form [41]

n 1 Y − βn λ2 Y β P ( λi ) = e 4 k λi λj , (2.1) { } Zβ,n | − | k=1 i

11 where n is the dimension of the random matrix, and β is known as Dyson index. For

random matrix class AI (GOE), A (GUE), AII (GSE), Dyson index are β = 1, 2, 4, respectively. Note that the probability density function Eq.(2.1) will vanish if any two eigenvalues coincide λi = λj. In RMT, the statistically improbable of coincidence of two eigenvalues property is known as level repulsion, and Dyson index β characterizing the strength of level repulsion in the bulk.

For the other seven random matrix ensembles with spectral mirror symmetry, energy levels appear in pairs λ and λ. Theirs joint probability density function for the (nonzero) − eigenvalues is [39]

n 1 Y βn 2 Y α − 4 λk 2 2 β P ( λi ) = λk e λi λj (2.2) { } Zβ,α,n | | | − | k=1 i

where β is Dyson index characterizing the strength of level repulsion in the bulk, and α is

an index describing the level repulsion from the hard-edge at λ = 0. On the contrary, the

soft-edge is referred to as the spectral edge at the maximal λ . Note that the α index is not | | deﬁned for the three original Wigner-Dyson classes due to the lacking of mirror symmetry,

which leads to no special point in the spectrum after unfolding procedure. Moreover, these

random matrix ensembles may have exact zero eigenvalues, whose number deﬁnes another

RMT index ν, and α is closely related to ν. However, as stated in the introduction, the SYK

models and supersymmetric SYK models discussed in this dissertation have no zero mode;

thus only ensembles with ν = 0 will be considered. When the zero eigenvalue is absent, it

is known [37, 45, 39] that there is one to one correspondence between each random matrix

ensemble and indices (β, α), which has been summarized in Table 2.3.

12 2.3 Wigner surmises of energy level spacing and two universal ratios

In the section, we will show that the random matrix indices (β, α) can be effectively extracted from energy level data. Let λn be an ordered set of eigenenergy obtained { } from the Hamiltonian, then the energy level spacing is sn = λn+1 λn, and the ratios of − nearest-neighbor energy level spacings are deﬁned as rn = sn+1/sn.

2.3.1 Wigner surmise of energy level spacing

By considering a 2 2 matrix system, Wigner derived a simple approximate probability × distribution function for the energy level spacing, known as Wigner surmise [41]:

2 β −bβ s PW,β(s) = aβs e (2.3)

2 18 6 3 where aβ = π/2, 32/π , 2 /(3 π ) and bβ = π/4, 4/π, 64/(9π) for β = 1, 2, 4, respec-

β tively. By taking the limit s 0, it is clear PW,β(s) will vanish as s , which is consistent → ∼ with level repulsion showed in the joint probability density function. The probability distribution function for independent random energy levels yields the Poisson distribution

−s PP (s) = e (2.4)

which is a typical result of integrable systems. By taking the limit s 0, it is clear PP (s) → is ﬁnite, and there is no level repulsion between energy levels. However, to compare data from different systems, the energy levels will need some sort of normalization, which is often called an unfolding procedure. It is not convenient to unfold energy levels when large enough statistics are not available, i.e., quantum many-body systems.

13 2.3.2 The ratio r and universality in the bulk

The nearest-neighbor ratio r ﬁrst introduced in [46] is designed to overcome the dif-

ﬁculties in unfolding. Because taking the two ratios can get rid of the dependence on the

local density of states, the further unfolding procedure becomes unnecessary. By consid-

ering a 3 3 matrix system, the authors in [47] obtained the Wigner-like surmises of the × ratio of the nearest-neighbor level spacing distribution

1 (r + r2)β PW,β(r) = 2 1+3β/2 , (2.5) Zβ (1 + r + r )

where Zβ = 8/27, 4π/81√3, 4π/729√3 for β = 1, 2, 4, respectively. Besides, the Poisson case yields distribution

1 P (r) = (2.6) P (1 + r)2

The distribution function PW,β(r) has the same level repulsion at small r as PW,β(s) at small

β −(2+β) s, namely, PW,β(r) r . However, the large-r asymptotic behavior PW (r) r is ∼ ∼

dramatically different from the fast exponential decay of PW (s). It was known [47] that

1 1 the nearest-neighbor ratio satisﬁes the functional equation P (r) = r2 P ( r ), which enables

us to restrict the study to the range [0, 1] by considering the variable r˜ = min r, 1/r . { }

The above surmises also yield an analytic expression for the averaged values r˜ w = h i 0.386, 0.536, 0.603, 0.676 for the Poisson, GOE, GUE, and GSE, respectively. Although

the Wigner-like surmises are originally derived for three Wigner-Dyson classes, they still

work very well for all ten random matrix classes. Thus, calculating r˜ value provides an h i efﬁcient way to determined Dyson index β, see Table 2.2.

14 Table 2.2: RMT index β and the nearest-neighbor ratio r˜ . h i

RMT β 0 1 2 4 Class Poisson GOE GUE GSE r˜ 0.386 0.536 0.603 0.676 h i

2.3.3 The ratio A and universality in the edge

For seven random matrix classes with mirror symmetry, in addition to the bulk ELS,

one may also need to focus on the universality near the hard-edge λ = 0. One way is to

3 integrate out all the energy levels in P ( λi ) except the smallest positive one , which re- { } sult in the probability distribution function of the smallest positive eigenvalues P (λ1). The hard-edge universality can be unveiled from the small λ1 asymptotic behavior of P (λ1),

α which takes form P (λ1) λ . ∼ 1

2.3.3.1 The known results for the three chiral classes

For the exact results of the distribution function of the smallest positive eigenvalue

λ1 of the three chiral classes AIII(chGUE), BDI(chGOE), and CII(chGSE) were known

[48, 49, 50, 51, 52]:

−λ2/4 PAIII(λ1) = (λ1/2)e 1 , (2.7)

2 −λ /8−λ1/2 PBDI(λ1) = [(2 + λ1)e 1 ]/4 , (2.8)

3/2 2 1/2 −λ1/2 PCII(λ1) = (π/2) λ1 e I3/2(λ1) , (2.9)

15 where In(λ) is the modiﬁed Bessel functions of the ﬁrst kind. Using the fact I3/2(λ1) =

q 2 3/2 9π λ1 + O(λ1), it is easy to show that these exact results indeed lead to asymptotic

α behavior P (λ1) λ for small λ1. ∼ 1

2.3.3.2 Our new results using the Wigner surmises for the four BdG classes

However, the exact analytical results 4 of the distribution function of the smallest positive eigenvalue of four BdG classes C(BdG), D(BdG), CI(BdG), and DIII(BdG) are not known yet. Instead of deriving the exact results, we propose Wigner surmises for the distribution function of the four BdG classes. This method was even proved to be very accurate for Hermitian and non-Hermitian chiral random matrices ensemble to describe QCD [53].

Here, we will show it can be used to determine the edge exponent α very effectively for the four BdG classes.

Starting from the joint probability density for the energy levels with n = 2,

2 2 β α −(λ2+λ2) P (λ1, λ2) λ λ λ1λ2 e 1 2 , (2.10) ∝ | 1 − 2| | | where means the normalization constant is ignored, one can obtain the distribution func- ∝ tion of the smallest positive eigenvalue:

Z ∞ P (λ1) = dλ2P (λ1, λ2) , (2.11) λ1

4However, for BdG classes D and C, the exact probability distribution function of the smallest positive eigenvalue can express in terms of Fredholm determinants.

16 which can be done analytically. By choosing (β, α) = (2, 0) for class D(BdG), (β, α) =

(2, 2) for class C(BdG), (β, α) = (1, 1) for class CI(BdG), (β, α) = (4, 1) for class

DIII(BdG), we obtain the following results:

1 −2λ2 3 λ2 2 4 P (λ1) = e 1 [6λ1 4λ + √πe 1 erfc(λ1)(3 4λ + 4λ )] , (2.12) D π − 1 − 1 1

2 −2λ2 2 3 λ2 2 4 P (λ1) = e λ [30λ 4λ + √πe erfc(λ)(15 12λ + 4λ )] , (2.13) C 3π − −

−2λ2 PCI(λ1) = PDIII(λ1) = 4λe , (2.14) where erfc(λ) is the complementary error function. Note that the random matrix class with

α = 1 is special, and P (λ1) is independent on β index upto to a rescaling factor [49].

Using the fact erfc(λ) = 1 2λ/√π + O(λ2), one can check all of these results have the − α asymptotic behavior P (λ1) λ when λ1 is small. In principle, the edge exponent α1 ∼ 1

can be obtained by studying the small λ1 behavior of P (λ1), but in practice, although this

method can be used to easily distinguish between α = 0 and α > 0 cases, it may not be

efﬁciently used to tell the differences among α = 1, 2, 3 cases.

Since λ1 may has a dimension of energy; thus, a rescaling of P (λ1) cP (cλ1) may be → R required when compared with numerical data. Typically, one can choose c dλ1λ1P (λ1) =

λ to ﬁx the rescale factor c, which needs to be the same for all P (λk), where λmin is h mini h i the average value of the smallest positive eigenvalue from the numerical data.

2.3.3.3 A new moment ratio A and its relation to the edge exponent α

Inspired by the r-parameter introduced to remove the dependence on the local density

of state and determine the bulk ELS efﬁciently, we ﬁnd it is more convenient to introduce

17 a new moment ratio A, which is a dimensionless quantity to get rid of the dependence on the rescale factor:

2 R ∞ 2 λmin 0 dλ1λ1P (λ1) A = h i2 = 2 1 , (2.15) λmin h R ∞ i ≥ h i 0 dλ1λ1P (λ1) which is obviously independent of the rescale factor. In Table 2.3, we list AW calculated from the exact P (λ1) of the three chiral classes or our surmises of the four BdG classes, and Anum calculated from the exact diagonalization of corresponding random matrices of

3 5 size n = 10 with Gaussian distributed entries, averaged over 10 samples. Since AW and

Anum are in good agreement, our surmises of the four BdG classes work well. Noticing ensembles with different α will result in different A values, which suggests one can extract the edge exponent α by calculating the ratio A from data and comparing it with surmises.

As a conclusion, we showed there is one-to-one mapping among the random matrix classes, the two indices (β, α), and the two ratios (˜r, A). This fact can be used to extract indices (β, α) from numerical data, and then determine the random matrix class from indices (β, α) by looking up Table 2.2 and 2.3.

Table 2.3: RMT indices (α, β) and the corresponding moment ratio A for the seven classes with a mirror symmetry (but no zero mode).

RMT α 0 1 2 3 RMT β 1 2 1 2 4 2 4 Class BDI D CI AIII DIII C CII

AW 1.6018 1.5795 1.2732 1.2732 1.2732 1.1753 1.1291

Anum 1.6021 1.5832 1.2745 1.2738 1.2719 1.1735 1.1296

18 CHAPTER 3

A UNIFIED MINIMAL SCHEME TO CLASSIFY ORDINARY AND

SUPERSYMMETRIC SYK MODELS

3.1 Representation and relevant operators

In order to give a uniﬁed classiﬁcation of the SYK model and supersymmetric SYK

model, it is convenient to introduce a compact notation

bq/2c X Qq = i Ci ···i χi χi χi (3.1) 1 q 1 2 ··· q i1<···

SYK Model with q-fermion interactions [6, 8]; when q is odd, Qq is a fermionic operator

which is the supercharge of = 1 supersymmetric SYK model [26, 20, 21]. Thus the N 2 Hamiltonian is H = Qq for even q and H = Qq for odd q.

Before classifying the SYK model and supersymmetric SYK model, it is good to choose a representation, and the result of classiﬁcation is independent of which representations we use. The conventional way to construct Majorana fermion is breaking one complex fermion into two Majorana fermions. Based on this idea, the RMT classiﬁcations

19 of the SYK models have been applied to q (mod 4) = 0 for all N [16, 17, 18, 23] and q

(mod 4) = 1, 2, 3 for the even case [20, 21] only. It was generally believed that the odd N

case need to be treated quite differently than the even N case. In fact, N odd case was ﬁrst

discussed in the context of the 1D symmetry protected topological phase [54], and then

applied to the SYK model with q = 4 in [17, 23]. Reference [54, 17] developed a special

procedure to treat the N odd case by adding an extra decoupled Majorana fermion into the

system, while such a procedure is not necessary in the N even case. So in all previous

works, the SYK and supersymmetric SYK, the odd N and even N cases were treated dif-

ferently, and classiﬁcations of SYK models with q (mod 4) = 1, 2, 3 and odd N are even

not known.

In the dissertation, we will develop a uniﬁed minimal scheme to classify random matrix

behaviors of the SYK models with generic q-body interaction and N site by constructing

Majorana fermion from the irreducible representations of Clifford algebra. Equation (3.1)

† contains N Majorana fermions satisfying the Clifford algebra χi, χj = 2δij and χ = χi, { } i thus admitting a 2bN/2c-dimensional matrix representations. In this representation, one can choose χi with odd i to be real and symmetric, and χi with even i to be pure imaginary and skew-symmetric. 1 By collecting real and imaginary represented Majorana fermions, one

QdN/2e can deﬁne two anti-unitary particle-hole symmetry operators: P = K i=1 χ2i−1 and

QbN/2c R = K iχ2i, where K is the complex conjugate operator, N/2 (integer ceiling) i=1 d e denotes the smallest integer greater than N/2, therefore N/2 + N/2 = N holds for b c d e any integer N.

1Note that one can not choose odd i to be pure imaginary and even i to be real when N is odd, because the number of real matrices equal to the number of real matrices plus one. See Chapter 4 for more details. 20 With helps from operator algebra identities presented in Appendix A, it can be shown

−1 dN/2e −1 bN/2c that P χiP = ( 1) χi and RχiR = ( 1) χi, thus − − −

−1 dq/2e qdN/2e PQqP = ( 1) ( 1) Qq , (3.2a) − −

−1 bq/2c qbN/2c RQqR = ( 1) ( 1) Qq . (3.2b) − −

One can also ﬁnd their squared values

P 2 = ( 1)bdN/2e/2c,R2 = ( 1)dbN/2e/2c . (3.3) − −

Multiplying two equations in Eq.(3.2) leads to the following uniﬁed classiﬁcation:

When q(1 + N) is even, P and R either both commute with Qq or both anti-commute with

Qq, thus Λ = PR is a conserved quantity (fermion number parity) satisfying [Λ,Qq] = 0 if

N is even, or an identity operator if N is odd. Then one only need one of the two operators acting as T+ or T−. When q(1 + N) is odd, one of P , R commutes with Qq and the other anti-commutes with Qq, thus Λ = PR is a chirality operator satisfying Λ,Qq = 0. Then { } one need both operators, one acting as T+, the other acting as T−.

3.2 The Bott periodicity and reﬂection relation

From Eq.(3.2) and Eq.(3.3), it is easy to tell that the classification will depend on both parameters q and N. In fact, we have the following two Lemmas on the properties of the classification. The first one reveals a periodicity of the classification, while the second one indicates a reflection relation within a period. Lemma 1 (Bott periodicity)

The SYK model with (q, N) and (4n + q, 8m + N) are always in the same random matrix class, where n and m are any two positive integers. 21 Proof: Since the classiﬁcation of SYK models is completely determined by operator alge-

bra Eq.(3.2) and (3.3), one needs to check that (q, N) case and (4n + q, 8m + N) case have

the same operator algebra. By replacing q with 4n + q and N with 8m + N in Eq.(3.2) and

(3.3), the commutation relations become

−1 d(4n+q)/2e (4n+q)d(8m+N)/2e PQ4n+qP = ( 1) ( 1) Q4n+q − −

2n+dq/2e 4m(4n+q)+4ndN/2e+qdN/2e = ( 1) ( 1) Q4n+q − −

dq/2e qdN/2e = ( 1) ( 1) Q4n+q (3.4) − −

−1 b(4n+q)/2c (4n+q)b(8m+N)/2c RQ4n+qR = ( 1) ( 1) Q4n+q − −

2n+bq/2c 4m(4n+q)+4nbN/2c+qbN/2c = ( 1) ( 1) Q4n+q − −

bq/2c qbN/2c = ( 1) ( 1) Q4n+q (3.5) − − and the squared values are

P 2 = ( 1)bd(8m+N)/2e/2c = ( 1)b(4m+dN/2e)/2c − − = ( 1)2m+bdN/2e/2c = ( 1)bdN/2e/2c, (3.6) − − R2 = ( 1)db(8m+N)/2c/2e = ( 1)d(4m+bN/2c)/2e − − = ( 1)2m+dbN/2c/2e = ( 1)dbN/2c/2e (3.7) − −

where we have used relation X + x = X + x and X + x = X + x for any integer b c b c d e d e X. Thus we showed SYK models with (q, N) and (4n+q, 8m+N) have the same operator

algebra, therefore in the same random matrix class.

By repeating the calculation for (an + q, bm + N) with a 3 and b 7, one can ≤ ≤ ﬁnd that the operator algebra does not stay the same as (q, N). Thus “4” and “8” are two 22 smallest positive numbers, which keep the algebra invariant under (q, N) (4n+q, 8m+ → N), and one can conclude the Bott periodicity of SYK models is 4 in q and is 8 in N.

Lemma 2 (reﬂection relation)

The SYK model with (q, N) and (4n q, 8m N) are always in the same random matrix − − class, where n and m are two integers.

Proof: By replacing q with 4n q and N with 8m N in Eq.(3.2) and (3.3), the commu- − − tation relations become:

−1 d(4n−q)/2e (4n−q)d(8m−N)/2e PQ4n−qP = ( 1) ( 1) Q4n−q − −

d−q/2e −qd−N/2e = ( 1) ( 1) Q4n−q − −

bq/2c qbN/2c = ( 1) ( 1) Q4n−q (3.8) − −

−1 b(4n−q)/2c (4n−q)b(8m−N)/2c RQ4n−qR = ( 1) ( 1) Q4n−q − −

b−q/2c −qb−N/2c = ( 1) ( 1) Q4n−q − −

dq/2e qdN/2e = ( 1) ( 1) Q4n−q (3.9) − − and the squared values are

P 2 = ( 1)bd(8m−N)/2e/2c = ( 1)bd−N/2e/2c − − = ( 1)b−bN/2c/2c = ( 1)−dbN/2c/2e = ( 1)dbN/2c/2e, (3.10) − − − R2 = ( 1)db(8m−N)/2c/2e = ( 1)db−N/2c/2e − − = ( 1)d−dN/2e/2e = ( 1)−bdN/2e/2c = ( 1)bdN/2e/2c (3.11) − − −

23 In above, we have used the following identities for integer ceiling and integer ﬂoor,

X/2 + X/2 = X, (3.12) d e b c X/2 + X/2 = 0 , (3.13) d e b− c

where X is an integer such as N and q. Actually, the second identity Eq.(3.13) still holds

when X/2 is a real number.

These we showed (q, N) and (4n q, 8m N) have exactly the same operator algebra − − if exchanging the two operators, P and R. So one can conclude that (q, N) and (4n − q, 8m N) are always in the same random matrix class. −

3.3 Application to SYK models

The classiﬁcation can be systematically done by evaluating Eq.(3.2), Eq.(3.3), and their

relations with Λ. Since we have proved that the classiﬁcation has N (mod 8), q (mod 4)

double Bott periodicity, we only need to consider SYK models with q (mod 4) = 0, 1, 2, 3

and N (mod 8) = 0, 1, , 7. Below, we apply our classify scheme to SYK models. ···

3.3.1 SYK model with q (mod 4) = 0

F Since q(1 + N) is always even, [P,Qq] = [R,Qq] = 0, Λ = ( 1) is a conserved − parity when N is even or Λ = 1 when N is odd. Depending on the commutation relation between P and Λ, the value of P 2, one reach the following classiﬁcation:

When N (mod 8) = 2, 6, both P and R swap the parity, thus Qq is in Class A(GUE), the energy level degeneracy is 1 + 1 which means there are two degenerate energy levels with opposite parities. When N (mod 8) = 2, 6, either P preserves the parity [N 6 24 2 (mod 8) = 0, 4 ] or no conserved quantity (N is odd), thus P = +1 means Qq belongs to

2 Class AI(GOE) with no degeneracy, P = 1 means Qq belongs to Class AII(GSE) with − double degeneracy. The classiﬁcation and the energy level degeneracy are summarized in the 4th and 5th row of Table 3.1 , respectively.

All these results for both even and odd N have been achieved before in [17, 23] in the enlarged Hilbert space by adding an extra Majorana fermion at when N is odd. ∞ However, here we achieved the same results in the minimal Hilbert space in both even and odd N in a uniﬁed scheme. Our new method may be the most powerful and convenient way to achieve new results in the other three cases, as shown in the following.

3.3.2 SYK model with q (mod 4) = 2

Since q(1 + N) is still always even, two operators always anti-commute with Hamilto- nian P,Qq = R,Qq = 0, and Λ = PR is the conserved parity if N is even or Λ = 1 { } { } if N is odd. Depending on commutation relation between P and Λ, the value of P 2, one

ﬁnd the following classiﬁcation:

When N (mod 8) = 2, 6, both operators swap the parity, thus Qq is in Class A(GUE); when N (mod 8) = 2, 6, either P preserves parity [N (mod 8) = 0, 4 ] or no conserved 6 2 2 quantity (N is odd), thus P = +1 means Qq belongs to Class D(BdG) and P = 1 − means Qq belongs to Class C(BdG). There is no degeneracy in all these cases. The clas- siﬁcation and the energy level degeneracy is listed in the 6th and 7th row of Table 3.1, respectively.

25 The classiﬁcations and degeneracy with even N are consistent with those in [21]. How- ever, our results with odd N are new. In order to justify these results, we need an exact diagonalization study of SYK model with q (mod 4) = 2. For example, one can compare

ELS from q = 6 SYK model with N = 17, 19, 21, 23, and and random matrix prediction from Table 3.1.

3.3.3 SYK supercharge with q (mod 4) = 1

Now q(1 + N) is odd for even N and even for odd N, the former leads to the chiral operator Λ = ( 1)F , the latter leads to Λ = 1, so neither will lead to a conserved quantity. − In the following, we discuss even N and odd N, respectively.

Even N case: When N (mod 8) = 0, 4, P,Qq = [R,Qq] = 0. From the values of { } 2 2 P and R in Table 3.1, we ﬁnd: N (mod 8) = 0 is class BDI(chGOE), thus Qq has no

2 degeneracy, but a mirror symmetry, so H = Qq has 2-fold degeneracy; N (mod 8) = 4 is

2 class CII(chGSE), thus Qq has double degeneracy and also a mirror symmetry, so H = Qq has 4-fold degeneracy. When N (mod 8) = 2, 6, [P,Qq] = R,Qq = 0. From the values { } 2 2 of P and R in Table 3.1, we obtain: N (mod 8) = 2 is class CI(BdG), thus Qq has no

2 degeneracy, but a mirror symmetry, so H = Qq has 2-fold degeneracy; N (mod 8) = 6 is

2 class DIII(BdG), thus Qq has double degeneracy and also a mirror symmetry, so H = Qq has 4-fold degeneracy.

Odd N case: When N (mod 8) = 1, 5, [P,Qq] = [R,Qq] = 0. From the values of

2 2 P and R in Table 3.1, we obtain: N (mod 8) = 1 is Class AI(GOE), thus Qq has no

2 degeneracy, no mirror symmetry either, so H = Qq has no degeneracy. N (mod 8) = 5 is

26 2 Class AII(GSE), thus Qq has double degeneracy and no mirror symmetry, so H = Qq have

2-fold degeneracy. When N (mod 8) = 3, 7, P,Qq = R,Qq = 0. From the values { } { } 2 2 of P and R in Table 3.1, we obtain: N (mod 8) = 3 is Class C(BdG), thus Qq has no

2 degeneracy, but a mirror symmetry, so H = Qq has 2-fold degeneracy. N (mod 8) = 7 is

2 Class D(BdG), thus Qq has no degeneracy, but a mirror symmetry, so H = Qq has 2-fold degeneracy.

The classiﬁcation for q (mod 4) = 1 and the energy level degeneracy are summarized

2 in the 8th, 9th (for Qq) and 10th row (for H = Qq) of Table 3.1, respectively. The results with even N are consistent with those in [21]. However, the results with odd N are new. In order to justify these results, we need an exact diagonalization study of SYK supercharge with q (mod 4) = 1. For example, one can compare ELS from q = 5 SYK supercharge with N = 17, 19, 21, 23, and and random matrix prediction from Table 3.1.

3.3.4 SYK supercharge with q (mod 4) = 3

The situation is similar to q (mod 4) = 1 case. In fact, the reﬂection relation in the operator algebra Eq.(3.2),(3.3) hints the SYK model with (q, N) and (4n q, 8m N) − − are in the same class, where n and m are integers. The classiﬁcation and degeneracy of q

(mod 4) = 3 case can be obtained from q (mod 4) = 1 results by replacing N 8 N. → − In the following, we also re-derive these results by directly checking its operator algebra.

Since q(1+N) is still odd for even N and even for odd N. The former leads to the chiral operator Λ = ( 1)F , the latter leads to Λ = 1, thus neither lead to conserved quantity. In − the following, we discuss even N and odd N, respectively.

27 Even N case: When N (mod 8) = 0, 4, [P,Qq] = R,Qq = 0, From the values of { } 2 2 P and R in Table 3.1, we obtain: N (mod 8) = 0 is class BDI(chGOE), thus Qq has no

2 degeneracy, but a mirror symmetry, so H = Qq has 2-fold degeneracy; N (mod 8) = 4 is

2 class CII(chGSE), thus Qq has double degeneracy and also a mirror symmetry, so Qq has

4-fold degeneracy. When N (mod 8) = 2, 6, P,Qq = [R,Qq] = 0. From the values { } 2 2 of P and R in Table 3.1, we obtain: N (mod 8) = 2 is class DIII(BdG), thus Qq has

2 double degeneracy and also a mirror symmetry, so H = Qq has 4-fold degeneracy; N

2 (mod 8) = 6 is class CI(BdG), thus Qq has no degeneracy, but a mirror symmetry, so Qq has 2-fold degeneracy.

Odd N case: When N (mod 8) = 1, 5, P,Qq = R,Qq = 0. From the values { } { } 2 2 of P and R in Table 3.1, we obtain: N (mod 8) = 1 is Class D(BdG), thus Qq has no

2 degeneracy, but a mirror symmetry, so H = Qq has 2-fold degeneracy; N (mod 8) = 5

2 is Class C(BdG), thus Qq has no degeneracy, but a mirror symmetry, so Qq have 2-fold

2 degeneracy. When N (mod 8) = 3, 7, [P,Qq] = [R,Qq] = 0. From the values of P

2 and R in Table 3.1, we obtain: N (mod 8) = 3 is Class AII(GSE), thus Qq has double

2 degeneracy, but no mirror symmetry, so H = Qq has 2-fold degeneracy; N (mod 8) = 7

2 is Class AI(GOE), thus Qq has no degeneracy and no mirror symmetry either, so H = Qq have no degeneracy.

The classification for q (mod 4) = 3 and the energy level degeneracy are summarized in the 11th, 12th and 13th row of Table 3.1, respectively. Classifications and degeneracy with even N are consistent with [21]. However, odd N results are new, which need ver- ified from an exact diagonalization study of SYK supercharge with q (mod 4) = 3. For

28 example, one can compare ELS from q = 3 SYK supercharge for N = 17, 19, 21, 23, and random matrix prediction from Table 3.1.

3.4 The Periodic Table of SYK Models

Since the classiﬁcation has N (mod 8), q (mod 4) double Bott periodicity, we can summarize all results as an periodic table, see Table 3.1, which are arranged by number of site N, q-body interaction, and corresponding random matrix classes. In the periodic table, we obtain 3 classes for SYK and 8 classes for supersymmetric SYK, thus all ten classes except class AIII show in the table.

Table 3.1: The periodic table of SYK for even q > 2 and supersymmetric SYK for odd q > 1.

N (mod 8) 0 1 2 3 4 5 6 7 P 2 value + + + + − − − − R2 value + + + + − − − − q (mod 4) = 0 AI AI A AII AII AII A AI

H = Qq degeneracy 1 1 1+1 2 2 2 1+1 1 q (mod 4) = 2 DDACCCAD

H = Qq degeneracy 1 1 1 1 1 1 1 1 q (mod 4) = 1 BDI AI CI C CII AII DIII D

Qq degeneracy 1 1 1 1 2 2 2 1 2 H = Qq degeneracy 2 1 2 2 4 2 4 2 q (mod 4) = 3 BDI D DIII AII CII C CI AI

Qq degeneracy 1 1 2 2 2 1 1 1 2 H = Qq degeneracy 2 2 4 2 4 2 2 1

29 CHAPTER 4

NUMERICAL SIMULATIONS

4.1 Algorithm for Simulation of SYK Models

In physics, the irreducible matrix representation of the Clifford algebra is often known

as gamma matrices γ. The d-dimensional gamma matrices can be iteratively constructed

(2) from the d = 2 case, and the base d = 2 case often choose three Pauli matrices: γ1 = σx,

(2) (2) (d+2) γ2 = σy, γ3 = σz. Then d + 2 case can be build by taking tensor products: γk =

(d) (d+2) (d+2) σx γ , k = 1, , d + 1, γ = σy 1 d/2 , γ = σz 1 d/2 , where d is an ⊗ k ··· d+2 ⊗ 2 d+3 ⊗ 2 1 d/2 d/2 even number, and 2d/2 stands for a 2 by 2 identity matrix. Then it can be shown that

(d) (d) these gamma matrices satisfy Clifford algebra γ , γ = 2δij1 d/2 . When comparing { i j } 2 with the commutation relation χi, χj = δij, it is easy to tell N Majorana fermions can { } bN/2c 1 (2bN/2c) 1 be represented by 2 -dimensional matrices χk = √ γ with k = 1, 2, ,N. 2 k ··· After obtaining the matrix representation of Majorana fermions, one can build the

Hamiltonian matrix by summing up all possible q-body interaction for each random re-

alization of disorder. Then, one can test the block structure of the Hamiltonian matrix,

and obtain the eigenvalues by diagonalizing each block. In order to get enough samples,

one may need to repeat this build-and-diagonalize Hamiltonian procedure for millions of

times.

1 (2 N/2 ) q/2 Equivalently, one may still choose χk = γk b c , but rescale the coupling Ci1 iq to its one 2 -th. ···

30 After enough samplings, the ELS can be obtained via extracting consecutive energy

level ratio r and the smallest positive energy level λ1 from all the eigenvalues of a given

block, as explained in chapter 2. One can generate histograms for r and λ1 to compare with predicted surmises P (r) and P (λ1), and one can also calculate the averaged values,

2 2 r˜ and A = λ / λ1 to compare with predicted surmises results. h i h 1i h i The algorithm can be implemented in C++, Matlab, or Mathematica, and the efﬁciency makes it possible to run on laptop, when dealing with N 34. To ﬁnish millions of ≤ samplings in a few days, we restrict N 24 in the dissertation. ≤

4.2 Numerical Results

It is non-trivial to compare numerical results of SYK models and RMT predictions from Table 3.1 because the RMT is designed for dense matrix instead of sparse matrix. To verify the periodic table 3.1, we perform simulation of SYK Models with q = 3, 4, 5, 6 and N = 17, 18, , 24, and obtain at least 104 samples for each. Our numerical data in ··· agreement with all known results q = 4 at all N, as well as q = 3, 5, 6 at even N. Below, we discuss on the new results on SYK model with q = 3, 5, 6 at odd N = 17, 19, 21, 23.

In Figure 4.1, we show many-body energy level statistics of q = 6 SYK model for

N = 17, 19, 21, 23. The random matrix indices β and α are extracted from the probability

2 distribution function P (r) and P (λ1), or r˜ and moment ratio A respectively. Both meth- h i ods lead to (β, α) = (2, 0), (2, 2), (2, 2), (2, 0) for N = 17, 19, 21, 23, respectively. Thus, the SYK model with q = 6 and N = 17, 19, 21, 23 responds to class D(BdG), C(BdG),

2 The smooth curves for λ1, λ2, λ3 are obtained from numerical diagonalization of corresponding random 3 6 matrix ensembles with size 10 averaged over 10 samples. In addition to P (λ1), we ﬁnd P (λ2) and P (λ2) are also in agreement with RMT predictions. 31 C(BdG),D(BdG), respectively. We also checked the degeneracy is always 1. These results

match the classiﬁcation and degeneracy listed in Table 3.1.

In Figure 4.2, we show many-body energy level statistics of q = 5 SYK supercharge for

N = 17, 19, 21, 23. Since the index α is only deﬁned for the cases with mirror symmetry, if

a mirror symmetry is absent, we plot the spectral density averaged from 1 sample and 104

samples. It shows the mirror symmetry is absent for every single realization of Ci1,··· ,iq , but still emerges after making disorder averages. The random matrix indices β and α, if exist,

are extracted from probability distribution P (r) and P (λ1) or r˜ and A. Both methods h i lead to (β, α) = (1, ), (2, 2), (4, ), (2, 0) for N = 17, 19, 21, 23 respectively. Thus, the − − SYK supercharge with q = 5 and N = 17, 19, 21, 23 responds to class AI(GOE), C(BdG),

AII(GSE), D(BdG), respectively. Both the random matrix classes and the energy level

degeneracy match our classiﬁcation and degeneracy listed in Table 3.1.

In Figure 4.3, we show many-body energy level statistics of q = 3 SYK supercharge

for N = 17, 19, 21, 23. These data show (β, α) = (2, 0), (4, ), (2, 2), (1, ), for N = − − 17, 19, 21, 23 respectively. Both the indices (α, β) and the energy level degeneracy match

our classiﬁcation and degeneracy listed in Table 3.1.

In conclusion, we have numerically proved our periodic table 3.1 of SYK models.

Despite its sparse nature in randomness, SYK models with general q and N can still be

described precisely by the RMT.

32 1.0 Poisson 1.5 10 3 P (r) GOE × 1 P (λ) λ 0.8 GUE A=1.5889 λ2 GSE q=6 1.0 λ3 0.6 α = 0 N=17 r˜ = 0.5997 0.4 GUE β = 2 0.5 Class D 0.2 r λ 0.0 3 0.5 1.0 1.5 2.0 2.5 1 2 3 4 5 6 10 − 1.5 × 0.8 q=6 A=1.1725 0.6 r˜ = 0.6003 1.0 N=19 α = 2 0.4 GUE β = 2 0.5 Class C 0.2 λ 0.0 3 0.5 1.0 1.5 2.0 2.5 1 2 3 4 10 − × 0.8 2.5 2.0 A=1.1725 0.6 r˜ = 0.5989 α = 2 q=6 1.5 0.4 GUE N=21 1.0 β = 2 Class C 0.2 0.5 λ 0.0 3 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 10 − 8 × 0.8 A=1.5785 6 q=6 0.6 α = 0 r˜ = 0.5998 N=23 4 0.4 GUE Class D β = 2 0.2 2 λ 0.0 3 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 10 − (a) (b) ×

Figure 4.1: Many-body energy level statistics of q = 6 SYK model for N = 17, 19, 21, 23. (a) Distributions of consecutive level spacing ratio r in q = 6 SYK model are presented to show level statistics in the bulk. (b) Distributions of the smallest 3 energy level in q = 6 SYK model are presented to show level statistics in the hard edge.

33 1.0 P (r) Poisson ρ(λ) no mirror symmetry GOE 1.5 0.8 GUE GSE q=5 0.6 1.0 r˜ = 0.5298 N=17 0.4 GOE β = 1 0.5 Class AI 0.2 1 Sample 10 4 Samples λ 0.0 0.5 1.0 1.5 2.0 2.5 -0.4 -0.2 0 0.2 0.4 2 q=5 0.8 8 10 P (λ) A=1.1740 λ1 × λ2 N=19 α = 2 0.6 r˜ = 0.5987 6 λ3 GUE Class C 0.4 4 β = 2 0.2 2 λ 0.0 3 0.5 1.0 1.5 2.0 2.5 1.5 1 2 3 4 5 6 7 10 − × q=5 0.8 ρ(λ) no mirror symmetry N=21 0.6 1.0 r˜ = 0.6743 Class AII 0.4 GSE β = 4 0.5 0.2 1 Sample 10 4 Samples 0.0 λ 0.5 1.0 1.5 2.0 2.5 5 -0.5 0 0.5 P (λ) 0.8 3 4 10 A=1.6016 × λ1 q=5 0.6 r˜ = 0.5994 3 α = 0 λ2 N=23 λ3 0.4 GUE 2 Class D β = 2 0.2 1 λ 0.0 3 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 10 − (a) (b) ×

Figure 4.2: Energy level statistics of q (mod 4) = 1 SYK supercharge. The same as ﬁgure 4.1 but for q = 5 SYK supercharge. The spectral density from 1 sample is plotted in (b) to show the spectral mirror symmetry is absent for a single

realization of Ci1,··· ,iq .

34 1.0 2.0 10 3 P (r) Poisson × λ1 GOE P (λ) λ2 0.8 q=3 GUE 1.5 λ3 GSE A=1.5818 N=17 0.6 α = 0 r˜ = 0.5986 1.0 Class D 0.4 GUE β = 2 0.5 0.2 r λ 0.0 3 0.5 1.0 1.5 2.0 2.5 1 2 3 4 10 − q=3 0.5 × 0.8 ρ(λ) no mirror symmetry N=19 0.4 0.6 r˜ = 0.6746 Class AII 0.3 0.4 GSE 0.2 β = 4 1 Sample 0.2 0.1 10 4 Samples λ 0.0 0.5 1.0 1.5 2.0 2.5 -1.5 -1.0 -0.5 0 0.51.0 1.5 3 q=3 0.8 3 10 P (λ) A=1.1759 × λ1 α = 2 N=21 0.6 r˜ = 0.5992 λ2 2 λ3 0.4 GUE Class C β = 2 1 0.2 λ 0.0 3 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 10 − × 0.8 0.5 ρ(λ) no mirror symmetry 0.4 q=3 0.6 r˜ = 0.5306 0.3 N=23 0.4 GOE β = 1 0.2 Class AI 0.2 0.1 1 Sample 10 4 Samples λ 0.0 0.5 1.0 1.5 2.0 2.5 -1.5 -1.0-0.5 0 0.5 1.0 1.5 2.0 (a) (b)

Figure 4.3: The same as ﬁgure 4.2 but for q = 3 SYK supercharge. Notice the RMT classes appear in a reversed order rather than q (mod 4) = 1 case.

35 CHAPTER 5

CONCLUSIONS

It is instructive to compare with the periodic tables of topological insulators and superconductors ﬁrst put forward in [42, 43]. Despite using the same Cartan labels and the same sets of antiunitary operators, it is on topological equivalent classes of non-interacting electrons. It classiﬁes gapped bulk states with gapless edge modes. Its Bott periodicity is in the space dimension d (mod 8). Later, adding more symmetries such as translational symmetries, point group symmetries or non-isomorphic symmetries lead to richer periodic tables for topological crystalline insulators [55].

The periodic table 3.1 is on quantum chaos in 0 + 1 dimensional gapless quantum spin liquids of interacting fermions. The most compact, systematic and complete classi-

ﬁcation method developed in this work can be applied to all kinds of generalizations of

SYK models such as the colored-SYK model [25, 38], SYK models with a global U(1) or

O(M),U(M) symmetry [56, 30] and two indices SYK model [4]. It may also be applied to study quantum chaos in the colored (Gurau-Witten) tensor model [32, 33], uncolored

(Klebanov-Tarnopolsky) tensor model [35] or cavity QED [57]. This is just putting more symmetries, leading to more conserved quantities and more operators, therefore generating

36 more complicated periodic tables. Its advantages over the usual/extended schemes may be

even more evident and dramatic as the periodic tables get more complicated.

As mentioned in the introduction, there are two complementary ways to characterize

the quantum chaos or quantum information scramblings. One way is to evaluate the OTOC

function at a ﬁnite temperature 1 βJ N by the 1/N expansion to extract the Lya- punov exponent at an early time β < t < β log N, namely, between the dissipation time td β = 1/T and the Ehrenfest (scambling) time ts td log N . It is insensitive to ∼ ∼ N (mod 8) Bott periodicity and also the ground state degeneracy. Another way is to use the RMT to characterize the ELS or spectral form factor in a ten-fold way at a ﬁnite and large enough N, which shows N (mod 8) Bott periodicity and also the ground state degeneracy. The RMT describes the energy level correlations at the Heisenberg time scale

N tH 1/∆ e /NJ, where ∆ is the mean many-body energy level spacing, but breaks ∼ ∼ down when the energy level spacing is beyond the Thouless energy E N 2∆. So it Th ∼ 2 fails when t < t 1/E tH /N . Although the 1/N expansion has been pushed from Th ∼ Th ∼

the scrambling time ts to the longer time scale N/J in [58], it remains unknown how to

push it to the Thouless time tTh. Because the wide separation between the two time scales

N/J and tTh, it remains challenging to explore the double periodicity and the reﬂection

symmetry in the table 3.1 by 1/N, 1/q or N/q2 expansions. The possible impacts on the

classiﬁcations of quantum black holes in the bulk are being studied [59, 60].

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42 APPENDIX A

SOME USEFUL OPERATOR ALGEBRA IDENTITIES

43 In the Appendix, we work with a gamma matrix representation of Majorana fermion

χ, in which χ is either real (symmetric matrix) or pure imaginary (antisymmetric matrix).

For a sequence of distinct χ with length n, i.e. Sn = χS , χS , , χS , we deﬁne { 1 2 ··· n }

#(Im χS ) ΓS = i χS χS χS , where #(Im χS) denotes the number of pure imaginary χ n 1 2 ··· n in the sequence Sn.

One can easily verify that [ΓSn ,K] = 0 and the squared value

2 bn/2c #(Im χ) (KΓS ) = ( 1) ( 1) , (A.1) n − − then arrive at an useful relation   n ( 1) KχkK, χk / Sn = χ1, χ2, , χn −1  − ∈ { ··· } (KΓSn )χk(KΓSn ) = (A.2)  n−1 ( 1) KχkK, χk Sn = χ1, χ2, , χn − ∈ { ··· }

The result of Eq.(A.2) can be easily memorized as: “If χk is a member of Sn, then

−1 n (KΓS )χk(KΓS ) = ( 1) KχkK, otherwise need an extra minus sign”. n n − #(Im χ) In addition, if we commute two operators ΓS = i χS χS χS and ΓS0 = n 1 2 ··· n m #(Im χ) i χS0 χS0 χS0 , then following relation holds 1 2 ··· m

mn #(χ in common) ΓS ΓS0 = ( 1) ( 1) ΓS0 ΓS (A.3) n m − − m n where #(χ in common) denotes the number of common χ shared by two sequences. Sim- ilarly, one can derive

mn #(χ in common) (KΓS )ΓS0 = ( 1) ( 1) ΓS0 (KΓS ) , (A.4) n m − − m n

In the dissertation, antiunitary symmetry operators such as P and R are just examples of

KΓSn ; conserved quantity such as Λ is an example of ΓSn . Hence Eq.(A.3) and Eq.(A.4) become extremely useful when evaluating (anti)commutators. 44