熱場の量子論とその応用
TQFT 2019
YITP, Kyoto University

多体量子系の時間相関からの量子カオスの特徴づけ

2 September 2019

Masaki TEZUKA （手塚真樹）(Kyoto University)Collaborators (in SYK-related papers) and references• Jordan Saul Cotlera, Guy Gur-Aria (Google), Masanori Hanada (YITPSouthampton)• Joseph Polchinskib, Phil Saada, Stephen H. Shenkera, Douglas Stanforda, Alexandre Streicherb• Ippei Danshita (YITPKindai), Hidehiko Shimada (OIST), Hrant Gharibyana, Brian Swingle (Maryland)• Antonio M. García-García (SJTU), Bruno Loureiro (Cambridge), Aurelio Romero-Bermúdez (Leiden)• Pak Hang Chris Lau (MITNTHU), Chen-Te Ma (SCNU & Cape Town), Jeff Murugan (Cape Town & KITP)
aStanford bUCSBDanshita, Hanada, and MT, PTEP 2017, 083I01 (arXiv:1606.02454)
Cotler, Gur-Ari, Hanada, Polchinski, Saad, Shenker, Stanford, Streicher, and MT, JHEP 1705, 118 (2017)(arXiv:1611.04650)
Hanada, Shimada, and MT, Phys. Rev. E 97, 022224 (2018) (arXiv:1702.06935)

García-García, Loureiro, Romero-Bermudez, and MT, PRL 120, 241603 (2018)(arXiv:1707.02197)
García-García and MT, Phys. Rev. B 99, 054202 (2019) (arXiv:1801.03204)
Gharibyan, Hanada, Shenker, and MT, JHEP 1807, 124 (2018) (arXiv:1803.08050)

Gharibyan, Hanada, Swingle, and MT, JHEP 1904, 082 (2019) (arXiv:1809.01671),
submitted (arXiv:1902.11086)
Lau, Ma, Murugan, and MT, Phys. Lett. B 795, 230 (10 August 2019) (arXiv:1812.04770)

How to characterize quantum chaos?
𝑑𝑖
𝜓 =𝐻𝜓
𝑑𝑡

𝐻 = const.

𝑡𝜓 𝑡=T exp −𝑖

𝐻 𝑡′ 𝑑𝑡 𝜓 𝑡 = 0= exp −𝑖 𝐻𝑡 𝜓 𝑡 = 00

Linear dynamicsUnitary time evolution• Out-of-time correlator• Energy level statistics
Correlation between levels, as in random matrices

Short range: Normalized level separation
distribution, gap ratio, …
Longer range: Number variance, spectral
form factor, …
cf. Bohigas-Giannoni-Schmit conjecture

Classically,
𝑥𝑖 𝑡 , 𝑝𝑗 02PB

=

𝜕𝑥𝑖 𝑡
𝜕𝑥𝑗 02→ 𝑒 2𝜆L 𝑡 for large t

Quantum version:

OTOC: 𝐶𝑇 𝑡 =𝑊 𝑡 ,𝑉 𝑡 = 02= 𝑊† 𝑡 𝑉† 0 𝑊 𝑡 𝑉 0 + ⋯Numerically Actual energy eigenvalues needed Hard to see exponential time dependence

Our proposal (1902.11086): Singular value statistics of two-point correlators
𝜙𝐺𝑎𝑏 𝑡 = 𝜙 𝜒𝑎 𝑡 𝜒𝑏 0 𝜙

The Sachdev-Ye-Kitaev model
𝐻=cf. Sachdev-Ye model (1993)

3!
𝑁 3/2

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁[A. Kitaev, talks at KITP (2015)]

𝜒𝑎=1,2,…,𝑁 : 𝑁 Majorana fermions ( 𝜒𝑎 , 𝜒𝑏 = 𝛿𝑎𝑏 )𝐽𝑎𝑏𝑐𝑑 : independent Gaussian random couplings ( 𝐽𝑎𝑏𝑐𝑑 2 = 𝐽2 = 1)
𝐽3567

𝐽1259

𝐽4567

𝐽1348

⋯Two versions of the SYK modeland large-N solvabilityN Majorana- or Dirac- fermions randomly coupled to each other[Dirac version][Majorana version]
𝐻=3!
𝑁 3/2

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1
𝐻=2𝑁 3/2

1≤𝑎<𝑏<𝑐<𝑑≤𝑁[A. Kitaev: talks at KITP

𝐽𝑖𝑗;𝑘𝑙 𝑐𝑖 † 𝑐𝑗 † 𝑐𝑘 𝑐𝑙
𝑖𝑗;𝑘𝑙[Kitaev’s talk][S. Sachdev: PRX 5, 041025 (2015)](Feb 12, Apr 7 and May 27, 2015)]“Two-body random ensemble” since 1970s

Both solvable in the large-N limit“Melon diagrams” dominate;
Reparametrization symmetry emerges
See [I. Danshita, M. Tezuka, and M. Hanada:
Butsuri 73(8), 569 (2018)] including our proposal
for experimental realization [PTEP 2017]

𝑂 𝑁0𝑂 𝑁 −2 for 𝑖 ≠ 𝑚

Definition of Lyapunov exponent
using out-of-time-order correlators
𝑊 𝑡 = e𝑖𝐻𝑡 𝑊e−𝑖𝐻𝑡

Classical:
Infinitesimally different initial states𝛿𝑥 𝑡 ~𝑒 𝜆L 𝑡 𝛿𝑥 𝑡 = 0𝜆L : Lyapunov exponent

t=0Real time t

𝜕𝑥 𝑡𝑥 𝑡 , 𝑝 0 PB =𝜕𝑥 022→ 𝑒 2𝜆L 𝑡Consider operators 𝑉 and 𝑊,𝐶 𝑡 = 𝑊 𝑡 ,𝑉 𝑡 = 0 2= 2 1 − Re 𝐹 𝑡quantifies strength of quantum
scrambling“Black holes are fastest quantum
scramblers”[P. Hayden and J. Preskill 2007] [Y. Sekino andL. Susskind 2008] [Shenker and Stanford 2014]

Chaos bound 𝜆L = 2π𝑘B 𝑇 ℏ[J. Maldacena, S. H. Shenker, and D. Stanford, JHEP08(2016)106]

Out-of-time-ordered correlators(OTOCs)Regularized OTOC can becalculated for large-N SYK model,
satisfies the chaos bound𝜆L = 2π𝑘B 𝑇 ℏ at low T limit

𝜒𝑖 𝑡1 𝜒𝑖 𝑡2 𝜒𝑗 𝑡3 𝜒𝑗 𝑡4

Γ 𝑡1 , 𝑡2 , 𝑡3 , 𝑡4 = Γ0 𝑡1 , 𝑡2 , 𝑡3 , 𝑡4 +SYKq: q-fermion interactions

𝑑𝑡𝑎 𝑑𝑡𝑏 Γ 𝑡1 , 𝑡2 , 𝑡𝑎 , 𝑡𝑏 𝐾 𝑡𝑎 , 𝑡𝑏 , 𝑡3 , 𝑡4

[Kitaev’s talks][J. Polchinski and V. Rosenhaus, JHEP 1604 (2016) 001]
[J. Maldacena and D. Stanford, Phys. Rev. D 94, 106002 (2016)]

Holographic connection to gravity[S. Sachdev,Phys. Rev. X 5,041025 (2015)]

NMR experiment for the SYK model“Quantum simulation of the non-fermi-liquidstate of Sachdev-Ye-Kitaev model” Zhihuang Luo,
Yi-Zhuang You, Jun Li, Chao-Ming Jian, Dawei Lu,
Cenke Xu, Bei Zeng and Raymond Laflamme,
npj Quantum Information 5, 53 (2019)1707.02197

SYK4 + SYK2: Large-N calculation for OTOC
𝑁

𝐻=

SYK4𝑁

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑 + 𝑖
1≤𝑎<𝑏<𝑐<𝑑normalized
Lyapunov exponent

𝐾𝑎𝑏 : standard deviation

𝐾𝑎𝑏 𝜒𝑎 𝜒𝑏𝐾𝑁

1≤𝑎<𝑏

SYK4 limit →SYKq + K SYK2
Large-q limit(𝐾 = 0)𝐾

Chaos bound[Maldacena,Shenker, andStanford 2016]

SYK2

non-chaoticchaotic

𝐾 = 0.20.512

𝛽 = 1 𝑘B 𝑇 inverse temperature
A. M. Garcia-Garcia, B. Loureiro, A. Romero-Bermudez, and MT, PRL 120, 241603 (2018)

Deviation from the chaos bound as SYK2 component is introduced

Contents
The Sachdev-Ye-Kitaev modelLarge-N solvability: conformal symmetry and maximal chaos
Deformation and suppression of maximal chaos 1707.02197• Characterization of chaos in random systems 1702.06935• Quantum Lyapunov spectrum 1809.01671• Singular value statistics of two-point correlators 1902.11086

Other related works: 1801.03204, 1812.04770

The Bohigas-Giannoni-Schmit conjecture
Assume quantum mechanical systems with a classical limit

Justifications:circular:integrable
Sinai billiard:
chaotic“Spectral statistics of chaotic
systems can be described as arandom matrix”
O. Bohigas, M. J. Giannoni, and C. Schmit,
Phys. Rev. Lett. 52, 1 (1984);
J. de Phys. Lett. 45, 1015 (1984).

Non-linear sigma-model(Andreev 1993, Altland 2015)
Gutzwiller trace formula interms of periodic orbits(Berry 1985, Gutzwiller 1990,Sieber, Richter, Braun, Muller,Heusler, …)

Also more examples
including systems without
clear classical versionGaussian random matrices
𝛽𝐾− 4 Tr𝐻 2𝛽𝐾42
𝐾𝑎𝑖,𝑗 𝑖𝑗

𝑎𝑖𝑗 = 𝑎𝑗𝑖∗Density ∝ 𝑒Gaussian distribution

Real (β=1): Gaussian Orthogonal Ensemble (GOE)
Complex (β=2): G. Unitary E. (GUE)
Quaternion (β=4): G. Symplectic E. (GSE)𝐾

𝑎𝑖𝑗 𝑖,𝑗=1= exp −

Level repulsion

Joint distribution function
for eigenvalues 𝑒𝑗
𝑝 𝑒1 , 𝑒2 , … , 𝑒𝐾 ∝𝐾𝑒𝑖 − 𝑒𝑗
1≤𝑖<𝑗≤𝐾𝛽𝑒 −𝛽𝐾𝑒𝑖

24𝑖=1• 𝑃 𝑠 : Distribution of normalized level separation
𝑒𝑗+1 −𝑒𝑗𝑠𝑗 =∆ 𝑒2

GOE/GUE/GSE: 𝑃 𝑠 ∝ 𝑠 𝛽 at small 𝑠, has 𝑒 −𝑠 tailUncorrelated (Poisson): 𝑃 𝑠 = 𝑒 −𝑠•𝑟 : Average of neighboring gap ratio𝑟=

min 𝑒𝑖+1 − 𝑒𝑖 , 𝑒𝑖+2 − 𝑒𝑖+1max 𝑒𝑖+1 − 𝑒𝑖 , 𝑒𝑖+2 − 𝑒𝑖+1

Uncorrelated𝑟

GOEGUEGSE

2log 2 – 1 = 0.38629… 0.5307(1) 0.5996(1) 0.6744(1)[Y. Y. Atas et al. PRL 2013]

N mod 8 classification of Majorana SYKq=4
𝐻=3!
𝑁 3/2

SPT phase classification for class BDI:ℤ  ℤ8 due to interaction

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁[L. Fidkowski and A. Kitaev, PRB 2010, PRB 2011]

Introduce 𝑁/2 complex fermions 𝑐𝑗 =𝑁 ≡ 0, 4𝑋

𝜒2𝑗−1 +i𝜒2𝑗2

𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑 respects the complex fermion parity
Even (𝐻E ) and odd (𝐻O ) sectors: 𝐿 = 2𝑁/2−1 dimensions(mod 8)𝐻E
𝑋
𝑁 ≡ 2, 6

𝐻O

𝑁/2𝑐𝑗† + 𝑐𝑗

𝑋=𝐾𝑁 mod 80246

𝜂-1+1+1-1

𝑋2+1+1-1-1

𝑋 maps 𝐻E to𝐻E

𝐻O

𝐻E

𝐻O

[You, Ludwig, and Xu, PRB 2017]ClassAI

A+AAII

A+AGaussianensemble

GOEGUEGSE GUE[Fadi Sun and Jinwu Ye, 1905.07694]
for SYKq, supersymmetric SYK𝑗=1𝑋𝑐𝑗 𝑋 = 𝜂𝑐𝑗† ;𝑋, 𝐻 = 0

SYK: sparse matrix, but energy spectral statistics strongly resemble[Cotler, …, MT, JHEP 2017]that of the corresponding (dense) Gaussian ensemble

The spectral form factor

𝑔 𝛽, 𝑡 =𝑍 𝛽, 𝑡2𝑍 𝛽2𝐽𝐽1611.04650

𝑍 𝛽, 𝑡 = 𝑍 𝛽 + i𝑡 = Tr e−𝛽𝐻−i𝐻𝑡plateau 𝑔 𝑡 = 𝑁𝐸 𝑍 2𝛽 /𝑍 𝛽β=1

dip

Exponentiallylong ramp 𝑔 𝑡 ~𝑡 12GOE

1GUE2GSE

2GUE2GOE

1GUE

2GSE

2GUE

2GOE

1GUE

2

Lyapunov growth of phase space

Coarse-grained•Just one direction?
•If more than one, what arerelations between λ?

1809.01671

Quantum Lyapunov spectrum
Finite-time classical Lyapunov spectrum: obeys RMT statistics for chaos[Hanada, Shimada, and MT: PRE 97, 022224 (2018)]
Singular values of 𝑀𝑖𝑗 =𝐿 = 𝑥𝑖 𝑡 , 𝑝𝑗 0

𝜕𝑥𝑖 𝑡
𝜕𝑥𝑗 02

PBOTOC: 𝐶𝑇 𝑡 ==at finite t:= 𝑒 𝜆𝑘𝑡

𝑠𝑘 𝑡

𝛿𝑥 𝑡
= 𝑀𝛿 𝑥 0

𝛿𝑥 0

𝜕𝑥𝑖 𝑡
𝜕𝑥𝑗 02→ 𝑒 2𝜆L 𝑡 for large t

𝑊 𝑡 ,𝑉 𝑡 = 0

20𝑡= 𝑊† 𝑡 𝑉† 0 𝑊 𝑡 𝑉 0 + ⋯Quantum Lyapunov spectrum: Define 𝑀𝑎𝑏 𝑡 as (anti)commutator of 𝑂𝑎 𝑡 and 𝑂𝑏 0𝑁𝐿𝑎𝑏 𝑡 = 𝑀 𝑡 † 𝑀 𝑡

𝑎𝑏

𝑀𝑗𝑎 𝑡 † 𝑀𝑗𝑏 𝑡=𝑗=1

For 𝑁 × 𝑁 matrix 𝜙 𝐿𝑎𝑏 𝑡 𝜙 , obtain singular values 𝑠𝑘 𝑡

The Lyapunov spectrum is defined as 𝜆𝑘 𝑡log 𝑠𝑘 𝑡
=2𝑡.

𝑁𝑘=1 .

Quantum Lyapunov spectrum for SYK model + modification
𝑁𝐻=6𝑁𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑 + 𝑖
1≤𝑎<𝑏<𝑐<𝑑𝐽𝑎𝑏𝑐𝑑 : s. d. = 𝑁3 2𝐾𝑎𝑏 𝜒𝑎 𝜒𝑏

𝐾𝑎𝑏 : s. d. =1≤𝑎<𝑏• Define 𝐿𝑎𝑏 𝑡 = 𝑁
𝑗=1 𝑀𝑗𝑎 𝑡 𝑀𝑗𝑏 𝑡 for time-dependent
anticommutator 𝑀𝑎𝑏 𝑡 = 𝜒𝑎 𝑡 , 𝜒𝑏 0 .
• Obtain the singular values 𝑎𝑘 𝑡

𝐾
𝑘=1 of• Quantum Lyapunov spectrum: 𝜆𝑘 𝑡 =(also dependent on state 𝜙)

𝜙 𝐿𝑎𝑏 𝑡 𝜙log 𝑎𝑘 𝑡
2𝑡

𝑘=1,2,…,𝐾

Other possibilities: see Rozenbaum, Ganeshan, and Galitski: PRB 100, 035112 (2019),
Hallam, Morley, and Green: 1806.05204

𝐾𝑁

Spectral statistics of quantum Lyapunov spectrum: SYK 1809.01671𝐾 = 0.01 (●): Remains GUE
for long time𝐾 = 10 (◆):
Approaches Poisson

Energy eigenstates
N/2 larger exponents𝑟 : average ofmin 𝑠𝑖 ,𝑠𝑖+1max 𝑠𝑖 ,𝑠𝑖+1(fixed-i unfolding: unfold eachgap 𝑔𝑖 = 𝜆𝑖+1 − 𝜆𝑖 using its
average 𝑔𝑖 𝐽 , 𝑠𝑖 = 𝑔𝑖 𝑔𝑖 𝐽 )QLS: The case of the random field XXZ model𝑁

𝐻=𝑁ℎ𝑖 𝑆𝑖𝑧

𝑺𝑖 ∙ 𝑺𝑖+1 +𝑖ℎ𝑖 : uniform distribution [−𝑊, 𝑊]𝑖

Many-body localization (MBL) transition at 𝑊 = 𝑊c ~ 3.5(though recently disputed; e.g. 𝑊c ≥ 5 proposed in E. V. H. Doggen et al., [1807.05051] using large
systems with time-dependent variational principle & machine learning)e.g. M. Serbyn, Z. Papic, and D. A. Abanin,
Phys. Rev. X 5, 041047 (2015) (arXiv:1507.01635)

MBL

Matrix element of local perturbation

chaotic
Energy separation ofneighboring energy eigenstates

𝜖c = 0.45

cf. MBL in short-range SYK [García-García and MT, Phys. Rev. B 99, 054202 (2019)]; Localization offermions on quasiperiodic lattice with attractive on-site interaction [Phys. Rev. A 82, 043613 (2010)]

1809.01671

Spectral statistics of QLS for random field XXZ𝑁

𝐻=𝑁ℎ𝑖 𝑆𝑖𝑧

𝑺𝑖 ∙ 𝑺𝑖+1 +𝑖ℎ𝑖 : uniform distribution [−𝑊, 𝑊]

𝑀𝑎𝑏 𝑡 = 𝑆𝑎+ 𝑡 , 𝑆𝑏− 0𝑖

W = 0.5(delocalized)

W = 4.0(Many-bodylocalized)

 Exponential
growth of thesingular values is
not observed,
but the statistics
approach GUE

Quantum Lyapunov spectrum distinguishes chaotic and non-chaotic phases

1902.11086

Singular value statistics of
two-point time correlators
𝜙𝐺𝑎𝑏 𝑡 = 𝜙 𝜒𝑎 𝑡 𝜒𝑏 0 𝜙 as a matrix
𝜙𝜆𝑗 𝑡 = log singular values of 𝐺𝑎𝑏 𝑡

SYK4
GUE at all times

Stronglyperturbed

GOE at late time

1902.11086
min 𝜆𝑖+1 −𝜆𝑖 , 𝜆𝑖+2 −𝜆𝑖+1
𝑟 : average of the adjacent gap ratio max
𝜆𝑖+1 −𝜆𝑖 , 𝜆𝑖+2 −𝜆𝑖+1

Uncorrelated (Poisson): 2 log 2 − 1 ≈ 0.386
Correlated: larger (GOE: 0.5307, GUE: 0.5996 etc. ) [Atas et al., PRL 2013]

SYK, larger N/2 exponents
𝜙: energy eigenstates
fixed-i unfolded

N mod 8 = 2, 4, 6: GUE
N mod 8 = 0: GOE(the matrix is symmetric)
𝜙𝐺𝑎𝑏 = 𝜙 𝜒𝑎 𝑡 𝜒𝑏 0 𝜙

At late time, for two-point correlator singular values,

Random matrix behavior  chaotic

Two-point time correlator: XXZ model
𝑁𝐻=𝑁ℎ𝑖 𝑆𝑖𝑧

𝑺𝑖 ∙ 𝑺𝑖+1 +𝑖𝑖ℎ𝑖 : uniform distribution [−𝑊, 𝑊]

Random matrix behavior  chaotic
for both early time and late timeSummary
The Sachdev-Ye-Kitaev modelLarge-N solvability: conformal symmetry and maximal chaos
Deformation and suppression of maximal chaos 1707.02197
𝐻 =𝑁
1≤𝑎<𝑏<𝑐<𝑑 𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑 + 𝑖𝑁1≤𝑎<𝑏 𝐾𝑎𝑏 𝜒𝑎 𝜒𝑏

Characterization of chaos in random systems 1702.06935
Quantum Lyapunov spectrum 1809.01671
 𝜙𝑁†
𝑗=1 𝑀𝑗𝑎 𝑡 𝑀𝑗𝑏 𝑡

𝜙 , 𝑀𝑎𝑏 𝑡 = 𝜒𝑎 𝑡 , 𝜒𝑏 0

Singular value statistics of two-point correlators 1902.11086
𝜙 𝐺𝑎𝑏 𝑡 = 𝜙 𝜒𝑎 𝑡 𝜒𝑏 0 𝜙
Other related works: 1801.03204, 1812.04770