引力全息与量子物质研讨会
Workshop on Holography and Quantum Matter
Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing

The Sachdev-Ye-Kitaev model as a
maximally chaotic lattice model:
study towards experimental realizationand new characterizations of chaos

26 August 2019

Masaki TEZUKA （手塚真樹）(Kyoto University)

Contents
𝐻SYK =

black
hole

𝑁 3/2

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁non-gravitational
quantum system

holography

maximally chaotic(chaos bound)3!

maximally chaotic(chaos bound)

𝜆Lyapunov
random matrix behavior of
finite-time Lyapunov spectrum[Classical] Hanada, Shimada and MT,
PRE 2018 [1702.02197]

modifications to study
chaos / integrable
transition

cold atom
realization
Danshita, Hanada and MT,
PTEP 2017 [1606.02454]

numerical analysis;
relation to otherscrambling systems
Cotler et al., JHEP 2017 [1611.04650]
Gharibyan et al., JHEP 2018 [1803.08050]& many-body localization[Quantum] Gharibyan, Hanada,
Swingle and MT, JHEP 2019 [1809.01671]

García-García et al., PRL 2018 [1707.02197]

random matrix behavior of
two-point correlators

García-García and MT, PRB 2019 [1801.03204]

Lau, Ma, Murugan, and MT, Phys. Lett. B 2019 [1812.04770]

Gharibyan, Hanada, Swingle and MT, 1902.11086

Collaborators (in SYK-related papers) and references• Jordan Saul Cotlera, Guy Gur-Aria (Google), Masanori Hanada (YITPBoulder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)

The Sachdev-Ye-Kitaev model𝐻=

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 model
N Majorana- or Dirac- fermions randomly coupled to each other[Dirac version][Majorana version]
𝐻=

3!
𝑁 3/2

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁[A. Kitaev: talks at KITP
(Feb 12, Apr 7 and May 27, 2015)]

1
𝐻=
2𝑁 3/2

𝐽𝑖𝑗;𝑘𝑙 𝑐𝑖 † 𝑐𝑗 † 𝑐𝑘 𝑐𝑙
𝑖𝑗;𝑘𝑙[A. Kitaev’s talk][S. Sachdev: PRX 5, 041025 (2015)]
Studied as “Two-body random
ensemble” since 1970s(The first paper by A. Kitaev on the SYK model:
Alexei Kitaev and S. Josephine Suh, arXiv:1711.08467 (JHEP05(2018)183);
First papers by J. Ye on the SYK model: arXiv:1809.06667 and arXiv:1809.07577)

Why solvable in the 𝑁 ≫ 1 limit?
(after sample average ⋯Free two-point function 𝐺0 𝑡 𝛿𝑖𝑗 = − T𝜓𝑖 𝑡 𝜓𝑗 0

𝐽 )

1
2

= − sgn 𝑡 𝛿𝑖𝑗

Perturbation expansion by interaction term

𝐻=

3!
𝑁 3/2

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁

𝐽𝑎𝑏𝑐𝑑 2 𝐽 = 𝐽2 , independent Gaussian distribution
𝐽𝑎𝑏𝑐𝑑 𝐽𝑎𝑏𝑐𝑒

𝐽= 0 if 𝑑 ≠ 𝑒  Most diagrams average to zero

“Melon-type” diagrams dominate in large N

“Melon” diagrams dominate in the 𝑁 ≫ 1 limit
Dotted line connects same couplings

Figure from [I. Danshita, M. Tezuka, and M. Hanada: Butsuri 73(8), 569 (2018)]

Feynman diagrams[J. Polchinski and V. Rosenhaus, JHEP 1604 (2016) 001]
[J. Maldacena and D. Stanford, PRD 94, 106002 (2016)]

𝐻=

3!
𝑁 3/2

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁

Sample average ⋯𝐽𝑎𝑏𝑐𝑑 2 = 𝐽2 = 1

𝐽

𝐽𝑖𝑗𝑘𝑙

𝐽𝑗𝑘𝑙𝑚
𝐽𝑖𝑗𝑘𝑙 𝐽𝑗𝑘𝑙𝑚

𝑗𝑘𝑙

𝜒𝑎 , 𝜒𝑏 = 𝛿𝑎𝑏

𝑁3
=
𝛿𝑖𝑚
𝐽
3!

𝑂 𝑁 0 contribution

𝐽𝑖𝑗𝑘𝑙

𝐽𝑖𝑗𝑘𝑙 𝐽𝑗𝑘𝑙𝑚 𝐽𝑚𝑗′𝑘′𝑙′ 𝐽𝑗 ′𝑘 ′𝑙′ 𝑖′𝑚≠𝑖 𝑗𝑘𝑙𝑗 ′ 𝑘 ′ 𝑙′𝐽𝑗𝑘𝑙𝑖′

𝐽𝑗𝑘𝑙𝑚 𝐽𝑚𝑗′𝑘′𝑙′
𝐽∝ 𝑁 4 𝛿𝑖𝑖′

𝑂 𝑁 −2 contributionReparametrization symmetry𝐺 1 − Σ𝐺0 = 𝐺0
𝐺 −1 = 𝐺0 −1 − Σ

𝐺 𝑖𝜔

−1

Σ = 𝐽2 𝐺 3= 𝑖𝜔 − Σ 𝑖𝜔

Low energy (𝜔, 𝑇 ≪ 𝐽): ignore 𝑖𝜔 and we have𝑑𝑡𝐺 𝑡1 , 𝑡 Σ 𝑡, 𝑡2 = −𝛿 𝑡1 , 𝑡2

Invariant under imaginary time reparametrization
τ
β

𝑓 𝜎
-β

σ

Large-N saddle point solution
𝑑𝑡𝐺 𝑡1 , 𝑡 Σ 𝑡, 𝑡2 = −𝛿 𝑡1 , 𝑡2

(Derived in replica formalism; assume replica symmetry)

Not invariant under arbitrary reparametrization,
but invariant under

Symmetry broken to SL(2, R).
cf. isometry group of AdS2

[see e.g. A. Strominger, hep-th/9809027][J. Maldacena and D. Stanford, Phys. Rev. D 94, 106002 (2016)]
Study of the Goldstone modes: e.g. [D. Bagrets, A. Altland, and
A. Kamenev, Nucl. Phys. B 911, 191 (2016)]

SYK: Nearly CFT1 “NCFT1”

emergent conformal gauge invariance[Sachdev, PRX 5, 041025 (2015)]

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 = 0
𝜆L : Lyapunov exponent

t=0

Real time t

𝜕𝑥 𝑡
𝑥 𝑡 , 𝑝 0 PB =
𝜕𝑥 0
2

2

→ 𝑒 2𝜆L 𝑡

Consider 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 and
L. 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 +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)]

Contents• The Sachdev-Ye-Kitaev modelLarge-N solvability: conformal symmetry and maximal chaos• Experimental proposal 1606.02454 (and realization)• 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

Towards quantum gravity
experiments using holography(d+1)-dimensional spacetime

d-dimensional spacetime

e.g.
AdS5 / CFT4

Supergravity in (4+1)danti-de Sitter space(×S5)

Supersymmetric Yang-Mills
Theory (without gravity)(Maldacena 1997) in (3+1)-dim

Theory withgravity(classical limit)

Theory without gravity
Strongly correlated
system◎ Not limited to classical limit
Several supporting evidences
e.g. check of the leading gravity
correction for the black hole mass
[M. Hanada, Y. Hyakutake, G. ishiki, andJ. Nishimura, Science 344, 882 (2013)]

Many “AdS/CMT”
applications

This work:
Theory with gravity(significant
quantum effects)

Theory without
gravityStrongly correlated,
experimentpossible

approach quantum gravity
by realizing corresponding
non-gravity models in coldgases

Our proposal: coupled atom-molecule model [arXiv:1606.02454]

Detuning ν: controlled by laser energy

Ωi,j: photoassociation (PA) laser
wm: molecular site wavefunctions
wa,i(j): atomic site wavefunctionss = 1, 2, …, ns

Sums of two single atom energies

Consider atomic levels i,j,… = 1, 2, …, N
coupled to a molecule state m1

Consider multiple molecular states; assume they are short-lived
 integrate them out and obtain the effective model for atoms

Realizing Dirac SYK models = 1, 2, …, ns(Gaussian distribution for g)(For simplicity we take 𝜈𝑠 = −1 𝑠 𝑛s 𝜎𝑠 )approaches the real-coupling version of
the Dirac SYK model as ns  ∞.

Gaussian J reproduced

Real SYK:
Physical quantities coincidewith those for complex SYK
in N∞ limitA double-well
optical lattice(no degeneracy
in the band levels)with 6Li(large recoil energy)

Two-atom band levels

Optical lattice setup in our proposal

Possible to satisfy required conditions

Danshita, Hanada, and MT,
PTEP 2017, 083I01(arXiv:1606.02454)

Out-of-time-order correlation measurement
Interferometric protocol proposed in
B. Swingle et al.: PRB 94, 040302 (2016)

SYK: Initial state of the probed system: states of the control qubit

𝑊 𝑡 = e𝑖𝐻𝑡 𝑊e−𝑖𝐻𝑡
Create the cat stateΨ = 𝑊 𝑡 𝑉 𝜓 𝑆 1 𝐶 + 𝑉𝑊 𝑡 𝜓 𝑆 0 𝐶
by applying
𝐼𝑆 ⊗ 0 0 𝐶 + 𝑉 ⊗ 1 1 𝐶 , 𝑊 𝑡 ⊗ 𝐼𝐶 ,and 𝑉 ⊗ 0 0 𝐶 + 𝐼𝑆 ⊗ 1 1 𝐶 in this order, thenmeasure the qubit to find Re 𝐹 𝑡 and Im 𝐹 𝑡 .

Time evolution with
𝐻′ = −𝐻 (𝜈 ′ = −𝜈)Our qubit C:
A single particle ina double well

Implementation of this protocol in our modelusing a qubit on additional optical double well [1606.02454]

Proposals for experimental realization
N quanta of magnetic flux
through a nanoscale hole

Zero energy states: Majorana fermionsInhomogeneous wave functions
due to the irregular shape of the hole
D. I. Pikulin and M. Franz,“Black Hole on a Chip: Proposal
for a Physical Realization of the
Sachdev-Ye-Kitaev model in a
Solid-State System”,
PRX 7, 031006 (2017)arXiv:1702.04426

Proposals for experimental realization

Anffany Chen, R. Ilan, F. de Juan, D.I. Pikulin,
M. Franz,“Quantum holography in a graphene flakewith an irregular boundary”,
PRL 121, 036403 (2018)arXiv:1802.00802

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=

SYK4

𝑁

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

𝐾𝑎𝑏 : standard deviation

𝐾𝑎𝑏 𝜒𝑎 𝜒𝑏

𝐾
𝑁

1≤𝑎<𝑏

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

Chaos bound[Maldacena,
Shenker, andStanford 2016]

SYK2

non-chaotic

chaotic

𝐾 = 0.2

0.5
12

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Experimental proposal 1606.02454 (and realization)
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 2

𝛽𝐾
4

2
𝐾
𝑎
𝑖,𝑗 𝑖𝑗𝑎𝑖𝑗 = 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 ∝𝐾

𝑒𝑖 − 𝑒𝑗
1≤𝑖<𝑗≤𝐾𝛽

𝑒 −𝛽𝐾𝑒𝑖2

4𝑖=1• 𝑃 𝑠 : Distribution of normalized level separation
𝑒𝑗+1 −𝑒𝑗
𝑠𝑗 =
∆ 𝑒

2

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+1
max 𝑒𝑖+1 − 𝑒𝑖 , 𝑒𝑖+2 − 𝑒𝑖+1

Uncorrelated
𝑟

GOE

GUE

GSE

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, 4𝑋𝜒2𝑗−1 +i𝜒2𝑗2𝜒𝑎 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/2

𝑐𝑗† + 𝑐𝑗

𝑋=𝐾

𝑁 mod 8

0

2

4

6

𝜂

-1

+1

+1

-1

𝑋2

+1

+1

-1

-1

𝑋 maps 𝐻E to

𝐻E

𝐻O

𝐻E

𝐻O

[You, Ludwig, and Xu, PRB 2017]

Class

AI

A+A

AII

A+A

Gaussianensemble

GOE

GUE

GSE GUE[Fadi Sun and Jinwu Ye, 1905.07694]
for SYKq, supersymmetric SYK

𝑗=1

𝑋𝑐𝑗 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, 𝑡

2

𝑍 𝛽

2

𝐽

𝐽

1611.04650

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

β=1

dip

Exponentiallylong ramp 𝑔 𝑡 ~𝑡 1

2

GOE

1

GUE

2

GSE

2

GUE

2

GOE

1

GUE

2

GSE

2

GUE

2

GOE

1

GUE

2

1611.04650(Non-)self-averaging

Consistent with RMT

1 sample = many samples
Self-averaging“Gravity”
1/N expansion

O(1) variance for 1 sample
Not self-averaging“Random matrix theory”
1/K 〜 e−N expansion
K = 2N/2

1812.04770

Note: dip-ramp-plateau structure does not require chaos“Randomness and Chaos in Qubit Models”
Pak Hang Chris Lau, Chen-Te Ma, Jeff Murugan, and MT, Phys. Lett. B 795, 230 (2019).

Lyapunov growth of phase space

Coarse-grained•Just one direction?
•If more than one, what are
relations 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 0
2

PB

OTOC: 𝐶𝑇 𝑡 ==at finite t:= 𝑒 𝜆𝑘𝑡𝑠𝑘 𝑡

𝛿𝑥 𝑡= 𝑀𝛿 𝑥 0

𝛿𝑥 0

𝜕𝑥𝑖 𝑡
𝜕𝑥𝑗 0

2

→ 𝑒 2𝜆L 𝑡 for large t

𝑊 𝑡 ,𝑉 𝑡 = 0

2

0

𝑡

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

𝐿𝑎𝑏 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 𝑡

log 𝑠𝑘 𝑡
=
2𝑡

.

𝑁
𝑘=1 .

Quantum Lyapunov spectrum for SYK model + modification
𝑁

𝐻=

6

𝑁

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑 + 𝑖
1≤𝑎<𝑏<𝑐<𝑑𝐾𝑎𝑏 𝜒𝑎 𝜒𝑏1≤𝑎<𝑏

𝐽𝑎𝑏𝑐𝑑 : s. d. = 𝑁3 2
𝐾𝑎𝑏 : s. d. =𝐾
𝑁

• 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 𝑡2𝑡

𝑘=1,2,…,𝐾

Other possibilities: see Rozenbaum-Ganeshan-Galitski, 1801.10591; Hallam-Morley-Green: 1806.05204

Spectral statistics of quantum Lyapunov spectrum: SYK 1809.01671

𝐾 = 0.01 (●): Remains GUE
for long time𝐾 = 10 (◆):
Approaches Poisson

Energy eigenstates
N/2 larger exponents𝑟 : average ofmin 𝑠𝑖 ,𝑠𝑖+1
max 𝑠𝑖 ,𝑠𝑖+1(fixed-i unfolding: unfold eachgap 𝑔𝑖 = 𝜆𝑖+1 − 𝜆𝑖 using its
average 𝑔𝑖 𝐽 , 𝑠𝑖 = 𝑔𝑖 𝑔𝑖 𝐽 )1809.01671

Growth of (largest Lyapunov exponent)*time

Almost linear growth oflog(singular values of L)

1809.01671

Kolmogorov-Sinai entropy vsentanglement entropy production
Initial state with 𝑆EE = 0:
𝜓 𝑡 = 0 = 000 … 000
in the complex fermion basis

Coarse-grained entropy= log(# of cells covering the region)~ (sum of positive 𝜆 ) 𝑡

𝑐𝑗 =

A
𝑒 𝜆𝑡

B

𝜌A 𝑡 = TrB 𝜌 𝑡 , 𝜌 𝑡 = 𝜓 𝑡

𝜒2𝑗−1 + i𝜒2𝑗

𝜓 𝑡

𝑆EE 𝑡 = −Tr 𝜌A 𝑡 log 𝜌A 𝑡

Similar time scales of growthfor SYK4

Kolmogorov-Sinai entropy ℎKS= (sum of positive 𝜆 )= entropy production rate2

1809.01671

Fastest entropy production?
SYK4 limit
1

1𝑁• 𝜆𝑁 and 𝜆OTOC = log𝑒 2𝜆𝑖𝑡𝑖=12𝑡𝑁approach each other; difference
decreases as 1 𝑁• Same for 𝜆𝑁 and 𝜆1 :all exponent  single peak
• All saturate the MSS bound at strong
coupling (low T) limit• Growth rate of entanglement entropy~ ℎKS = sum of positive (all) 𝜆𝑖 [conjecture] SYK model: not only the fastest scramblers,
but also fastest entropy generators

QLS: The case of the random field XXZ model
𝑁

𝐻=

𝑁

ℎ𝑖 𝑆𝑖𝑧𝑺𝑖 ∙ 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)
Matrix element of local perturbation

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 XXZ
𝑁

𝐻=

𝑁

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

𝑀𝑎𝑏 𝑡 = 𝑆𝑎+ 𝑡 , 𝑆𝑏− 0𝑖

W = 0.5(delocalized)

W = 4.0(Many-body
localized)

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

Quantum Lyapunov spectrum distinguishes chaotic and non-chaotic phasesTwo-point correlation function𝜙𝐺𝑎𝑏𝑡 = 𝜙 𝜒𝑎 𝑡 𝜒𝑏 0 𝜙

1902.11086

Singular value statistics of
two-point time correlators𝜙

𝐺𝑎𝑏 𝑡 = 𝜙 𝜒𝑎 𝑡 𝜒𝑏 0 𝜙 as a matrix𝜙

𝜆𝑗 𝑡 = log singular values of 𝐺𝑎𝑏 𝑡

SYK4

GUE at all times

Strongly
perturbed

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)𝜙

𝐺𝑎𝑏 = 𝜙 𝜒𝑎 𝑡 𝜒𝑏 0 𝜙

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

Random matrix behavior  chaotic

Summary
𝐻SYK =

black
hole

𝑁 3/2

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁non-gravitational
quantum system

holography

maximally chaotic(chaos bound)3!

maximally chaotic(chaos bound)

cold atom
realization
Danshita, Hanada and MT,
PTEP 2017 [1606.02454]

𝜆Lyapunov
random matrix behavior of
finite-time Lyapunov spectrum[Quantum] Gharibyan, Hanada,
Swingle and MT, JHEP 2019 [1809.01671]

modifications to study
chaos / integrable
transition& many-body localization

García-García et al., PRL 2018 [1707.02197]
random matrix behavior of
two-point correlators
Gharibyan, Hanada, Swingle and MT, 1902.11086

P. H. C. Lau, Chen-Te Ma, J. Murugan,and MT, Phys. Lett. B 2019 [1812.04770]