The Sachdev-Ye-Kitaev
model, scrambling andchaos
东南大学高能理论物理青年研讨会
Youth Symposium on theoretical high energy physics
in Southeast University
20 August 2019
Masaki TEZUKA （手塚真樹）

Contents• The Sachdev-Ye-Kitaev model• Large-N solvability: conformal symmetry and maximal chaos• Experimental proposal 1606.02454 (and realization)• Random matrices (RM) and spectral form factor 1611.04650• Deformation and suppression of maximal chaos 1707.02197• Onset of RM behavior in scrambling systems 1803.08050• k-local and local systems• Random circuits• 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

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 : 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)](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)

Note on the Dirac SYK model[Dirac version]𝐻=

1
2𝑁 3/2

𝐽𝑖𝑗;𝑘𝑙 𝑐𝑖 † 𝑐𝑗 † 𝑐𝑘 𝑐𝑙𝑖𝑗;𝑘𝑙Studied for long time in the nuclear theory context• [J. B. French and S. S. M. Wong, Phys. Lett. B 33, 449 (1970)]• [O. Bohigas and J. Flores, Phys. Lett. B 34, 261 (1971)]“Two-body Random Ensemble”

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

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”

t=0

Real time t

𝜕𝑥 𝑡
𝑥 𝑡 , 𝑝 0 PB =𝜕𝑥 0
2

[P. Hayden and J. Preskill 2007] [Y. Sekino and L. Susskind 2008][Shenker and Stanford 2014]2→ 𝑒 2𝜆L 𝑡

Chaos bound 𝜆L = 2π𝑘B 𝑇 ℏ [J. Maldacena, S. H. Shenker, and D. Stanford, JHEP08(2016)106]
saturated by large-N SYK model [Maldacena and Stanford, PRD 94, 106002 (2016)]

Out-of-time-ordered correlators(OTOCs)
𝜒𝑖 𝑡1 𝜒𝑖 𝑡2 𝜒𝑗 𝑡3 𝜒𝑗 𝑡4

Regularized OTOC can becalculated for large-N SYK model,
satisfies the chaos bound𝜆L = 2π𝑘B 𝑇 ℏ at low T limit

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)]

Different models with similarsolutions“An SYK-Like Model Without Disorder”
E. Witten, arXiv:1610.09758“Uncolored Random Tensors, Melon Diagrams, and the SYK Models”
I. R. Klebanov and G. Tarnopolsky, PRD 95, 046004 (2017).
Y. Gu, X.-L. Qi, and D. Stanford, “Local criticality, diusion and chaos
in generalized Sachdev-Ye-Kitaev models,” JHEP05 (2017) 125;
X.-Y. Song, C.-M. Jian, and L. Balents, “Strongly Correlated Metal
Built from Sachdev-Ye-Kitaev Models,” PRL 119, 216601 (2017).

See review: V. Rosenhaus: “An introduction to the SYK model”
arXiv:1807.03334
cf. K. Okuyama: “Replica symmetry breaking in random matrix
model: a toy model of wormhole networks” arXiv:1903.11776

Proposal for experiment(d+1)-dimensional spacetime
Supergravity in (4+1)danti-de Sitter space(×S5)

d-dimensional spacetime

e.g.
AdS5 / CFT4

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: space-dependent photoassociation laser
wm: molecular site wavefunction
wa,i(j): atomic site wavefunctions = 1, 2, …, ns

Sums of two single atom energies

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

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

Optical lattice setup in our proposal(no degeneracy
in the band levels)with 6Li(large recoil energy)

Two-atom band levels

A double-well
optical lattice

Possible to satisfy required conditions

Realizing real Dirac SYK models = 1, 2, …, ns(For simplicity we take 𝜈𝑠 = −1 𝑠 𝑛s 𝜎𝑠 )Can be shown to approach the real Diracversion of the SYK model as ns  ∞.
Gaussian J reproduced
Real SYK:
Physical quantities coincidewith those for complex SYK
in N∞ limit(Gaussian g)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
arXiv:1702.04426

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)

Proposals for experimental realization
arXiv:1802.00802

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

NMR experiment for the SYK model“Quantum simulation of the non-fermi-liquid state of SachdevYe-Kitaev model” Zhihuang Luo, Yi-Zhuang You, Jun Li, ChaoMing Jian, Dawei Lu, Cenke Xu, Bei Zeng and RaymondLaflamme, npj Quantum Information 5, 53 (2019)

The Sachdev-Ye-Kitaev model
N Majorana- or Dirac- fermions randomly coupled to each other[Majorana version]3!
𝐻 = 3/2
𝑁

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁[A. Kitaev: talks at KITP
(Feb 12, Apr 7 and May 27, 2015)][Dirac version]
1𝐻=
2𝑁 3/2

𝐽𝑖𝑗;𝑘𝑙 𝑐𝑖 † 𝑐𝑗 † 𝑐𝑘 𝑐𝑙

𝑖𝑗;𝑘𝑙[A. Kitaev’s talk][S. Sachdev: PRX 5, 041025 (2015)]• Solvable in the large N limit, Sachdev-Ye “spin liquid” ground state• Nearly conformal symmetric at low temperature (“emergent …”)• Realizes the Maldacena-Shenker-Stanford chaos bound 𝜆L = 2π𝑘B 𝑇 ℏ• Holographically corresponds to a quantum black hole?
• Experimentally realized for small N
Generalizations: q-fermion interactions “SYKq”, supersymmetric SYK, lattice of SYK lands; etc.

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

Random matrices: level repulsion andspectral rigidity
Assume unfolded spectrum (rescaled so that average distance = 1)
Spectrum unfolding𝑠 = 𝜆𝑘+1 − 𝜆𝑘𝜌 𝜆𝜆=𝜆𝑘 𝜆𝑘+1

𝜌 𝜆𝜆𝜆𝜌 𝜆′ 𝑑𝜆′−∞Short range• 𝑃 𝑠 : Distribution of normalized
level separation 𝑠•𝑟 : Average of neighboring gap ratio𝑟=

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

𝜆𝑘 𝜆𝑘+1

𝜆Longer range• Σ2 statistics: variance of number oflevels in the energy range with 𝑀levels on average• Spectral form factor 𝑔 𝛽, 𝑡 : Fouriertransform of the density of states

SYK: Exact diagonalization and fermion parity
𝐻=

3!

𝑁 3/2

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁Consider 𝑁D = 𝑁 2 complex fermions 𝑐𝑗 =

𝜒2𝑗−1 +i𝜒2𝑗2, 𝑗 = 1, 2, … , 𝑁D𝜒𝜒𝜒𝜒 preserves parity of complex fermion number
Each (even, odd) sector: 2𝑁D −1 (= 65 536 for 𝑁 = 34) states
𝑁 = 34:
17
17
17
+
+
= 2517 (~ 3.8 %) non-zero matrix elements on each row
024

232 ~ 4 billion complex matrix elements: 64 GiB of memory

𝐻E

0

0

𝐻O

Can be fully diagonalized numerically*(248 ~ 281 trillion) complex number operations, ~ 5 samples / day on a single node(~ 10 RMB / one 𝑁 = 34 sample or 212 𝑁 = 26 samples)

cf. Lanczos code for up to 𝑁 = 46 by G. Gur-Ari https://github.com/guygurari/syk
using DMRG-like ideas (see JHEP 1811, 070 (2018))

N = 32 SYK model“Contrasting SYK-like
Models”,
Chethan Krishnan, K.V.
Pavan Kumar, Dario Rosa,
JHEP01(2018)064
arXiv:1709.06498

Diagonalization of the Hamiltonian  Eigenvalue spectrum

cf. Large N [A. M. García-García and J. J. M. Verbaarschot: 1610.03816]

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)𝐿

𝑎𝑖𝑗 𝑖,𝑗=1= exp −

Joint distribution function for
eigenvalues 𝑒𝑗
𝑝 𝑒1 , 𝑒2 , … , 𝑒𝐾 ∝Level repulsion
𝐾𝑒𝑖 − 𝑒𝑗

𝛽1≤𝑖<𝑗≤𝐾

𝑒 −𝛽𝐾𝑒𝑖2

4𝑖=1• 𝑃 𝑠 : Distribution of normalized level separation 𝑠𝑗 =GOE/GUE/GSE: 𝑃 𝑠 ∝ 𝑠 𝛽 at small 𝑠, has 𝑒
Uncorrelated (Poisson): 𝑃 𝑠 = 𝑒 −𝑠
•
𝑟=

−𝑠 2

𝑒𝑗+1 −𝑒𝑗∆ 𝑒

tail𝑟 : Average of neighboring gap ratio

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]

Time-dependent partition function and energy scale𝑁ℋM =𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑𝑎,𝑏,𝑐,𝑑𝑍 𝛽, 𝑡 = 𝑍 𝛽 + i𝑡

−𝛽
𝐻−i
𝐻𝑡
= Tr ee−𝛽𝐸𝑔 𝑡 ∝ 𝑍 𝛽, 𝑡Density of eigenstates
𝜌 𝐸

Re e−i𝐻𝑡

2

𝐽

𝐸
2𝜋 𝑡

• Analytical continuation of partition function 𝑍 𝛽• Fourier transform of 𝜌 𝐸 modified by temperature

Spectral form factor

𝑔 𝛽, 𝑡 =𝑍 𝛽, 𝑡

2

𝑍 𝛽

2

𝐽

𝐽

𝑍 𝛽, 𝑡 = 𝑍 𝛽 + i𝑡 = Tr e−𝛽𝐻−i𝐻𝑡

slope

plateaudip

ramp

Similar to dense random matrix𝑔 𝛽, 𝑡 =Plateau height
determined by 𝑍 𝛽

2𝑍 𝛽, 𝑡
𝑍 𝛽

𝐽

𝐽

2

For each sample, consider the long time average of𝑍 𝛽, 𝑡2e−𝛽 𝐸𝑚 +𝐸𝑛 ei 𝐸𝑚 −𝐸𝑛 𝑡

=
𝑚,𝑛

1 𝑇
lim
𝑑𝑡 𝑍 𝛽, 𝑡
𝑇→∞ 𝑇 0𝑁𝐸 2 e−2𝛽𝐸 = 𝑁𝐸 𝑍 2𝛽2 =𝐸

(if degeneracy of 𝐸: 𝑁𝐸 is independent of E)
𝑍 2𝛽𝑎𝑆
Because 𝑍~e (𝑎 > 0), long-time average 𝑁𝐸
will
2
𝑍 𝛽
be ~e−𝑎𝑆 (non-perturbative in 1/N)

Late time: governed by gc(t)[You, Ludwig, Xu: arXiv:1602.06964]
BDI class, Nχ Majorana fermions

Dense random matrix reproduces
the late-time ramp & plateau behaviorgc(t)𝑔c 𝛽, 𝑡 =𝑍 𝛽, 𝑡2𝐽−𝑍 𝛽, 𝑡𝑍 𝛽

2𝐽

Ramp from short-range
correlation(spectrum rigidity) ~ 𝑡 𝐿2

2

𝐿: matrix dimension𝐽𝛿𝜌 𝜆1 𝛿𝜌 𝜆1 − 𝜆

GUE = −

sin2 𝐿𝜆
1
+
𝛿 𝜆
2
𝜋𝐿𝜆
𝜋𝐿

F. T.
𝜋𝐿 −1

𝑡 2𝜋𝐿20𝑡

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 𝜒𝑑[L. Fidkowski and A. Kitaev, PRB 2010, PRB 2011]1≤𝑎<𝑏<𝑐<𝑑≤𝑁

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

𝜒2𝑗−1 +i𝜒2𝑗2𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑 respects the complex fermion parity
Even (𝐻E ) and odd (𝐻O ) sectors: 𝐿 = 2𝑁/2−1 dimensions(mod 8)𝐻E
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

1611.04650

The SYK spectral form factor: N dependence

plateau 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

Dip timeCorrelation function
𝐺 𝑡 = 𝜒𝑎 𝑡 𝜒𝑎 0
Dip-ramp-plateau structure similar to𝑔 𝛽, 𝑡 for 𝑁 ≡ 2 mod 8

𝑁/2

𝑐𝑗† + 𝑐𝑗 , 𝑐𝑗 =

𝑋=𝐾
𝑗=1

𝜒2𝑗−1 + i𝜒2𝑗
2

𝑁 ≡ 0 mod 8 : 𝑋 maps each charge parity sector
to itself and 𝑋 2 = 1 (no protected degeneracy)
𝑁 ≡ 2 mod 8 : 𝑋 maps each sector to the otherand even 𝜒 odd finite
𝑁 ≡ 4 mod 8 : 𝑋 maps each charge parity sector
to itself and 𝑋 2 = −1 (only internal degeneracy)
𝑁 ≡ 6 mod 8 : 𝑋 maps each sector to the otherbut even 𝜒 odd = 0

N = 14, GUE

N = 20, GSE

N = 16, GOE

N = 22, GUE

N = 18, GUE

N = 24, GOE

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)• Random matrices (RM) and spectral form factor 1611.04650• Deformation and suppression of maximal chaos 1707.02197• Onset of RM behavior in scrambling systems 1809.01671• k-local and local systems• Random circuits• Characterization of chaos in random systems• Quantum Lyapunov spectrum 1809.01671• Singular value statistics of two-point correlators 1902.11086

Where does the ramp start?

Ramp ∝ 𝑡‘Slope’ hides the beginning of the ramp

Energy spectrum of the SYK model‘Slope’ depends on the edge of the density of states

Spectral form factor𝑡min = 12.5 for 𝛼 = 2.9

𝑡min : almost constant? (8 < 𝑡min < 15 for 10 ≤ 𝑁 ≤ 34)

Scrambling

t=0

t>0

After some time, non-local measurements areneeded for information on the local perturbation at
t = 0 (“information scrambling”)

Scrambling

t=0

0<t<tst=ts

After t=ts , information has been scrambled with the entire system‘scrambling time’Diffusion

t=0

Conserved quantity (e.g. charge) diffuses, eventually(after diffusion time td , also called the Thouless time)
will be uniformly distributed

Scrambling or diffusion?
G. Gharibyan, M. Hanada, S. H. Shenker, and MT,
JHEP 1807, 124 (2018) (arXiv:1803.08050)

In this talk, we show the following examples:
0<t<tst=ts• Known case: band matrix (single
particle hopping)• Numerical results on spin systems

see also: Random circuit-based discussionin our paper•

RMT universality observed after ‘ramp time’ tramp•

Physical interpretation?

•

Relationship to BH information paradox?
scrambling? diffusion?

• Our results: ramp time seems to be

determined by diffusion, not by scrambling

Random band matrix(Single particle hopping: diffusion is defined)

see L. Erdös and A. Knowles, “The Altshuler-Shklovskii
Formulas for Random Band Matrices I: theUnimodular Case,” Comm. Math. Phys. 333, 1365
(2015) for derivation of the scalingLw𝑡scrambling ~ 𝐿 𝑤 1
𝑡diffusion ~ 𝐿 𝑤 2
𝑡ramp ~ 𝐿 𝑤 2

g(t)𝑆 = 1 2 spin chain

N=16
𝐽: normal distribution

Better overlap with diffusion time (N2)

Contents• The Sachdev-Ye-Kitaev model• Large-N solvability: conformal symmetry and maximal chaos• Experimental proposal 1606.02454 (and realization)• Random matrices (RM) and spectral form factor 1611.04650• Deformation and suppression of maximal chaos 1707.02197• Onset of RM behavior in scrambling systems 1803.08050• k-local and local systems• Random circuits• 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

How to characterize quantum chaos?
𝑑
𝑖
𝜓 =𝐻𝜓
𝑑𝑡

𝐻 = const.

𝑡𝜓 𝑡=T exp −𝑖

𝐻 𝑡′ 𝑑𝑡 𝜓 𝑡 = 0= exp −𝑖𝐻𝑡 𝜓 𝑡 = 00Linear dynamics

Unitary time evolution• Long time: energy level statistics• Shorter time: out-of-time correlator

Correlation between levels, as in random matrices

Classically,

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

𝑥𝑖 𝑡 , 𝑝𝑗 0

2

PB

=

𝜕𝑥𝑖 𝑡
𝜕𝑥𝑗 0

2

→ 𝑒 2𝜆L 𝑡 for large t

Quantum version:

OTOC: 𝐶𝑇 𝑡 =𝑊 𝑡 ,𝑉 𝑡 = 0

2= 𝑊† 𝑡 𝑉† 0 𝑊 𝑡 𝑉 0 + ⋯Numerically Hard to see exponential time limited to small systems
dependence1702.06935

Finite-time Lyapunov spectrum
in classical chaotic systems
Classical systems with 𝐾 degrees of freedom
Time t = 0  t𝜆𝑡eInfinitesimal deviation
in initial condition

Deviation at t : linear in initial deviation

𝛿𝜙𝑖 𝑡 = 𝑇𝑖𝑗 𝛿𝜙𝑗 0
Singular values of 𝑇𝑖𝑗 : 𝑎𝑘 𝑡 𝐾
𝑘=1
Time-dependent Lyapunov spectrum

𝛿𝜙𝑖 0

𝜆𝑘 𝑡 =

log 𝑎𝑘 𝑡𝑡

𝑘=1,2,…,𝐾• Usually we consider the 𝑡 → ∞ limit, chaotic if max 𝜆𝑘 > 0• We focus on finite time behavior
Spectrum 𝜆𝑘 𝑡 𝐾
𝑘=1 depends on system details

 Any universality for chaotic cases?
In many chaotic systems, for large 𝐾, the Lyapunov spectrum
behaves like that of a Gaussian random matrix at some time scale.

1702.06935

Numerical evidences
Level separation distribution 𝑃 𝑠 for the unfolded Lyapunov spectrum approaches
that of random matrix eigenvalues 𝑃RMT 𝑠 at some time scale if the degree of
freedom 𝐾 is large, so does the spectral form factor.
Unfolding
𝜌 𝜆𝜆𝑘 𝜆𝑘+1

Spectral form factor𝑠 = 𝜆𝑘+1 − 𝜆𝑘

𝜌 𝜆 =1

𝜆𝜆𝑘 𝜆𝑘+1
𝜌 𝜆 : Smoothed density of states 𝜆 =Characterizes the spectral rigidity“slope”𝜆“plateau”𝜆𝜌 𝜆′ 𝑑𝜆′−∞K=3600, 104 samples~𝜏 1 “ramp”

𝑒 −𝑖𝜏𝜆𝑗 ;

𝑍 𝜏 =
𝑗

𝑍 𝜏 2
𝑔 𝜏 =
= 𝐾 −2𝑍 0 2

Models: logistic map, Lorenz attractor,
D0 brane matrix model (without fermions) andits mass deformation, random band matrix products𝑒 𝑖𝜏 𝜆𝑗 −𝜆𝑘
𝑗𝑘cf. Lyapunov spectrum for random coupling[S. K. Patra and A. Ghosh]• Kuramoto model [PRE 93, 032208 (2016)],• Map networks [EPL 117, 60002 (2017)]
Strong coupling  GOE
Weak coupling  Poisson

1809.01671

Quantum Lyapunov spectrum
Finite-time classical Lyapunov spectrum: obeys RMT statistics for chaos
Singular values of 𝑀𝑖𝑗 =𝐿 = 𝑥𝑖 𝑡 , 𝑝𝑗 0

𝜕𝑥𝑖 𝑡
𝜕𝑥𝑗 0
2
PB

=

OTOC: 𝐶𝑇 𝑡 == 𝑒 𝜆𝑘 𝑡

at finite t: 𝑠𝑘 𝑡
𝜕𝑥𝑖 𝑡
𝜕𝑥𝑗 0

2

2

𝛿𝑥 𝑡
= 𝑀𝛿 𝑥 0

𝛿𝑥 0→ 𝑒 2𝜆L 𝑡 for large t

𝑊 𝑡 ,𝑉 𝑡 = 0[Hanada, Shimada, and MT:
PRE 97, 022224 (2018)]

0𝑡= 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=

𝑁

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

𝐽𝑎𝑏𝑐𝑑 : s. d. =𝐾𝑎𝑏 : s. d. =

6
𝑁3 2

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 𝑡=2𝑡

𝑘=1,2,…,𝐾

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

1809.01671

Growth of (largest Lyapunov exponent)*time

Almost linear growth oflog(singular values of L)

1809.01671

Kolmogorov-Sinai entropy vsentanglement entropy （纠缠熵） 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

Spectral statistics of quantum Lyapunov spectrum: SYK 1809.01671

𝐾 = 0.01 (●):

Remains GUE for long time𝐾 = 10 (◆):

Approaches Poisson𝑟 : average ofEnergy eigenstates
N/2 larger exponents

min 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 )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

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

holography

maximally chaotic(chaos bound)

3!
𝑁 3/2

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁non-gravitational
quantum system
maximally chaotic(chaos bound)

𝜆Lyapunov
random matrix behavior of
finite-time Lyapunov spectrum[Classical] Hanada, Shimada and MT,
PRE 2018 [1702.02197][Quantum] Gharibyan, Hanada, Swingle
and MT, JHEP 2019 [1809.01671]

random matrix behavior of
two-point correlators
Gharibyan, Hanada, Swingle and MT, 1902.11086

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

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]

García-García et al., PRL 2018 [1707.02197]
García-García and MT, PRB 2019 [1801.03204]

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

 Talk by Chris on ThursdaySummary• The Sachdev-Ye-Kitaev model• Large-N solvability: conformal symmetry and maximal chaos• Experimental proposal 1606.02454 (and realization)• Random matrices (RM) and spectral form factor 1611.04650• SYK4 + SYK2: suppression of maximal chaos 1707.02197• Onset of RM behavior in scrambling systems 1803.08050• Diffusion, rather than scrambling, controls ramp time scale• 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