SYK模型の準位統計とLyapunov spectrum
Level statistics and Lyapunov spectrum of the SYK model「離散的手法による場と時空のダイナミクス」研究会2018
東北大学、2018年9月11日 14:00 – 15:15
京都大学理学研究科

手塚真樹 (Masaki TEZUKA)

Contents• Introduction: the Sachdev-Ye-Kitaev (SYK) model• Experiment proposal [Danshita-Hanada-Tezuka PTEP 2017]• Energy spectrum of the SYK model and quantum chaos• [Cotler et al., JHEP 2017], [Gharibyan et al., JHEP 2018]• Extension of the model [Garcia-Garcia et al., PRL 2018, arXiv:1801.03204]• Lyapunov spectra of chaotic systems• Classical systems [Hanada-Shimada-Tezuka PRE 2018]• Quantum systems e.g. SYK [Gharibyan-Hanada-Swingle-MT, arXiv:1809.01671]• SummaryPlan of talk
𝐻SYK =

blackhole

maximally chaotic(chaos limit)

holography

𝜆Lyapunov

random matrix behavior of
finite-time Lyapunov spectrum

3!
𝑁 3/2

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

maximally chaotic(chaos limit)
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[Classical] Hanada, Shimada and MT,
PRE 2018 [1702.02197]
Cotler et al., JHEP 2017 [1611.04650][Quantum] Gharibyan, Hanada, Swingle
García-García et al., PRL 2018 [1707.02197] Gharibyan et al., JHEP 2018 [1803.08050]
and MT, 1809.01671
García-García and MT, 1801.03204

Collaborators
Jordan Saul Cotler, Guy Gur-Ari, Masanori Hanada, Joseph Polchinski,
Phil Saad, Stephen H. Shenker, Douglas Stanford, Alexandre Streicher
JHEP 1705, 118 (2017) (arXiv:1611.04650)

Antonio M. García-García, Bruno Loureiro, Aurelio Romero-Bermúdez
arXiv:1801.03204
PRL 120, 241603 (2018) (arXiv:1707.02197)
PTEP 2017, 083I01 (arXiv:1606.02454) also with M. Hanada
Ippei Danshita
日本物理学会誌 「最近の研究から」 2018年8月号 p. 569-574
Hidehiko Shimada
Hrant Gharibyan

Phys. Rev. E 97, 022224 (2018)
also with M. Hanada

JHEP 1807, 124 (2018) (arXiv:1803.08050)
also with M. Hanada and S. H. Shenker;
arXiv:1809.01671 also with M. Hanada and Brian Swingle

Plan of talk
𝐻SYK =

blackhole

maximally chaotic(chaos limit)

holography

𝜆Lyapunov

random matrix behavior of
finite-time Lyapunov spectrum

3!
𝑁 3/2

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

maximally chaotic(chaos limit)
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[Classical] Hanada, Shimada and MT,
PRE 2018 [1702.02197]
Cotler et al., JHEP 2017 [1611.04650][Quantum] Gharibyan, Hanada, Swingle
García-García et al., PRL 2018 [1707.02197] Gharibyan et al., JHEP 2018 [1803.08050]
and MT, 1809.01671
García-García and MT, 1801.03204

Introduction: the SYK model
𝐻=

3!
𝑁 3/2

Subir Sachdev

叶锦武 Алексей Китаев

𝜒𝑎 , 𝜒𝑏 = 𝛿𝑎𝑏Gaussian random
𝐽𝑎𝑏𝑐𝑑 2 = 𝐽2 = 1

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁Analytically solvable in 𝑁 ≫ 1 limit, and in the T0 limit,• Satisfies the “chaos bound” [Kitaev’s talks at KITP; S. Sachdev, PRX 5, 041025 (2015); …]
2𝜋𝑘B 𝑇• “chaos bound”: Under “physically reasonable” assumption, 𝜆L ≤ℏ
[J. Maldacena, S. H. Shenker, and D. Stanford, JHEP 1608, 106 (2016)]

Lyapunov exponent 𝜆L : Defined using out-of-time ordered correlator (OTOC)[N. Wiener 1938][Larkin & Ovchinnikov 1969]
𝐶𝑇 𝑡 = − 𝑥 𝑡 , 𝑝 0 2
𝛿𝑥 0
2
𝜕𝑥
𝑡
2
~ 𝑥 𝑡 , 𝑝 0 PB =→ 𝑒 2𝜆L 𝑡
𝜕𝑥 0

𝛿𝑥 𝑡
0

𝐶 𝑡 = 𝑊 𝑡 ,𝑉 𝑡 = 0 2= 𝑊 𝑡 𝑉 0 𝑊 𝑡 𝑉 0 +⋯𝑡

• Corresponding system with gravity?: Not conclusive so far• Models of 1+1D gravity: Jackiw-Teitelboim model, CGHS model, etc.

Holographic connection to black hole (BH) physics[Sachdev PRL 2010, PRX 2015; Maldacena-Stanford, Hosur-Qi-Roberts-Yoshida; Polchinski-Rosenhaus, …][S. Sachdev,
Phys. Rev. X 5, 041025(2015)]

Feynman diagrams
𝐻=

Sample average ⋯𝑁 3/2

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁𝜒𝑎 , 𝜒𝑏 = 𝛿𝑎𝑏𝐽𝑎𝑏𝑐𝑑 2 = 𝐽2 = 1

𝐽

𝐽𝑖𝑗𝑘𝑙

𝐽𝑖𝑗𝑘𝑙 𝐽𝑗𝑘𝑙𝑚
𝑗𝑘𝑙

3!

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

𝐽𝑗𝑘𝑙𝑚

𝑁3
=
𝛿𝑖𝑚
𝐽
3!

𝑂 𝑁 0 contribution

𝐽𝑖𝑗𝑘𝑙

𝐽𝑗𝑘𝑙𝑖′

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

𝑂 𝑁 −2 contribution

Large-N: “Melon diagrams” dominate∝ 𝑁 4 𝛿𝑖𝑖′

[J. Polchinski and V. Rosenhaus, JHEP 1604 (2016) 001]
[J. Maldacena and D. Stanford, PRD 94, 106002 (2016)]0
O(N ) terms
𝐺 1 − Σ𝐺0 = 𝐺0
𝐺 −1 = 𝐺0 −1 − Σ

𝐺 𝑖𝜔

−1

= 𝑖𝜔 − Σ 𝑖𝜔

Σ = 𝐽2 𝐺 3

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

𝐽2

𝑑𝑡𝐺 𝑡1 , 𝑡 𝐺 𝑡, 𝑡2 3 = −𝛿 𝑡1 , 𝑡2

1
𝐺 𝑡 =−
4𝜋𝐽2

1/4

sgn 𝑡

𝑡

Holography: how we use it
d-dimensional spacetime
Supersymmetric YangMills theory (without
gravity) in (3+1)-dim

Strongly correlated
system(d+1)-dimensional spacetimee.g.
CFT4 / AdS5Supergravity in (4+1)danti-de Sitter space(×S5)(Maldacena)

Theory withgravity(classical limit)◎ 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 without gravity
Strongly correlated,
experiment possible

Theory with gravity(significant quantum
effects)

approach quantum gravity byrealizing corresponding
non-gravity models in coldgases

(complex fermion) Sachdev-Ye-Kitaev model

A. Kitaev, KITP talks (2015);
S. Sachdev, PRX (2015)

is a model of Q spin-polarized fermions on N sites:whereis a complex random coupling constant
15
that satisfies
and• No supersymmetry (bosons  fermions) needed• Not relativistic, no antiparticles• Spinless fermions can be used

Experimental realization?

Different from other approachesto (classical) gravity in cold gases
Hawking radiation in sonic analogue of BEC[J. Steinhauer, Nature Phys. 10, 864 (2014); 12, 959 (2016)]
cf. theory [W.G. Unruh, PRL 46, 1351 (1981)]

Sakharov Oscillations in quenched BEC[C.-L. Hung, V. Gurarie, and C. Chin, Science 341, 1213 (2013)]

Creating the SY state of the SYK model in experiment:
equivalent to creating the dual quantum black hole!

Experiment proposal for SYK

arXiv:1606.02454[I. Danshita, M. Hanada, and MT, PTEP 2017]

Here we take the complex fermion version:
Trace out molecular d.o.f.:

OTOC measurement protocol

2 atoms

1 molecule

Optical lattice of non-degenerate sites“Interferometric protocol”(Swingle et al., PRA 2016)y/a

Photoassociation / dissociation

SYK model (S)site
Qubit atom potential

B. Butscher et al., NP (2010)

Sum of the energies
of two atoms

x/aOther proposals (after our arXiv:1606.02454)
Review: M. Franz and M. Rozali, “Mimicking black hole event horizons in atomic andsolid-state systems” arXiv:1808.00541 to appear in Nature Materials
arXiv:1607.08560

arXiv:1702.04426

arXiv:1703.06890

arXiv:1802.00802

L. García-Álvarez, I. L. Egusquiza,
L. Lamata, A. del Campo, J.
Sonner, and E. Solano,“Digital Quantum Simulation of
Minimal AdS/CFT”, PRL 119,040501 (2017)

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)

Aaron Chew, Andrew Essin, andJason Alicea,“Approximating the Sachdev-YeKitaev model with Majorana
wires”, PRB 96, 121119(R) (2017)

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

Plan of talk
𝐻SYK =

blackhole

maximally chaotic(chaos limit)

holography

𝜆Lyapunov

random matrix behavior of
finite-time Lyapunov spectrum

3!
𝑁 3/2

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

maximally chaotic(chaos limit)
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[Classical] Hanada, Shimada and MT,
PRE 2018 [1702.02197]
Cotler et al., JHEP 2017 [1611.04650][Quantum] Gharibyan, Hanada, Swingle
García-García et al., PRL 2018 [1707.02197] Gharibyan et al., JHEP 2018 [1803.08050]
and MT, 1809.01671
García-García and MT, 1801.03204

“Black Holes and Random Matrices”[Cotler, Gur-Ari, Hanada, Polchinski, Saad,
Shenker, Stanford, Streicher, and MT,
JHEP 1705, 118 (2017) (arXiv:1611.04650)]

Collaborators
Jordan Saul Cotler, Guy Gur-Ari, Masanori Hanada, Joseph Polchinski,
Phil Saad, Stephen H. Shenker, Douglas Stanford, Alexandre Streicher

Videos (on “videosfromIAS”)
Natifest – 17 September 2016 “Black Holes and Random Matrices” by Stephan Shenker

https://youtu.be/3qEoMBZLf10
26 October 2016 “The Sachdev-Ye-Kitaev quantum mechanics model, black holes, andrandom matrices” by Douglas Stanford

https://youtu.be/hK2S-pyAf0c

Here we present our numerical results for finite N, focusing on the spectral statistics

Information loss problem of black holes

Assume gauge / gravity correspondence

Study the real-time dynamics of the gauge theory side
Finite size: What to learn from the discrete energy spectrum?

𝛽 = 𝑘B 𝑇 −1

Spectral form factor
Spectral form factor 𝑔 𝛽, 𝑡Consider some real-time correlator:

𝑍: analytically continued partition function

Chaotic, non-degenerate spectrum
Long-time average will be 𝛿𝑚𝑛 .
Initial decay:
exponential

𝑔 𝛽, 𝑡Maldacena

Oscillates after long time;
Exponentially small but
no longer decaying
Expected to approach zero inclassical gravity;
Should reflect features ofquantum gravity𝑡(The initial behavior may differ from correlators)
Papadodimas-Raju; RMT literature𝑡

𝑍 𝛽 2 at 𝑡 = 0 → 𝑍 2𝛽 at late time
(Larger value than naively estimated fromthe correspondence between Super Yang-Millsand superstring theories)

SYK, not too large N: Numerically diagonalize the Hamiltonian for energy spectrum
𝑁 3/2

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑

𝐽𝑎𝑏𝑐𝑑 : Gaussian and variance 𝜎 2 = 𝐽2

1≤𝑎<𝑏<𝑐<𝑑≤𝑁

𝑁𝜌 𝜖

𝐻=

3!

Small N: clear repulsion
between levels𝜖

𝑁𝐽

Large N limit: the entropy
remains finite at low T limit

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

𝑎𝑖𝑗 = 𝑎𝑗𝑖∗𝑎𝑖𝑗𝐿Density ∝ 𝑒

𝛽𝐾
Tr𝐻 2
4

−

= exp

𝑖,𝑗=1
Eigenvalue distribution
𝑝 𝑒1 , 𝑒2 , … , 𝑒𝐾 ∝Level repulsion
𝐾𝑒𝑖 − 𝑒𝑗
1≤𝑖<𝑗≤𝐾

2
𝛽𝐾 𝐾−
𝑖,𝑗 𝑎𝑖𝑗
4[F. J. Dyson, J. Math. Phys. 3, 1199 (1962)]𝛽𝑒 −𝛽𝐾𝑒𝑖24𝑖=1

𝑃 𝑠 : level separation distribution
Uncorrelated: Poisson (𝑒 −𝑠 )2𝛽−𝑠GOE/GUE/GSE: ∝ 𝑠 at small 𝑠, 𝑒
tail𝑟=

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

𝑟 : mean neighboring gap ratio
Uncorrelated: Poisson (2 log 2 − 1 ≈ 0.386)
GOE/GUE/GSE: larger (e.g. 0.599 for GUE [Y. Y. Atas et al. PRL 2013])

Spectral form factor 𝑔 𝛽, 𝑡 =slopeplateau
ramp ∝ t1
dip

𝑍 𝛽, 𝑡

2

𝑍 𝛽

2

𝐽

𝐽

𝑍 𝛽, 𝑡 = 𝑍 𝛽 + i𝑡
= Tr e−𝛽𝐻−i𝐻𝑡
𝛽 = 𝑘B 𝑇 −1

Slope-dip-ramp-plateau structure of 𝑔 𝛽, 𝑡Early time: ~ 𝑡 −3𝑍 𝛽, 𝑡 = Tr e−𝛽𝐻−i𝐻𝑡(depends on the overall density of states)2𝑔 𝛽, 𝑡 =𝑔c 𝛽, 𝑡 =Connection to plateau:
Symmetry dependent

Slope: gd dominant𝑔d 𝛽, 𝑡 =2

𝑍 𝛽, 𝑡
𝑍 𝛽

𝐽

𝐽
2

∼

𝑍 𝛽, 𝑡𝑍 𝛽

𝑍 𝛽, 𝑡𝐽 − 𝑍 𝛽, 𝑡
𝐽

2
𝐽

2

𝑑𝜆1 𝑑𝜆2 𝛿𝜌 𝜆1 𝛿𝜌 𝜆2 𝑒 i𝑡 𝜆1−𝜆2

𝜋𝐿 −1sin2 𝐿𝜆1=−+
𝛿 𝜆𝜋𝐿𝜆 2 𝜋𝐿
F. T.

in RMT𝑡 2𝜋𝐿2

Long, linear ramp ~ 𝑡 1
 Strong spectral rigidity in the SYK model

2

2

𝑍 𝛽

Plateau height:
𝑅 𝜆 = 𝛿𝜌 𝜆1 𝛿𝜌 𝜆1 − 𝜆degeneracy~ 𝑡 1 ramp: gc dominant

𝐽

𝐽

0

2𝐿

𝑡

𝑔 𝑡 : N dependence (nonperturbative in 1/N)
𝐻=

3!
𝑁 3/2

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁plateau 𝑔 𝑡 = 𝑁𝐸 𝑍 2𝛽 /𝑍 𝛽 2

β=1Exponentiallydeep dip

Exponentially longramp 𝑔 𝑡 ~𝑡 1

𝑁𝐸
GOE

1

GUE

2

GSE

2

GUE

2

GOE

1

GUE

2

GSE

2

GUE

2

GOE

1

GUE

2

SPT (Symmetrically Protected Topological) phase classification for class BDI: Z  Z8 due to interaction[L. Fidkowski and A. Kitaev: PRB 81, 134509 (2010); PRB 83, 075103 (2011)]

Level statistics of the many-body system  Corresponding (dense) random matrix[Y.-Z. You, A. W. W. Ludwig, and Cenke Xu, PRB 95, 115150 (2017)]

[L. Fidkowski and A. Kitaev: PRB 83, 075103 (2011)][W. Fu and S. Sachdev: PRB 94, 035135 (2016)][Y. Z. You, A. W. W. Ludwig, and C. Xu: PRB 95, 115150 (2017)]

degeneracy
𝑁D

𝑐𝑗† + 𝑐𝑗

𝑃=𝐾

, 𝑐𝑗 =

𝜒2𝑗−1 + i𝜒2𝑗
2

𝑗=1

𝑐𝑗† + 𝑐𝑗

2= 𝑐𝑗† , 𝑐𝑗 = 1,𝑃2 = −1 𝑁D 𝑁D−1

2=

, 𝑁D = 𝑁 2

𝐾: Complex conjugate
𝑐𝑗 : Complex (Dirac) fermion

𝑐𝑗† + 𝑐𝑗 𝑐𝑘† + 𝑐𝑘 = − 𝑐𝑘† + 𝑐𝑘 𝑐𝑗† + 𝑐𝑗
+1
−1

𝑁D mod 4 = 0, 1
𝑁D mod 4 = 2, 3

𝑃𝑐𝑗† 𝑃 = 𝜂 𝑐𝑗 , 𝑃𝑐𝑗 𝑃 = 𝜂𝑐𝑗† , 𝜂 = −1

∴ 𝑃𝜒𝑃 = 𝜂𝜒, 𝑃𝐻𝑃 = 𝜂4 𝐻 = 𝐻𝑗≠𝑘

𝑁D −1 𝑃2 =+1
−1

𝑁D mod 4 = 1, 2
𝑁D mod 4 = 0, 3

𝐻, 𝑃 = 0𝜒𝜒𝜒𝜒 preserves the parity of complex fermions𝑁 ≡ 0 mod 8 : 𝑃 maps ee, oo and 𝑃2 = 1
𝑁 ≡ 2 mod 8 : 𝑃 maps eo and even 𝜒 odd finite
𝑁 ≡ 4 mod 8 : 𝑃 maps ee, oo and 𝑃2 = −1
𝑁 ≡ 6 mod 8 : 𝑃 maps eo but even 𝜒 odd = 0(not degenerate)
Twofold degenerate eigenstates

Correlation functions
𝐺 𝑡 = 𝜒𝑎 𝑡 𝜒𝑎 0
𝑁 ≡ 2 mod 8 : dip-ramp-plateau
similar to 𝑔 𝛽, 𝑡𝑁 ≡ 0 mod 8 : 𝑃 maps ee, oo and 𝑃2 = 1 (no degeneracy)
𝑁 ≡ 2 mod 8 : 𝑃 maps eo, even 𝜒 odd finite
𝑁 ≡ 4 mod 8 : 𝑃 maps ee, oo and 𝑃2 = −1 (degeneracy)
𝑁 ≡ 6 mod 8 : 𝑃 maps eo but even 𝜒 odd = 0
𝑍 𝛽, 𝑡𝑔 𝛽, 𝑡 ∼𝑍 𝛽, 𝑡 = 0
𝐺 𝑡 = 𝜒𝑎 𝑡 𝜒𝑎 0

𝑁/2

𝑐𝑗† + 𝑐𝑗 , 𝑐𝑗 =

𝑃=𝐾
𝑗=1

𝜒2𝑗−1 + i𝜒2𝑗
2

𝛽 =

2

=

1
𝑍 𝛽, 𝑡 = 0 2

1𝑍 𝛽, 𝑡 = 0e−𝛽 𝐸𝑚 +𝐸𝑛 ei 𝐸𝑚 −𝐸𝑛 𝑡𝑚,𝑛e−𝛽𝐸𝑚 𝑚 𝜒𝑎 𝑛 2 ei 𝐸𝑚 −𝐸𝑛 𝑡𝑚,𝑛N = 14, GUE

N = 20, GSE

N = 16, GOE

N = 22, GUE

N = 18, GUE

N = 24, GOE(Non-)self-averaging

Consistent with RMT
1 sample = many samples
Self-averaging

O(1) variance for 1 sample
Not self-averaging

(Non-)self-averaging“Gravity”
1/N expansion“Random matrix theory”
1/L〜e−N expansion L = 2N/2

ここまでのまとめ𝑎

Majorana 𝑑
𝐽𝑎𝑏𝑐𝑑

𝑏

𝑐

𝑎

Dirac

𝑑

𝐽𝑎𝑏;𝑐𝑑𝑏𝑐• q=4 Sachdev-Ye-Kitaev (SYK) 模型: 全ての組み合わせでのランダムな q 点相互作用• 実験提案 [I. Danshita, M. Hanada, and MT PTEP 2017] など[J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski,
P. Saad, J. Saul, S. H. Shenker, D. Stanford, and MT]• 有限の N に対する数値的な結果 [1611.04650; JHEP 2017]• スペクトル形状因子 𝑔 𝛽, 𝑡 ∝ 𝑍 2 𝐽 , 𝑔c 𝛽, 𝑡 ∝ 𝑍 2 𝐽 − 𝑍 𝐽 2 を分配関数を拡張した 𝑍 𝛽, 𝑡 から求めた。
2• 𝑔: 初期の急速な減衰 (slope): 𝑍 𝐽 の減衰による• Ramp + plateau: ランダム行列的な準位相関を反映; N mod 8 (GOE, GUE, GSE)
• 相関関数 𝜒 𝑡 𝜒 0 : 同様に slope-dip-ramp-plateau を示す (N mod 8 = 2 のとき)
• Ramp 以降は、自己平均化しない• Dip time と plateau time の比: N とともに指数関数的に増大 (長距離相関を反映)
 AdS5/CFT4 (N=4 SYM) でも同様に指数関数的に長い ramp を期待 (see our paper)

“Onset of Random Matrix Behavior in
Scrambling System”
Hrant Gharibyan, Masanori Hanada, Stephen H. Shenker, and MT, arXiv:1803.08050
JHEP 1807, 124 (2018)

Where does the ramp start?

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

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

Energy spectrum of the SYK modelSpectral form factor𝑡min : almost constant? (8 < 𝑡min < 15 for 10 ≤ 𝑁 ≤ 34)

𝑡min = 12.5 for 𝛼 = 2.9

Scrambling

t=0
t>0

After some time, non-local measurements are needed for informationon 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

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

Scrambling or diffusion?

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 discussion• RMT universality observed after ‘ramp time’ tramp
in our paper•

Physical interpretation?

•

Relationship to BH information paradox?
scrambling? diffusion?

•

Our results: ramp time seems to be determinedby 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: the Unimodular
Case,” Comm. Math. Phys. 333, 1365 (2015) for
derivation of the scalingLw𝑡scrambling ~ 𝐿 𝑤𝑡diffusion ~ 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)

Hrant Gharibyan, Masanori Hanada, Stephen H. Shenker, and MT, JHEP 1807, 124 (2018) (arXiv:1803.08050)

この節のまとめ• SYK模型のようなスクランブリング系: ランダム行列理論的なふるまいは、保存量が充分拡散した時間スケールで現れる

cf. D.A. Roberts, D. Stanford and A. Streicher, “Operator growth in the SYK model”, JHEP 06 (2018) 122[arXiv:1802.02633]; A. Altland and D. Bagrets, Quantum ergodicity in the SYK model, Nucl. Phys. B 930(2018) 45 [arXiv:1712.05073]• 蒸発するブラックホール

𝑡ramp ~ log 𝑁 < 𝑁 # ~𝑡evaporation• 重力側での導出?

 もう1つの universality — 短時間でのリアプノフスペクトル?
(Hanada-Shimada-MT PRE 2018, Gharibyan-Hanada-Swingle-MT, arXiv:1809.01671)

Plan of talk
𝐻SYK =

blackhole

maximally chaotic(chaos limit)

holography

𝜆Lyapunov

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

3!
𝑁 3/2

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

maximally chaotic(chaos limit)
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][Quantum] Gharibyan, Hanada, García-García et al., PRL 2018 [1707.02197] Gharibyan et al., JHEP 2018 [1803.08050]
Swingle and MT, 1809.01671 García-García and MT, 1801.03204

Preparation: Stability of chaos in the SYK model
Antonio M. García-García, Bruno Loureiro, Aurelio Romero-Bermúdez, and Masaki Tezuka, Phys. Rev.
Lett. 120, 241603 (2018) (arXiv:1707.02197)

Various modifications of SYK have been studied
e.g.
• Supersymmetric SYK [Fu, Gaiotto, Maldacena, and Sachdev 2016]• Non-random couplings [Witten, 1610.09758]• Higher-dimensional generalizations [Gu, Qi, and Stanford 2017][Davison, Fu, Georges, Gu, Jensen, and Sachdev 2017][S. Banerjee and E. Altman,
Phys. Rev. B 95, 134302 (2017)]

Addition of new Fermi species can induce a transition to a Fermi liquid[Banerjee and Altman 2017] or metal-insulator transition [S.-K. Jian and H. Yao 1703.02051],
additional interaction can induce a metal-insulator transition [C.-M. Jian, Bi, and Xu 1703.07793], …

Motivation and our model
Q.: Minimum requirements for chaotic behavior? ( gravity interpretation?)
Here we study a simple model with analytical + numerical methods
𝑁

𝐻=

SYK4
𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑 + 𝑖

1≤𝑎<𝑏<𝑐<𝑑

SYK2

𝑁𝐾𝑎𝑏 𝜒𝑎 𝜒𝑏1≤𝑎<𝑏6𝐽
𝐽𝑎𝑏𝑐𝑑 : average 0, standard deviation 3 2
𝑁

Gaussian random couplings 𝐾 : average 0, standard deviation 𝐾
𝑎𝑏

𝑁

SYK4 as unperturbed Hamiltonian,
𝐾 controls the strength of SYK2 (one-body random term, solvable)
Both terms respect charge parity in complex fermion description
 Numerical exact diagonalization (ED) of 2N/2-1-dimensional matrix, 𝑁 ≤ 34 possible

𝐽 = 1: unit of energy

Entropy per fermion
In SYK4, entropy S per fermion  0.2324 in large-N limit

SYK4[Cotler, …, and MT, JHEP05(2017)118]

Large K (weak SYK4):
S / N almost
independent of N,
and vanishing as T  0
𝑁

𝐻=

𝑁

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑 + 𝑖
1≤𝑎<𝑏<𝑐<𝑑𝐽𝑎𝑏𝑐𝑑 : average 0, std. dev.

𝐾𝑎𝑏 𝜒𝑎 𝜒𝑏
1≤𝑎<𝑏

6

𝑁3 2
𝐾
𝐾𝑎𝑏 : average 0, std. dev.
𝑁Specific heat and large-N analysis
Low T: specific heat C(T)  linear in T
Large N: replica fields + Hubbard-Stratonovich transf.
[cf. Maldacena and Stanford 2016, Sachdev 2015]

Slope of C(T)

N=34

← SYK4 limit

𝜋 6
SYK2 limit →Out-of-time order correlator (OTOC)

Chaos bound [Maldacena,
Shenker, and Stanford 2016]

SYK4 limit →Large-q limit𝑒 𝜆L 𝑡1 +𝑡2

𝐾=0

𝐾 = 0.2

2𝑓 𝑡 − 𝑡
1
2

0.5
1
2

Lyapunov exponent is obtained by solving
𝛽 = 1 𝑘B 𝑇

Deviation from the chaos boundas SYK2 component is introduced

Small κ: RMT-like behavior of energy spectrum
𝑃 𝑠 : level spacing distribution
Ratio of consecutive level spacing 𝐸𝑖+1 − 𝐸𝑖to the local mean level spacing Δ(requires unfolding of the spectrum
using a fitting of the spectral density)

SYK4 limit (small K): RMTSYK2 (large K): Poisson (𝑒 −𝑆 )(GUE if 𝑁 ≡ 2 (mod 4))

Central 10 % of eigenvalues

Small κ: RMT-like behavior of energy spectrum𝑟 : average adjacent gap ratio
Average ofmin 𝐸𝑖+1 −𝐸𝑖 ,𝐸𝑖+2 −𝐸𝑖+1
max 𝐸𝑖+1 −𝐸𝑖 ,𝐸𝑖+2 −𝐸𝑖+1(does not require unfolding)Central 10 % of eigenvalues

SYK4 (κ = 0): random matrix
SYK2 (large κ): Poisson (2 log 2 − 1 ≈ 0.386)(≈ 0.599 for GUE [Y. Y. Atas et al. PRL 2013])

Also: spectral form factor 𝑍 𝛽 + 𝑖𝜏 2 /𝑍 𝛽 2shows robust ramp reflecting level rigidity

Time-dependent Lyapunov spectra inquantum chaotic systems
arXiv:1809.01671 “Quantum Lyapunov Spectrum”
Hrant Gharibyan (Stanford),
Masanori Hanada (Boulder  Southampton),

Brian Swingle (Maryland),and Masaki Tezuka (Kyoto)

Statistics of classical Lyapunov spectrum
Classical system with 𝐾 degrees of freedom

Deviation at t : linear in the initial deviation

Time t = 0  te

M. Hanada, H. Shimada, and M. Tezuka,
Phys. Rev. E 97, 022224 (2018)

𝛿𝜙𝑖 𝑡 = 𝑀𝑖𝑗 𝛿𝜙𝑗 0

𝜆𝑡

Infinitesimal deviation
in initial condition

Singular values of 𝑀𝑖𝑗 : 𝑠𝑘 𝑡 𝐾
𝑘=1
Time-dependent Lyapunov spectrum

𝜆𝑘 𝑡 =

𝛿𝜙𝑖 0

log 𝑠𝑘 𝑡𝑡

𝑘=1,2,…,𝐾• Often the 𝑡 → ∞ limit is studied,
𝜆𝑘 𝑡𝐾
𝑘=1 depends on the details of the system

while we focus on the finite-time behavior

 Probability distribution of the unfolded gap approaches that of Gaussian random matrices(Examples: logistic map, Lorenz attractor, BFSS matrix model & mass-deformed version  GOE)
Similar behavior observed for singular values of Real matrix  GOE
Complex matrix → GUE
random band matrix products

Out-of-Time-Ordered Correlation

Our definition for quantum systems (cf. OTOC)
𝛿𝑥𝑖 𝑡 = 𝑀𝑖𝑗 𝛿𝑥𝑗 0[Norbert Wiener 1938][Larkin & Ovchinnikov 1969]

Canonically conjugate variables 𝑥, 𝑝 at different times

𝜕𝑥 𝑡
2
𝑥 𝑡 , 𝑝 0 PB =
𝜕𝑥 0

2

𝑀𝑖𝑗 =

𝛿𝑥𝑖 𝑡
𝛿𝑥𝑗 0

𝛿𝑥𝑖 𝑡
𝐿=
𝛿𝑥𝑗 0

→ 𝑒 2𝜆L𝑡

Singular values: 𝑠𝑘 𝑡
2

Corresponding quantity in quantum systems: for bosonic 𝑉, 𝑊 : 𝐶𝑇 𝑡 = − 𝑉 𝑡 , 𝑊 0
Out-of-time correlator is included
𝑉 𝑡 𝑊 0 𝑉 𝑡 𝑊 0 etc.

For systems of fermionse.g. Sachdev-Ye-Kitaev (SYK) model?

𝐻=

3!

𝑁 3/2

= 𝑒 𝜆𝑘 𝑡

2

𝐽𝑎𝑏𝑐𝑑 𝜒𝑎 𝜒𝑏 𝜒𝑐 𝜒𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁𝜒𝑎 : Majorana fermion, 𝑀𝑎𝑏 𝑡 = 0 = 𝜒𝑎 , 𝜒𝑏 = 𝛿𝑎𝑏

Anticommutator
𝑀𝑎𝑏 𝑡 = 𝜒𝑎 𝑡 , 𝜒𝑏 0
𝑁

𝐿𝑎𝑏 𝑡 =

𝑀𝑗𝑎 𝑡 𝑀𝑗𝑏 𝑡
𝑗=1

Our definition: for state 𝜙 (e.g. eigenstate)𝑁
𝑘=1
log 𝑠𝑘 𝑡=.
2𝑡

For matrix 𝜙 𝐿𝑎𝑏 𝑡 𝜙 , obtain singular values 𝑠𝑘 𝑡
and the Lyapunov spectrum is defined as 𝜆𝑘 𝑡

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

SYK: dependence on the SYK2 coefficient K
Sample- and state-averaged full Lyapunov
spectrum: time dependence

Almost linear growth oflog(singular values of L)
Close to constant between red lines(20 % and 80 % of the saturated value of 𝜆L 𝑡)

Classical KS entropy vs entanglement entropy production
Initial state: 𝜓 𝑡 = 0

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

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

𝑒

𝜆𝑡

A

B

= 000 … 000 in the complex fermion basis
𝑁 2

𝜌A 𝑡 = TrB 𝜌 𝑡 ,𝜌 𝑡 = 𝜓 𝑡 𝜓 𝑡𝑐𝑗 =𝜒2𝑗−1 + i𝜒2𝑗
2

𝑆EE 𝑡 = −Tr log 𝜌A 𝑡

Classical KS entropy vs entanglement entropy production
Coarse-grained entropy= log(# of cells covering the region)~ (sum of positive 𝜆 ) 𝑡

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

Similar time scale for saturation in SYK model; other models?

Spectral statistics: SYK

Close to SYK4 :remains GUE even after long time

(fixed-i unfolding: unfold each gap by its average)

Strong one-body hopping term:
approaches GUE

Spectral statistics: SYK

min 𝜖𝑖+1 −𝜖𝑖 , 𝜖𝑖+2 −𝜖𝑖+1
𝑟 : average of the adjacent gap ratio max
𝜖 −𝜖 , 𝜖 −𝜖
𝑖+1

𝑖

𝑖+2

𝑖+1

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

Summary of this part• SYK model: an all-to-all random two-body (SYK2) term destroys chaos of
SYK4 in the low temperature / long-time limits• Proposed a definition of the quantum Lyapunov spectrum• SYK model: linear growth of 𝜆L 𝑡;
KS entropy ~ entanglement entropy production rate
• Statistics of Lyapunov spectra: random-matrix like in SYK4 limit, becomes
Poisson-like as SYK2 term is introduced (also studied XXZ: MBL)
 Do we have to use out-of-time ordered operator products?

Summary
𝐻SYK =

blackhole

maximally chaotic(chaos limit)

holography

𝜆Lyapunov

random matrix behavior of
finite-time Lyapunov spectrum

3!
𝑁 3/2

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

maximally chaotic(chaos limit)
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[Classical] Hanada, Shimada and MT,
PRE 2018 [1702.02197]
Cotler et al., JHEP 2017 [1611.04650][Quantum] Gharibyan, Hanada, Swingle
García-García et al., PRL 2018 [1707.02197] Gharibyan et al., JHEP 2018 [1803.08050]
and MT, 1809.01671
García-García and MT, 1801.03204