Wormholes, Sachdev-Ye-Kitaev model and universal
dynamics of dissipative quantum chaotic matter
Antonio M. García-García
Shanghai Jiao Tong University

Verbaarschot Bermúdez Loureiro
Stony Brook Leiden
Paris

Victor Godet
Amsterdam

Lucas Sa
LisbonTezukaKyoto

Zheng
Shanghai

Zacarias
Shanghai

ZiogasShanghai

Yiyang Jia Rosa,Nosaka
Stony BrookKIAS

PRX 12, 021040 (2022)
PRL128, 081601 (2022)
arXiv:2210.01695

Summary:
Part I: Introduction

1. Universality and quantum chaos
2. Sachdev-Ye-Kitaev model and quantum chaos
3. Why SYK is interesting in quantum gravity?
Part II: SYK and wormholes
1. Traversable wormholes and SYK (Maldacena andQi). Why is it relevant beyond quantm gravity?
2. Euclidean wormholes and SYK (Godet, AGG)
3. Kelydish wormholes and real time evolution of SYK(AGG, Sa, Verbaarschot, Zheng)Interacting quantum many-bodyproblems are very difficult to solve
QCD
NucleiStrongly correlated electrons
Quantum gravity?
Solution

Search for universal featuresof the quantum dynamics

What universality means?
Universality can be a quite misleading name!
For some, it is related to the behavior of systems close tothermal second order phase transitions controlled by
critical exponents.
For others, it is related to systems
close to quantum phase transitions.
More boldly, it is claimed that one
can use CFT to describe stronglyinteracting phases of quantum
matter close to these critical points.

This talk is not about this!

What universality
means?
Details of the Hamiltonian are not important.
For sufficiently long times, the system relax to astate which is only controlled by the global
symmetries of the system
What dynamical feature is related to universality?
Maybe easier in simpler systems:
Non-interacting quantum chaotic systems
Non-interacting quantum disordered systems

Quantum Chaos
What is quantum
chaos?

Quantum mechanics ofclassically chaotic systems

Why isquantum chaos
important?

Universality

All happy families are alike; each unhappy
family is unhappy in its own way.
Anna Karenina, Toltstoy

Butterfly effectClassical chaos

Lorenz 60’s
Meteorology

Hadamard 1898
Lyapunov 1892

𝛿𝑥(𝑡) = 𝑒 𝜆𝑡 𝛿𝑥(0)𝜆>0ℎ𝐾𝑆 > 0Pesin
formulaDifficult to compute!

Random Matrix Theory characterizes
universality of Quantum Chaos
Bohigas-Giannoni-Schmit conjecture PRL 52, 1 (1984)
Proof Periodic Orbit Theory Sieber, Richter, Altland, Haake, ~2000
Oriol Bohigas

Quantum Chaos

Level statistics is
Random Matrix

Universality 𝑡 ∼ 𝑡𝐻 Heisenbserg time

Efetov
Similar results for a particle in a random potential (d>2) 1984

NuclearPhysics 60’s:
Wigner

How to model theatomic nucleus?

The ultimate
approximation “A randommatrix as an effective
nuclear Hamiltonian”
P(s)O. Bohigas, R.U. Haq, and A. Pandey, in Nuclear Data for
Science and Technology, (1983)
Coceva and Stefanon, Nuclear Physics A, 1979

When is it
valid?

𝑡𝐻 ∼ 1/Δ

Flores, Bohigas, French 1970

Heisenberg time

Δ ≡Mean level spacing

Poisson
spectrum

Nuclearexcitations

Level
repulsion

Spectralrigidity

Random
Matrix

𝑎11
⋮
𝑎𝑁1

⋯ 𝑎1𝑁
⋱
⋮
⋯ 𝑎𝑁𝑁

𝜌 𝐸 ∼

𝐸02 − 𝐸 2

𝑎𝑖𝑗 = 𝑎𝑗𝑖 ∈ ℝ
𝑎𝑖𝑗 = 𝑎𝑗𝑖∗ ∈ ℂ

GOE
GUE

Dyson-Mehta
Semicircle law

Only for Gaussian variables. No universal

Two level universal correlations: 𝑅2 𝑠

sin2 𝜋𝑠∼ 1 − 2 2 GUE
𝜋 𝑠

𝛽 −𝐴𝑠 2 𝑠 = (𝐸Level Repulsion

𝑃 𝑠 ∼𝑠 𝑒

Spectral rigidityΣ2 (𝑁) Number variance related to ∫ 𝑅2

Long range correlations

Σ2 𝑁 = 𝑛 𝑁 2 − 𝑛 𝑁

1
𝑅2 𝑠 ∼ 2 𝑠 ≫ 1
𝑠𝑖+1 −𝐸𝑖 )/Δ∼ log 𝑁 𝑁 ≫ 1
Universality

2Spectral form factor ≡ Fourier transform of Σ2 (𝑁)1611.04650

𝑡𝐻

𝑡𝑇ℎ

Universality

Universal ramp
related to spectral
rigidity
𝑅2 𝑠 ∼

𝑡𝑇ℎ ≡ Thouless time. Minimum time scale for which
the system feels the spectrum is discrete

1
𝑠≫1
𝑠2

Multiple applications: Quantum chaos-RMT
Mesoscopic physics: Interactions, disorder
Deterministic chaotic systems

k-body random ensembles

Quantum gravity?

BohigasFrench, Bohigas, Before
Ma, Flores 70’s Sachdev-Ye!

QCD: Spectrum Dirac operator
Chiral ensembles, lattice simulations

Efetov

Verbaarschot, ShuryakAnother window of universality𝑡𝐸 (exponential) growthof quantum uncertainty

Kicked Rotors: Zaslavsky, Berman,
Physica 91A 450 (1978)
Disorder: Larkin, Ovchinnikov,
Soviet Physics JETP 28, 1200 (1969)

Quantum Chaos
𝑂෠ 𝑡 , 𝑂෠ 0

𝑡𝐸 ~ log ℏ−1 Τ𝜆2≈∝ ℏ2 exp 𝝀𝑡

𝝀 Classical Lyapunov
exponent

The exponential growth at Ehrenfest time 𝑡𝐸
is Universal for quantum chaotic systems
Non-quantum chaotic motion Quantum effects
2
develop faster in𝑂෠ 𝑡 , 𝑂෠ 0≈∝ ℏ2 𝑡𝛽𝑡𝐸

∝ ℏ𝛼 , 𝛼 > 0

Non-universal

quantum chaos!

What quantum chaos has to
do with (quantum) gravity?

Chaos in black-hole physics
Quantum Black Holes are fastest scramblers in nature
P. Hayden, J. Preskill, JHEP 0709 (2007) 120

Sekino, Susskind,JHEP 0810:065,2008

Membraneparadigm

𝑡𝑆 ~𝑡𝐸 ~ log ℏHandwaving GR +Heisenberg principle ….
−1~log(𝑁)

Rindler geometry is important
AdS/CFT
Field theory dual also fastest scramblers

A bound on chaos
Maldacena, Shenker, Stanford

arXiv:1503.01409

OTOC

𝑡𝑑 ≪ 𝑡 < 𝑡∗𝜆 ≤ 2𝜋𝑘𝐵 𝑇/ℏℏ2

𝑝𝑧 𝑡 , 𝑝𝑧 0 2 ≈𝑝𝑧 𝑡 , 𝑝𝑧 0 2 ∝ ℏ2 exp 𝜆𝑡Black holes and its field theory dual saturate theboundHolographic dualities (AdS/CFT): Certain(quantum) asymptotically AdS gravity
theories in d+1 are dual to field theories ind dimensions. N=4 Super-Yang Mills theory
2πkB T
Maldacena et. al, proposes that λ =inℏthese theories but no example is given!!!
Kitaev,
KITP 2015
talks“I have got
one example”

Kitaev: “A simple model of quantum holography”
http://online.kitp.ucsb.edu/online/entangled15/kitaev/

𝐻=෍
𝑖𝑗𝑘𝑙

𝐽𝑖𝑗𝑘𝑙 𝜓𝑖 𝜓𝑗 𝜓𝑘 𝜓𝑙

q=4

Majoranas

𝜓𝑖 , 𝜓𝑗 = 𝛿𝑖𝑗

Gaussian2
𝐽𝑖𝑗𝑘𝑙∼ J 2 /N 3

Strong coupling𝛽𝐽 ≫ 1 𝜏𝐽 ≫ 1

A solvable field theoy with a quantum gravity dual

SYK = Sachdev-Ye-Kitaev
Motivation: Test Maldacena bound on chaos

Before Kitaev …. only Dirac fermions, technicallymore challenging… Kitaev idea of Majoranas very
cleverN fermions French, Bohigas,
m levels
Flores, Wong,70’s
k-random body ensembles
2-bodyFrench, Mon, Annals ofPhysics 95, 90 (1975).

𝑚≫𝑁

𝐻𝑝

→ 𝜌 𝐸

2 /𝜎 2−𝐸∝𝑒Very popular in 1980-2000’s!
Thermalization

Metal-insulator transitions

Level statistics

Transport, quantum dots

Kota

Heisenberg Spin-Chain

Holography dual?

Sachdev, Ye, PRL. 70, 3339 (1993)

S. Sachdev PRL 83, 74408 (2010)

Why is SYKinteresting?

Conformal (reparametrization)
symmetry in the IR limit… like gravity
𝐽𝜏, 𝐽𝛽 ≫ 1 𝜕𝜏 → 0

Finite Zero
Temperature
entropy

Δ = 1/𝑞

Georges, Parcollet 90’s

Kitaev 2015

Conformal symmetry spontaneously and explicitly
broken to SL(2,R) due to finite T, N effects

Conformal symmetry broken to SL(2,R) in the𝛽𝐽 ≫ 1 limit
Schwarzianaction𝜌 𝐸 ∝ sinh(𝛾𝐸 − 𝐸0 )Quantum black holes𝜏 around𝑡𝐸 ≪ t 𝐻

SYK is quantum chaotic and analytically tractable
SYK saturates Maldacena bound
SYK in the low T is dual to NCFT/NAdS2, why?

Jackiw-Teitelboim gravity

1606.01857
1809.08647

Simple non-trivial model in (near) AdS2Classical EOM:
Poincare𝑅 = −2
disk 𝑑 = 2
Boundary action

K extrinsic
curvature𝜙 ≡ dilaton

Schwarzian low energy action
𝜖→0

Diffs invarianceSL(2,R)

The physics is in the motion of the physicalboundary of AdS2 inside a rigid AdS2 space.

Quantum Chaotic
SYK dual* to a quantum JT gravity

SYK and RMT
RMT in SYK?
Is q=4 SYK spectral
density consistent withthat of a black hole ?

Yes

AGG, Verbaarschot Phys.
Rev. D 94, 126010 (2016)(a few weeks later: 1611.04650)

Yes

AGG, Verbaarschot, Phys.
Rev. D 96, 066012 (2017)

Universality
classes in SYK?

Yes

AGG, Jia, Verbaarschot, PRD
97, 106003 (2018) Before:
Wettig et al., Ludwig et al.,Is RMT a feature of JT
quantum gravity?

Yes

Saad et al. 1903.11115
AGG, S. Zacarias Phys. Rev.
Res. 2 (2020) 4, 043310

Symmetries of SYK depend on N,q

𝐾 ≡ Charge conjugation. 𝐶1 , 𝐶2 anti-unitary symmetries
Why? Clifford algebra representations in N dimensions

You, Ludwig,
Xu1604.06964

Why relevant?
If SYK is quantum ergodic for long times, then it can be a
toy model for different systems that share the same
global symmetries: Tenfold way of RMTSYK spectral density is the same as that of quantum
black holes
J. Riordan, Mathematics of

q= 4, 𝜂(𝑁) Computation 29, 215 (1975)𝑞 ∝ 𝑁 1/2

Erdos, et. al, 1407.1552

AGG, Verbaarschot, PRD 96, 066012 (2017)

Density = Q-Hermite polynomials
Bagrets, et al., 1702.08902
Agreement with the exact Schwarzian path integral Stanford, et al. 1703.04612

Exponential
increase
Cardy’s formula
Bethe’s formula

Black holes density

Sqrt edge
Typical of
randommatricesLevel statistics

RMT

Close ground stateHigh excitations

Relevantfor gravity
AGG, Verbaarschot Phys. Rev. D 94, 126010 (2016), later Cotler et al.
1611.04650

Random matrix correlations characterize
quantum black holes?

Tenfold way in black hole physics?

Yes!

Supersymmetric SYK Model

1610.08917
Fu, Gaiotto, Maldacena, Sachdev

chGOE

CI

DIII

chGSE

AGG, Jia, Verbaarschot,
PRD 97, 106003 (2018))

Li, Liu, Xin, Zhou
1702.01738

Wettig, Kanazawa
1706.03044

Even more, the non-Hermitian SYK is also a
toy model for strongly interacting open and/or
dissipative quantum chaotic matter

𝐻𝑁𝐻−𝑆𝑌𝐾 = ෍ 𝐽𝑖𝑗𝑘𝑙 𝜓𝑖 𝜓𝑗 𝜓𝑘 𝜓𝑙𝑖𝑗𝑘𝑙

𝐽𝑖𝑗𝑘𝑙 → 𝐽𝑖𝑗𝑘𝑙 + 𝑖𝑀𝑖𝑗𝑘𝑙
Depending on q and N, we identify 19 out of the 38 Phys. Rev.
X 9, 041015 (2019) non-Hermitian universality classes in thenHSYK model,
AMGG, L Sa, JJM Verbaarschot, PRX 12, 021040 (2022)

Global symmetriesSYK Operators

K charge conjugation

𝑆2 = 𝕀𝑆 ∼ Γ5

Action of symmetry operators on the SYKSymmetry Classification for nhSYKThis completes thetenfold way and add a
few more universality
classesSymmetry Classification for chiral nhSYK

𝐻, 𝐻′ Hermitian𝐻, 𝐻′ non-HermitianComplex spacing ratios

q=2

q>2

Integrability

Quantum Chaos

Radial 𝜌(𝑟) and angular 𝜌(𝜃) distributions of
the complex spacing ratioNo free parameters!!

Excellent quantitative agreement withrandom matrix theory
Assuming that BGS conjecture applies, it means that
random matrix theory can be used to describe
many-body dissipative quantum chaos
SYK, a toy model for many-body dissipative quantum
chaos. Can all universality classes be identified?

Relation to quantum gravity? See next
Long range spectral correlations and limits of dissipative
quantum chaos? See AMG, Sa, Verbaarschot
2211.01650

SYK beyond quantum black
holes: traversable/
Euclidean/Keldysh(?)
wormholes

SYK and traversable wormholes

Gao,Jafferis,WallarXiv:1608.05687

𝐻𝑊 = 𝐻𝐿𝑆𝑌𝐾 + 𝐻𝑅𝑆𝑌𝐾 + 𝑖𝜆 ෍ 𝜓𝑖𝐿 𝜓𝑖𝑅
Maldacena and Qi 1804.00491

𝑖

A two-site SYK model with a weak 𝜆 ≪ 1 coupling is dual, in
the low T limit, to a traversable wormhole in a near 𝐴𝑑𝑆2
background

Why?
Both models share the same Schwarzian effective action

Where this gravity action comes from?

Global 𝐴𝑑𝑆2
2
2
−𝑑𝑡
+
𝑑𝜎
𝑑𝑠 2 =, 𝜎 ∈ [0, 𝜋]2
sin 𝜎It covers the full spacetime (two boundaries) which
has a 𝑆𝐿 2, 𝑅 group of isometries
The idea of MQ paper is to find solutions where the boundaries
corresponds to lines of constant 𝜎 in these coordinates. In thissetting, a t-translational invariant dilaton would grow towards
both boundaries This is not possible in pure 𝐽𝑇gravity. It is also
not possible if matter obeys the integrated null energy
condition in the bulk. However, it becomes possible (Gao,
Jafferis 1608.05687) if we introduce a double trace deformationthat (weakly) couple the two boundaries explicitly.

𝑁

𝑆𝑖𝑛𝑡 = 𝑔 ෍ ∫ 𝑑𝑢 𝑂𝐿𝑖 𝑢 𝑂𝑅𝑖 (𝑢)𝑖𝑁 operators of conformal weight Δ corresponding to 𝑁
bulk matter fields. Large 𝑁 is needed to enhance its effect.
For technical reasons we need 𝑔 ≪ 1 so𝑒𝑖
𝑖𝑔 σ𝑖 𝑑𝑢𝑂𝐿𝑖 𝑢 𝑂𝑅𝑢∼𝑒𝑖
𝑖𝑔 σ𝑖 ∫ 𝑑𝑢⟨𝑂𝐿𝑖 𝑢 𝑂𝑅
𝑢 ⟩This is equivalent to a perturbative re-summation of ladder
diagrams, see next figure, that is dominant in the large 𝑁,
small 𝑔 limit with 𝑁𝑔 fixed.

After coupling this to the gravity modes by performing a
reparametrization of the left and right times. More specifically,
this is a map between the boundary time 𝑢 and the global
times 𝑡𝑙 𝑢 , 𝑡𝑅 𝑢 at the two boundaries.

In this way, we describe the theory describes the position of
the physical boundary.

Why is this interesting?

Interesting features 𝜆 ≪ 1
Gapped system 𝐸𝐺 ∼ 𝜆2/3
Not 𝐸𝐺 ∼ 𝜆 as expectedfrom a explicit coupling
Important!
Non trivial interplay of interactions
in each site and intersite-couplingleads to a novel form ofinteraction-enhanced tunneling

Ground state close toa TFD state⟨𝑇𝐹𝐷𝛽(𝑘) |𝐺𝑆⟩

Hawking-Page transition between
wormhole and two blackhole configurations
𝜆Mapping to a Liouville like QM problem withbound states related to wormhole excitations

Quantum chaos andtraversable wormholes

Two site SYK dual to atransversable wormhole

Maldacena, Qi 1804.00491

Hawking Page transition (weakcoupling) is also a quantumchaos integrable transition
AGG, Rosa, Nosaka, Verbaarschot, PRD 100, 026002 (2019), 28

Non-Hermitian two-site SYK and Euclidean
wormholes

𝐻𝐸𝑊 = 𝐻𝐿𝑆𝑌𝐾 + 𝐻𝑅𝑆𝑌𝐾 + 𝑖𝜆 ෍ 𝜓𝑖𝐿 𝜓𝑖𝑅𝑖

𝜆=0
𝐻𝑆𝑌𝐾 = ෍ 𝐽𝑖𝑗𝑘𝑙 𝜓𝑖 𝜓𝑗 𝜓𝑘 𝜓𝑙𝑖𝑗𝑘𝑙

𝐽𝑖𝑗𝑘𝑙 → 𝐽𝑖𝑗𝑘𝑙 + 𝑖𝑀𝑖𝑗𝑘𝑙
Complex couplingsBecause of dominance of replica off diagonal configurations,the properties of the SYK system after ensemble average are
qualitatively different.

SYK free energy undergoes a first ordertransition that in the gravity dual is interpreted
as wormhole/black hole phase transition innear A𝑑𝑆2

AMG, Jia, Rosa, Verbaarschot,
PRL, 128, 081601 (2022)

Spectrum of the averaged system is gapped𝐸𝑔 ∼ 𝑘 2

But there is no gap for asingle disorder realization!

Interactions 𝑞 = 4 are important
The free energy for a 𝑞 = 2 two site SYK is
qualitatively different

Gravity dual?
Euclidean Wormholes
We find solutions in near AdS2
background (JT gravity) with additional
matterRelevant for the problem of factorization inholography and, in a different context, for
the information paradox
V. Godet, AGG, Phys. Rev. D 103, 046014 (2021)

Complexsourcesb fixed by dynamics

Euclidean wormhole solutions of JT gravity

How is the physics of wormholes related to theout of equilibrium dynamics of dissipative
quantum chaotic matter?

arXiv:2210.01695

AdS/CFT Takayanagi 1911.07861
Overlap with Ryu. et al. 2210.04093 that came out a few days later

The real time dynamics of a SYK at 𝑇 → ∞ weakly
perturbed by a Markovian environment is equal to
the Euclidean dynamics of a two-site non-Hermitian
SYK in the 𝑇 → 0 limit
The late time dynamics is characterized by the decay
rate/gap to a steady state which is computed by the
exponential decay of certain Green’s functions.
Quantum chaos is important. Only for 𝑞 > 2, the decayrate is finite even if there is no coupling with the bath.

The free energy of the Euclidean problem undergoes
a first order transition as in wormholes settings.
We speculate that our SYK is dual to a bra-ket 2007.16091
wormhole: a perturbed double cone in near 𝑑𝑆2

Real time evolution of a SYK coupled toa Markovian bathSingle site SYK
Jump operators that characterize the bath

We study the dynamics after a weak perturbation(linear response) of a TFD state at T = ∞Keldysh formalism
We are interested in the following retarded Green’s
function

In order to proceed, it is useful to employ the Keldysh
path integral that involves the doubling of degrees of
freedom and the vectorization of the Liouvillian
The partition function related to the path integral
is given by,

The degree of freedom are doubled in the
Keldysh action 𝑆𝐾
∞

1
1
𝑖
𝑖
𝑆𝐾 = න 𝑑𝑡(− ෍ 𝜓𝐿 𝜕𝑡 𝜓𝐿 − ෍ 𝜓𝑅𝑖 𝜕𝑅𝑖 + ℒ)
2
2
∞
𝑖

𝑖

Vectorized Liouvillian 𝐻 = 𝐻𝐿 ⊗ 𝐻𝑅
𝑞
1
𝑆𝑌𝐾
𝑆𝑌𝐾
𝑖
𝑖
ℒ = −𝑖𝐻𝐿 + 𝑖 −1 2 𝐻𝑅 − 𝑖𝜇 ෍ 𝜓𝐿 𝜓𝑅 − 𝜇𝑁
2𝑖

We assume the system is sufficiently close to the steady state so
we neglect initial conditions. We assume 𝛽 = 0Note the similarities are differences with theprevious SYK related to wormholesSolving numerically the saddle point equations for
𝑞 = 4 and analytically in the large 𝑞 limit and for
𝑞 = 2, we have found that for 𝑞 > 2, the retarded
Green’s function decays exponentially even if there
is no coupling (𝜇 = 0) to the environment.

𝑅𝑖𝐺 𝑡−Γ(𝜇)𝑡∝𝑒cos Ω 𝜇 𝑡

Γ is the typical (inverse) relaxation time to thesteady state.
We interpret Ω as an imaginary part of Γ indicating
metastability though it also occurs at 𝜇 = 0!

The Euclidean problem

Note that the SYK’s are anti-Hermitian
The ground stateof this Hamiltonian is the thermofield-double(TFD) state at 𝛽 = 0

𝑖𝐺𝐿𝑅 𝜏 ∝ 𝑒−Γ(𝜇)𝑡cos Ω 𝜏 𝑡

This is identical to the real time results with 𝜏 = 𝑡provided that the two Green’s functions 𝐺𝑅 (t) and𝐺𝐿𝑅 (𝜏) are identified
Therefore the original one-site SYK coupled to an
environment is mapped to a two-site non
Hermitian SYK at zero temperature

Equivalence of 𝐺𝐿𝑅 𝜏 and 𝐺𝑅 (𝑡)𝑞=4The existence of a gap/decay rate (𝐸𝑔 , Γ) does not require
the coupling to a bath. Only for 𝜇 large, we recover the
expected Γ ∼ 𝜇 dissipation driven decay. Transition at 𝜇 ∼0.14

The frequency Ω vanishes in what looks a second
order phase transition for sufficiently strong
coupling to the environment.

Analytical large 𝑞 calculation

It is important to impose
2𝛼𝑒 𝑔𝐿𝐿 = 𝑒 𝑔𝐿𝑅 = 2
𝐽 cosh2 (𝛼 𝜏 + 𝛾)

𝛼 2 /𝐽2 cosh2 𝛾 = 1, 2𝛼 tanh 𝑦 = 𝜇Ƹ

2𝛼 2𝐽
𝐸𝑔 =
=
𝑞
𝑞

𝜇ො
2𝐽

2

+1

This result is only about 20% off the numerical one for 𝑞 = 4.
It agrees with 𝐸𝑔 ∼ 𝜇 for 𝜇 ≫ 1 but it does predict oscillations

Non-chaotic case 𝑞 = 2
The previous results require that the dynamics is quantum chaotic.
For 𝑞 = 2, where the dynamics is not quantum chaotic, Γ = 0 if nocoupling to the bath 𝜇 = 0. For 𝜇 > 0 , Γ = 𝜇 so the relaxation is always
dissipation driven.

𝜇=0

𝜇>0

Excellent agreement with analytical results. 𝐸𝑔 (0) =0 so quantum chaos (𝑞 > 2), especially for 𝜇 ≪ 1,
alters qualitatively the way the system approaches
the steady state.

Dissipation driven relaxation 𝐸𝑔 = 𝜇

Gravity dual of a dissipative SYK: Keldysh dS
wormholes

Free energy of the Euclidean SYK setting at
finite T is similar to that of wormholes in near
𝐴𝑑𝑆2

SD eqs.

Exact diagonalization

Good agreement with numerical results

Gravity dual cannot be traversable or Euclidean wormholein near 𝐴𝑑𝑆2 because the Schwarzian action is different

However, thedifference is just thecomplexification of
the couplings
𝐽 → 𝑖𝐽𝐿 , J → −𝑖𝐽𝑅
Cotler et al. JHEP 2020, 6, (2020), Maldacena et al., JHEP 2021, 1 (2021)

Double-trumpet configuration in a near de Sitterbackground in two dimensions (dS2) in Lorentzian timeAfter a matter perturbation (𝜇 in our case)we speculate that 𝑑𝑆2 is unstable to theformation of bra-ket wormholes

Maldacena et al, 2007.16091

Looking for
gravityspecialists totest thishypothesis!

Conclusions
Quantum ergodicity (quantum chaos) and random matrix
behaviour seems to be distinctive features of quantum black
holes and SYK models
SYK models reproduce most, if not all, universality classes ofstrongly interacting systems
Both traversable and Euclidean wormhole have SYK duals whichfacilitates the study or wormhole physics in condensed mattersystems
Wormhole physics seems to play a role in the process ofequilibration of strongly interacting quantum matter bothisolated or in contact with an environment.

Thanks!