A semiclassical ramp in SYK and in gravity
Douglas Stanford
IASJune 25, 2018

Based on work with Phil Saad and Steve Shenker

In a black hole geometry, correlators like hφ(T )φ(0)i seem to justdecay with time T . However, AdS/CFT sayshφ(T )φ(0)iβ =1 X −( β −iT )Em −( β +iT )Ene 2e 2|hn|φ|mi|2 .
Z (β) m,nWhich is NOT asymptotically decaying as a function of T
[Maldacena, Barbon/Rabinovici, Dyson/Lindesay/Kleban/Goheer/Susskind].

A similar story applies to [Papadodimas/Raju]
Xββ|Z (β/2 − iT )|2 =e −( 2 −iT )Em e −( 2 +iT )En ,Z (x) ≡ Tr[e −xH ]m,nThis is essentially the “spectral form factor,” heavily studied in thequantum chaos community [Berry, ...]. E.g. for SYK [CGHPSSSST]
single disorder realization
avg. 90 realizations

10 13peslo|Z(β+iT)|210 12

10 1110102SI slope ∼ e 3TI ramp ∼ T

plateauramp

I plateau ∼ e S .

10 910 810 0X

10 110 210 310 410 510 6TRandom matrix universality =⇒ ramp and plateau should looksomewhat similar for many systems, e.g. N = 4 SYM.

Challenge: Compute this curve using the collective field
description of a large N quantum system.

The ramp in SYKThe SYK model is a quantum mechanics of N Majorana fermions
with HamiltonianX
H=Ja1 ...a4 ψa1 ψa2 ψa3 ψa4
a1 <...<a4We can write the spectral form factor (start with β = 0) using two
replicas of the system:|Z (iT )|2 = Tr [e −iHT ] · Tr [e iHT ]| {z } | {z }“L00“R 00

Both “L” and “R” have the same random couplings Ja1 ...a4 . We
will consider the disorder average of the product
ZN3 22h|Z (iT )| i ≡ dJa1 ...a4 e − J 2 Ja1 ...a4 |Z (iT )|2 .

This can be computed by an effective theory of collective fields:
Zh|Z (iT )|2 i = DGij (t, t 0 )DΣij (t, t 0 )e −N I [G ,Σ]Here Σij (t, t 0 ) are Lagrange multipliers enforcing the relation
Gij (t, t 0 ) =1 X (i)(j)ψa (t)ψa (t 0 ).
N at=TGLL(t,t')
GRR(t,t')
GLR(t,t')"L""R"

t=0

1. Action:
I = − log Pf (∂t δij − Σij )+12

Z TZ T00
J2dtdt 0 Σij Gij − sij Gij4
4

where sLL = sRR = −1 and sLR = 1.
2. Ansatz:
Gij (t, t 0 ) = Ĝij (t − t 0 ),

Σij (t, t 0 ) = Σ̂ij (t − t 0 ).

3. Saddle point equations:hi−1Ĝij (ωn ) = − iωn δij + Σ̂ij (ωn ),Σ̂ij (t) = sij J 2 Ĝij3 (t)Two types of saddle point
Correlated saddle pt.

Uncorrelated saddle pt.
0.50.5^^Re(GLL)= Re(GRR)^ )Im(GLR00^^ )Im(GLL)= -Im(GRR-0.5 0-0.5t

I GLR = 0, ĜLL →TT

I exp. decaying in t

const.
1)2(iT sin πt
T

I N× action → −2S0
I describes “slope” regionh|Z (iT )|2 i ⊃t0

e 2S0T3

I action → 0I two zero modes (see below)
I describes “ramp” regionh|Z (iT )|2 i ⊃ TThe important zero mode
Given one solution with GLR nonzero, can find a family:
ĜLR (t) → ĜLR (t − ∆t).
0.5

0.5

000.5ΔtΔt-0.50-0.50102030-0.501020300102030

The parameter ∆t is an exact Goldstone zero mode for thespontaneous breaking(time translation)L × (time translation)R → (time translation)diag.
The integral over this zero mode gives the ramp factor of TZ Td(∆t) = T0

Where does the correlated saddle pt. come from?
Consider an auxiliary problem
Tr[e −bH e −iHT e −bH e iHT ] = Tr[e −2bH ]and cut and paste the solution from the long Lorentzian portionsof the contour (need to add image to make correctly antiperiodic).
b
Lorentziantime
EuclideantimeTTr[e-bH e-iHTe-bHeiHT] = Tr[e-2bH]

Tr[e-iHT]Tr[eiHT]

Answer to first problem is independent of T , so saddle point action
along Lorentzian part is zero.

The second zero mode

The parameter b was arbitrary. This is a second zero mode,
reflecting that all energies contribute equally to the ramp in|Z (iT )|2 .
To focus on a given energy band, can study instead amicrocanonical version
Z2 2YE ,∆ (T ) ≡dβe βE +β ∆ Z (β + iT )+i RThis fixes b in terms of E .

Translating to gravity

At low temperatures, there is a correspondence betwen the SYKmodel and Jackiw-Teitelboim (JT) gravity
ZZ√
1 √S0gR −g φ(R + 2) + boundary terms
IJT = −
2π2Solutions to SYK can be compared to solutions to JT by matching
the correlation functions G of SYK fermions to those of free 2dfermions in the bulk JT space.
Z (β)Z (β) = two separate hyperbolic disks (Euclidean BHs)

βZ (β + iT )Z (β − iT ) = two separate complexified hyperbolic disks

Can one find a solution in JT gravity that represents the nontrivial“ramp” saddle points?

timeidentify

ds 2 = − sinh2 (ρ)dt 2 + dρ2 ,t ∼ t + T.

The action is zero, and we get a factor of T from a zero mode.
This is the JT gravity version of the SYK ramp.

Is the double cone singular?
When t is compactified t ∼ t + T in Rindler space:ds 2 = −ρ2 dt 2 + dρ2
we have to say what happens at ρ = 0.
Things can go badly: a naive regularization
ds 2 = −(ρ2 + 2 )dt 2 + dρ2
is singular as  → 0, R = −22 /(2 + ρ2 )2 . Worse, due to purely
Lorentzian periodicity, the partition function of even a free field on
this space is ill-defined.
Contour in microcanonical transform suggests:
ds 2 = −(ρ + i)2 dt 2 + dρ2
this space is smooth, and in fact R = 0. In at least some cases,
quantum field theory seems OK on this type of space.

More general black holes
Can consider a similar solution for any stationary black hole in anydimension.

tL=0identify

ΔttR=0T

BHs with all charges and angular momenta will contribute equally,so we should sum over them (subject to the energy constraint).
So there are solutions in the supergravity theory dual to N = 4
SYM. This leads to puzzles (e.g. in this case expect an erratic
ramp, not a smooth one).

SummaryIn the SYK model, we can understand the ramp as arising from a
nontrivial saddle point that correlates the two systems.
This saddle can be translated to JT gravity.
In more general gravity theories, one can also consider similar“double cone” saddle points. The interpretation is not clear but
see short talk on Friday for some discussion of this.