Black holes and random matrices

Stephen Shenker

Stanford University

KITP October 12, 2017

Stephen Shenker () Black holes and random matrices KITP October 12, 2017 1 / 19 A question

What accounts for the finiteness of the entropy–from the bulk point of view in AdS/CFT? The stakes are high here. Many approaches to understanding the bulk: Eternal black hole ↔ Thermofield double state Ryu-Takayanagi Geometry from entanglement Tensor networks ER = EPR Bulk reconstruction and error correction Complexity Bit threads ... suggest that any complete bulk description of quantum gravity (if one exists) must be able to describe these states. (e.g., what do the virtual indices in a tensor network represent?)

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 2 / 19 A diagnostic

A simple diagnostic of a discrete spectrum [Maldacena]. Long time behavior of hO(t)O(0)i.(O is a bulk (smeared boundary) operator) X X hO(t)O(0)i = e−βEm |hm|O|ni|2ei(Em−En)t / e−βEn m,n n At long times the phases from the chaotic discrete spectrum cause hO(t)O(0)i to oscillate in an erratic way. It becomes exponentially small and no longer decreases. (See also [Dyson-Kleban-Lindesay-Susskind; Barbon-Rabinovici]) To focus on the oscillating phases remove the matrix elements. Use a related diagnostic: [Papadodimas-Raju] X e−β(Em+En)ei(Em−En)t = Z(β + it)Z(β − it) = Z(t)Z ∗(t) m,n The “spectral form factor”

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 3 / 19 Properties of Z(t)Z ∗(t)

X Z(t)Z ∗(t) = e−β(Em+En)ei(Em−En)t m,n Z(β, 0)Z ∗(β, 0) = Z(β)2 (= L2 = e2S for β = 0) Assume the levels are discrete (finite entropy) and non-degenerate (generic, implied by chaos) At long times, after a bit of time averaging (or J averaging in SYK), the oscillating phases go to zero and only the n = m terms contribute. Z(β)2 → Z(2β). (= L = eS for β = 0) e2S → eS , an exponential change. How does this occur?

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 4 / 19 SYK as a toy model

The Sachdev-Ye-Kitaev model can serve as a toy model to address these questions. X 2 2 3 H = Jabcd ψaψbψc ψd , hJabcd i ∼ J /N abcd Maximally chaotic, discrete spectrum

Has a sector dual to AdS2 gravity 0 1 0 0 Has a collective field description: G(t, t ) = N ψa(t)ψa(t ), Σ(t, t ). Reminiscent of a bulk description: O(N) singlets nonlocal Nonperturbatively well defined (two replicas) Z ∗ hZ(t)Z (t)i = dGabdΣab exp(−NI (Gab, Σab))

(Of course many differences with bulk descriptions...)

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 5 / 19 ZZ ∗(t) in SYK

Finite dimensional Hilbert space, D = L = 2N/2, amenable to numerics Guidance about what to look for

[Jordan Cotler, Guy Gur-Ari, Masanori Hanada, Joe Polchinski, Phil Saad, Stephen Shenker, , Alex Streicher, Masaki Tezuka] ([CGHPSSSST])

See also [Garcia-Garcia–Verbaarschot]

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 6 / 19 Results

100 SYK, Nm = 34, 90 samples, β=5, g(t ) The Slope ↔ Semiclassical -1 10 quantum gravity -2 10 The Ramp and Plateau ↔ ) t

( -3

g 10 Random

10-4 ([see You-Ludwig-Xu]) The Dip ↔ crossover time 10-5

10-1 100 101 102 103 104 105 106 Time t /J -1

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 7 / 19 Slope, contd.

Slope is determined by semiclassical quantum gravity – nonuniversal.

100 SYK, Nm = 34, 90 samples, β=5, g(t ) 3 10-1 In SYK slope ∼ 1/t . From sharp edge in

10-2 DOS. One loop exact Schwarzian result: ) t

( -3

g 10 S 1/2 ρ(E) ∼ e 0 (E − E0) . -4 10 ([Bagrets-Altland-Kamenev; CGBPSSSST; 10-5 Stanford-Witten]) 10-1 100 101 102 103 104 105 106 -1 Time t /J In BTZ summing over geometries gives oscillating slope with power law envelope:

1090 1046 nonperturbatively small oscillations in the 1044 gn(β,t) 70 α 10 42 10 t3 1040 1 5 10 50100 density of states [Dyer–Gur-Ari] 50 10 t gBTZ(β,t) g(β,t) gn(β,t)

30 10 gn(β,2πn) In AdS5 “ gas” → constant slope ZBTZ(2β) 1010 ([CGBPSSSST])

10-10 0.5 1 5 10 50 100 aS t In each case the dip time is ∼ e (with different a), exponentially shorter than the plateau time Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 8 / 19 The Ramp and Plateau

0 The Ramp and Plateau are signatures of 10 SYK, Nm = 34, 90 samples, β=5, g(t ) 10-1 Random Matrix Statistics, believed to be 10-2 universal in quantum chaotic systems ) t

( -3 g 10 ∗ 10-4 hZZ (t)i is essentially the Fourier transform (2) 0 10-5 of ρ (E, E ), the pair correlation function

10-1 100 101 102 103 104 105 106 Time t /J -1 sin2(L(E − E 0)) ρ(2)(E, E 0) ∼ 1 − (L(E − E 0))2 [Dyson; Gaudin; Mehta] The decrease before the plateau is due to repulsive anticorrelation of levels (“The correlation hole”) Conjecture that this pattern is universal in quantum black holes

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 9 / 19 N versus L

(2) 0 sin2(L(E−E 0)) ρ (E, E ) ∼ 1 − (L(E−E 0))2

(2) 0 1 t  tp, ρ (E, E ) ∼ 1 − L2(E−E 0)2 , “spectral rigidity”

1 1 −cN 1 L2 perturbative in RMT, L2 ∼ e , nonperturbative in N , SYK. sin2(L(E − E 0)) → exp(−2L(E − E 0)), Altshuler-Andreev

∼ exp(−ecN ) in SYK (!)

These effects must be realized in the G, Σ formulation. A research program...

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 10 / 19 Onset of RMT behavior

[Hrant Gharibyan (Stanford), Masanori Hanada (Kyoto), SS, Masaki Tezuka (Kyoto)] [In progress]

100 SYK, N = 34, 90 samples, β=5, g(t ) At how large an energy eigenvalue separation m 10-1 does spectral rigidity end? 10-2 ) t

( -3 g 10 At what time tr does the ramp begin?

-4 10 The Thouless time [Garcia-Garcia–Verbaarschot] 10-5

10-1 100 101 102 103 104 105 106 The dip is just a crossover: edge versus bulk Time t /J -1 dynamics, tr 6= td

0.6 Follow the ramp below the slope: use Gaussian N = 34, 90 samples

0.5

) filter [Stanford] ε (

ρ 0.4

0.3 X 2 2 0.2 ∗ −α(En +Em) +i(Em−En)t Density of states Y (α, t)Y (α, t) = e e 0.1

0 m,n -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Energy ε

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 11 / 19 YY ∗

100 SYK, Nm = 34, 90 samples, β=5, g(t ) Dip time td ∼ 200, N = 34 10-1 10-2 Onset of ramp tr . 10, N = 34 ) t

( -3 g 10 (The ramp is an exponentially subleading 10-4 ∗ 10-5 effect in ZZ and correlation functions

10-1 100 101 102 103 104 105 106 before the dip) Time t /J -1 An upper bound. Very little variation in N 100 g(β = 0) * 2 10-1 (α=2.5) for N ≤ 34

) -2 α 10 >( 2 10-3 log N? scrambling? =0) t (

Y -4 /

* 10

YY Maybe; no. 10-5

-6

= 0), < 10

β Simplify problem by looking at nearest , t -7 ( 10 g

10-8 neighbor qubit chain, random couplings, n

10-9 100 101 102 103 qubits. Scrambling time ∼ n, easier to Time J t study.

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 12 / 19 Brownian circuits

Scrambling describes the growth of a simple operator [Roberts-Stanford-Susskind; Lieb-Robinson] Generic. Also happens in Brownian circuit e−iHt → e−iHm∆t e−iHm−1∆t ... e−iH1∆t

Hm drawn from an ensemble

Unitary gates U = UmUm−1 ... U1 (random quantum circuit) Can analyze dynamics including scrambling analytically [Oliveira-Dahlsten-Plenio; Lashkari-Stanford-Hastings-Osborne-Hayden; Harrow-Low; Brandao-Harrow-Horodecki; Brown-Fawzi ...] Theory of approximate unitary k designs. Approximations to Haar ensemble that accurately compute monomials of k U’s , k U†’s [Denkert et al. ...]

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 13 / 19 Markov chain

† P Study UρU , ρ = p γpσp, σp a of Paulis (Need k copies for k design) Defines a Markov process on on Pauli strings e.g., IIIZZIIXII ... Two qubit Haar random gates and k = 2: II → II; AB → 15 other possibilities, uniformly [Harrow-Low] Initial condition for an OTOC: ZIIIIIIIIII Time to randomize last qubit ∼ n, scrambling time [Nahum-Vijay-Haah; Keyserlingk-Rakovszky-Pollmann-Sondhi]

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 14 / 19 Markov chain, contd.

For spectral statistics study htr(Uk )tr((U†)k )i, k = 1, 2 ...

RMT statistics htr(Uk )tr((U†)k )i → Haar average value

∗ ∗ For k = 2 (two design) slowest terms are like UaaUaaUaaUaa (no sum) Study U|aiha|U† where |ai = |00 ... 00i

1 n ⊗n |00 ... 00ih00 ... 00| = ( 2 ) (I + Z) ZIZZIIZZI ... Easy to equilibrate

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 15 / 19 Markov chain, contd.

ZIZZIIZZI ... Length of longest run of I s in typical string ∼ log n. Equilibrates in ∼ log n n rare strings IIIIZII .... Contribution decays like ne−t . Order one at t ∼ log n. At long times the system relaxes at a rate determined by the gap of the Markov chain. The gap is independent of n [Brandao-Harrow-Horodecki] Equilibration time ∼ log n, shorter than scrambling !

Correlation functions of very complicated operators [Roberts-Yoshida; Cotler-Hunter-Jones-Liu-Yoshida] For nonlocal pair interactions (2 local, analogous to SYK), get a time of order log log n, (for non Haar random gates goes back to log n )

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 16 / 19 Hamiltonian systems (geometrically local)

Local QSG, N = 15 100

10-2 n geometrically local qubits

-4 > 10 2 H = P Jαβσασβ , J random -6 g(t)=<|tr U(t)| 10 i i i i+1

10-8 Gaussian density of states → 2 10-10 10-2 10-1 100 101 102 103 104 105 106 slope ∼ exp(−Nt ), rapid decay t

Local QSG Scrambling time ∼ n 250 2 200 tr ∼ n

150 Slower than scrambling ?? Ramp Time 100

50

0 6 8 10 12 14 16 N

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 17 / 19 Diffusion

Crucial difference between Hamiltonian and random quantum circuit systems – conserved quantities (energy) 2 tr ∼ n is the time for something to diffuse across the n qubit chain. For a single particle described by a random hopping H (a banded matrix) the ramp time is just the time to for the particle to diffuse across the system, the Thouless time [Altshuler-Shklovskii, Efetov...]. Same here, for energy?

A new tool: Random quantum circuit with a conserved quantity Sz . Analytically tractable [Khemani-Vishwanath-Huse]. Small diffusive corrections to OTOCs. Plan: compute spectral form factor quantities using this circuit. Large effect because signal is so small.

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 18 / 19 The Thouless time for black holes

Assume geometrically local d dimensional Hamiltonians have tr 2/d governed by diffusion, tr ∼ n q-local systems like SYK correspond to d → ∞. Then n2/d → log n [Susskind]

tr ∼ log n but with a different coefficient than the scrambling time. Important because the black hole evaporation time is of order S ∼ n  log n. So these phenomena would appear in small black holes as well, although as an exponentially subleading effect We need to know what they mean in quantum gravity...

Stephen Shenker (Stanford University) Black holes and random matrices KITP October 12, 2017 19 / 19