Black holes and random matrices

Transcription

1 Black holes and random matrices Stephen Shenker Stanford University Kadanoff Symposium Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 1 / 18

2 Black holes and chaos Strong chaos underlies thermal behavior Sensitive dependence on initial conditions Lyapunov behavior Quantum black holes are thermal They have entropy [Bekenstein] They have temperature [Hawking] Suggests a connection between quantum black holes and Lyapunov behavior Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 2 / 18

3 AdS/CFT Make precise using Gauge-Gravity duality: AdS/CFT Thermal state of field theory on boundary chaotic Black hole in bulk [Maldacena, Sci. Am.] dim boundary dim bulk Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 3 / 18

4 Quantum Lyapunov exponents Define quantum Lyapunov exponents λ L [Larkin-Ovchinnikov; Almheiri-Marolf-Polchinski-Stanford-Sully] Compute λ L in theories dual to Einstein Gravity by studying very high energy collisions near the horizon of the black hole λ L = 2π β = 2πT [SS-Stanford; Kitaev] Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 4 / 18

5 A bound on chaos This gravitational value is a universal bound on the rate of development of quantum chaos [Maldacena SS Stanford] λ L 2π β + O( 1 N 2 ) What systems saturate the chaos bound? Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 5 / 18

6 Sachdev-Ye-Kitaev Model A variant of the Sachdev-Ye model! (The Sachdev-Ye-Kitaev model) [Kitaev] Quantum mechanics of N species of Majorana fermions {χ a, χ b } = δ ab H = a,b,c,d J abcdχ a χ b χ c χ d J abcd random, gaussian distributed around 0, J 2 abcd = 1 N 3 J 2 dim H = 2 N/2 = L Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 6 / 18

7 Sachdev-Ye-Kitaev model H = a,b,c,d J abcdχ a χ b χ c χ d Vectorlike model. Solvable in a large N expansion, but not almost integrable! λ L 2π β βj, N Gravitational sector in the bulk, but many other states as well [ Maldacena-Stanford-Yang] A model black hole What can we do with this new tool? Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 7 / 18

8 Finite black hole entropy from the bulk Black holes have finite entropy equal to the area of the horizon in Planck units What accounts for this finite entropy from the bulk point of view? Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 8 / 18

9 A diagnostic A simple diagnostic [Maldacena]. Let O be a bulk (smeared boundary) operator. ( ) O(t)O(0) = tr e βh O(t)O(0) /tre βh = e βem m O n 2 e i(em En)t / m,n n e βen At short times can treat the spectrum as continuous. O(t)O(0) generically decays exponentially. Perturbative quantum gravity quasinormal modes. [Horowitz-Hubeny] Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 9 / 18

10 A diagnostic, contd. O(t)O(0) = m,n e βem m O n 2 e i(em En)t / n e βen But we expect the black hole energy levels to be discrete (finite entropy) and generically nondegenerate (chaos). Then at very long times O(t)O(0) oscillates in an erratic way. It is exponentially small and no longer decreasing. What accounts for the end of smooth relaxation from the bulk point of view? A nonperturbative effect in quantum gravity. (See also [Dyson-Kleban-Lindesay-Susskind; Barbon-Rabinovici]) Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 10 / 18

11 Another diagnostic, Z(t)Z (t) To focus on the oscillating phases remove the matrix elements. Use a related diagnostic: [Papadodimas-Raju] e β(em+en) e i(em En)t = Z(β + it)z(β it) = Z(t)Z (t) m,n The spectral form factor How does Z(t)Z (t) decrease to its exponentially small asymptotic value? Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 11 / 18

12 SYK model The Sachdev-Ye-Kitaev model is a promising system in which to investigate these questions [Jordan Cotler, Guy Gur-Ari, Masanori Hanada, Joe Polchinski, Phil Saad, Stephen Shenker, Douglas Stanford, Alex Streicher, Masaki Tezuka] [in progress] See also [Garcia-Garcia Verbaarschot] Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 12 / 18

13 SYK Z(t)Z (t) 10 0 SYK, N m = 34, 90 samples, β=5, g(t ) g(t ) The Slope The Dip The Ramp The Plateau What do they mean? Time t /J -1 Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 13 / 18

14 Random Matrix Theory Chaotic quantum systems typically have fine grained energy level statistics described by Random Matrix Theory (RMT) [Wigner; Dyson; Bohigas-Giannoni-Schmit, Berry...] Consider a simple model where H M, an L L (hermitian) random matrix [You-Ludwig-Xu] The spectral form factor of RMT [Brezin-Hikami] Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 14 / 18

15 Random Matrix Theory Z(t)Z (t) 10 0 Random, N dim = 4096, 1200 samples, β=5, g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) g(t ) 10-2 g(t ) Time t /J Time t /J -1 Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 15 / 18

16 Meaning g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) The Slope gravity The Plateau Eigenvalue repulsion The Ramp Spectral rigidity The Dip TBD Time t /J -1 Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 16 / 18

17 Spectral rigidity 10 0 SYK, N m = 34, 90 samples, β=5, g(t ) g(t ) Energies roughly evenly spaced eigenvalue repulsion, no crossing rule ( Dyson gas long range logarithmic repulsion) Time t /J -1 Long range fluctuations suppressed δe n δe m log n m Nearest neighbor balls and springs δe n δe m n m Spectral rigidity Conjecture that these phenomena generically describe the long time behavior of generic (large AdS) black hole horizon fluctuations Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 17 / 18

18 Lessons What lessons do these phenomena teach us about nonperturbative quantum gravity? The SYK model can be exactly rewritten using Hubbard-Stratonovich style fields This description is a rough proxy for the bulk gravitational theory How do RMT effects show up in this description? Stephen Shenker (Stanford University) Black holes and random matrices Kadanoff Symposium 18 / 18