Black holes and random matrices

Transcription

1 Black holes and random matrices Stephen Shenker Stanford University Strings 2017 Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

2 A question What accounts for the finiteness of the black hole entropy from the bulk point of view? Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

3 A question What accounts for the finiteness of the black hole entropy from the bulk point of view? The stakes are high here. Many approaches to understanding the bulk TFD/Eternal black hole Ryu-Takayanagi Geometry from entanglement Tensor networks ER = EPR Code subspaces... suggest that any complete bulk description of quantum gravity must be able to describe these states. Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

4 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) ho(t)o(0)i = X m,n e E m hm O ni 2 e i(em En)t / X n e E n Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

5 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) ho(t)o(0)i = X m,n e E m hm O ni 2 e i(em En)t / X n e E 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]) Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

6 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) ho(t)o(0)i = X m,n e E m hm O ni 2 e i(em En)t / X n e E 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 e m+e n) e i(em En)t = Z( + it)z( it) =Z(t)Z (t) m,n Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

7 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) ho(t)o(0)i = X m,n e E m hm O ni 2 e i(em En)t / X n e E 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 e m+e n) e i(em En)t = Z( + it)z( it) =Z(t)Z (t) m,n The spectral form factor Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

8 Properties of Z(t)Z (t) Z(t)Z (t) = X m,n e (E m+e n) i(em En)t e Z(, 0)Z (, 0) = Z( ) 2 (= L 2 = e 2S for = 0) Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

9 Properties of Z(t)Z (t) Z(t)Z (t) = X m,n e (E m+e n) i(em En)t e Z(, 0)Z (, 0) = Z( ) 2 (= L 2 = e 2S for = 0) Assume the levels are discrete (finite entropy) and non-degenerate (generic, implied by chaos) Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

10 Properties of Z(t)Z (t) Z(t)Z (t) = X m,n e (E m+e n) i(em En)t e Z(, 0)Z (, 0) = Z( ) 2 (= L 2 = e 2S for = 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. Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

11 Properties of Z(t)Z (t) Z(t)Z (t) = X m,n e (E m+e n) i(em En)t e Z(, 0)Z (, 0) = Z( ) 2 (= L 2 = e 2S for = 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( ) 2! Z(2 ). (= L = e S for = 0) Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

12 Properties of Z(t)Z (t) Z(t)Z (t) = X m,n e (E m+e n) i(em En)t e Z(, 0)Z (, 0) = Z( ) 2 (= L 2 = e 2S for = 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( ) 2! Z(2 ). (= L = e S for = 0) e 2S! e S, an exponential change. How does this occur? Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

13 SYK as a toy model SYK can serve as a toy model to address these questions: [see Stanford s talk] Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

14 SYK as a toy model SYK can serve as a toy model to address these questions: [see Stanford s talk] has a sector dual to AdS 2 dilaton gravity Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

15 SYK as a toy model SYK can serve as a toy model to address these questions: [see Stanford s talk] has a sector dual to AdS 2 dilaton gravity Chaotic, discrete spectrum Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

16 SYK as a toy model SYK can serve as a toy model to address these questions: [see Stanford s talk] has a sector dual to AdS 2 dilaton gravity Chaotic, discrete spectrum G(t, t 0 ), (t, t 0 ) description has aspects reminiscent of a bulk description: O(N) singlets nonlocal Nonperturbatively well defined (two replicas) Z hz(t)z (t)i = dg ab d ab exp( NI(G ab, ab )) Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

17 SYK as a toy model SYK can serve as a toy model to address these questions: [see Stanford s talk] has a sector dual to AdS 2 dilaton gravity Chaotic, discrete spectrum G(t, t 0 ), (t, t 0 ) description has aspects reminiscent of a bulk description: O(N) singlets nonlocal Nonperturbatively well defined (two replicas) Z hz(t)z (t)i = dg ab d ab exp( NI(G ab, ab )) Finite dimensional Hilbert space, D = L =2 N/2,amenableto numerics Guidance about what to look for Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

18 ZZ (t) in SYK [Jordan Cotler, Guy Gur-Ari, Masanori Hanada, Joe Polchinski, Phil Saad, Stephen Shenker, Douglas Stanford, Alex Streicher, Masaki Tezuka] ([CGHPSSSST]) See also [Garcia-Garcia Verbaarschot] Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

19 Meaning SYK, N m = 34, 90 samples, β=5, g(t ) Results 10-2 g(t ) Time t /J -1 Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

20 Meaning g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) Results The Slope $ Semiclassical quantum gravity Time t /J -1 Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

21 Meaning g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) Results The Slope $ Semiclassical quantum gravity The Ramp and Plateau $ Random Matrix Theory Time t /J -1 Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

22 Meaning g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) Results The Slope $ Semiclassical quantum gravity The Ramp and Plateau $ Random Matrix Theory The Dip $ crossover time Time t /J -1 Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

23 α t 3 Slope, contd. g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) Slope is determined by semiclassical quantum gravity nonuniversal Time t /J gn(β,t) t gbtz(β,t) g(β,t) gn(β,t) gn(β,2πn) ZBTZ(2β) t Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

24 α t 3 Slope, contd. g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) Slope is determined by semiclassical quantum gravity nonuniversal Time t /J -1 In SYK slope 1/t 3. One loop exact Schwarzian result: (E) e S 0 (E E 0 ) 1/2 ([Bagrets-Altland-Kamenev; CGBPSSSST; Stanford-Witten]) gn(β,t) t gbtz(β,t) g(β,t) gn(β,t) gn(β,2πn) ZBTZ(2β) t Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

25 α t 3 Slope, contd. g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) Slope is determined by semiclassical quantum gravity nonuniversal Time t /J -1 In SYK slope 1/t 3. One loop exact Schwarzian result: (E) e S 0 (E E 0 ) 1/2 ([Bagrets-Altland-Kamenev; CGBPSSSST; Stanford-Witten]) g(β,t) gn(β,t) t gbtz(β,t) gn(β,t) gn(β,2πn) ZBTZ(2β) In BTZ summing over modular transforms of blocks gives oscillating slope with power law envelope: nonperturbatively small oscillations in the density of states t [Dyer Gur-Ari] Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

26 The Ramp and Plateau SYK, N m = 34, 90 samples, β=5, g(t ) The Ramp and Plateau are signatures of Random Matrix Statistics, believed to be universal in quantum chaotic systems 10-2 g(t ) Time t /J -1 Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

27 The Ramp and Plateau g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) The Ramp and Plateau are signatures of Random Matrix Statistics, believed to be universal in quantum chaotic systems hzz (t)i is essentially the Fourier transform of (2) (E, E 0 ), the pair correlation function Time t /J -1 Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

28 The Ramp and Plateau g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) The Ramp and Plateau are signatures of Random Matrix Statistics, believed to be universal in quantum chaotic systems hzz (t)i is essentially the Fourier transform of (2) (E, E 0 ), the pair correlation function Time t /J -1 (2) (E, E 0 ) 1 [Dyson; Gaudin; Mehta] sin 2 (L(E E 0 )) (L(E E 0 )) 2 Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

29 The Ramp and Plateau g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) The Ramp and Plateau are signatures of Random Matrix Statistics, believed to be universal in quantum chaotic systems hzz (t)i is essentially the Fourier transform of (2) (E, E 0 ), the pair correlation function Time t /J -1 (2) (E, E 0 ) 1 [Dyson; Gaudin; Mehta] sin 2 (L(E E 0 )) (L(E E 0 )) 2 The decrease before the plateau is due to anticorrelation of levels Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

30 The Ramp and Plateau g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) The Ramp and Plateau are signatures of Random Matrix Statistics, believed to be universal in quantum chaotic systems hzz (t)i is essentially the Fourier transform of (2) (E, E 0 ), the pair correlation function Time t /J -1 (2) (E, E 0 ) 1 [Dyson; Gaudin; Mehta] sin 2 (L(E E 0 )) (L(E E 0 )) 2 The decrease before the plateau is due to anticorrelation of levels Conjecture that this pattern is universal in quantum black holes Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

31 The Ramp and Plateau g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) The Ramp and Plateau are signatures of Random Matrix Statistics, believed to be universal in quantum chaotic systems hzz (t)i is essentially the Fourier transform of (2) (E, E 0 ), the pair correlation function Time t /J -1 (2) (E, E 0 ) 1 [Dyson; Gaudin; Mehta] sin 2 (L(E E 0 )) (L(E E 0 )) 2 The decrease before the plateau is due to anticorrelation of levels Conjecture that this pattern is universal in quantum black holes Some evidence for this in melonic models [Witten; Gurau; Carrozza-Tanasa; Klebanov-Tarnopolsky; Krishnan-Kumar-Sanyal] Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

32 NversusL (2) (E, E 0 ) 1 sin 2 (L(E E 0 )) (L(E E 0 )) 2 Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

33 NversusL (2) (E, E 0 ) 1 sin 2 (L(E E 0 )) (L(E E 0 )) 2 t t p, (2) (E, E 0 1 ) 1, spectral rigidity L 2 (E E 0 ) 2 Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

34 NversusL (2) (E, E 0 ) 1 sin 2 (L(E E 0 )) (L(E E 0 )) 2 t t p, (2) (E, E 0 1 ) 1, spectral rigidity L 2 (E E 0 ) perturbative in RMT, L 2 L 2 e cn, nonperturbative in 1 N,SYK. Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

35 NversusL (2) (E, E 0 ) 1 sin 2 (L(E E 0 )) (L(E E 0 )) 2 t t p, (2) (E, E 0 1 ) 1, spectral rigidity L 2 (E E 0 ) perturbative in RMT, L 2 L 2 e cn, nonperturbative in 1 N,SYK. sin 2 (L(E E 0 ))! exp( 2L(E E 0 )), Altshuler-Andreev instanton Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

36 NversusL (2) (E, E 0 ) 1 sin 2 (L(E E 0 )) (L(E E 0 )) 2 t t p, (2) (E, E 0 1 ) 1, spectral rigidity L 2 (E E 0 ) perturbative in RMT, L 2 L 2 e cn, nonperturbative in 1 N,SYK. sin 2 (L(E E 0 ))! exp( 2L(E E 0 )), Altshuler-Andreev instanton exp( e cn )insyk(!) Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

37 NversusL (2) (E, E 0 ) 1 sin 2 (L(E E 0 )) (L(E E 0 )) 2 t t p, (2) (E, E 0 1 ) 1, spectral rigidity L 2 (E E 0 ) perturbative in RMT, L 2 L 2 e cn, nonperturbative in 1 N,SYK. sin 2 (L(E E 0 ))! exp( 2L(E E 0 )), Altshuler-Andreev instanton exp( e cn )insyk(!) How are these e ects realized in the G, formulation? Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

38 NversusL (2) (E, E 0 ) 1 sin 2 (L(E E 0 )) (L(E E 0 )) 2 t t p, (2) (E, E 0 1 ) 1, spectral rigidity L 2 (E E 0 ) perturbative in RMT, L 2 L 2 e cn, nonperturbative in 1 N,SYK. sin 2 (L(E E 0 ))! exp( 2L(E E 0 )), Altshuler-Andreev instanton exp( e cn )insyk(!) How are these e ects realized in the G, formulation? q = 2 SYK in progress [Saad, SS] Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

39 Onset of RMT behavior [Gharibyan-Hanada-SS-Tezuka, in progress] SYK, N m = 34, 90 samples, β=5, g(t ) At what time does the ramp begin? 10-2 g(t ) Time t /J N = 34, 90 samples 0.5 Density of states ρ(ε) Energy ε Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

40 Onset of RMT behavior [Gharibyan-Hanada-SS-Tezuka, in progress] g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) At what time does the ramp begin? The dip is just a crossover: edge versus bulk dynamics Time t /J N = 34, 90 samples 0.5 Density of states ρ(ε) Energy ε Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

41 Onset of RMT behavior [Gharibyan-Hanada-SS-Tezuka, in progress] g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) At what time does the ramp begin? The dip is just a crossover: edge versus bulk dynamics The Thouless time [Garcia-Garcia Verbaarschot] Time t /J N = 34, 90 samples 0.5 Density of states ρ(ε) Energy ε Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

42 Onset of RMT behavior [Gharibyan-Hanada-SS-Tezuka, in progress] g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) At what time does the ramp begin? The dip is just a crossover: edge versus bulk dynamics The Thouless time [Garcia-Garcia Verbaarschot] Time t /J -1 Single particle hopping, n sites: di usion time, t th n N = 34, 90 samples 0.5 Density of states ρ(ε) Energy ε Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

43 Onset of RMT behavior [Gharibyan-Hanada-SS-Tezuka, in progress] g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) At what time does the ramp begin? The dip is just a crossover: edge versus bulk dynamics The Thouless time [Garcia-Garcia Verbaarschot] Time t /J -1 Single particle hopping, n sites: di usion time, t th n 2 Density of states ρ(ε) N = 34, 90 samples Follow the ramp below the slope: use Gaussian filter [Stanford] Y (, t)y (, t) = X m,n e (E 2 n +E 2 m) e +i(em En)t Energy ε Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

44 YY SYK, N m = 34, 90 samples, β=5, g(t ) Dip time t d 200, N = 34 g(t ) Time t /J -1 g(t, β = 0), <YY * /Y(t=0) 2 >(α) g(β = 0) <YY * /Y(t=0) 2 >(α=2.5) Time J t Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

45 YY g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) Dip time t d 200, N = 34 Onset of ramp t r. 10, N = Time t /J -1 g(t, β = 0), <YY * /Y(t=0) 2 >(α) g(β = 0) <YY * /Y(t=0) 2 >(α=2.5) Time J t Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

46 YY g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) Dip time t d 200, N = 34 Onset of ramp t r. 10, N = 34 g(t, β = 0), <YY * /Y(t=0) 2 >(α) Time t /J -1 g(β = 0) <YY * /Y(t=0) 2 >(α=2.5) Time J t (The ramp is an exponentially subleading e ect in ZZ and correlation functions before the dip) Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

47 YY g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) Dip time t d 200, N = 34 Onset of ramp t r. 10, N = 34 g(t, β = 0), <YY * /Y(t=0) 2 >(α) Time t /J -1 g(β = 0) <YY * /Y(t=0) 2 >(α=2.5) Time J t (The ramp is an exponentially subleading e ect in ZZ and correlation functions before the dip) An upper bound. Very little variation in N for N apple 34 Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

48 YY g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) Dip time t d 200, N = 34 Onset of ramp t r. 10, N = 34 g(t, β = 0), <YY * /Y(t=0) 2 >(α) Time t /J -1 g(β = 0) <YY * /Y(t=0) 2 >(α=2.5) Time J t (The ramp is an exponentially subleading e ect in ZZ and correlation functions before the dip) An upper bound. Very little variation in N for N apple 34 log N? scrambling? Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

49 YY g(t ) SYK, N m = 34, 90 samples, β=5, g(t ) Dip time t d 200, N = 34 Onset of ramp t r. 10, N = 34 g(t, β = 0), <YY * /Y(t=0) 2 >(α) Time t /J -1 g(β = 0) <YY * /Y(t=0) 2 >(α=2.5) Time J t (The ramp is an exponentially subleading e ect in ZZ and correlation functions before the dip) An upper bound. Very little variation in N for N apple 34 log N? scrambling? Maybe; no. Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

50 Geometrically local qubits 10 0 Local QSG, N = n geometrically local qubits g(t)=< tr U(t) 2 > t 250 Local QSG 200 Ramp Time N Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

51 Geometrically local qubits 10 0 Local QSG, N = 15 g(t)=< tr U(t) 2 > n geometrically local qubits H = P i J i i i+1, J random t 250 Local QSG 200 Ramp Time N Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

52 Geometrically local qubits 10 0 Local QSG, N = 15 g(t)=< tr U(t) 2 > n geometrically local qubits H = P i J i i Scrambling time n i+1, J random t 250 Local QSG 200 Ramp Time N Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

53 Geometrically local qubits 10 0 Local QSG, N = 15 g(t)=< tr U(t) 2 > t Local QSG 250 n geometrically local qubits H = P i J i i Scrambling time n i+1, J random Gaussian density of states! slope exp( Nt 2 ), rapid decay 200 Ramp Time N Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

54 Geometrically local qubits 10 0 Local QSG, N = 15 g(t)=< tr U(t) 2 > t Local QSG n geometrically local qubits H = P i J i i Scrambling time n i+1, J random Gaussian density of states! slope exp( Nt 2 ), rapid decay t r n 2?di usion? Ramp Time N Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

55 Geometrically local qubits 10 0 Local QSG, N = 15 g(t)=< tr U(t) 2 > Ramp Time t Local QSG n geometrically local qubits H = P i J i i Scrambling time n i+1, J random Gaussian density of states! slope exp( Nt 2 ), rapid decay t r n 2?di usion? Maybe not scrambling N Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

56 Brownian circuits Scrambling describes the growth of a simple operator [Roberts-Stanford-Susskind; Lieb-Robinson] Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

57 Brownian circuits Scrambling describes the growth of a simple operator [Roberts-Stanford-Susskind; Lieb-Robinson] Generic. Also happens in Brownian circuit Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

58 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 ih m 1 t...e ih 1 t Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

59 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 ih m 1 t...e ih 1 t H m drawn from an ensemble Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

60 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 ih m 1 t...e ih 1 t H m drawn from an ensemble Unitary gates U = U m U m 1...U 1 Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

61 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 ih m 1 t...e ih 1 t H m drawn from an ensemble Unitary gates U = U m U m 1...U 1 Can analyze dynamics including scrambling analytically [Oliveira-Dahlsten-Plenio; Lashkari-Stanford-Hastings-Osborne-Hayden; Harrow-Low;...] Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

62 Markov chain Study U U, = P p p p, p a string of Paulis Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

63 Markov chain Study U U, = P p p p, p a string of Paulis (Need k copies for k design) Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

64 Markov chain Study U U, = P p p p, p a string of Paulis (Need k copies for k design) Defines a Markov process on on Pauli strings e.g., IIIZZIIXII... Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

65 Markov chain Study U U, = P p p p, p a string of Paulis (Need k copies for k design) Defines a Markov process on on Pauli strings e.g., IIIZZIIXII... Random two qubit gates: I I! I I; AB! 15 other possibilities, uniformly (for k = 2) [Harrow-Low] Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

66 Markov chain Study U U, = P p p p, p a string of Paulis (Need k copies for k design) Defines a Markov process on on Pauli strings e.g., IIIZZIIXII... Random two qubit gates: I I! I I; AB! 15 other possibilities, uniformly (for k = 2) [Harrow-Low] Initial condition for an OTOC: ZIIIIIIIIII Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

67 Markov chain Study U U, = P p p p, p a string of Paulis (Need k copies for k design) Defines a Markov process on on Pauli strings e.g., IIIZZIIXII... Random two qubit gates: I I! I I; AB! 15 other possibilities, uniformly (for k = 2) [Harrow-Low] Initial condition for an OTOC: ZIIIIIIIIII Time to randomize last qubit n, scramblingtime[nahum-vijay-haah; Keyserlingk-Rakovszky-Pollmann-Sondhi] Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

68 Markov chain, contd. For spectral statistics study htr(u k )tr((u ) k )i, k =1, 2... Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

69 Markov chain, contd. For spectral statistics study htr(u k )tr((u ) k )i, k =1, 2... RMT statistics htr(u k )tr((u ) k )i!haar average value Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

70 Markov chain, contd. For spectral statistics study htr(u k )tr((u ) k )i, k =1, 2... RMT statistics htr(u k )tr((u ) k )i!haar average value For k = 2 (two design) slowest terms are like U aa U aau aa U aa (no sum) Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

71 Markov chain, contd. For spectral statistics study htr(u k )tr((u ) k )i, k =1, 2... RMT statistics htr(u k )tr((u ) k )i!haar average value For k = 2 (two design) slowest terms are like U aa UaaU aa Uaa (no sum) Study U aiha U where ai = 0000i Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

72 Markov chain, contd. For spectral statistics study htr(u k )tr((u ) k )i, k =1, 2... RMT statistics htr(u k )tr((u ) k )i!haar average value For k = 2 (two design) slowest terms are like U aa UaaU aa Uaa (no sum) Study U aiha U where ai = 0000i 0000ih0000 =( 1 2 )n (I + Z) n Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

73 Markov chain, contd. For spectral statistics study htr(u k )tr((u ) k )i, k =1, 2... RMT statistics htr(u k )tr((u ) k )i!haar average value For k = 2 (two design) slowest terms are like U aa UaaU aa Uaa (no sum) Study U aiha U where ai = 0000i 0000ih0000 =( 1 2 )n (I + Z) n ZIZZIIZZI... Easy to equilibrate Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

74 Markov chain, contd. For spectral statistics study htr(u k )tr((u ) k )i, k =1, 2... RMT statistics htr(u k )tr((u ) k )i!haar average value For k = 2 (two design) slowest terms are like U aa UaaU aa Uaa (no sum) Study U aiha U where ai = 0000i 0000ih0000 =( 1 2 )n (I + Z) n ZIZZIIZZI... Easy to equilibrate Equilibration time log n, shorterthanscrambling! Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

75 Markov chain, contd. For spectral statistics study htr(u k )tr((u ) k )i, k =1, 2... RMT statistics htr(u k )tr((u ) k )i!haar average value For k = 2 (two design) slowest terms are like U aa UaaU aa Uaa (no sum) Study U aiha U where ai = 0000i 0000ih0000 =( 1 2 )n (I + Z) n ZIZZIIZZI... Easy to equilibrate Equilibration time log n, shorterthanscrambling! Correlation functions of very complicated operators [Roberts-Yoshida; Cotler-Hunter-Jones-Liu-Yoshida] Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

76 Evaporation and RMT For geometrically local Hamiltonian systems (in contrast to Brownian circuits) it looks like some kind of propagation (di usion?) is occurring: t n p Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

77 Evaporation and RMT For geometrically local Hamiltonian systems (in contrast to Brownian circuits) it looks like some kind of propagation (di usion?) is occurring: t n p For q-local systems like SYK n p! log n [Susskind] Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

78 Evaporation and RMT For geometrically local Hamiltonian systems (in contrast to Brownian circuits) it looks like some kind of propagation (di usion?) is occurring: t n p For q-local systems like SYK n p! log n [Susskind] A plausible guess Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

79 Evaporation and RMT For geometrically local Hamiltonian systems (in contrast to Brownian circuits) it looks like some kind of propagation (di usion?) is occurring: t n p For q-local systems like SYK n p! log n [Susskind] A plausible guess Important because the black hole evaporation time is S n. Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

80 Evaporation and RMT For geometrically local Hamiltonian systems (in contrast to Brownian circuits) it looks like some kind of propagation (di usion?) is occurring: t n p For q-local systems like SYK n p! log n [Susskind] A plausible guess Important because the black hole evaporation time is S n. So these phenomena would appear in small black holes as well, although as an exponentially subleading e ect Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17

81 Evaporation and RMT For geometrically local Hamiltonian systems (in contrast to Brownian circuits) it looks like some kind of propagation (di usion?) is occurring: t n p For q-local systems like SYK n p! log n [Susskind] A plausible guess Important because the black hole evaporation time is S n. So these phenomena would appear in small black holes as well, although as an exponentially subleading e ect We need to know what they mean in quantum gravity! Stephen Shenker (Stanford University) Black holes and random matrices Strings / 17