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Black Holes and Random Matrices Black holes and random matrices Stephen Shenker Stanford University Natifest Stephen Shenker (Stanford University) Black holes and random matrices Natifest 1 / 38 Stephen Shenker (Stanford University) Black holes and random matrices Natifest 2 / 38 Finite black hole entropy from the bulk What accounts for the finiteness of the black hole entropy{from the bulk point of view? For an observer hovering outside the horizon, what are the signatures of the Planckian \graininess" of the horizon? Stephen Shenker (Stanford University) Black holes and random matrices Natifest 3 / 38 A diagnostic A simple diagnostic [Maldacena]. Let O be a bulk (smeared boundary) operator. hO(t)O(0)i = tr e−βH O(t)O(0) =tre−βH X X = e−βEm jhmjOjnij2ei(Em−En)t = e−βEn m;n n At short times can treat the spectrum as continuous. hO(t)O(0)i generically decays exponentially. Perturbative quantum gravity{quasinormal modes. [Horowitz-Hubeny] Stephen Shenker (Stanford University) Black holes and random matrices Natifest 4 / 38 A diagnostic, contd. X X hO(t)O(0)i = e−βEm jhmjOjnij2ei(Em−En)t = e−βEn m;n n But we expect the black hole energy levels to be discrete (finite entropy) and generically nondegenerate (chaos). Then at long times hO(t)O(0)i oscillates in an erratic way. It is exponentially small and no longer decreasing. A nonperturbative effect in quantum gravity. (See also [Dyson-Kleban-Lindesay-Susskind; Barbon-Rabinovici]) Stephen Shenker (Stanford University) Black holes and random matrices Natifest 5 / 38 Another diagnostic, Z(t)Z ∗(t) 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 Natifest 6 / 38 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 Assume the levels are discrete (finite entropy) and non-degenerate (generic, implied by chaos) In the long time average hZ(β; t)Z ∗(β; t)i = 1 R T0 dt Z(β; t)Z ∗(β; t) T0 2T0 −T0 the oscillating phases go to zero and only the n = m terms contribute. ∗ In the limit T0 ! 1; hZ(β; t)Z (β; t)iT0 ! Z(2β) Z(β)2 ! Z(2β). Roughly e2S ! eS . An exponential change. We will focus on the nature of this transition. Stephen Shenker (Stanford University) Black holes and random matrices Natifest 7 / 38 A model system 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] Stephen Shenker (Stanford University) Black holes and random matrices Natifest 8 / 38 SYK Model Sachdev-Ye-Kitaev model: Quantum mechanics of N Majorana fermions with random couplings X H = Jabcd χaχbχc χd a;b;c;d fχa; χbg = δab 1 hJ2 i = J2 abcd N3 dim H = 2N=2 = L (We will often set J = 1.) Stephen Shenker (Stanford University) Black holes and random matrices Natifest 9 / 38 SYK as a good model system Maximally chaotic [Kitaev] Gravitational sector with horizon (with enhanced amplitude), but stringy states in bulk [Maldacena-Stanford] J average plays role of time average. ∗ hZ(t)Z (t)iJ is a smooth function of t. Z = R dG(t; t0)dΣ(t; t0) exp(−NI [G; Σ]) Proxy for a bulk theory. 1=N ∼ GN . Finite dimensional H. N = 34 ! L = 217 = 128K. Numerics feasible.... Stephen Shenker (Stanford University) Black holes and random matrices Natifest 10 / 38 SYK Z(t)Z ∗(t) 100 SYK, Nm = 34, 90 samples, β=5, g(t ) 10-1 The Slope 10-2 The Dip ) The Ramp t ( -3 g 10 The Plateau 10-4 What do they mean? 10-5 10-1 100 101 102 103 104 105 106 Time t /J -1 Stephen Shenker (Stanford University) Black holes and random matrices Natifest 11 / 38 Random Matrix Theory Chaotic quantum systems typically have fine grained energy level statistics described by Random Matrix Theory (RMT) [Wigner; Dyson; Bohigas-Giannoni-Schmit...] Consider a simple model where H ! M, an L × L (hermitian) random matrix R L 2 Z ∼ dMij exp(− 2 trM ) GUE ensemble ∗ Compute hZ(t)Z (t)iM The spectral form factor of RMT (β = 0) RMT connection and N mod 8 ensemble variation for near neighber eigenvalue statistics [You-Ludwig-Xu]. We will focus on longer range correlations, in fourier transform. Stephen Shenker (Stanford University) Black holes and random matrices Natifest 12 / 38 Random Matrix Theory Z(t)Z ∗(t) 0 Random, N = 4096, 1200 samples, β=5, g(t ) 10 dim 10-1 ) t ( g 10-2 10-3 10-1 100 101 102 103 104 105 106 Time t /J -1 Stephen Shenker (Stanford University) Black holes and random matrices Natifest 13 / 38 RMT spectral form factor, contd. 0 Random, N = 4096, 1200 samples, β=5, g(t ) 10 dim 10-1 ) t ( g 10-2 X Z(t)Z ∗(t) = e−β(Em+En)ei(Em−En)t m;n 10-3 The plateau: n = m only nonzero contribution -1 0 1 2 3 4 5 6 What about10 the10 rest of10 the features:10 10 the ramp,10 and10 the10 slope? Time t /J -1 Stephen Shenker (Stanford University) Black holes and random matrices Natifest 14 / 38 Lightning review of RMT theory Z L Z = dM exp(− trM2) ij 2 Z Y L X = dE (E − E )2 exp(− E 2) j i j 2 i ij I Z = Dρ(E) exp(−I [ρ(E)]) Z Z E 2 I [ρ(E)] = −L2( dEdE 0ρ(E)ρ(E 0) log((E − E 0)2) + dEρ(E) ) 2 ρ(E) is the density of eigenvalues. The Dyson gas. Stephen Shenker (Stanford University) Black holes and random matrices Natifest 15 / 38 The Wigner semicircle Z Z E 2 I [ρ(E)] = −L2( dEdE 0ρ(E)ρ(E 0) log((E − E 0)2) + dEρ(E) ) 2 n The Dyson gas. 0.6 Random, dim = 4096, 1200 samples Saddle point at large L gives 0.5 ) ε p ( 2 ρ ρW (E) ∼ 1 − 4E 0.4 The Wigner semicircle law. 0.3 0.2 Smallest energy spacing Density of states δE ∼ 1=L. 0.1 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Energy ε/J Stephen Shenker (Stanford University) Black holes and random matrices Natifest 16 / 38 The slope in RMT 0 Random, N = 4096, 1200 samples, β=5, g(t ) 10 dim 10-1 n 0.6 Random, dim = 4096, 1200 samples ) ∗ ∗ At earlyt times we can approximate hZ(t)Z (t)i ∼ hZ(t)ihZ (t)i ( g -2 0.5 ) 10 R −(β+it)E ε ( hZ(t)i ∼ L dE ρW (E)e ρ 0.4 Long time behavior dominated by sharp0.3 edge of semicircle 10-3 R 1 −(β+it)E 2 0.2 hZ(t)i ∼ L dE(E − E0) e Density of states At long times, hZ(t)i ∼ L=t3=2 (β =0.1 0) ∗ 2 3 0 hZ(t)Z (t-1)i ∼ L0 =t 1 2 3 -2 -1.5 4 -1 -0.5 5 0 0.5 61 1.5 2 10 10 10 10 10 10 10Energy ε/J 10 Time t /J -1 Stephen Shenker (Stanford University) Black holes and random matrices Natifest 17 / 38 The ramp At later times fluctuations become important. X hZ(t)Z ∗(t)i = h e−β(En+Em)+it(En−Em)i n;m X = Z(2β) + h e−β(En+Em)+it(En−Em)i n6=m Z 0 0 = Z(2β) + L2 dEdE 0ρ(2)(E; E 0)e−β(E+E )+it(E−E ) ρ(2)(E; E 0) is the eigenvalue pair correlation function. hZ(t)Z ∗(t)i is essentially the fourier transform of ρ(2), the spectral form factor. Stephen Shenker (Stanford University) Black holes and random matrices Natifest 18 / 38 Spectral form factor in RMT (take β = 0 for convenience) Near the center of the semicircle ρ(2)(E; E 0) has a simple universal form given by the Sine kernel (GUE) [Dyson; Gaudin; Mehta] sin2(L(E − E 0)) ρ(2)(E; E 0) ∼ 1− (L(E − E 0))2 Stephen Shenker (Stanford University) Black holes and random matrices Natifest 19 / 38 Fourier transform of ρ(2) Its fourier transform (accounting for the n = m terms) gives hZ(t)Z ∗(t)i in GUE (β = 0) Plateau height is ∼ L, tp ∼ L, ramp is ∼ t. Stephen Shenker (Stanford University) Black holes and random matrices Natifest 20 / 38 Physical meaning of ramp For times shorter than tp ∼ L, can average over the rapidly oscillating sin2(L(E − E 0)) factor. ρ(2)(E − E 0) ∼ 1 − 1=(L(E − E 0))2. This gives the ramp. The ramp is lower than the plateau because of long range anticorrelation. Long wavelength fluctuations in the Dyson gas are strongly suppressed by the long distance log interaction. For example, near neighbor eigenvalue couplings would yield ρ ∼ jE − E 0j Spectral rigidity Analog of incompressibility in FQHE fluid Stephen Shenker (Stanford University) Black holes and random matrices Natifest 21 / 38 The dip time in RMT 0 Random, N = 4096, 1200 samples, β=5, g(t ) (β = 0)10 dim The slope in RMT is ∼ L2=t3 The ramp is ∼ t 10-1 They meet at the dip time ) 1=2 t td ∼( L g 10-2 The plateau time tp ∼ L −1=2 −S=2 td =tp ∼ L ∼ e -3 An exponentially10 long period dominated by spectral rigidity 10-1 100 101 102 103 104 105 106 Time t /J -1 Stephen Shenker (Stanford University) Black holes and random matrices Natifest 22 / 38 SYK analysis We now try to repeat this analysis for the SYK model.
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