Title Black Holes and Random Matrices Author(S)

Title Black holes and random matrices

Cotler, Jordan S.; Gur-Ari, Guy; Hanada, Masanori; Polchinski, Author(s) Joseph; Saad, Phil; Shenker, Stephen H.; Stanford, Douglas; Streicher, Alexandre; Tezuka, Masaki

Citation Journal of High Energy Physics (2017), 2017

Issue Date 2017-5

URL http://hdl.handle.net/2433/226332

Type Journal Article

Textversion publisher

Kyoto University JHEP05(2017)118 Springer May 9, 2017 May 22, 2017 : d,e a,d April 26, 2017 : : , Accepted Received Published 10.1007/JHEP05(2017)118 doi: Joseph Polchinski, , , Alexandre Streicher f [email protected] a,b,c , Published for SISSA by Douglas Stanford, [email protected] [email protected] Masanori Hanada, a , , a [email protected] [email protected] , , . 3 g Guy Gur-Ari, a 1611.04650 The Authors. c 1/N Expansion, AdS-CFT Correspondence, Field Theories in Lower Dimen-

We argue that the late time behavior of horizon fluctuations in large anti- Stephen H. Shenker, as well as correlation functions as diagnostics. Using numerical techniques we , a 2 [email protected] | ) it + β ( [email protected] Department of Physics, KyotoKyoto University, 606-8501, Japan E-mail: [email protected] [email protected] Santa Barbara, CA 93106, U.S.A. Kavli Institute for Theoretical Physics,Santa University Barbara, CA of 93106, California, U.S.A. Institute for Advanced Study, Princeton, NJ 08540, U.S.A. Stanford, CA 94305, U.S.A. Yukawa Institute for Theoretical Physics,Kyoto Kyoto 606-8502, University, Japan The Hakubi Center forKyoto Advanced Research, 606-8502, Kyoto Japan University, Department of Physics, University of California, Stanford Institute for Theoretical Physics, Stanford University, b c e g d a Z f Open Access Article funded by SCOAP estimate of the crossover time.understanding the We late make some time dynamics preliminary from comments about aKeywords: bulk challenges point to of view. sions, Random Systems ArXiv ePrint: | establish random matrix behaviorexactly at in late a times.to double random scaling We matrix determine limit, behavior. thelarge giving AdS early We us black use time holes, a these behavior like ideas plausible those to estimate dual formulate to for a 4D the conjecture super-Yang-Mills theory, about crossover giving general time a provisional Abstract: de Sitter (AdS) blackquantum holes chaotic is systems. governed Our by mainuse tool the as is random the a matrix Sachdev-Ye-Kitaev simple (SYK) dynamics model, model which characteristic of we of a black hole. We use an analytically continued partition function Jordan S. Cotler, Phil Saad, and Masaki Tezuka Black holes and random matrices JHEP05(2017)118 42 44 47 43 38 13 46 17 24 11 27 Σ variables 40 24 – i – 9 G, 33 34 5 45 and plateau height ) 15 t p t ( d g = 2 model 47 ) q E 3 21 ( 28 6 ρ ), and 37 t ( c g N , plateau time ), t d − ( t 1 g 30 Σ integral vs. 2 q G, − N H.1 Plots of H.2 Dip time H.3 Comparison ofH.4 factorized and unfactorized Density quantities of states 7.1 The ramp in more general theories 9.1 The ramp in SYM 5.1 The ramp and the dip time 3.1 The ramp and the eightfold way 1.1 Summary of results F On G Constraints on saddle point originsH of Data the ramp C A toy D Subleading saddle points in the E Saddle points and the 10 Discussion A Particle-hole symmetry of SYK B The double-scaled SYK theory 8 Single realization of random9 couplings Conjecture about super-Yang-Mills 5 Spectral form factor in random matrix theory 6 Spectral form factor in7 the SYK Correlation model functions 3 Spectral form factor 4 Thermodynamics of the SYK model Contents 1 Introduction 2 The Sachdev-Ye-Kitaev model JHEP05(2017)118 (1.2) (1.1) ]. The decay 3 . t ) n E are energy eigenstates with − ) and analytically continuing m i β E n ( ( | . i Z e 2 ) vanishes, which means that at late ) to oscillate rapidly and erratically. i i| ] applied these ideas to the study of t n iHt ( 2 is given by | − (0) β, t G ( O β | O βH Z ) t m − ( e |h O m ) cause – 1 – βH βE Tr − − 1.1 e e ≡ perspective of holographic duality. For large black h ) ) will generically decay exponentially in time, but t ( m,n Tr X β, t G ) ) ( bulk β β ) also oscillates erratically. 1 1 Z ( ( Z Z β, t ( ]. We define = is the partition function and 4 Z ) = t ) at inverse temperature ( t ( G βH O In fact, the time average of − ) can be obtained by starting with e 1 2 ] use this idea in a closely related context. ] pointed out a signature of a discrete energy spectrum that can (in 5 β, t 1 ( Z . At early times we can replace the sum over eigenvalues by a coarse grained . At late times ) = Tr n it β E ( + Z β The time average of an observable and its moments is a simple way to quantify its Holographically the coarse grained approximation is equivalent to a perturbative grav- There is a somewhat simpler diagnostic of a discrete energy spectrum, introduced in To understand the way in which a two-point function diagnoses a discrete energy Maldacena [ We assume that in a quantum fieldThe authors theory of the [ operator is suitably smeared to eliminate any short 1 2 → late time behavior. times this observable fluctuates around zero. The typicaldistance size divergences. of the fluctuations can be The quantity β ity calculation, and thecontinues exponential forever decay in to this approximation. quasinormal mode behavior [ the black hole context by [ energies integral over a smooththe decay density. does notbecomes continue important, indefinitely. and the At phasesThe late in correlation times ( function is the exponentially discreteness small of and the no spectrum longer decays. Here, spectrum we can expressHermitian it operator in the energy basis. The two-point correlation function of a like branes, is still mysterious. principle) be computed inat the very bulk late — time. thecorrelators lack in Dyson, of de Lindesay, decay Sitter and space. of Susskind two-point [ functions evaluated One of the deepblack hole questions microstates, in from quantumholes the gravity the is AdS/CFT the dualityboundary makes origin field the of theory answer the on clearBut discrete a from its spectrum compact the origin of boundary space from generically perspective bulk has — gravity a a or discrete string spectrum theory, even of including states. nonperturbative effects 1 Introduction JHEP05(2017)118 ) 9 1.5 (1.3) (1.4) (1.5) -fermion q =4 SYM, is given by N S ]. Either way, CFTs. These 7 ) by an integral , d ). (See section 6 1.3 1.5 ) by making a coarse , . t 1.5 ) βE survive. It is given by n 2 E n − 3 e − . E 2 E m ) 2 E ) = β N is a positive constant. So ( ( i β ( e m (2 a ) E Z E Z X n E 2 ] in the context of 2 ) = + . If the spectrum has no degeneracies β 1 m 2 (

E E 11 ) ( , Z ) is the black hole entropy which scales as β Majorana fermions with random ] is a good laboratory to explore these ques- β − 10 = ( β, t . e S 7 ( N , 2 Z

6 in matrix theories like super-Yang-Mills (SYM) Z – 2 – /N )

m,n X ) 2 2 β dt N ) ( β, t ( ) root-mean-square (RMS) height of the correlator. T β 1 Z is the entropy and ∼ 0 ( Z aS Z

Z S S − 1 T e dt are the string coupling and Newton constants of the bulk = ( T 2 O N 0

→∞ Z ) lim G ) where T 1 ] pointed out that such an instanton might not describe the T β 9 ( β, t aS ( and Z e Z s →∞ lim

g T in vector theories like the Sachdev-Ye-Kitaev model [ ) is non-perturbative in 1 N . In the holographic context ∼ ) is nonperturbative in the bulk coupling. For large black holes, 1.5 , where aS S N 1.5 is the degeneracy of the energy level − ].) Therefore, by studying how the long-time decay of the partition function (or e 8 /G E 1 N ∼ = 1), the long-time average becomes The Sachdev-Ye-Kitaev (SYK) model [ From the bulk perspective, Maldacena initially suggested that an instanton might be Now, suppose we attempt to compute the left-hand side of ( It is sometimes said that this problem is related to the question of why a black hole has finite entropy. 2 s E 3 generically scales as /g N tions. It is a quantum mechanical model of Indeed, in standard QM,theories, finite or entropy implies in a adensity discrete thermodynamic of spectrum, approximation, states. but for we example, note one that can in effectively disorder-averaged have a smooth but finite Barbon and Rabinovicidetails [ of the irregularin long-time fluctuations correlation expected functions in wasquestions the have also correlator. been studied Information difficult loss to indue address to [ in the standard difficulty holographic in contexts analyzing like the chaotic boundary theory with sufficient precision. and also [ of the correlator) is avoidedhole in spectrum gravity — we a are basic in characteristic fact of probing its the quantumresponsible discreteness nature. of for the the black analogous the quantity ( grained approximation. If weover replace a the smooth discrete densityanalytically sum continuing over we saddle states find points in that we ( also the find long-time disagreement average with vanishes. ( In holography, by 1 theory, so ( the thermal entropy of theof boundary freedom. field In theory, and particular, it wetheory, and have scales with the number of degrees ( Z scales as oscillating phases average to zero and only terms with where As in the case of thecomplicated. two-point function, the One late time can behavior make of this progress quantity is by generically taking the long-time average, where terms with studied by considering the squared quantity JHEP05(2017)118 5 ]. 35 we 2 ) singlet = 0, the ”/“nearly t N 2 ( O ]. Other related , appears in [ i 26 2 | ) ) average to zero = 4 SYM system. β, t 1.3 ( N Z ), which characterizes h| ), ( / . Starting at 5 1.5 1 1.1 4 ]. A supersymmetric generalization 24 , ]. Recent discussion of a connection be- 23 34 we write down the spectral form factor, 3 ]. (After we had finished our numerical analysis , where we show the relation between the choice – 3 – 32 , 3.1 ] that the spacing statistics of nearby energy lev- 31 ], a property that is characteristic of black holes in Ein- 36 . The theory is highly chaotic: at strong coupling it averaged over the random couplings. At late times this N 15 ]. A multiflavor version has been constructed in [ 2 ) 25 β ( /Z 2 | ]. As is the case for other vector models, there is an exact rewrite ) 22 – β, t . ( ]. It realizes a (highly curved) description of a “nearly AdS 19 Z 4 , | ]. 18 – ] the chaos bound [ 14 30 – 16 14 ] appeared. It has significant overlap with our numerical results.) 27 – 33 . Further evidence for the relation between the late time behavior and random 12 time, making these quantities smooth functions of time. This makes them more Σ that presumably are related to the bulk description. 2 G, ” system [ 1 fixed It is a widely held conjecture [ Building on these observations we can make a plausible conjecture about the behavior One goal of this paper is to explore the late time behavior of the SYK model. We The SYK model has several other properties that make it useful in the study of late time Higher dimensional versions of SYK have been constructed in [ Another discussion of random matrices in black hole physics is [ 4 5 in SYK. See figure of the model has beenwork constructed includes in [ [ tween chaotic systems and random ensembles, including observables generalizing fashion (we call this regionexplained if the we ‘ramp’), approximate joining the SYK ontoin Hamiltonian the figure by plateau. a Gaussian This randommatrix behavior matrix, theory as is (RMT) shown readily is given inof section RMT ensemble (GUE, GOE, or GSE) and the detailed shape of the late time behavior which is given by quantity goes over tothe a discreteness plateau of value theits given spectrum. late approximately time by By behavior ( spectral exhibits numerically form an computing factor interesting this first feature, dips quantity see below we figure its find plateau that value and then climbs back up in a linear of more complicated holographic systems, like the Type1.1 IIB AdS Summary of results Here we give anintroduce outline the of SYK the model. paper and Then summarize in the section main results. In section els in quantum chaoticmatrix systems ensemble. should Since be latenatural well times one. approximated corresponds by to an small appropriate energy random differences our result is a the paper [ present numerous numerical results aboutusing such a behavior variety of in analytic the andtionship conceptual model, between arguments. and the One late interpret of time them our behavior key of findings the is model a and close the rela- behavior of random matrices. properties. The averagesufficiently over to the make random the couplingsat rapidly should a oscillating rattle terms the inamenable energy equations to study. ( eigenvalues In addition, themethods model can is yield computationally significant simple insight enough [ that numerical saturates [ stein gravity [ CFT of the disorder-averaged modelfields in terms of a functional integral over bilocal couplings that is soluble at large JHEP05(2017)118 ] βJ 20 , effect . This 14 1 /N , large . N 10 . . We also argue 2 | N ) β, t ( , but we argue that by Z | N . We argue that the subsequent limit a sector of the model [ 2 Σ path integral, which is an exact βJ G, = 4 SYM, giving a preliminary estimate N , large Σ formulation of the model. We point out the N – 4 – G, ] black hole in AdS ]. 38 , 39 , 37 32 = 2 of it. We explain why this infinite family of saddle points q mod 8 [ we consider a toy model of the we review the particle-hole symmetry of the SYK model, whose prop- we again consider the we discuss the double-scaled limit of SYK, where the disorder-averaged N we discuss a similar ramp that appears in SYK correlators. We work C A D B we make a connection with we make a digression to discuss the thermodynamic properties of SYK. we review the analytical origin of the ramp and plateau in RMT, and the we consider the behavior of the spectral form factor for a single realization we explain the early-time power-law decay in SYK visible in figure 7 9 4 5 8 6 limit. This serves as an incisive check both on our results and on existing we find excellent agreement with existing analytical calculations carried out in N N In appendix In appendix In appendix In appendix We conclude and discuss futureSeveral directions appendices and contain ongoing additional work results in and section discussion. In section In section In section In section In section In section fermionic behavior can arise from these bosonic variables. existence of a family ofin subleading the saddle integrable points that version showdoes up not both significantly in affect the the SYK thermodynamics model of and the model at large erties depend on density of states can be computed exactly. rewrite of the SYK model in terms of bosonic bilocal fields. We explain how the original that there should bedominated a subsequent by long ‘ramp’ period physics,SYM of theory. folded time where against this the quantity coarse-grained is growing density and of states of the spectral form factortime exhibits averaging large (and no fluctuations disorderbrought even averaging) into the at view. underlying large ramp/plateau structure can be of the gravity saddle points that give the early-time decay of out the conditions under whichstructure the of fermion the two-point spectral function form exhibits factor, the and ramp/plateau check theseof results the numerically. random couplings.Yang-Mills which The do motivation not here involve an is averaging to over make couplings. contact For with a theories single realization such the as exact in a double scalingis limit. dual In the to large alinearly growing dilaton ramp gravity and the [ between plateau the should survive late in time this behavior limit, of suggesting black a holes connection and random matrix theory. relation of the rampbe to understood the phenomenon as of ain spectral perturbative SYK, rigidity. effect as in We we show RMT explain). that (though the not ramp as can is a related perturbative to 1 thelimit, low that energy is portion described of by the the spectrum, Schwarzian theory dominant of in reparametrizations. the large We argue this is We compute the entropyinfinite and energy numerically,the and large by extrapolatinganalytic these calculations. results to JHEP05(2017)118 i c . (2.2) (2.3) (2.4) (2.1) Q . q mod 2). Q = 2 version q . The Hamilto- Dirac fermions i c . i 2 N ¯ c d ψ d c = =1 N i ψ . d b P N ψ a = 0 . , = ψ ) 2 } i j is the number of fermions interacting ¯ c ¯ Q c N/ abcd 2 q , i J − ¯ c √ i { c ( i , X = a

) as follows. 8 2 ) = 0, at early times the 1.3 β ( ) for the SYK model. In t ) remain well within a percent t β ( /Z ; g , t 2 ( | g J ) i ) survive, and the height of the ) , β, t ( 1.3 J . β, t i Z ( ) ) | ∗ 2 J β

i ; Z ) t 2 J β, t ( β i ( · h d ( ) ∗ g J β Z Z i ( h ) ) − Z ) h – 6 – β β, t β, t requires just one), whereas quenched quantities ; ( ( t ) at length, so let us point out the main features 7 d ( Z Z t = 34, computed numerically. h h g ( g g we considered, and the difference appears to decrease with terms in the sum ( N ≡ ≡ ≡ β ) ) ) m ). As discussed in the introduction, the late-time behavior β β β ; ; ; E t t t ( ( ( , and beyond this we have an almost constant value of 1.2 c d = p g t g g ) and in quantum field theory in general. n = 5) for E βJ β ; t ( g . Next comes a period of linear growth that we will call the ‘ramp’. ) is smooth, and does not exhibit the large fluctuations that one expects d t t ( g requires two replicas, g ) was defined in ( ) drops quickly along what we will call the ‘slope’, until it reaches a minimum t denotes the disorder average — the average over the ensemble of random cou- β, t ( ( g J we present Z 1 h·i We will be discussing the curve Notice that Now we present one of the central results of this work, Numerically, we find that the quenched and annealed versions of All numerical results in this paper were computed by fully diagonalizing the SYK Hamiltonian for 8 7 of each other for(At all infinite times temperature and the valuesis annealed of independent and of quenched quantities the are random in couplings.) fact equalindependently because generated Gaussianthe random mean. couplings, computing the relevant quantity, and then taking at the ‘dip time’ It ends at thethat plateau we time call theOn ‘plateau’. the plateau The only plateau the height is equal to the long-time average of fluctuations are apparent at lateof times, samples used but in these the are computation. an artifact We will due discuss toin this the this point finite plot further number in andvalue section of introduce some nomenclature. Starting with plateau and then climbsimplications back of up. this behavior, One and goalvalues to of of estimate the this how parameter work prevalent it is is to both understand in the SYKat source (for various late and timessmooths in out a the fluctuations typical exhibited quantum by each theory. realization of This the random is couplings. due (Some to the disorder average, which calculations ( require an arbitrary number of replicas. figure early times does not simply join onto the late-time plateau, but instead dips below the of these quantities probes theior discreteness of of two-point the functions. spectrum,that similar Notice to we that the we are late-time are behav- takingThis working is with the in annealed contrast disorder with quantities, averageworking quenched meaning with quantities separately annealed such as quantities in is the that they numerator require a and finite denominator. number of replicas in analytic Here plings. We define disorder-averaged analogs of the quantity in equation ( 3 Spectral form factor JHEP05(2017)118 , ramp and it is 7 = 34. Here we use the 10 N 6 10 spectral form factor ) . We call this increase the t ( ). The factor of 2 is due to a 4 g 5 10 10 1.4 =5, × β 4 ). = 3 10 A tJ tJ 3 ]: Gaussian Unitary Ensemble (GUE), Gaus- 10 . After that the value increases roughly linearly, Time 40 – 7 – = 0) is called the 2 dip = 34, 90 samples, = 34, 90 samples, m β 10 N = 5), plotted against time for are studied extensively in the field of quantum chaos. 1 β c SYK, ; 10 t g , in accordance with ( ( g aS 0 − 10 e 9 , and for time. Initially the value drops quickly, through a region we call d . The data was taken using 90 independent samples, and the disorder ∼ g -1 tJ , ) 10 g 0 -2 -3 -4 -5 -1 β

( 10

10 10 10 10 10 2

) t ( g /Z ) (typically used with ) t β plateau time ( g (2 Z ) by this name. It supplies information about the correlations of eigenvalues at ].) The particular ensemble to use depends on the symmetries of the original t ( . A log-log plot of SYK , to a minimum, which we call the g 41 ] first discussed the quantum chaotic properties of SYK by studying the distribu- , 32 36 One of the basic conjectures in the field of quantum chaos is that the fine grained en- Quantities such as slope The spectral form factor contains information about the pair correlation between well-separated eigen- , until it smoothly connects to a plateau around 9 t observable. They showed that the distribution is consistent with RMT predictions. values that the (perhaps morenot. familiar) diagnostic Conversely, of the thecorrelation nearest-neighbor nearest-neighbor functions energy of spacing level nearby distribution spacing does eigenvalues that distribution the contains spectral information form about factor does multi-point not. see [ Hamiltonian. Random matrix theory canas then the be spectral used form to factor) computeXu that certain [ quantities are (such sensitive totion eigenvalue of correlations. spacings You, between Ludwig nearest-neighbor and energy levels, another standard quantum chaos different energy separations. ergy eigenvalue structure of a chaoticfrom system one is of the the same standard assian Dyson that Orthogonal ensembles of [ Ensemble a (GOE), random matrix or chosen Gaussian Symplectic Ensemble (GSE). (For reviews, 2-fold degeneracy in the spectrum (see appendix In particular, a standard diagnostic of therefer pair to correlation function of energy eigenvalues. We will often and the time atplateau which level the the extrapolated linearaverage fit was of taken for the the ramp numerator in and the denominator log-log separately. plot crosses the fitted plateau is 2 Figure 1 dimensionless combination the ∼ JHEP05(2017)118 . , 12 12 and = 2 = 2 L . On the RMT form, after RMT /L | /N L m − n | . For comparison, if | 6 m 10 − n 5 | ]. The log 10 ) t 41 ( log , g 4 denotes the deviation of an energy 40 =5, , i ∼ 10 β n m 36 δE δE 3 n is apparent. 10 tJ 1 δE h ) is determined by eigenvalue correlations on t 2 Time ( – 8 – g 10 , the inverse mean level spacing. 1 = 4096, 1200 samples, = 4096, 1200 L = 0 the height of the plateau is of order 1 10 RMT β L GUE, , while SYK is governed by an expansion in 1 0 = 5) for the GUE ensemble of matrices of rank = 5) against time for GUE random matrices, dimension 10 /L β β ; ; t t )). At ( ( ]. More quantitatively, if g g -1 , a much less rigid behavior [ | 5.1 of order 41 10 0 , -2 -3 -4 -5 -1 t m

10 10 10 10 10

− ) t ( g 40 , n 36 2 (see ( i ∼ | we present m 2 δE < λ < . A log-log plot of n 2 − δE h What is the physical origin of the ramp in RMT? Eigenvalues of generic matrices In figure Note the similarity between the RMT result and the SYK result, and in particular from its average value, thenthe eigenvalues at formed leading a one order dimensionalthen crystal with harmonic near neighborsuitable interactions, processing we will discuss below,ramp accounts lies for below the the linear behavior plateau of because the repulsion ramp. causes The the eigenvalues to be anticorrelated. repel, so nearof degeneracies onset are of extremely theThe unlikely. plateau ramp, is This though, determined is causesspectrum. by due the the to This plateau. the scale repulsion,gives repulsion when of The rise between balanced near to time a eigenvalues against neighbor very that the rigid eigenvaluespectral are eigenvalue effects rigidity spacings. structure. far [ that This apart phenomenon keep in is the referred the energy to as finite, long-range a perturbative expansion in 1 other hand, at times wellscales beyond much the smaller dip, thanagreement the between total SYK width and RMT. of the spectrum, and there one expects to find the presence of thethe ramp spectral and form the factortime plateau. in behavior We SYK of will RMT cantypical argue differs be eigenvalue that from explained density the SYK, has by late-time althoughleads different random behavior it to dependence matrix of is somewhat on theory. not different energy initial obvious The in decays. from early the the Moreover, two plots. at systems, early The which times RMT is governed by computed numerically, with a normalizationrange such that thethe eigenvalues typically plateau lie time is in at the Figure 2 A dip, ramp and plateau structure similar to figure JHEP05(2017)118 (3.4) ]. These 1. The between 43 − s = 2 P ) is sensitive to t ( g ] for details). energy levels [ 39 A , 32, and we see excellent , = 1. The corresponding 32 2 , P = 30 31 unfolded N , ) i maps the even and odd parity sectors c + P i c , while those of a chaotic theory generally (¯ s d ) determines the scale of the energy differences − =1 t N e i Y – 9 – ( The nearest-neighbor statistics of an integrable g K 10 = ]. P 42 mod 8) as follows (see appendix ]. N again maps each sector to itself but now 41 maps each sector to itself and ] studied the nearest-neighbor level spacing distribution in P P parameter in 32 t mod 8 = 4, mod 8 = 2 or 6, the symmetry mod 8 = 0, N N N is an anti-linear operator. The properties of this operator determine the class of shows the nearest-neighbor statistics of SYK with K 3 When corresponding ensemble is GSE. to each other.corresponding ensemble Individual of sectors each do sector is not GUE. When have any anti-linear symmetry,ensemble is and GOE. the When While the nearest-neighbor spacing distribution is sensitive to correlations between The SYK model has a particle-hole symmetry given by [ You, Ludwig and Xu [ The Hamiltonian of a chaotic theory is generally believed to resemble a random matrix More precisely, one considers the distribution of spacings between • • • 10 structure of these correlationsexhibit depends a on ramp the and a ensemble. plateau The but three withare RMT slightly the levels ensembles different one shapes: all everywhere. obtains For by in further making details, GUE a see the change [ of (unfolded) variables such that the mean level spacing becomes one adjacent energy levels, theat spectral larger form separations. factor The probesbeing correlations probed. between As energy discussed levels are above, probed, beyond while thecorrelations plateau at between time earlier levels only that times individual are energy (and much levels farther in apart particular than the on mean the level ramp) spacing. The Figure agreement with RMT predictions. where RMT statistics of each charge parityare sector determined of by the the Hamiltonian. value In of particular, ( the statistics SYK. They made themented important in point the that model all as three we Gaussian now RMT review. ensembles are imple- pairs of neighboring energytheory levels follow [ an exponentialfollow one distribution of the threedepends reference ensembles on GUE, the GOE, symmetries and of GSE. the The Hamiltonian. particular ensemble We now present furtherpresence evidence of of the the ramp relation in between the SYK random spectral matrix formwhen theory factor. studied and at the sufficiently fineis energy their resolution. nearest-neighbor One level basic statistics, property namely of the random distribution matrices of the distance 3.1 The ramp and the eightfold way JHEP05(2017)118 is s (3.5) = 12870 34 L GUE , the SYK spec- 32 GOE A 30 32 = = N N GUE GOE mod 8) = 0 there is no . These facts are consis- 30 to within a few percent. GUE N /L results (correcting the Wigner 1 . The corresponding RMT t . In particular, they explain 6= 0 due to the particle-hole L N 3.0 4 ∼ 28 ) GSE ) t in detail. They are equal to the ( 1.4 g 4 2.5 mod 8) = 0. When ( 26 become significant). We found a power N GUE β . 6 2.0 2 E at N 2 . As explained in appendix 24 L E /L ], but we computed the RMT curves from E GOE – 10 – s 1.5 P 45 28 are a signature feature of the ramp in the GSE , , = 0 is given by ( 44 5 for various values of = 2) when ( β , 22 1 1.0 E GUE = 20 , N 32 are reduced by a factor of 2 compared with the rest. N ]. , = 0 41 24 20 0.5 β , GSE ), which at t ) are available [ ) at 11 = 16 ( s t 0.0 g ( (

18 . As an initial test we ﬁtted the ramp at times well before the plateau

0.6 0.4 0.2 0.0 0.8

N g P

) ( GUE s P 6 = 34 (GUE) a careful comparison that confirms the RMT ramp shape is N 16 shows GOE 4 = 0 the early time behavior exhibits oscillations, which will not play a role in is the degeneracy of energy level . Unfolded nearest-neighbor level spacing distribution for SYK vs. RMT. Here β E N N class Let us now consider the plateau heights of figure For Figure See, for instance, figure 10 in [ 11 trum has a doublesymmetry, leading degeneracy to ( a plateauprotected height degeneracy, and of in 2 thosetent cases with the the plateau pattern height is ofwhy 1 the plateau plateaus heights of exhibited by figure Here time-average value of These are strong piecesrandom of matrix evidence theory. that the ramp structure in SYKthis can work. be The attributed oscillations to form are factor is due sensitive to to the the fact hard that, edges at at infinite both temperature, ends of the the spectral energy spectrum. particular, the kinks visible for ensemble. For described in section time (where unfolding effects discussedbehavior in section agreeing with the expected GUE behavior The shape of the ramp in each case agrees with the RMT prediction outlined above. In ensembles are surmise) for the RMT exact diagonalization data. ramp and plateau connect atthey a connect sharp at corner, a in kink. GOE they connect smoothly, and in GSE Figure 3 measured in units of the mean spacing. Semi-analytical exact large JHEP05(2017)118 is N E ) t ( (4.2) (4.1) N g = 24), ). The =1, β 2 = 16 = 18 = 20 = 22 = 24 = 26 = 28 = 30 = 32 = 34 N N N N N N N N N N N , τ . 7 1 10 . Here τ ) 2 6 10 H.2 , τ 1 5 τ 10 ( q ) and Σ( 2 4 G = 22), 48,000 ( 10 2 , τ q 1 = 34). J N tJ τ 3 ( ) 10 t N ( − Time g G ) =5, 2 β 2 = 16 = 20 = 22 = 24 = 26 = 28 = 30 = 32 = 34 = 18 10 N N N N N N N N N N , τ 7 1 1 10 τ 10 ( mod 8) = 0. As explained in the main 6 G = 32), 90 ( 0 ) 10 N as discussed in appendix 10 2 values. The value at late times, which is 2 N = 20), 120,000 ( 5 ) , τ 10 -1 ) correction. This serves as an important 1 N ] β 10 τ ( 0 N -1 -2 -3 -4 -5 -6 -7 10

10 10 10 10 10 10 10

4 t g ) ( /N 46 /Z Σ( 10 , ) 7 (1 β .) We are led to the reasonable conjecture that tJ 2 is that it allows us to learn about the large the dip time grows quickly, but the plateau time 3 O ) – 11 – (2 10 t ( 4 Time dτ g N Z = 30), 516 ( 1 6= 0 and 1 for ( =0, H.2 2 β E = 16 = 18 = 20 = 22 = 24 = 26 = 28 = 30 = 32 = 34 10 N N N N N N N N N N dτ N N β 7 5 and various theory, and that the dip time is a new time scale in the 0 1 10 , 10 Z 1 = 18), 240,000 ( N , mod 8) 6 1 2 0 10 N 10 N = 0 5 10 β -1 Σ) + 10 0 -2 -3 -4 -5 -1 10

10 10 10 10 10 = 28), 1,000 (

4 t g ) ( − , 10 , matches with I τ N p ∂ g − tJ ) with 3 limit. We ﬁnd excellent agreement with existing analytic results, both e 10 we will present an analytic argument that supports this conclusion. Time Σ t, β limit and for the leading 6 N ( 2 = 16), 600,000 ( g 10 N log det( N 1 DGD 1 2 10 Z − = 26), 3,000 ( 0 . SYK 10 = = N Z -1 I N 10 0 We begin with a brief review of the known analytic results. There is an exact rewrite One important consequence of ﬁgure -6 -7 -1 -2 -3 -4 -5

10 10 10 10 10 10 10

t g ) ( path integral is given in Euclidean time by [ olate to the large for the infinite cross-check both on our results and on existing results. of the SYK model in terms of bi-local anti-symmetric variables theory. In section 4 Thermodynamics of theIn SYK this model section we compute the thermodynamic properties of SYK numerically, and extrap- behavior of the ramp. Asgrows we go even to faster, larger resultingof in the a numerical more evidence, and seethe appendix more ramp prominent is ramp. a feature (For of further the discussion large the eigenvalue degeneracy, 2 for ( text, the shape ofwith the the ramp counterparts and in theare the 1,000,000 plateau RMT ( with depends GUE, on10,000 GOE, the ( and symmetry GSE class, are and good. the The agreements numbers of samples Figure 4 equal to plateau height JHEP05(2017)118 ) + 2 τ /N ) is ( β ( (4.3) a ]. In ψ 4.2 Z ) h 1 46 , of degree τ 7 ( ) gives the a ψ /N ], and can be 4.2 =1 ∞ 47 [ N a = = 32 (the largest , computed by exact P N ) is manifest with a N 1 N → ∞ 4.1 . ) , introducing a Hubbard- βJ answer. The mean energy Saddle point τ , compared with the result 28 30 32 18 20 22 24 26 ( ======ijkl 1 N N N N Extrapolation to N N N N N N N corrections to the free energy J − q G 0 and result, the next term is the 1 as follows. At fixed temperature /N 2 J → N 1.0 N ) = βJ τ for different values of saddle point equations. We find excellent Σ( 0.8 /N i . Plugging the result back in ( N ) , . The convergence of ( T N ) ( βJ – 12 – values and fit to a polynomial in 1 >/ ω αβ E E 0.6 h . Σ < = 4 in most of our analysis. The action ( Σ( N , T C q − αβ G one now writes the saddle point equations for the bi-local iω and taking the leading term. This is almost indistinguishable 0.4 ]. Majorana − ]. Certain perturbative 1 N /N = 48 49 , ) 0.2 14 at different ω 1 ( G Σ become /N ) coefficient is then the infinite result obtained by solving the Schwinger-Dyson equations numerically. i 0 ) G, 0.0 ) should be thought of as the fermion bi-linear N T N 2 ( ( 2. 0.05 0.01 0.02 0.03 0.04 E , , τ - - - - - we compute the mean energy and other thermodynamic quantities numer- O h 1 we need two copies (called replicas) of the fermion fields labelled by replica τ ( N J shows the mean energy extrapolated to infinite = 1 ) as a Lagrange multiplier enforcing this identification. To compute i G 2 ) 5 it , τ . Shown are SYK thermodynamic thermal free energy [ 1 α, β τ − N β Figure At finite To solve the theory at large ( to a three-tem expansion in 1 we compute Z ) correction, and so on. obtained from a directagreement solution between the of two the methods large ofvalue computation, considered although here) even the at result is not close to the infinite ically by fully diagonalizinglation, the we Hamiltonian. extrapolate the ToT numerical make results contact to with large 2. the analytic The calcu- leading The first equationequations is can in be frequency solvedsolved analytically space, numerically in for and the arbitrary the limits large values second of is inhave also Euclidean been time. computed [ These contour choice described in appendix fields. Stratonovich field for theparticular, fermion bi-linear, andand integrating Σ( out theit fermions [ indices N from the exact large We remind the readerobtained by that performing we the set disorder average over couplings Figure 5 diagonalization. We also plot the point-wise extrapolation obtained by fitting the eight values of JHEP05(2017)118 ]. 0. N 14 (4.9) (4.6) (4.4) (4.5) (4.7) (4.8) → saddle, T N (the large c ] reads term. 14 4 . This must be T . 3 2 . /N = 77 . and are given by a 3 T 12 : the late time behavior . We again find excellent ··· ... 37 2 is within fifteen percent of . . . N + + c | 3 ). The coefficients . + 0 ··· theory 198 0 T . K 3 3 (2) 2 0 c 0 + α N T N T k S ( | 9 the entropy goes to zero as . 2 ≈ α Nc 52 ) O 2 . c 2 π J 0 + N NcT 2 ) is subleading in 1 + 2 ) term in the entropy, which becomes β 2 ( − 2 T , T 2 4.4 − T ) + 3 2 4 piece of the extrapolated energy and find T 2 T S . Nc 3 α = J 22 2 . β π /N + − log( 2 as expected, while – 13 – a T + 0 + aT 6 , a N . + 0 T ] from a one-loop fluctuation correction to the large s + 0 0 + 49 , 0406 calculation. At low temperature the entropy is given by term in the free energy, which in the notation of [ results slightly better if we replace this with a Ns . J 0025 . N 14 0 23 + 1 2 β . 0 N N 0 /β = ) = 6 is fairly close to the expected value − − . i = # ≈ − T ) 592) is the analytic zero-temperature entropy density (in the ( . 0 04 Z T . S = 1 ( N 0 log a E − have been computed in the large ). Next, we fit the 1 h log(1 term to account for the first nontrivial operator dimension in the model [ a = 1 2 4.5 77 has not been reported in the literature, but we believe it should be . E 3 shows the entropy extrapolated to infinite ≈ N was computed in [ 14 2 T ]. We derive two of the main features of figure Notice that the linear term in ( a 6 c is suppressed at infinite 41 13 a 2324 , . 0 36 419. limit). Notice that the numerical extrapolation correctly captures the large . ≈ 0 0 N s − Next, figure This is based on a conjectured 1 We included a The coefficient specific heat) and = 13 14 12 2 Surprisingly, the fit agrees with large ensemble [ of the slope andand the from early there time we behavior get of an the estimate ramp. of the Bothor dip are equivalently time from described in summing by diagrams RMT. power formed by laws, bending ladder diagrams around into a loop. large zero-temperature entropy, even though at any fixed 5 Spectral form factorIn in this random section we matrix review theory properties of the spectral form factor in the GUE random matrix agreement with a direct infinite Here the expected value ( Here the fitted value of negative at finite temperature.numerical Let results: us now compare these coefficients to the extrapolated We see that The coefficient c the case because this term corresponds to a log( The normalization is such thatN all coefficients scale as can be written at low temperature as JHEP05(2017)118 L )– . | GUE 2 (5.2) (5.4) (5.5) (5.3) (5.1) 3.1 λ h·i − . We will . 1 ρ λ 5 | ∞ ) log = 2 λ . ( ) ρ . , with ensemble aver- )˜ λ 1 ( L ˜ ρ λ ( L iMt . ˜ ρ are then defined by ( Saddle point − 28 30 32 18 20 22 24 26 2 c ======) g N N N Extrapolation to N N N N N N βM ) = 2 16 dλ − λ of rank 1 e ( M . and dλ M is defined by Tr d Tr( Z ) 1.0 g 2 2 2 M L iMt , ρ M L − − Tr( + 2 L βM 2 , whereas in RMT we typically expand in ) = 1 0.8 − − ), one obtains λ e λ , analyzed in the same way as figure e ) 15 λ exp ( /N ( ij λ ˜ ). ρ ( ρ ij /N N ˜ ) ρ – 14 – Tr dM dλ >/ 5.1 T by ˜ 0.6 S ( dM dλ ≡ Z < i Z S ) λ T for the unit normalized density: i,j Z Y 1 2 GUE ρ 2 β, t Z Z L ( is analogous to the SYK Hamiltonian, and the rank 0.4 L, Z − Majorana = = over the random couplings is replaced by the average M = ) = limit the eigenvalues can be described by a density J GUE and the related quantities i GUE λ and change variables from its matrix elements to its eigenvalues ) 0.2 ( g L ) should be imposed (for example) by a Lagrange multiplier. The resulting h·i Z λ ,S ( β, t M ( ρ S dλρ Z − h e Z 0.0 ) λ . SYK thermodynamic 0.15 0.10 0.35 0.30 0.25 0.20 ( ˜ ρ D . Z N = − Figure 6 e for the physical density, and ˜ ∼ Let us diagonalize The partition function for a given realization of Consider the GUE ensemble of Hermitian matrices ρ ), where the average GUE For example, for the partition function we have The normalization of Z 15 16 /L 3.3 saddle point equations are solved subject to this constraint. Replacing the individual eigenvalues and a unitary change of basis.eigenvalues. In This introduces the a large Jacobianuse that describes repulsion between The spectral form factor ( over random matrix elements, given by ( In this context,corresponds the to matrix the dimensionnatural of perturbative the parameter Hilbert in1 space. SYK is One 1 important difference is that the aging defined by JHEP05(2017)118 ) t ( ) is d t (5.9) (5.6) (5.7) (5.8) g ( , and (5.10) g ρ at times ) given by 2 t -independent λ ) before the L ( L t s ( ρ g ∼ ) , t (0 . c is a small g ) t , (2 t t ) 1 2 λ ). Notice that the average where . LJ − λ 2 1 ( λ s λ L = ( ˜ ρ i . e ∼ − L 3 ) . iλt 1 t t 4 2 − = e ) after the dip time. We will discuss ∼ p , λ ) t 1 ( GUE i λ π 2 c 1 i ) λ ( | ) are almost equal, both exhibiting the 2 ) g ( s t λ J 2 2 ˜ ( ( ρ i c λ ≡ ) ρ R ( = 0), and show that g h 2 ) ˜ , t ρ 2 L λ δ β dλL the ramp extends to very early times, giving ( ) follows from the semicircle law. Working for ) (0 ; dλ 2 ) is dominated by the disconnected piece s t 1 2 – 15 – t 1 c Z ( ρ − λ ( g c ). Therefore, the late time decay of β, t ( |h Z ) and g t g ( dλ ˜ t = ˜ ρ ( ( δ d = Z ≡ g Z g ) . This particular power is a consequence of the fact that t ≡ h 3 GUE ( . i d ) GUE ) vanishes as a square root near the edge of the spectrum. ) /t g i 2 /L λ ) ; 0) = ( t 5.6 , λ ˜ , t ρ ( 1 h c 17 λ g ( = 0 , and therefore at late times we have behavior of 2 2 / β R L 3 ). A basic result of RMT is that, near the center of the semicircle, ( Z h ) is the fluctuation around the mean eigenvalue density ˜ 5.6 L/t λ . Roughly speaking, ( s ˜ 2 ρ perturbation theory remains valid up to times − saddle point of the above is given by the Wigner semicircle law, ) . Beyond the dip time, /L λ L L ( is the connected pair correlation function of the unit-normalized density ˜ is a Bessel function of the first kind. At late times we find that the partition ˜ ρ 2 ≤ 1 ≡ J R t ) The connected spectral form factor can be written as We now turn to discuss the slope and ramp that appear in the spectral formThe factor, leading large In fact, 1 λ ( 17 ˜ ρ Here δ the semicircle law ( parameter. We now review howon to the derive connected the spectral1 presence form of factor aramp/plateau ramp structure. in RMT. However, Webetter for focus perturbative control. for simplicity dominated by the disconnected part dip time is also proportional tothe 1 mean eigenvalue density ( 5.1 The ramp and the dip time function decays as This is true also at finite temperature. Before the dip time, the spectral form factor Here shown in figure before the dip time,each and in by turn. the connected piece simplicity at infinite temperature, we have The physical eigenvalue densityeigenvalue spacing is is given of order by 1 The large JHEP05(2017)118 ). ). ). 6.3 5.10 5.11 (5.15) (5.12) (5.14) (5.16) (5.11) (5.13) ) about the saddle ) ) following Altshuler . ]. . 4 5.5 ) . − F 2 | 5.5 52 . L , 2 λ The ramp lies below the ( ) λ ] plus a delta function at 51 4 − O − 18 − 1 50 + L . 1 λ , ( | ) in [ ( . λ 2 t . s | perturbation theory to compute δ ( O ) | )) g 45 s s , L ) 2 1 L L + 2 Lπ − 2 2 λ λ ( 44 2 ) in RMT is the formalism developed ) log [ ˜ t ρ − 2 + − ≥ ( )) 1 δ λ 1 2 λ g | 1 ( ( )] 2 s 1 λ i ˜ λ ) and find ρ 2 | , t < ( )] δ , t ) λ ) beyond the dip time: there is a ramp up − ) 2 s ) 2 1 isλ dse 1 ( λ 1 are strongly suppressed: this is the spectral πL − 2 ˜ λ ρ λ ( 1 ρ πL πL − ( 2( Z signature of spectral rigidity discussed earlier. ˜ ( ρ λ | 1 2 ( / δ (2 – 16 – sine kernel ds δ and expanding the action ( m λ πL 1 ) exp( 2 L L ( t/ [ s E 2 1 ˜ ≈ − ρ Z ( 2( 2 π δ ( dλ ˜ ) − ρ 2 πL 4 2 − 1 δ [ 2 n + L ∼ sin π E , λ s = = | dλ ) ds 2 − 1 t ρ = i ( R λ ) Z c log ( 2 = ˜ ) gives Then we find 2 g 2 δS ) = λ L ρ ( 2 R ) = i ∼ 19 ˜ 5.9 ρ − λ m δ , λ ( . We discuss this technology in appendix . We will show how to include regions with different mean densities in ( ) 1 ˜ ρ = 1 δE λ δ M λ n ( , and a constant plateau value beyond. ( L/π 2 δS ˜ L ρ ] which uses standard ‘t Hooft large δE R h δ h 54 ]. Writing ˜ 53 ) is given by the square of the 2 , λ 1 A calculationally more efficient way of studying This is a good explanation of the ramp and plateau, but it requires an appeal to the λ Our analysis here only applies to the contribution from eigenvalues nearBy the observing center that of the the local eigenvalue semicircle, density is the inverse of the level spacing one can read off from ( 18 19 2 the double resolvent of where the mean density is Brezin and Hikami derive remarkable nonperturbative formulas for the following result the by Brezin and Zee [ Notice that long-wavelength fluctuations of rigidity referred to in RMT. We can now carryWe out go the to Gaussian Fourier space integral to determine the two-point function ( We now review how toand derive Shklovskii this [ perturbatively frompoint, the we action find ( the quadratic term sine kernel. In fact,the one sine can kernel. derive Notice the thatby ramp the approximating in initial the linear a sine time more kernel dependence basic by of way the without ramp knowing can about be obtained This explains the observedto behavior the in plateau figure timeplateau 2 because the eigenvalues are anticorrelated as reflected in the minus sign in ( coincident points: Fourier transforming as in ( R JHEP05(2017)118 ). ), is 20 x aN (6.1) (6.2) (6.3) spec- e limit. 5.12 , expo- − L N L and divide . . √ 2 d L , , ∼ t i ) ) = d 2 ] t λ E x ( − 2 i ] 1 ρ ) h λ x dλ ρ ( i , E i R ) ( π − t ρ E e 2 h ) ( are of order exp( 2 π ρ [ λ h L 2 + , also exponential in the en- π − [ 1 L e λ min sin ( √ β βE − − 2 ∼ e 1 − i d ) 2 , capturing the ramp part of ( . Notice that in this expression and /t i , λ 2 ) p 2 i ( t dE e 2 λ ∗ ρ suggests that the SYK model possesses 2 λ + ) ( t/L 1 Z 1 3 ) gives the RMT dip time ρ λ λ ZZ + ( h ih ) is a perturbative result in RMT at order ) ρ = 2 h 1 5.12 i| 2 λ ) curve obtained from exact diagonalization. E ) ( t 5.16 – 17 – ( it ρ dλ ) by the appropriate spectral form factor. Second, in cases ]. The oscillating term comes from continuing to h g 1 + and 6.3 dλ 56 β ) + 2 , ( x λ Z Z ( 55 δ − |h = i ) 1 i = ) λ E . ) is proportional to -fold degenerate, we should multiply the ramp term by i ( . We find that the ratio t it d d ) ( ρ = ) and the ramp ( L c h it . Therefore, perturbative RMT effects are nonperturbative in − g x 5.8 = β − aN ( i − β ) Z , the physical eigenvalue density, normalized so ( e 2 ) . Nonperturbative RMT effects of order ρ Z λ it ) ( ∼ aN ρ 2 it + ) expression, which can be obtained from a RMT instanton expression ) − 1 β + /L e λ 21 ( ( β 5.11 Z ρ ( h h so 1 Z with imaginary energy [ h 2 N/ inside the second term by imag ). Starting with the general definition of . Its contribution to i t Now, for late times we assume that the integral is dominated by regions where First, let us explain how an assumption of spectral rigidity produces the ramp observed This has important consequences for the application to the SYK model. For SYK, This derivation makes it clear that ( Equating the slope ( ) 2 ( In GUE the ramp is the full perturbative result, whileWe should in make other a RMT few ensembles comments (such about as this GOE formula. and First, if the local statistics are GOE or GSE, LE g = 2 E 2 21 20 ( /L − ρ GSE) the ramp receivesexist higher-order in perturbative all corrections. cases. Non-perturbative corrections tothen the we ramp would replace thewhere ramp the function in spectum ( h is uniformly which leads to sufficiently small that weor can GUE approximate statistics. the For density-density simplicity, we correlator take by GUE GOE, statistics GSE, in it is convenient to define below, we are using The presence of the rampspectal in the rigidity, results even of section forspacing. eigenvalue By spacings combining far thistrum, assumption larger we with reproduce than coarse-grained reasonably the features well the mean of the nearest-neighbor large SYK, of order an extremely small nonperturbative effect. 6 Spectral form factor in the SYK model Indeed, the appearance ofsine the plateau kernel depends ( one the oscillating factor inreal the energy more and exact extracting the appropriate part of theL result. nential in the “entropy” log tropy, and therefore the ramp in the RMT spectral form factor1 survives in the large In other words the ramp is a perturbative RMT effect. By contrast, the plateau is not. JHEP05(2017)118 1 + C β ( (6.6) (6.4) (6.5) is not . One Z ; these t β h q ]. configuration. , giving a p t J 57 . ]: 2 0 τ ] 17 ) is a reparametrization , − u 14 ( 14 . In fact, with the correct 2 . 2 τ 0 00 , the coefficient multiplying Z τ τ it S + , which can be captured by the configuration to β J it du Nα 2 β J π + . 2 π 0 2 saddle point that determines 2 β N q Z SYK. First we discuss the disconnected · N S N to be strongly peaked around a maximum. J J exp π ). One source of such fluctuations would be β β E q 2 . The classical and one-loop contributions to 4 πNα / S ], with the result 3 − ) α /N → 2 – 18 – J # 14 J π sets the scale of the Hamiltonian in a way appro- ) will have large fluctuations. Naively, this inval- S 4 β u ) for large is a numerical coefficient that depends on ( ) will smoothly approach zero at ( exp J J t = τ S ( β ) 6.3 c c )] α πNα R g ) = u , ( , we also need to consider fluctuations about this saddle. β by ) to SYK, but there is an important subtlety. The second ( τ , where the energy that maximizes the integral is simply it [ ) c β 6.3 D + , and SL(2 (2 q β 1-loop S Sch e Z Z = ) = terms will always be smaller than either the first term or the second p β t q ( . Then the coefficient in the action becomes [ q is the physical time variable of the model, and /N (more precisely of order 1 Sch π Z ) becomes small, and N 2 ) is simply summing up these individual ramps which then yields another ) can be interpreted as follows: we approximate the spectrum by bands over 6.4 ), so the formula still gives a reasonable picture of SYK. We will return to this ) should be understood as exponentially smaller than the first, as long as 6.3 ) varies very little. From each band, we get a GUE ramp. The integral over 6.3 6.3 E 6.3 ( < u < ρ ), the energy that dominates the canonical ensemble at inverse temperature 2 Our derivation is based on an intermediate step where we think about the SYK model Let us now attempt to evaluate ( One would like to apply ( Eq. ( β , but for large values of i (2 ) derivation of this fact. A direct proof isfor also large possible, values and of will appear in [ However, notice that when wethe continue to action large ( values of idates a perturbative analysis,measure, making the it theory difficult turns to out evaluate to be one-loop exact. We will present a somewhat indirect of the thermal circle.priate The for parameter general valuesare of related to the specificthis heat action have been studied previously [ There are a setpartition of function modes of that the become effective soft Schwarzian for derivative theory large [ Here, 0 term in ( point below. first term. We canit numerically evaluate the large too large. In SYK, wesuppressed also by expect correlations between eigenvaluesthe that overall are scale only power-law of theSuch Hamiltonian, terms which would varies dominate from over thehope ramp that contribution these at 1 short times. However, we might plateau at the time E can check thattransition the onto derivative the plateau of even though individual bands haveterm a in kink. ( which energy in ( smoothed ramp. Thisdegrees of is freedom, the we expect inverse theIn integral of over general, an the “unfolding” location process. of In this a maximum theory will with depend many on time. The ramp will join the JHEP05(2017)118 7 , we (6.8) (6.9) (6.7) E . ) is actually N π p ) contributes ). In figure t ) 2 0 6.5 6.3 6.3 E < 2 marks the boundary π 0 π − t 2 in ( N Ns . large and the energy E ( < ∼ E , and we have a power c 2 . We can evaluate the 2 N 0 q < e 1 ) N q < t. ]. 2 = C t π Ns t ). Away from the holo- √ p 61 2 2 fixed, then the scrambling time function, by evaluating the – + N α & 6.3 . ) free energy in the holographic 2 59 , e t 2 0 N cNt β / ( 0 E N q/N i β To isolate the contribution of the ) − Ns . Therefore J 0 π − e 1 2 E . We can therefore take a “double- (2 22 Ns ( . 2 ∼ e t/ N q S keeping d α > B α t ) determined by solving the Schwinger- exp 2 4 E 2 ( )) / log 3 → ∞ → S ) 2 held fixed. It is clear that the Schwarzian part β t 3 cN N B.15 2 – 19 – ) , q, N β N q , we learn that the one-loop result ( + 0 saddle point expression for the partition function − i 2 J E , t βE 2 β β β N cN cN − q eq. ( ( − i e with ), which gives ) β − − B E = cN saddle point action, which can be computed numerically. q 4 . Notice that this function is ( E 3 0 0 ) 2 ( 6.9 β β ( i| = . Neglecting one-loop factors from the integral over ), one finds that the disconnected term in ( − ) S Ns Ns 2 β 0 c 2 2 i N 2 it dEρ ) ) ) πe t − − 0 β R ( + Ns h h , while it is of order 1 when + , z ( ) and ( E g 1 2 0 β − and large = Z = 2 ( s h . Using the expression for the large h z − 6.8 2 exp exp 2 Z Z β N / + α < 2 cN E π π 3 t t |h √ ]. We sketch this in appendix ) should be an exact result for the Schwarzian theory. Computing the ( 2 2 ) exp ): β + c t π 0 0 it ( 58 2          6.7 g ( + . Away from the holographic limit, one would replace the piece in the expo- Ns p ∼ 3 β ) when N − ) πe + sinh t N t = ( 0 ∝ ∼ Z = 2 ) Ns ) above the ground state small, with the product held fixed. In this limit, we find saddle point action numerically, and doing the integral over p ramp 0 E t g ( β E ρ One can also make a more exact analysis of the large Now, we would like to evaluate the second term in ( The conclusion of this discussion is that we can include the effect of the soft mode ) = Notice that if we take a double scaling limit − E 22 ( E finite is of order log( between the behaviors expected for k-local and nonlocal Hamiltonians [ where dip time by equating ( graphic limit, one hasDyson to equations. use However, the weS can numerical give a simplehave formula the contribution in to the holographic limit, where The time dependence of thelaw exponent decay becomes negligible at nential by the appropriate finite by a factor of ( limit, log the following to scaled limit, so ( partition function via the exact answer for the Schwarzian theory. integral by simply dividing the large ( the density of states (see appendix We expect that the Schwarzian sector is the only part of the theory that survives this triple- scaled” limit of large of the theory will survive inand this limit, it but becomes the rest possible oftechniques the to SYK from exactly theory [ simplifies compute significantly, theSchwarzian, we disorder-averaged take density a of further states “triple-scaled” using limit where we take In particular, the coefficient is only a function of JHEP05(2017)118 23 but = 34 2 / 2) N − q ( effect that . Another N 2 / q 0 density of states /N Ns e N looks reasonable, it = 34 to compare to 6 7 N 10 ) decreasing more rapidly t ( ) at times of order 5 g t ( 10 = 5 large N numerical ) must be present in a single sample. d t β g ( g 4 ) would differ in important ways. t 10 ( g 3 10 . Another possibility is that the very bottom 2 Time tJ q – 20 – 10 edge in the spectrum, leading to a faster decay. N free energy (without a proper one-loop term) gets the answer for 0 ∼ N E 1 N d 10 t − partition function.) ) (blue/solid). The discrepancies in the ramp and plateau t E ( N ). The agreement is reasonably good, although the ramp 0 g √ 10 effect is the sample-to-sample variation of the edge of the eigenvalue Large N vs. numerical g(t) at N = 34, numerical g(t) at N = 34, Large N vs. 6.8 ) evaluated for the SYK model using the large q 6.3 /N -1 0 10 -1 -2 -3 -4 -5 -6

10 ], and may also be present at exponentially low temperatures in the

10 10 10 10 10 10 g(t) 48 = 34 plus the Schwarzian partition function (red/dashed) against the = 34 and evaluate the numerical finite temperature saddle for the slope ]. Such effects may also lead to a softer edge in the spectrum, again leading N ). Roughly speaking, effects that cause a crash in t N ( 62 saddle point. (Presumably this factor would be mostly resolved by a complete g N . Comparision of ( We caution the reader that although the agreement in figure A simple example of a 1 23 SYK model [ to a faster decay of the slope and a correspondingly shorter dipspectrum. time. This causescancels out a in gaussian crash in the partition function tends to smooth out theIn sharp this situation, thea slope short would dip crash time,of and perhaps the intersect of spectrum the order would ramp beSachdev-Ye model controlled much [ by sooner, a leading spin-glass to phase that was argued to exist in the is very possibleIn that particular, the true wethe are large simple not Schwarzian confident effectivepossibility that theory is the out that slope to someat very region effect an long continues leads earlier times to to timescale. of the be order slope For described example, portion by this of could be the result of some 1 portion, slightly correcting ( and plateau are offthe by large factors that representone-loop the correction discrepancy to in the the large exact free energy vs. the plateau we are roughly dividing by it once. we show the resultthe exact of diagonalization doing data.spectrum this We for computation also take and into plugging account the in two-fold degeneracy in the Figure 7 extrapolated to exact diagonalization answerregions for are due topartition the function fact wrong that by the an large order one factor. In the ramp we are dividing by this twice, and in JHEP05(2017)118 , i (7.3) (7.1) (7.2) n | , such ψ m | E m , based on h . 2 ) = / 2 0 N n i| . ( E Ns η m e | A ψ = , because at times | p n t i |h than m n | to write down a selection βE ,P − can be nonzero. We have i P , later e i J m i | m n | E ψ i ψ = ψ | n,m X ) n ) . This allows us to conclude that t J p m E ( N i h g in the random couplings realization, i ( ) ) ) equals zero and the plateau vanishes. ψ η β β n ( 1 ( E = βH 7.2 Z Z h − e i = . (We neglect the effect of degeneracies for i m i | ψ denote degenerate states with Tr can never be larger than – 21 – h i (0) ], which is reviewed in appendix ,P d i t ψ . m N =1 ) | 0 i m X t 39 | , , ( ) is only sensitive to individual energy levels, with i Ns t 1 ψ N ( e n | 32 g , βH ≡ P ψP − 31 ) = ) e t ( h ∞ = G Tr = i β dt m ( | o p t t 0 Z ≤ , ψP o ) 1 i t , which is enough to establish a parametrically long ramp at non-zero . Then we are interested in whether 0 i m ∞ | n ] to estimate matrix elements. As we will see, the answer depends on the Ns | →∞ = e lim o ). In this case the matrix element in ( t = 64 = ) is equal to the constant plateau height β , is the energy eigenbasis with energies t ) ( i 2 is odd then we can use the particle-hole operator i A ( ], but for our purposes it will be enough to consider the average of the two-point d β i 63 n t the spectral form factor g 1 m ( | stands for any one of the fermions | 9 n N/ , , | Z ψ p ≤ 2 p | P ψ If Let us first determine whether the two-point function has a nonzero plateau. This can As before, we focus on the annealed (‘factorized’) two-point function We are fairly confident that the dip time should be no ) mod 8) symmetry pattern [ m 2 is even then there is no degeneracy between the charge parity odd and even sectors (see β h ( N d rule for the matrixthat element. Let Here, and simplicity.) We expect a non-zeroN/ plateau to appear unless theappendix matrix element vanishes. If be determined by considering therandom following couplings. long-time average in a single realization of the This quantity is easierpassing that to it study is analyticallytion sometimes than [ useful to the consider quenchedfunction the correlator. itself. average of the We squared note two-point in func- ( in which the disorder average is taken separately in the numerator and the denominator. 7 Correlation functions In this sectionplateau we structure at will late(ETH) discuss [ times. when We two-point will correlation use functions the Eigenstate exhibit Thermalization ramp Hypothesis + the edge oft the spectrum. > t Noteall that correlations between differentt levels > getting t washed outt by the oscillating terms. For the idea that neglectedAs effects an are extreme not fallback position,is likely one to less can make than argue thetemperature. without spectrum any vanish calculation To more that make sharply. monotonic the the and dip roughly argument, time independent one of temperature, assumes based that on the the idea slow that decay it in comes from the slope is JHEP05(2017)118 2 is ) in (7.5) (7.4) 7.2 N/ factors ρ . J . E , ETH predicts 2 i J

i E 0 m | 2

E ). We conclude that | i 0 i mod 8) = 2. ψ E | and A.3 | N i E i ψ h | n

| is a smooth function of the ) E 0 h J

E E D ( 2 · ρ ) i| for clarity. In the second equality J 0 mod 8). E

i gives the spectral form factor. It will lead to a g , connects eigenstates from two diﬀerent charge e – 22 – i J ≈ 1 i i 0 | βE ) J 0 − E E | e . For typical eigenstates E ) = 1, or equivalently when ( i 2 0 2 (

ψ i ρ N | i| 0 ) ( E η m E E h | | ( i dEdE ψ , as in ETH, and we approximate it by a constant. The value ρ , and in the third equality we used ( ψ | | h 0 | n P determines whether there is a non-zero plateau, as discussed Z E E |h 0 h

i − E ) = X 4 then there will be no ramp or plateau. 0 E , | 2 J = E | i ( ) ρ E β nm ) ( ) are de-correlated for sufficiently small energy differences (corresponding 1 ψ E | Z ( h 7.4 ρ D N mod 8) = 6 then there will be a ramp but no plateau (the two-point function show a numerical computation of the two-point function that bears out these mod 8) = 0 mod 8) = 2 then there will be a ramp and a non-zero plateau. ) is the energy spectrum in a given realization of the random couplings. Notice 9 2 is odd then the two charge parity sectors are degenerate, so effectively there 2 is even then there is no degeneracy between the two sectors. The two ) = , N N N t E ( 8 ( N/ N/ ρ G If ( If ( will vanish at late times). If ( Finally, let us estimate the height of the correlator plateau written down in ( To summarize, the two-point function will display the following combinations of a ramp If If Next we ask when will a ramp appear in the two-point function. To answer this • • • conclusions. the cases where itmatrix does elements not vanish.that the This matrix requires elements us will to estimate be the of typical the size same of order the as for random states, which would Figures ramp, just as in the case of the observable and a non-zero plateau, depending on the value of ( Furthermore, for a Gaussian randomsmall energy matrix difference of this function at above. The remaining factor differences in the spectrum, whereGaussian we random expect matrix. each For sector suchfactorize, of a and matrix, the we the Hamiltonian can averages to over approximate resemble eigenvalues a and eigenstates even or odd. that appear in ( to late times), and we do not expect ais ramp only to one appear. sector. As discussed above, at late times the correlator probes small energy Here, again that the matrixparity sectors. element We will again consider two separate cases, depending on whether we used the antiunitarity of a plateau can only appear when question, we put the disorder-averaged two-point function in the following form. where we used inner product notation JHEP05(2017)118 = 5. β 004. . with a i m 5 24. A ramp | 400 , 1000 1000 10 4 = 5 = 5 24 (right), = 22 β β , 10 ), is also of order = 5, GSE = 5, GUE = 5, GOE 800 800 320 β = 24 = 20 = 22 β β N N N N 22 , 1.4 5. A slope, dip, ramp, 3 samples, samples, , 4 2 10 for 600 240 600 tJ 2 = 20 = 0 tJ tJ tJ 2 = 18, 10 = 26, 10 Time 10 N N N β . The height of the plateau Time Time Time 160 400 400 /L 1 1 10 are related by the action of the ∼ i mod 8) symmetry pattern. . 0 2 80 = 0, given by ( 200 200 = 20, and 10 | 10 m | 4. A non-zero plateau appears only for N /L β , N nm 18 (left) and , -1 ψ 26, plotted for | 0 0 0 10 , for -1 -2 -3 16 and 0 0 0

10 10 10

) ( Re Re 0.01

G i

0.002 0.005 0.002 0.001 0.004 -0.001 -0.002 ) ( Re ) ( Re ) ( Re t G t G t G n | = 18 = 18 data, where the correlator plateau height = 14 5 400 200 1000 – 23 – N mod 8 = 0 N 18, 10 10 N ≤ N for 4 = 0 = 0 β β ) = 5, GUE = 5, GUE = 5, GOE N 10 320 160 800 = 16 = 14 = 18 β β ) for β N N N t ( ≤ 7.1 ( G 3 samples, samples, 4 2 10 600 240 120 tJ tJ tJ tJ for 14 2 = 18, 10 = 26, 10 Time 10 N N Time Time Time 4 , so we are actually considering a diagonal expectation value 6 but not for i 80 , 400 160 1 m . Then ETH instructs us to estimate this by replacing | 10 2 P i| 0075, and the spectral form factor plateau is at approximately 0 . However, in our case, 80 40 . 200 0 m | = 10 /L i mod 8 = 2 ψP n | -1 | 0 0 0 = 1 0 0 10 -2 -3 -1

0.02 0.01 0.05 0.01 ) (

10 10 Re 10 t

G 2

0.015 0.005

) ( Re ) ( Re ) ( Re Re |h t G t G t G | . SYK two-point function . SYK two-point function = nm ψ 2 | | Notice that the spectral form factor plateau at . We therefore expect the correlator plateau and the spectral form factor plateau to be operator, mod 8 = 2. These properties are all explained by the ( nm /L ψ in the correlation function should then be of order1 1 of the same order. Thisis holds approximately true 0 for the give P | random state. One can check that this also gives Figure 9 The number of samplesappears is 10 for N Figure 8 and plateau can be seen. JHEP05(2017)118 . ), 8 7.6 (7.7) (7.6) . t ) n E ; the difference − 1 L m E − ( i 2 t e L 2 | ∼ nm p in such theories. We will — much smaller than the 1 L O G | i m − /L i− 2 (0) βE t L − O (0) ) e t O ( · m ) t ) O E ( h β 6= ), computed for a single realization of ( n,m X O n are of order unity then we find a plateau h E Z 2 3.1 . The remaining sum then describes the · | ) = ) t 2 are of order 1 β | ( )( /L nn . 1 ( 2 . 1 β | G O Z ; | 1 L t – 24 – diag ∼ ( nm + − . We would like to make sure that ramp behavior g − 2 O 2 | | | off . 2 /L t O in the double energy sum, exactly gives the plateau 1 L nn | diag O m ∼ | + + − n E p p S off − G G βE O = e | − ∼ ∼ e ) encodes correlations between different energy levels. Beyond n ) t E n ( 7.6 X G ) as expected. β 1 ( ) factor in front of the parentheses is needed because the correlator is Z /L β ( = Z i 1 as discussed above. ) is given schematically by (0) ∼ 7.6 O . If the diagonal matrix elements ) is a simple operator, we expect the two-point function to approach its time average p p t ( G G O O h The second sum in ( If The correlator can be written in the energy basis as plus fluctuations of order ) and get a smooth ramp. This type of averaging will be discussed further in section t ( p 8 Single realization ofIt random is couplings importantdip, to the ask ramp and whether thea the plateau) disorder appear late average. in time ordinaryconsider As chaotic features the quantum a of field SYK first theories spectral the without step form spectral towards factor form addressing factor this question, (the in this section we are imagining that we are averagingG over time somewhat, in orderIn to any supress case, fluctuations the of conclusionis of this suppressed analysis by is 1 that Note that the normalized differently than the spectral form factor. In writing the above expression, we can be treated asramp of constant, the spectral formfunction factor, ( sans the plateau contribution. Altogether, the two-point height the dip time, itthat is the responsible off-diagonal for matrix thediagonal elements ones. linear time To get dependence an estimate of for the the second ramp. sum we ETH assume predicts that these matrix elements Here we assumed thecoming spectrum from is terms non-degenerate with height for simplicity. The first sum in ( G is consistent with this expectation. One goal of thisquantum work field is theories. to evaluate Inappear how this generic in section the we two-point ramp/plateau ask functionsmake structure whether of two is it assumptions: the in is chaotic form plausible thatincludes for a ETH this ramp, holds structure as for to predicted the by theory, RMT. and that the late time behavior 7.1 The ramp in more general theories JHEP05(2017)118 ave self t β, J (8.1) ]. and large 65 . N i 2 24 | ) dt the signal is essen- the autocorrelation + t , . We expect ordinary 6 2 | β, t decay 10 ) ( t 10 Z β, t ( ih| 2 Z | | ) 1 (We have suppressed the . After 4 β, t ∼ ( 10 Z decay t decay on the plateau and at large t i − h| t 2 ] to behave similarly. | ) a typical realization of the couplings does not 66 2 ) is plotted together with the average of 90 samples TIme Jt dt t – 25 – ( 10 N g For this to work we need to be able to take + = 5, 90 samples vs. one sample = 5, 90 samples 26 compares the single sample result with the disorder , β β, t ( 25 on the ramp 10 Z | t 2 | 0 ) 10 β, t ( . For SYK, N = 34, SYK, N = 34, . Figure with a typical time scale Z N 0 h| ijkl -8 -6 -4 -2 dt √ J -10 10 /

10 10 10 10

10 1 g(t) ) = β ∼ ; t, dt ]. This implies that at large limit and at early times, a typical sample should give a good approximation ( decay t h 65 at early times. . A single sample (red, erratic) of N ) decays with β ; to be a fixed time greater than the dip time. At such fixed Despite the large fluctuations, the underlying dip, ramp and plateau are still clearly At late times, and in particular after the dip time, the spectral form factor is not self t We also expect theSuch model time recently averaging discussed and inAnother estimates [ possible compatible way with to ours reduce fluctuations have in already a been CFT discussed is in to introduce [ a weak form of disorder averaging, , smearing out the fluctuations. , consider the auto-correlations in the random variable we have t, dt 24 25 26 ( ave ave β dependence here). by averaging slightly over the value of a marginal coupling. Set h tially uncorrelated. As we will see shortly, for visible. These features can bet made clear by averaging overparametrically a shorter sliding than time the window of lengthin width of will the not ramp, get sot smeared that out the along features with we the are interested fluctuations. To estimate the required size of averaging [ give a result thattypical approaches realization the exhibits disorder-averaged largequantum value. fluctuations field as In theories (with shown particular, no in at disorder figure late average) to times have a similar behavior. averaged spectral form factor.single Before sample the and dip thethe time averaged large there results. isto good This the agreement is disorder-averaged between consistent spectral the with formaveraging factor. our expectation We say that that in the spectral form factor is Figure 10 (blue, smooth). the random couplings JHEP05(2017)118 q ]. 36 /N (8.3) (8.2) . In the β . To have . For the . 4 ) i ) power law is ave dt t n the averaging . decay t ( 3 E ) ) / the exponential − β N q dt β m , cancels with the ( ; 2 i E n / ) )( dt n E ( to be no greater than dt is small, the first sum E g can be chosen to make + − = · t ( and large fixed dt m ave 2 i t i m E e ave . The typical decay time ) ) t )( N E n β 3 dt , E ) ) decays like 1 l (2 + + β , gives the approximate answer t required to remove the fluctua- E ( dt Z m i m (2 = 0. For a chaotic spectrum we ( h E / E ( h = is parametrically smaller than the )+ ave n 2 l t β = at large ) to show that for most of the ramp k E = E − , and when i . Therefore, at large t ) − e ave l E . These power laws are integrable and t k n N − 6.3 N 3 ih E E E cNdt would require ) ) ( √ , l m − 2 − / it n E dt m E 1 e ( − E ) E / k n ( /N + ∼ 1 E E = ( l exp idt + 1. This means that it ) provides a good approximation to the spectral k ∼ – 26 – e E . For such e ) m E p ∼ ) ) and still leave exponentially many data points on ∼ l E m t decay − 6.8 t E ) dt + E l k ( + N β + E h k E n + E ; 2 , decay E ( k ( t 0 β β dt E ( 2 ( − Ns β g e − power law decay. By this time the spectral form factor (and h − e 3 h e ) is already exponentially suppressed. A 1 ) h , equation ( h t < e dt dt m,n X X ( k,l,m,n / , as advertised above. After a time of order a few X ≈ k,l,m,n − N ) β √ ; / is as large as 1. Such a value of ) = times. 1 we can estimate the minimal value of β of order 1 the autocorrelation function ; and small t, dt ∼ ( β decay dt h early t t, dt decay ( t can be taken to be parametrically smaller than the length of the ramp. decay h t the time dependence of the autocorrelation is given by the spectral form factor ) at ave t t β be greater than the plateau time On the ramp the analysis is more subtle. First we make the plausible assumption At large Now we show that on the plateau Given t ; 2 , even if 4 dt ( earliest part of theso ramp, we estimate thatthe on error the smaller ramp thanthe any exponentially power long of ramp.decrease Numerics of are error not after conclusive time here, averaging. but do show a systematic that the leading multipointsmall energy scales eigenvalue appropriate correlation tocorrections. functions the In at ramp GUE the are theseThen exponentially the correlators we same factorize can into as use sums aafter the of procedure a RMT products like time correlators, of that up leading sine to to kernels ( [ 1 form factor, whichscales decays as as decay is replaced by ahence 1 the autocorrelation integrable so it does not alter our above estimate for the required time averaging window. (Here we assumed that theretimes are no degeneracies forg simplicity.) We find that at very late generally expect only twodisconnected solutions. part. The One second solution, solution, Let is dominated by terms which obey width energy eigenbasis the autocorrelation function can be written as tions from the single-sample data. Wethe expect standard the deviation fractional standard to deviation the (the mean)a ratio for of curve such averaged with data fluctuations to smallerN behave like than, say, 1 length of the ramp, which is exponentially large in JHEP05(2017)118 . β (9.2) (9.3) (9.4) (9.1) is the where 4 I − ) and cor- e is the bulk t ( CFT g D ) via is determined by ]. The black hole 9.1 where term is specific to + + 1 = 4 67 the entropy becomes r 4 n l 3 8 T , M /CFT n . 5 ) should then be given by an- G . . it corresponds to adding a small 4 ) . The ∼ l E + . 2 2 ) l l β 3 8 1 µ β it 2 N becomes complex. For small real ( 2) + n − + Z β n + . ( 2 − ∼ is maximally chaotic, i.e., it saturates r l 4 − l 2 q n , λ e i where 3 8 − N ) it n + · 2 G E + ( ) by saddle point and use the bulk gravita- r ( + β /l 2 + → − 2 + 9.1 β N r C . We will assume that the fine grained spectral + n + nr G dEρ 3 ( r r + → – 27 – / S , ) between the relatively short times governed by 2 − ∞ ) t = t 0 β − ( + Z n r g β 2 l AdS ( N I µ/r ) = C π G − we evaluate ( ]. The thermal AdS action in this scheme is β, t = = 4 2 ( . At large ). The initial behavior of 67 Z is the AdS radius. The horizon radius β N + ) is given by E r BH ) = 1 l ) as discussed in the previous section. We can relate this to the ( I I r ρ − ( ) e β, t V E ( ( = ) in the canonical AdS/CFT duality AdS Z ρ t is a positive constant and ]. We also assume that there are no new intervening nonperturbative ( Z ( G 15 C I, . In other words, we do not include the Casimir term. . Adding a small positive imaginary part to AdS I = 5. These Casimir energy type terms are missed without thinking about /β − 1 ,D ∼ ) = 0. BH Now we analytically continue. As The action The inverse temperature is determined by finding the periodicity of time of the At early times and large In the following we use the results and follow the notation of [ There is no ensemble of Hamiltonians in this system so we want to describe the time I + + r r = 4 SU(N) super Yang-Mills theory on = 4 ( = , We find it convenient toI subtract the ground state energy, and study ( β negative imaginary part to Here and below n holographic renormalization [ spacetime dimension, and V Euclidean signature metric, tional action to determine alytically continuing the large euclidean AdS-Schwarzschild black hole action to complex metric has warp factor averaged behavior of full density of states the chaos bound [ time scales governing thegravity behavior and of the veryof long this times system governed compared by toparametrically random SYK large matrix is and theory. that the at A plateau very distinctive parametrically high aspect high temperatures relative to the dip. The above ideas make itrelation possible functions to give aN conjecture about thestatistics behavior of of this system areble described given that by this random system matrix at theory. large ‘t This Hooft seems coupling highly plausi- 9 Conjecture about super-Yang-Mills JHEP05(2017)118 . , , 2 2 3 2 t cN N = 9 and cN c/β e e l e n ∼ ∼ ∼ 2 ) oscillates 2 | . | 2 ] indicates it | Z to diverge as Z N | + | Z 68 2 | becomes less β | ( ) = 0 coalesce. Z 2 to | Z | 1 or increases again, + β r Z . Using the term is present to | ∼ ( 2 l | 2 , θ ∼ | V Z β | t ) to behave like Z | + becomes of order ∼ 9.1 r t + r ) leading to a slow long-time , l scaling E ( ∼ N ρ enhancement). This apparently t 3 1 β 1, evolving at it does seem like the most plausible 28 denotes positive constants of order one.) 2 | c is very large. This reflects the very large = 6 the dominant saddle causes Z this feature will be smoothed out, so we do | β increases to AdS scale (with no N ,D t 2 – 28 – = 5 drops very quickly. At 1 again. But at cN transition it may produce a more extreme form of becomes large the solutions of n 27 e 2 | t N Z ∼ | 2 is maximally chaotic according to the out of time order | , 2 | weakly interacting gravitons of AdS energy and so Z it λ | Z 3 1 | β which presumably can be analytically continued around. If + which for small β Z 3 . Taken at face value these large corrections invalidate the saddle → /β 2 2 β N cN e ∼ t ∼ Z in the slope region leading up to the dip have 2 | Z = 0, | t . This is inconsistent so presumably this saddle eventually leaves the integration contour. In general is increased and -independent much smaller value. Although this analysis is far from conclusive Although it is never thermodynamically dominant, the recent analysis of [ Other saddle points could be relevant here. For example, at high temperature, small But this is not the whole story. As But there is another wrinkle. As At t To be precise, there are of order As an example of the subtleties here, for N 27 28 , there is a 10D small Schwarzschild black hole saddle point with → ∞ with AdS frequency. t we have not attempted to decide which saddles are on or off the steepest descent contour. 9.1 The ramp in SYM SYM at large ‘t Hooftcorrelator diagnostic, coupling so it is plausibleare to conjecture described that its by fine random grained matrixsymmetries eigenvalue statistics of theory. the The system. relevant For ensemble simplicity will let be us determined imagine that by a the nonzero values of We will assume these values andnow compute turn. the dip by matching onto the ramp, to which we falloff. If this transitionbranch is point caused by singularity a in singlethere mode is becoming a tachyonic more thensingularity. serious it In kind produces any of a event, large atnot large expect but it finite to produce significant long-time effects past times polynomial in and then rapidly decreasingAdS to corrections become important. A more careful analysis wouldthat be there required. issaddles. a At Gregory-Laflamme-type first glance 5D this could to produce localized a singularity 10D in transition in the space of becoming large at point analysis for times larger than this. β version of the above formulas gives an initial eventually becoming of order dominates over the thermal AdS again. This causes the fluctuation corrections to the saddle point in ( entropy at high temperatures. (HereAs and below than one. Then anotherloop saddle, determinant the around thermal thisan AdS saddle solution, representing a dominates. gas Including of the gravitons one we find JHEP05(2017)118 . ). ). 8 t , /l ( 8 s g 6.3 (9.9) (9.5) (9.6) (9.7) (9.8) r 8 2 / N radius. 7 ) e 3 t S = t (log ): l β E 4 ( / 1 S , where the relevant N 2 − N such that . # ], but in fact these black s # with Schwarzschild radius r 3 . 71 exp / 4 – E 4 3 / 2 ) t > e / 3 ) t 69 For simplicity, we will discuss 2 β t E ( 2 N / 29 Z 1 (log 3 , N ≈ / ) vanishes. 4 3 1 0 / o c β 1 0 T 8 , . 3 c 3 9.8 2 / /l 4 5 N N 8 s / N 2 4 r 3 0 2 is small compared to the spatial 3 c − /G /G N 7 8 s s " β r r = = , t, e – 29 – 4 3 = = n exp T T as denoting a time average rather than a disorder 2 2 S where (in the remainder of this section we suppress 2 E E/T ) l we need to work out the behavior of the ramp at N N β t min h·i 0 0 − ( 8 d c c ) will be sharply peaked, and the band that makes the t /l S Z 7 s . The procedure is equivalent to “unfolding” the spectrum, 6.3 = = 3 = 4 βr 8 will be the band which is just reaching the plateau at time 2 ) = S Z E t N t ( − e log where the derivative of ( s 3 ramp . Using the above equations we then have ) g dr t β (2 Z / 2 is in 10D. The contribution of such black holes to the ramp would be 2 ) N 2 0 β the integral in ( 1 c ) = log ( 4 /N e E Z N symmetry. Then we expect GUE statistics. 8 ( l = S ∼ T ) = ) β t ( N (2 S G e To understand the dip time At large We can outline a simple expectation for the ramp behavior using the formula ( We thank Alex Maloney for this observation. ramp much smaller than the AdS radius = g 29 s p . That is, where we used thatThe the dip time integral is is the dominated first by time such the that value this of contribution is larger than the contribution of Here holes give contributions toTo the see ramp this, we that consider are 10Dr smaller Schwarzschild black than holes of the mass slopenumerical contribution factors) to earlier times. Itramp, is but possible we that focus weakblack our interactions hole attention in states. on theare the AdS The determined region gas by smallest microscopic where could black parameters, the lead holes as ramp to discussed that by would a dominate [ be small the associated microcanonical to ensemble The growth ist somewhat slower than linear, and the ramp joins the plateau at time largest contribution at time t together. First, we studyenergies the will ramp be at high reasonably enough late that time, we can use planar SYM formulas for the case of a high temperature state, where We interpret the expectationaverage, value as discussed in section analyzing the ramp and plateau for each narrow energy band, and then adding them up break the JHEP05(2017)118 . 0 ]. c 8 effects (9.10) . The q a, b /N , but not β we can write ) first exceeds to be at most ∼ The early time s β a, b 9.9 r 30 for an order one 2 N 0 . Eq. ( c 2 e ) β 1 ( ∼ Z d t . So the overall heights of the S . This effect is subleading to the S . , or equivalently at a time 8 β /l 8 β = 2 s N r e – 30 – is = s d r t , leading to a parametrically long ramp. 3 instead, then we would expect contains the sum over energies in the fixed sectors ) . If we denote the separate sectors by indices β 2 S ab (2 is under greater analytic control and has been analyzed in [ g / cN 2 e 2 2 N ) 0 β c 1 ( 4 The dip time is exponentially late, and the ramp region, controlled by Z e /CFT 31 ) where 3 = ∼ t ( p t ab g slope g a,b P Using numerical techniques we established random matrix behavior at late times. We There is a subtlety in these estimates. SYM and many other theories have global It would be interesting to consider observables that probe the ramp during earlier times In fact here the hierarchy is more dramaticAs than noted in earlier SYK there could because be the new plateau phenomena can at be early made times, like arbitrarily spin-glass behavior, or 1 ) = t 30 31 ( these two curves. high by increasing the black hole temperature. absent in the double scalingso limit. make the It dip seems time quite earlier. plausible these would cause the slope to decay faster and black holes is governed bywhich random we used matrix as dynamics. a Our simple main model tool of was awere the black able SYK hole, to model, adequate determine for theenabled such early us qualitative time questions. to behavior give precisely a in plausible the estimate double for scaling the limit. dip This time by computing the intersection of in the Dirac SYK model which contains a U(1) charge. 10 Discussion In this paper we have argued that the late time behavior of horizon fluctuations in large AdS g diagonal terms in thisterms sum have contribute vanishing as contribution usual atramp to late and the time plateau ramp and and areexponential large plateau; effects suppressed we the by are off-diagonal polynomials interested in in and so we ignore it. We have confirmed these ideas symmetries (like the SO(6)have R chaotic symmetry). RMT correlations, We but expectWe the the expect different sectors spectrum the would within number bepolynomial each of essentially in sector uncorrelated. thermodynamically the to significant entropy sectors at fixed behavior of AdS There is also an exponential hierarchy, although not aswhere large, microscopic in this black system. holesa are microcanonical relevant. partition One function possibility that would selects be this part to of directly the consider spectrum. coefficient at high temperature.time behavior Note of that the this slope,if conclusion so we is have this rather identification sensitive ofEither to the way, the dip at time long- high is temperatureplateau tentative. we time For have example, a large hierarchy between the dip time and the this value when the dominant value of So we see thatmicroscopic. the first The black associated holes dip that time are is relevant exponential, are but small, with with a parametrically small the slope. The expected slope contribution is no smaller than JHEP05(2017)118 2 34 33 ). t ( g /CFT ] about 3 74 – scale of the 72 S , 2 − 58 e ). These are signaled t ( d g ) one needs two copies t ( g ]. There is also an exponential 8 ). ). For 0 t ( g t, t The early time behavior of AdS Σ( expansion get large as time is increased? , 32 ) 0 /N t, t ( – 31 – G ) provides an example of this. As discussed above, the gaussian . This softening cancels out in the time dependence of t ( k ) d 2 g − q /N 2 t possibilities are present then there is a new physical phenomenon at the dip. q /N . But perhaps one could take an ensemble of SYM theories with slightly 8 we make some preliminary remarks about the origin of the F Perhaps the central question this work raises is the nature of the bulk interpretation of Another set of ideas that might be useful are developments in the theory of sparse To understand the SYM situation better it would be useful to understand more about A more indirect strategy would be to look for precursors of the ramp starting from In all of these situations the dip time does not signal a new physical phenomenon It will be useful to have analytic insight into the ramp behavior in theWe SYK model. used these In ideas to formulate a conjecture about more general large AdS black In fact here the hierarchy isIf the more spin dramatic glassThe or because disconnected 1 partition the function plateau can be made arbitrarily high by 32 33 34 increasing the black hole temperature. fluctuations of the edge ofby the a eigenvalue series distribution produce of a terms gaussian of falloff the in form ( of various types ofmore sparse general contexts. matrices. This These is a might question give we clues wouldthe about like random to the matrix return behavior. SYK toin model, The in disorder future terms and averaged research. of SYK model the can bilocal be rewritten collective exactly fields different parameters. This possibility may be easier to implementrandom in matrices. calculations. From this point ofmatrix view with the correlated SYK model randomness is inthe a the certain universality entries. type of of Insights sparse dense have random emerged random [ matrix behavior in the fine grained eigenvalue statistics an exponentially smallKnowing error this sufficient would be to helpful accommodate in looking the for ramp signalsthe in and averaging SYM plateau procedures of that these signals? cussed phenomena. are in available. section Averaging over time windows has been dis- controlled by string scale blackit holes, might and be eventually useful the to chaotic use graviton a gas. more refined Toshort probe do times. this than Do the individualOr terms is in the there 1 just a factorial growth of coefficients signaling an asymptotic expansion with hierarchy in this system. — it is just thethe time new when physics the of ramp thesee becomes ramp what larger it happens in would at size be than early interesting the to times. slope. follow For it To instance “underneath” understand the in slope SYM to one would start accessing regions holes, like those dualthat to the 4D fine SYM grainedmatrix structure theory. of theory. Here energy we levels We relyprovisional in estimate then chaotic on of systems the the estimated is analytically widely theAgain described continued accepted 5D we by time find AdS-Schwarzschild intuition black random an at hole exponential metric. hierarchyis which of under scales. the greater analytic ramp control and appears has been by analyzed making in [ a behavior sets in. appendix height of the ramp in this model. long-range spectral rigidity, is exponentially long, stretching until the asymptotic plateau JHEP05(2017)118 ) 0 N 2. , − t, t ( effect. = 1 ). This αβ N N Σ − . e , . e ) 0 α, β − F effect, which C 2 t, t ( /L αβ G , quenched correlators E ] must have an explanation . This functional integral is a ) singlet objects and in some 39 , N C ( 32 O , ) which is of order exp ( Σ are discussed in appendix 31 L ). But various auxiliary quantities like G, effect and the plateau is an fields carry replica indices − t ( 2 , so new saddle points are a natural mech- G αβ /N N Σ , αβ – 32 – Σ integral is another mystery. After continuing G ]. This map would be related to reconstituting the becomes crucial. The ramp is a 1 we point out obstacles to a possible single saddle G, 5 This is a challenging proposition but the SYK model 56 , G = 2, as explained in appendix 35 55 q 2 the situation is qualitatively different. Here the interplay can be computed by saddle point, giving the desired 2 F Σ action includes q > G, effects. For Σ to new effective random matrix degrees of freedom with effective discussed in section N G, − . It is unclear whether this has anything to do with an actual saddle point effect. In appendix e N k whose dynamics would give the plateau as a standard Andreev-Altshuler N mod 8 “eightfold way” pattern noted in [ − and e /L Σ integral. In some ways it seems analogous to the behavior of the Haldane spin ] as the spin varies from integer to half integer. There the explanation in the N L discussed in appendix 75 G, k This work was partially supported by JSPS KAKENHI Grant Numbers JP25287046 The origin of the plateau in the The For the interacting case The coefficient of the This functional integral must contain the ramp and plateau behavior — the question f Some ways in which fermionic properties are coded into 35 The authors thank Tom Banks, Ethan Dyer,Dan Alexei Roberts, Kitaev, Juan Lenny Maldacena, Susskind Alex and Maloney, Edward Witten for discussions. (M.H.), JP15H05855 (M.T.), JP26870284 (M.T.), JP15K21717 (M.T.) and JP17K17822 fermions from the collective fields. provides the most concrete arena known in which to explore it. Acknowledgments is an unusually smallof nonperturbative effect, multi-instanton the corrections. sizea of mapping A the from error more incoupling natural an 1 way asymptotic series toinstanton explain nonperturbative effect these [ effects would be continuum is a topologicalthe term question in is the what action.find topology the is That origin being would of be probed. this effect a As in natural an the guess moment initialto here, calculations step imaginary discussed and it energy in will this appendix be is an important effect to of order exp ( value for large description of the ramp involving a sum over manyin saddle the points. chain [ between means an point explanation of the rampthe in the correlator is how. We cannot yetHere answer this we question will — just it make will some continue to preliminary be comments. a focusanism of for our such research. do seem tocorrections. be Here described the by ramp is sums a over perturbative 1 new saddle points with appropriate fluctuation An appropriate contour canis be nonperturbatively chosen well-defined, so as the discussedrough functional in proxy appendix integral for over arough bulk way description, bulk because fields it should involves be able to be reconstructed from the bilocal singlet fields. (“replicas”) of the fermions and so the JHEP05(2017)118 to be (A.2) (A.3) (A.4) (A.5) i ¯ c , ] i c 39 , 32 . , ) has two sectors maps a state with 31 2.1 P , a ηψ (here we choose = mod 4 = 0 mod 4 = 1 mod 4 = 2 mod 4 = 3 . d d d d C P k 1 a ∈ − ,N ,N ,N ,N d z P ψ 2 1 1 N , 3 ) (A.1) +1 +1 − − ¯ z j i c          1) . ⇒ → . The Hamiltonian ( + − i = i z c c c i i 2 (¯ ¯ c = ( / d i ] = 0 d ηc 2 =1 N i Y N P b P – 33 – = 1 K 1) H,P = − [ P d i − = ˆ ¯ c N Q P 1) maps the even and odd charge parity sectors to each = ( − 1) P (in our convention the Fock space vacuum has fermion ,P is a symmetry, − i d ¯ = ( c 2 ) the Hamiltonian has conserved charge parity, where the Q N P η ( η d − 2.3 = N d 1) P i this leads to a degeneracy in the spectrum. N − ). One can check that d P c K to N = ( 2 is odd then 2 Q P N/ = d is the anti-linear operator that takes N K other, and so the two sectors are degenerate. If One can now check that The theory also has a particle-hole symmetry under the operator [ 1. For some values of fermion number number 0). The action on the fermions is given by where where real with respect to In the Diraccharge description (fermion ( number) operatorfor is charge parity even and odd. Physics Workshop (June 9,(September 2016) 17, and at 2016). “Natifest”with at institutional We policy the thank the Institute these datascientists for used institutions on in Advanced for request. the Study preparation their of hospitality. this paper In is available accordance toA other Particle-hole symmetry of SYK Graduate Fellowship program.Foundation. G.G. is The opinions supported expressednot by in necessarily a this grant reflect publication fromby the are NSF the views those grant John of of PHY-1316699. Templeton Preliminary the the D.S. versions authors John is of and Templeton supported this do Foundation. by work the were S.S. Simons presented Foundation at is grant the supported 385600. Yukawa Institute for Theoretical (M.T.). J.C. is supported by the Fannie and John Hertz Foundation and the Stanford JHEP05(2017)118 ] 58 (B.5) (B.2) (B.3) (B.4) (B.1) . q a . 2 k/ . . . ψ . Then, we appeal k 1 mod 8) = 0 there is 2 a . H 2 λ/ ψ 2 N N q J λe q 2 . b = = is both anti-linear and obeys . . . ψ 2 1 b = fixed , λ k/ P ψ λ 2 N fixed is that the expected number of q − e q ! N = m /N m λ 2 . Taking the product over the pairs and i q ]. Now, of course we also have to consider 1 2 2 q J i 2 1. Since J 14 k = h [ − , λ i H ψ 1)! 2 i – 34 – = = tr − − h ψ q q ( 2 → ∞ 2 N = 1. In this case there is no protected degeneracy. P J # fermions in common 2 1 P − 1) q maps each charge parity sector to itself. q , since 2 − factors and contracting the flavor indices of the fermions integral by Wick contractions. This involves pairing up , q q 1 P 2 = ] to get the distribution for which these are the moments. J H = ( i 76 q → ∞ q b 6= 0 there is 2-fold degeneracy, while for ( ...i fermions in common) = 1 2 N i J m tr 1 h ( . . . ψ 1 P b 1 it cannot map energy eigenstates states to themselves, and we have mod 4) = 2 then mod 8) ψ mod 4) = 0 then − 2 is even then d d N q = N a N N/ 2 k assuming all pairs next to each other = double degeneracy within each sector. P H d If ( If ( . . . ψ is defined by 1 N tr a (a) (b) J even. We evaluate the ψ If Let’s consider what happens when we move one product of fermions past another. The argument goes as follows: first, we compute theFirst, moments we tr discuss the computation of the moments. We would like to evaluate k 2. The important feature of the limitfermions where in we common hold stays ofdistributed, order with one distribution in this limit. More precisely, the number is Poisson past each other until thelines contracted do pairs not are adjacent cross. or nested, so that Wick-contraction Notice that where cases where Wick-contracted pairs are not adjacent. The procedure is to commute the terms for the various terms in different that appear in thewe pair. could If evaluate all each pair ofsumming as the over the Wick-contracted possible pairs were fermion adjacent flavors that in can the occur product, in each pair, we get Majorana operators). to a combinatoric result in [ double-scaled limit The computation is afor closely small related modification systems of (composed the of Pauli analysis matrices by with Erd˝osand random Schröderin couplings [ instead of Therefore, for ( no protected degeneracy. B The double-scaled SYKIn theory this appendix we compute the disorder-averaged spectrum of the SYK theory in the JHEP05(2017)118 . ]. q c 76 ψ (B.8) (B.9) (B.6) (B.7) . The ), and (B.10) (B.11) λ ··· − 1 c e B.3 ψ . = 2 2 Q . E J ) ) λ a 2 ]. Specifically, this − . e 58 − ) . 2 λ (1 2 2 nλ 2 2arccos − ( λ 2 2 e λ/ kπ a − fermions past each other, we λe π ( q cross(pairing) cosh sinh ≡ λ λ kπ k 2 2 − − 1 e 2 enhancement, so we ignore it in the ) cosh k/ 2 , a . nλ log q 2 − ( -Hermite polynomials, with 2 2 2 a 1 ) Q N/ 2 λ/ 1 πikn 2 ≥ J nλ λe cosh k X ( − 2 – 35 – 2 2 a + − 2 1 dn e fermions in common) = ) cosh a ∞ , without a m −∞ − ( log Z ] that the distribution with these moments is known [ 1 as follows P /N X m 58 ρ −∞ −∞ (arcsin ∞ ∞ 1) X X Wick pairings = =0 = ∞ Y 1 λ k n n − . Anyhow, doing this sum independently for each set of fermions might also be shared with a third copy containing ( = 2 1 2 1 2 3 − a q k =0 b λ/ ∞ = = = 4 X m − ψ H N − ) tr 1 1 e tr E ··· √ N ( 1 ρ b ψ ) = log E ( and ρ 1 and zero otherwise. The normalization factor can be determined from the q a < ψ | a It is convenient to rewrite The final step is to notice [ So now, each time we have to commute a product of ··· | 1 integral by contour integration, summing over a geometric series of cuts of finite length a In the second linen we used the Poisson resummation formula. In the last line we did the for constraint that the total number of states is 2 of crossings of Wick contraction lines. It is related to thedistribution integration measure is for given the by Here cross() gives the numberso of commutations that required they to get arethat the all lines pairs connecting adjacent arranged the in or Wick a nested. pairs way will We not can cross. describe Then this cross() is target just situation the by initial saying number is where the fact thatanalogous we sum have gives Majoranas insteadthat of spins we is need relevant. to Inwe commute the find past spin each case, other, the we can now correct the expression ( get a factor Notice that this step differs somewhat from the case considered by [ that the same fermions thatψ are shared between twoOr, copies more of the generally, Hamiltonian thatprobability containing the is number proportional ofdouble-scaled to limit. such 1 This terms is might the be key point correlated. that makes However, it the possible to solve. Now, in principle, things will get complicated because we have to consider the possibility JHEP05(2017)118 ], 14 (B.19) (B.16) (B.17) (B.18) (B.12) (B.13) (B.14) (B.15) . ) . . , /λ . . . 2 ) 2 N q π , which leads to ) λ ( q 2 λ π − ) ) e = . O J 2 , τ ( 2 q G q 1 , τ 2 + π τ O ) λβ (2 ( 1 1 z 2 J τ 2 N/ g 2 + ( 2 2 q , τ g J 2 4 , λ = fixed √ ) ) τ = ) 2 λ ( π 1 + ) 0 − ( 0 σ exp , τ O − E N ) 1 G E 2 ) + J τ / 2 J z λ 34 − Σ ( 3 λ ]. π 12 2 − τ π 4 λ ) σ 2 τ 2 √ E ) 2 2 14 E J ( π = √ ( 2 12 β dτ sgn( τ ) and we have ( 2 − 1 sgn( = ) 3 e ≡ 0 3 )) S λ dτ a ( − + , τ sgn( 0 ) = 2 O 1 Z λ 2 τ − fixed, the first term dominates, and exactly ( βE + , z 2 ) , τ N σ , z − 2 a 1 2 ) 0 e – 36 – τ ,τ z − ( 1 (arcsin( . This agrees with the 1-loop calculation of [ 12 zλ τ 2 = 2 τ ( λ = log → 2 4 g π √ Σ) e 0 π ) βE 2 1 + − − E 0 with a sgn( − − J τ e 4 ,G J − ) ∂ ) 2 → = the normalization factor is 2 2 E E sinh log(2) ( , a , τ λ λ dτ ) 2 λ q 1 . . . dτ 1 4 π λ N 2 τ 2arccos 1 ( ( λ − dτ dEρ kπ e , σ = log det( O − dτ 0 0 Z axis. This formula is now in a convenient form for discussing the Z π N 2 + thermodynamics computed in [ N ( Z S → z + ) = n q sinh Σ action for the disorder-averaged partition function: λ 2 2 − = kπ . ) = λ k ) is a symmetric function of its two arguments that is constrained to √ , τ λ ) = 2 I β G, 2 and 1 ( 2 E τ q = , τ − N ( Z λ J 4 1 ρ ) cosh τ Σ( ( a − = g − I = 1 λ − 0 1 ≥ E k X arccos( Finally, we will mention that there is another way to analyze the double-scaled limit, For example, if we take Our primary goal is to use this to evaluate the partition function of the Schwarzian vanish when they coincide. The action in the double-scaled limit is To take the double-scaled limit, we write where now starting from the where but here we conclude thatSchwarzian. it is the exact answer in the triple-scaled limit that isolates the One can check that for small We conclude that in the triple-scaled limit we have In this limit, we approximate behavior at small reproduces the large theory, so we take a further “triple-scaled” limit along the imaginary JHEP05(2017)118 ) Σ = σ G, (C.7) (C.2) (C.3) (C.4) (C.5) (C.6) (C.1) along Nσg (B.20) g − e p . (2) q (2)] q dσσ g i q i 2 =0 ψ q , R J g

...ψ (1) q ) πi ]+ N q g 2 2 i (2) 2 q Ng 1 J , τ i ...ψ − 1 ψ N τ e ( (2) (2) i g 1 N i 2 ψ (1)+ ) ψ τ q g i ∂ (1) . Concretely, the integral we ∂ ) i (1) . 1 2 1 ψ ...ψ q i − i ψ , τ (2). This leads to the expression (1) 1 i N P q (2)] [ 1 i τ i ψ σ ( ψ we have a simple Gaussian integral. ψ ,...,N e [ g 2 q 1 (1) (1) <... N G g variables. There are integral was supported in a neighborhood of the origin, and the calculation reduced with g In the A confusing aspect of the One can also discuss saddle points for this integral. For these purposes we go back to However, we can also change the contour and make the integral more manifestly well- g, σ p g with solutions Choosing frequencies will lead to subdominant saddles. The difference in saddle point action induced by comparison to the simpler decouple, and we have D Subleading saddle pointsBesides in the the standard saddlethere point are a that family givesexplicit of the form subleading themodynamics for saddles discussed for in the path section integral ( This seems inconsistent with theindeed fact studying that saddle we points are with integratingof over nonzero values ofthe ˜ fact that we area at term most of taking degree the deformation of the integration contourdoes in in detail, but fact we give observe the that right this leading large saddle Grassmann variables, and we could askFor example, how the this fact is that consistent the with square a of representation a by Grassmann vanishes should imply that the There is one real solution, and this is the one that naively dominates. We have not analyzed where we integrate ˜ first over ˜ final rather trivially to a direct fermionic computation of ( defined. We rotate the to define new variables ˜ This is the right answer, and we got it from an integral over bosonic variables, but the JHEP05(2017)118 = 2 q (D.5) (D.3) (D.4) one would = 4 action 5 where we q = 4 solution, N ) is 2 n . q n ω ) ω ,..., π + − 2 = 2 2 = 0 β 1 βJ J βJ . J 4 n 2 ! q = 4 p gap in the action even at | O n 3 2 N ) ω + | (assuming 1 = 4 cases is that the actions τ βJ N q ( 2 + 1 + n O + + 1 + 1 + 2 2 ) n n iπ 0 1 2 3 4 5 6 = 2 and 1 2 0 6 flipped (orange/dotted). At right we have /ω /ω 0.4 0.3 0.2 0.1 . ,

-0.1 -0.2 -0.3 -0.4 2 2 + ) G( τ π N 0 and the corresponding J J βJ n = 2 4 4 ( ≈ > π n ) = 20 p p = 2 solution and iterating the Schwinger-Dyson 4 – 39 – 2) q / J − flipped, we find numerically that the , but preliminary investigation suggests that this + β β 1 + 1 n . For the simplest case, where we start with a = 2 solution there is a corresponding + 1 ω iπ from two to four. We give a plot of some solutions in q n log βJ subleading ( q π G N N β = 2 standard saddle (blue/solid) and a subleading saddle with 2 ( I q = ) = π ) = n − ) + = 2 ω β G ( I q = 2 standard ) + = 4 solutions. We use G + ( (assuming q τ G (for both I ( I − − , we see that the saddles become almost degenerate. Naively, this would − G = 4 model we do not have explicit formulas, but we can find subleading saddles . q βJ . At left we show the E . An important difference between the 0 1 2 3 4 5 6 0 11

0.4 0.3 0.2 0.1

In the

-0.1 -0.2 -0.3 -0.4

) G( τ thermodynamics. A logical possibility is that the relative dominance of these saddles = 1 flipped (red/dashed) and one with both were able to check. Thevery important low point temperature. is that This there explainsN is why a these large additional saddlescould do change not when disturb the weis large study not complex the case, and that the gap remains. is given by We are not surepercent that or so this of simple the expression numerical is answer for exactly the correct, first only few that frequencies it is within a which can be found byequations starting while with slowly the increasing figure do not become degenerate atsolution large with a single frequency pair order one amount, wehave do to not sum have over suchappendix all a of mode. these However, saddles. at large We willnumerically. see It seems that that they for play each an important role in For large suggest a soft mode connecting the saddles, but because the imaginary part differs by an the corresponding by choosing Figure 11 n JHEP05(2017)118 ]. , we 36 , (E.2) (E.3) (E.1) Q 26 , we expect /L is a skew Her- ). The averaged t ( ij d g iJ ]. It is not a chaotic . In this appendix we 2, since it is equivalent . L 78 , instead of 1 ∼ q > 1 2 77 belonging to particle number p , /N t − 1

k 26 c m ∼ | † k , c

2. Notice that n , | k contribute to this behavior. j √ λ ψ / corrections. Eigenvalue (mass) pair cor- i D 2 ) instead of ! k ψ =1 2 k N/ k X ij 0 2. Then the Hamiltonian can be written as ˜ N /N λ ψ J i . = k N/ ∼ k N λ − 0 2 i

p k − = t 2 k ˜ 2 ψ H ˜ ψ = 2 model k λ = ( q 2 random matrix leads to a “mini-ramp” and “mini-plateau” , are only nonzero for † k = 2 model is

=1 c N/ k X q N m i | i × = running from 1 to ψ | N k n H 2 and

, , and we made Dirac fermions out of these pairs of Majoranas, √ i i / ) ψ ) k ) are described by a modified sine kernel whose short distance behavior 2 0 Qψ ˜ ψ i λ, λ to a block diagonal form with each block given by + ( 0 and with = ( the spectrum is a semicircle, with 1 2 1 ij is a real antisymmetric matrix. Conjugating with an orthogonal matrix > R i − J N k ˜ ij k ψ 2 J λ ˜ = 2 model is qualitatively different than the model with ψ q The simplest observable in this model with a ramp is the (quenched) disorder averaged Part of the reason that the correlation functions are easier to calculate is that the It follows that eigenvalues in the single particle sector will repel, because of the usual The Hamiltonian of the Finally, we will mention that in the = ( k squared correlation function. It turns outcorrelation this function is easier (not to squared) calculate does than not have a ramp. matrix elements of is that of GUE. eigenvalue repulsion of a random matrix.from However, nearby sectors multiparticle eigenvalues coming witheigenvalues very that repel different each particle other havethat numbers an the average will plateau spacing time repel in only this model weakly. is Because the Where c mitian matrix, not aAt GUE large Hermitian matrix. Itsrelations eigenvalue statistics are known [ where where can take show how the saddle points discussed in appendix E Saddle points andThe the to a model ofsystem, free but the fermions explicit within a certain random quantities, mass with matrix plateau [ time that depend nontrivially onwords, both we of have the saddle timehave arguments, points not not that studied just spontaneously the theselarger break difference. systematically, action time but than In translation the other the invariance. standard examples saddle. We that we found numerically had JHEP05(2017)118 ) 0 =0 =0 t, t } m ( (E.8) (E.7) (E.4) (E.6) ˜ z ˜ . Let z , 2 c { n , n } z z G ). After z ) for the

) 0 { t , n

0 ( ) (E.5) ) c ω J 0 This is why ( g

λ, λ τ J i ) (

ψ − ) 2 λ, λ m 36 ) ( ( } z λτ R n ˜ 2 (˜ z to the operators G − { ω ) m R ( e τ = 2. ) n Z − Z ( z ( λ i q ( G log ˜ ψ ρ log

−

J =0 J J

+ 1 ∞ n

) ) 1 n ) with } ] 0 P z βλ j z ( τ ) in the GUE by saddle point 0 { − 2 n ψ 4.1 ( e =1 λ ) N i Z 0 Z + τ τ λ, λ dλ over only one of the Matsubara frequency P 1 ( log − log λ ) 2 1 2 (

Z j

− R e ψ − 4.1 ⊃ ) − = = 0 it is precisely equal to ( 0 J J J

S βH β

) ) − λ, λ 2 m } e ] ( ) i ˜ z z 2 (˜ β { ψ , because of the presence of ( ( ) m R 0 ]Tr[ – 41 – i Z τ Z Z it ( ψ ) i ] calculated ) + 1 β ψ → τ ( 1 56 ) log ) log ( 0 i } βλ n Z βH τ z z − ψ − { ( ). We can see that the real time correlation function, e ( e n λ βH Z ( in the above expression, will not have a ramp or plateau. Z and − ρ Tr[ + 1 e it log log it 1

βλ Tr[ → − ), can be calculated by coupling sources ) = ˜ → N =1 i e X λ ), and since the logarithms of these products turn into sums over m m τ 0 m τ ˜ ∂ z } ˜ ∂ z − ) separately). 1 =1 N n ∂ ( ∂ , ω N z ] calculated is closer to the integral X ρ n n i,j n ) with a term in the action { ∂ E.5 ω ∂ dλdλ ( 56 n 2 ( ∂z ∂z ) = n 1 2 c ω 2 τ N 2 ( Z Z 1 ( i 1 G N N ψ G = =0 ) ) = ∞ n n 0 ) = ) = ω Q m m ) is the average mass density. In the above integral we are extending it to a − τ, τ ( λ ( i , ω , ω ( 2 c ) = ) be the partition function with the source term included, n ψ n ρ } } ω G ω The quenched disorder averaged squared correlation function in Matsubara fre- This simple expression for the square of the averaged correlation function in terms of ( z z ( The integral that [ =1 2 c 2 c { { N i 36 ( ( G G the different frequencies, the derivatives simplify. We find modes. The key simplification isZ that the partition function is a product over all the frequencies, and numerator in ( quency space, P Z the mass pair correlatorsaddle suggests point. that it Kamenev may andwith be Mezard an [ possible integral to that calculatewe is in want very a to similar simple to considerannealed way the disorder the by path average quenched correlation integral function disorder ( (where averaged we would correlation J function average the instead denominator of the Note that the annealedanalytically correlator continuing cannot bewill written have simply a in ramp terms andensemble of of plateau. skew In Hermitian particular, matrices. at function will have a ramp and plateau Here ˜ symmetric function ˜ obtained by continuing However, the connected part of the quenched disorder averaged square of the correlation only receives contributions fromsector energies, energy making differences the thatcorrelation calculation function are much is simpler. equal Explicitly, to the the Euclidean single quenched particle sectors differing by a particle number of one. This means that the correlation function JHEP05(2017)118 , q 37 ]. N ]. At (F.1) 56 ) and n 54 ω ( + G ) “straight across” F.1 . i scale of the ramp comes for each replica gives the 2 ) gives the sine kernel type S + H n 2 ] will be helpful. We hope to across the real axis, one gets ω G − tr ( h e 79 − w 1 , which gives rise to the ramp. L G i ) = ) using saddle points will be more w and t 2 ( ( ρ z g ) ) and z n ( ω ρ ( h , σ + i ) and ), except that we now account for the source 2 t G k ( 2 – 42 – k Z H D.1 1+ tr k ), and thus gives us the plateau. 1 . k σ m n 2 z H L , ω + n tr makes it possible to evaluate the double resolvent h ω n ( 2 2 c ≡ iω ,k G 2 1 and then Wick-pair the Hamiltonians in ( k ,k 1 F k → − k ) in 0 F n N = − , one considers planar graphs only: most of the Wick contractions do 2 iω ) for one frequency ( 2 k λ, λ n − ( by dressed propagators. All that remains is a special class of graphs ω 2 /L = , n 1 R vs. 2 . By taking discontinuities in both ω k i q z, w ). Choosing a replica symmetric solution with H − 1 − w D.2 N ) and Σ( n tr )( ω n H ( 1 This procedure has been carried out for the GUE ensemble by Brezin and Zee [ We start by defining the quantity As we noted above, calculations of For the average of the product of logarithms, the saddle point equations have a Now we evaluate the averaged logarithms of the single frequency factors of the sourced − ω G Their calculation applied to the GUE, while ours applies to the ensemble of skew Hermitian matrices. z ( 37 − tr by replacing where we take The difference between ourdifferent result integrals at comes order from 1/N. the reality constraint on the Majoranas, which gives a In principle,h knowing an expression for the pair correlation function leading order in 1 not contribute, and many of the remaining graphs for the double resolvent can be summed exponential can emerge from such diagrams. It would be niceAs to a have a first directfrom. step, analytical one argument Naively, would for this thetwo like is ramp replicas, to puzzling, and and understand plateau because innot where in simple the the exponential SYK. diagrams ramp factors. such arises correlations from In are correlations this suppressed between appendix, by the powers we of make a simple comment about how the complicated. It appears thatreturn the to Itzykson-Zuber this integral issue [ in future work. F On These contributions correspond topair the correlation semicircle function, part andmetry of the breaking the fluctuations saddle mass give point distribution the involvingcontribution both ramp. and to mass Considering a replica sym- for with a shift of mixing term. TheseG equations are quadraticdominant and contribution thus to have the two solutions, integrals, the fluctuation is the first term that survives. partition function. This isThey almost use exactly the the replica calculationof trick of to the Kamenev rewrite replicated and the partition Mezardby average function. the of [ saddle They the point then logarithm approximation.logarithm in evaluate Their is the terms saddle equivalent replicated of point to the partition equation the average function for equation the obtained average by of combining a the single saddle point equations JHEP05(2017)118 . . , k k . is k k f N /L k 1) − k f 1 . −   , N p . − k 1) +1 coefficient = 1 when q kf − for large relevant for ( − N +1) α . We suspect k k − q and 2 q f m m p − q − − m − − N q − q, N N N − Γ( ) and we Wick-contract m, p m − F.1 ( 1 q =0 F expansion instead of the 1 p in ( X 2 2   factor in the first trace is paired )! k /N ), based on random matrix theory and so it may be a more tractable m H N = − m N e approaches a constant value of 2 N 1 ! − N − k k e f N )!( =0 , which is the origin of the 2 , N binary constraints. The exact formula is q X m k N − − N N N − ramp. ( – 43 – = 2 N ≡ = 2 k − f 2 k . In particular, for the large values of dominates the sum. This is the value   q ∼ N i in the first trace with the corrresponding (reflected, as α − S , as expected for a two-replica correlation. For example, 1 2 q 2 , α − H . . . x k e N ) ≈ 1 i coefficients. k x f 2 q , the late time plateau is a highly non-perturbative effect in SYK ,k 1 of the possible terms that appear in the Hamiltonian. Then k 5 N, m, q the largest X F ( <... = 16 – 45 – ) on a log-log scale. The oscillation observed for N t = 18 ( ) = 0. For d t 32 for g ( , N ) for β d t ( g 24 d are due to the finite number of samples. We expect the true , g d g ), and t = 16 ( c ), and ), and g t t N ( ( c ), c g t g ( ), g t 32 (GOE), for ( ), , t g ( 24 g , we plot = 16 0 (mod 8) (GUE and GSE) cases appear shifted up compared to those for . Plots of 12 N ) decays quickly to typically much smaller values than 6≡ t ( d 0 (mod 8) (GOE) cases, for all values of Around the plateau time, the curves for g N ≡ = 20 and 28 (GSE), a kink for = 5 the peak is broadened and the kink is less visible. However, the plateau heights = 0 before the dip time is also visible for β for N are higher than those for interpretation, symmetry considerations and smoothing due to unfolding effects. seems to rebound.cancellation is This not is perfect.) just because the numberN of samples istion finite for and hence the β due to interference between the upper and lower edgesdip of the time. eigenvalue distribution. Thisfluctuations is in consistent the with edge thesomewhat of theoretical large the expectation coefficient). eigenvalue of distribution Such a at effects gaussian times cancel falloff of out due order in to disorder average to continue decreasing rapidly. H.1 Plots of In figure Figure 12 The noisy part of the curves for JHEP05(2017)118 , p g 7 32 when (H.1) . = 34. ) past t 2 ( . In the ) N p g 28 β t ( ) = 0 . However, e g 24 t /Z ( ) N =0 =1 =5 g β β β β 20 (2 Z 5.3 exp(0.25 N) exp(0.25 5.3 3.1 exp(0.31 N) exp(0.31 3.1 3.5 exp(0.30 N) exp(0.30 3.5 16 does not exhibit clear is defined by fitting the 12 d p κ . )). Also, as explained in t 5 4 3 2

. 10 p 10 10 10

Plateau time time Plateau J t , otherwise we use the time p ··· (2 N g d S κ + . Middle: comparison of ﬁts of the 7 e . 0 = 5. Right: plot of the plateau time 0 N min 32 ) 5. The lower and upper limits of the exp( β g are consistent with the estimate in , . ) corrections, which is close to our < 28 1 d N t t 2 , 2.18 ··· ) 04 for N =5, . − β d 24 ) reaches a plateau at exponentially late 1 t β + 0.12 N 0.12 t ( N = 0 const ( < 20 g (5 O /β β ) exp(2.51 + .091N) + exp(2.51 g t ∼ N ( . g 16 p if 198 t . N d – 46 – , for t 04. N and plateau height . 12 ) cannot be ruled out from our data up to 1 p d t α = 5 23 + 0 × . s N against t 0 0 10

100

is 0.27 up to t min against p

d t Dip time time Dip J t d , we expect ∼ t ) = (0 N < g d β 6 t ) t 014. (2 . ( S 0 g 5. 32 values are insufficient for a conclusive analysis here. . The end of the fitting range is the time at which , we plot p 1 =0 =1 =5 β β β g , N 28 with an exponential function of , plateau time 13 4 d 249 . from our data (although we expect a weak dependence theoretically). . = 0 t )) by a linear fit and find the time at which the line reaches log 24 dip t β β t ( N g which give ) = 0 20 s t t , for ( . Left: the dip time . Then, the lower and upper limits of the error bar are estimated as the smallest 16 N g . A power-law fit ( , the expression for entropy at low temperature is with exponential and power-law functions of min 4 6 = 5 the coefficient of 12 g d t As explained in section As discussed in the main text, the function We can fit β 10

against

100

Dip time time Dip J t p At numerical result 0 and we fit log( right panel of figure section we take an average withramp sufficiently many by samples. a power-law The of plateau constant, the and time finding (linear the functionthe crossing in fitting point the log-log of range plot) the forat and two the the which lines. ramp plateau as We by choose a the starting point of The error bars aresection large but theAgain, results the for available larger time. Numerically, we find that the height agrees with the expectation the dip. Therefore, wevalue estimated the error barand as largest follows. Firstly we found the minimum dependence on H.2 Dip time Intuitively, the dip time canwith be finite statistics, determined the by error finding is the large minimum because value of of the non-self-averaging nature of Figure 13 error bar indicate the rangeSYK of data points with t JHEP05(2017)118 . ) 4 t 0. N , ( c 14 and g > g ], with d κ 58 , otherwise ). Hence − 2 will not be β p mod 8 = 2 N/ κ (2 . For large ), unless there N − S 32 N N , consistent with ∼ 1.5 for all time. = N . These two choices ). Therefore, ,( 28 E 2 q N 2 34. Right: plot of the p ) ) = 0) = 2 β β ), and the unfactorized, /N ( ( 24 β ( Z ,..., < κ N 3.3 /Z ) 12 /Z d , 20 β κ = 0 this equals the inverse of (2 ) there. = 5 = 0 = 5 = 0 t = 5) = 0) Z β β β β β = 10 β β 16 ( , , , , ( ( ), and ( . We observe that 2 2 2 2 p p ) ) ) ) g g ), averaging the spectrum obtained β β β β ( ( ( ( uc norm sense to a Gaussian [ N Z Z Z Z g )/ )/ )/ )/ ) = E 3.2 /N β β β β 2 ( 12 β (2 (2 (2 (2 , where L Z Z Z Z ρ ( for . A clear mod-8 pattern can be seen. For = 0 equals 1 . For 2 2 ), ( p N 2 . We can see nice agreement in figure N β g β d ) for energies of order 0 -1 -2 -3 -4 -5 -6 2 κ after dividing by 10 ) β 3.1

10 10 10 10 10 10

p ( ρ β β β ) ( Z )/ (2 Z ), ( g Plateau height height Plateau β J ) is ( t /Z ∼ ( ) /Z g – 47 – ) β d t 5 against β (2 , ) for many disorder parameters. Almost periodic , which for (2 =20 =22 =24 =26 =28 =30 =32 =34 =12 =14 =16 =18 =10 Z 2 N N N N N N N N N N N N N ) are not large enough to make deﬁnitive statements. Z 2.1 = 0 β and ( N β 10 N ) = /Z ) = 2 p ) ) β β κ for β ( , all eigenvalues are doubly degenerate when ( e 8 E p J 2 p β (2 ( ) 7 g g mod 8 = 0, on the other hand, we do not expect eigenvalue Z β ρ ∼ , there are two options for deﬁning the spectral form factor. ( 6 should be constant up to 1 N 3 p and /Z d t ) κ β due to the degeneracy in the eigenvalue of the SYK Hamiltonian. 4 3.1 (2 2 − ) equals Inverse temperature ) Z . For p β β 2 ( 2 ( κ . However, the small tails of ˜ c g N/ /Z N ∼ ) 0 √ 0 β -3 -4 -5 -1 -2 we plot the normalized density of states ˜

, the distribution will converge in e.g. an

10 10 10 10 10

p (2 . Left: plot of the plateau height against β ) ( g Plateau height height Plateau q /N ∼ . Of course our values of ) Z 15 d 6 E /t = 0) = 2 0 (mod 8), p Theoretically the height of plateau of As we have seen ) = 2 t must agree at early time. Numerically we find they agree at large β β ≡ ( ( u c by diagonalizing the Hamiltonianoscillations due ( to leveland repulsion fixed are clearly observedwidth for small values of described by a Gaussian, and will contain an exponentially large number of states. is not self averaging at early time and so differs from H.4 Density of states In figure As explained inNamely, section the factorized, or annealed, quantitiesor ( quenched, versions where one averagesagree over when the quantity ofg interest is self-averaging (up to order 1 Z degeneracy and thus expect H.3 Comparison of factorized and unfactorized quantities section is degeneracy in the eigenvaluesdiscussed of in the sections model Hamiltonian. Inor the 6. SYK model, Therefore as has we been expect g log( Therefore, the length of the ramp seems to increase exponentially in Figure 14 plateau height and N JHEP05(2017)118 , 08 = 18), N 34. The bin JHEP ]. = 28), 1 000 , N ,..., SPIRE 12 Phys. Rev. Lett. , , IN ]. arXiv:1611.04592 ][ , = 10 = 16), 600 000 ( 0.08 N N (2003) 021 SPIRE = 26), 3 000 ( IN . The numbers of samples are 04 , 0.06 ][ N SYK NJ = 10 = 32 = 34 = 12 = 14 = 16 = 18 = 20 = 22 = 24 = 26 = 28 = 30 . N N N N N N N N N N N N N 0.04 JHEP hep-th/9909056 , [ , talks at KITP, 7 April 2015 and 27 May ]. ]. 0.02 = 14), 1200000 ( N = 24), 10 000 ( 0 SPIRE SPIRE E/N N cond-mat/9212030 IN IN – 48 – [ Is there really a de Sitter/CFT duality? ) for the SYK model with ][ Disturbing implications of a cosmological constant ][ ]. (2000) 024027 E ( -0.02 ρ Quasinormal modes of AdS black holes and the approach to SPIRE D 62 IN Local Operators in the Eternal Black Hole -0.04 ), which permits any use, distribution and reproduction in CFT Partition Functions at Late Times = 12), 5400000 ( (1993) 3339 ][ ]. = 34). D = 22), 48 000 ( N 70 N -0.06 N Gapless spin fluid ground state in a random, quantum Heisenberg hep-th/0208013 SPIRE

[ Phys. Rev. 0 5

arXiv:1502.06692

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ρ IN ,

(E) N N [ ~ CC-BY 4.0 ][ Eternal black holes in anti-de Sitter This article is distributed under the terms of the Creative Commons = 32), 90 ( hep-th/0202163 A simple model of quantum holography [ N (2002) 011 . Notice that the energy is measured in units of ]. Phys. Rev. Lett. J = 10), 10800000 ( 3 , = 20), 120 000 ( 10 − . Normalized density of states ˜ N http://online.kitp.ucsb.edu/online/entangled15/kitaev/ (2015) 211601 N SPIRE IN hep-th/0106112 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/ [ 115 JHEP magnet (2002) 045 thermal equilibrium [ A. Kitaev, E. Dyer and G. Gur-Ari, 2 L. Dyson, M. Kleban and L. Susskind, S. Sachdev and J. Ye, L. Dyson, J. Lindesay and L. Susskind, G.T. Horowitz and V.E. Hubeny, K. Papadodimas and S. Raju, J.M. Maldacena, = 30), 516 ( [7] [8] [5] [6] [2] [3] [4] [1] N References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. 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