Open accessJournal Article

# Spectral form factor in the double-scaled SYK model

02 Mar 2021-Journal of High Energy Physics (Springer Science and Business Media LLC)-Vol. 2021, Iss: 3, pp 1-38
Abstract: In this note we study the spectral form factor in the SYK model in large q limit at infinite temperature. We construct analytic solutions for the saddle point equations that describe the slope and the ramp regions of the spectral form factor time dependence. These saddle points are obtained by taking different approaches to the large q limit: the slope region is described by a replica-diagonal solution and the ramp region is described by a replica-nondiagonal solution. We find that the onset of the ramp behavior happens at the Thouless time of order q log q. We also evaluate the one-loop corrections to the slope and ramp solutions for late times, and study the transition from the slope to the ramp. We show this transition is accompanied by the breakdown of the perturbative 1/q expansion, and that the Thouless time is defined by the consistency of extrapolation of this expansion to late times.

Topics: Saddle point (53%), Extrapolation (50%)
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7 results found

Open accessJournal Article
17 Sep 2021-
Abstract: For a generic quantum many-body system, the quantum ergodic regime is defined as the limit in which the spectrum of the system resembles that of a random matrix theory (RMT) in the corresponding symmetry class. In this paper, we analyze the time dependence of correlation functions of operators. We study them in the ergodic limit as well as their approach to the ergodic limit, which is controlled by nonuniversal massive modes. An effective field theory (EFT) corresponding to the causal symmetry and its breaking describes the ergodic phase. We demonstrate that the resulting Goldstone-mode theory has a topological expansion, analogous to the one described by Altland and Sonner [SciPost Phys. 11, 034 (2021)] with added operator sources, whose leading nontrivial topologies give rise to the universal ramp seen in correlation functions. The ergodic behavior of operators in our EFT is seen to result from a combination of RMT-like spectral statistics and Haar averaging over wave functions. Furthermore, we capture analytically the plateau behavior by taking into account the contribution of a second saddle point. Our main interest is quantum many-body systems with holographic duals, and we explicitly establish the validity of the EFT description in the Sachdev-Ye-Kitaev class of models, starting from their microscopic description. By studying the tower of massive modes above the Goldstone sector, we get a detailed understanding of how the ergodic EFT phase is approached, and we derive the relevant Thouless timescales. We point out that the topological expansion can be reinterpreted in terms of contributions of bulk wormholes and baby universes.

Topics: Ergodic theory (57%), ,  ... show more

7 Citations

Open accessJournal Article
Abstract: We describe the non-equilibrium dynamics of the Sachdev-Ye-Kitaev models of fermions with all-to-all interactions. These provide tractable models of the dynamics of quantum systems without quasiparticle excitations. The Kadanoff-Baym equations show that the final state is thermal, and their numerical analysis appears consistent with a thermalization rate proportional to the absolute temperature of the final state. We also obtain an exact analytic solution of the non-equilibrium dynamics in the large $q$ limit of a model with $q$ fermion interactions: in this limit, the thermalization of the fermion Green's function is instantaneous.

5 Citations

Open accessJournal Article
Abstract: For a generic quantum many-body system, the quantum ergodic regime is defined as the limit in which the spectrum of the system resembles that of a random matrix theory (RMT) in the corresponding symmetry class. In this paper we analyse the time dependence of correlation functions of operators. We study them in the ergodic limit as well as their approach to the ergodic limit which is controlled by non-universal massive modes. An effective field theory (EFT) corresponding to the causal symmetry and its breaking describes the ergodic phase. We demonstrate that the resulting Goldstone-mode theory has a topological expansion, analogous to the one described in arXiv:2008.02271 with added operator sources, whose leading non-trivial topologies give rise to the universal ramp seen in correlation functions. The ergodic behaviour of operators in our EFT is seen to result from a combination of RMT-like spectral statistics and Haar averaging over wave-functions. Furthermore we analytically capture the plateau behaviour by taking into account the contribution of a second saddle point. Our main interest are quantum many-body systems with holographic duals and we explicitly establish the validity of the EFT description in the SYK-class of models, starting from their microscopic description. By studying the tower of massive modes above the Goldstone sector we get a detailed understanding of how the ergodic EFT phase is approached and derive the relevant Thouless time scales. We point out that the topological expansion can be reinterpreted in terms of contributions of bulk wormholes and baby-universes.

Topics: Ergodic theory (58%), ,  ... show more

1 Citations

Open accessPosted Content
Matteo A. Cardella1Institutions (1)
Abstract: We find a late times approximation for the SYK spectral form factor from a large $N$ steepest descent version of the path integral over two replica collective fields Main ingredients are a suitable uv regularization of the two replica kinetic operator, the property of its Fourier transform and some spectral analysis of the four point function two replica ladder kernel

Topics: Fourier transform (53%), Replica (52%), Kernel (statistics) (50%)

Open accessPosted Content
Abstract: We consider multi-energy level distributions in the SYK model, and in particular, the role of global fluctuations in the density of states of the SYK model The connected contributions to the moments of the density of states go to zero as $N \to \infty$, however, they are much larger than the standard RMT correlations We provide a diagrammatic description of the leading behavior of these connected moments, showing that the dominant diagrams are given by 1PI cactus graphs, and derive a vector model of the couplings which reproduces these results We generalize these results to the first subleading corrections, and to fluctuations of correlation functions In either case, the new set of correlations between traces (ie between boundaries) are not associated with, and are much larger than, the ones given by topological wormholes The connected contributions that we discuss are the beginning of an infinite series of terms, associated with more and more information about the ensemble of couplings, which hints towards the dual of a single realization In particular, we suggest that incorporating them in the gravity description requires the introduction of new, lighter and lighter, fields in the bulk with fluctuating boundary couplings

Topics: Boundary (topology) (51%)

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76 results found

Open accessJournal Article
Juan Maldacena1, Douglas Stanford1Institutions (1)
26 Apr 2016-Physical Review D
Abstract: The authors study in detail the quantum mechanical model of $N$ Majorana fermions with random interactions of a few fermions at a time (Sachdev-Ye-Kitaev model) in the large $N$ limit. At low energies, the system is strongly interacting and an emergent conformal symmetry develops. Performing technical calculations, the authors elucidate a number of properties of the model near the conformal point.

Topics: Conformal symmetry (59%), Fermion (53%),  ... show more

1,497 Citations

Open accessJournal Article
Subir Sachdev1, Jinwu Ye1Institutions (1)
Abstract: We examine the spin-S quantum Heisenberg magnet with Gaussian-random, infinite-range exchange interactions. The quantum-disordered phase is accessed by generalizing to SU(M) symmetry and studying the large M limit. For large S the ground state is a spin glass, while quantum fluctuations produce a spin-fluid state for small S. The spin-fluid phase is found to be generically gapless---the average, zero temperature, local dynamic spin susceptibility obeys \ensuremath{\chi}\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\omega})\ensuremath{\sim}ln(1/\ensuremath{\Vert}\ensuremath{\omega}\ensuremath{\Vert})+i(\ensuremath{\pi}/2)sgn(\ensuremath{\omega}) at low frequencies.

Topics: Heisenberg model (67%)

1,378 Citations

Open accessJournal Article
Abstract: We study a two dimensional dilaton gravity system, recently examined by Almheiri and Polchinski, which describes near extremal black holes, or more generally, nearly $AdS_2$ spacetimes. The asymptotic symmetries of $AdS_2$ are all the time reparametrizations of the boundary. These symmetries are spontaneously broken by the $AdS_2$ geometry and they are explicitly broken by the small deformation away from $AdS_2$. This pattern of spontaneous plus explicit symmetry breaking governs the gravitational backreaction of the system. It determines several gravitational properties such as the linear in temperature dependence of the near extremal entropy as well as the gravitational corrections to correlation functions. These corrections include the ones determining the growth of out of time order correlators that is indicative of chaos. These gravitational aspects can be described in terms of a Schwarzian derivative effective action for a reparametrization.

Topics: Symmetry breaking (60%), ,  ... show more

927 Citations

Open accessJournal Article
Kristan Jensen1Institutions (1)
Abstract: We revisit two-dimensional holography with the Sachdev-Ye-Kitaev models in mind. Our main result is to rewrite a generic theory of gravity near a two-dimensional anti-de Sitter spacetime throat as a novel hydrodynamics coupled to the correlation functions of a conformal quantum mechanics. This gives a prescription for the computation of n-point functions in the dual quantum mechanics. We thereby find that the dual is maximally chaotic.

Topics: Quantum chaos (59%), Gravitation (53%), Correlation function (52%) ... show more

638 Citations

Open accessJournal Article
Abstract: We investigate a dilaton gravity model in AdS2 proposed by Almheiri and Polchinski [1] and develop a 1d effective description in terms of a dynamical boundary time with a Schwarzian derivative action. We show that the effective model is equivalent to a 1d version of Liouville theory, and investigate its dynamics and symmetries via a standard canonical framework. We include the coupling to arbitrary conformal matter and analyze the effective action in the presence of possible sources. We compute commutators of local operators at large time separation, and match the result with the time shift due to a gravitational shockwave interaction. We study a black hole evaporation process and comment on the role of entropy in this model.

Topics: , Effective action (55%), Dilaton (54%) ... show more

557 Citations

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