Analytical spectral density of the Sachdev-Ye-Kitaev model at finite N

Abstract

We derive an approximate analytical formula for the spectral density of the q-body This spectral density reproduces the known result for the free energy in the large q and N limit at arbitrary values of the temperature. In the infrared region, where the SachdevYe-Kitaev model is believed to have a gravity-dual, the analytical spectral density is given. It therefore has a square-root edge, as in random matrix ensembles, followed by an exponential growth, a distinctive feature of black holes and also of low energy nuclear excitations. Results for level-statistics in this region confirm the agreement with random matrix theory. Physically this is a signature that, for sufficiently long times, the SYK model and its gravity dual evolve to a fully ergodic state whose dynamics only depends on the global symmetry of the system. Our results strongly suggest that random matrix correlations are a universal feature of quantum black holes and that the SYK model, combined with holography, may be relevant to model certain aspects of the nuclear dynamics.

Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev model

“…In the nonsupersymmetric SYK model, it was found [32,33] that in both the bulk of the spectrum and the region close to the ground state, spectral correlations for small eigenvalue separations are well described by random matrix theory. This is an indication that for sufficiently long times, of the order of the Heisenberg time, the SYK model is still quantum chaotic and has reached an ergodic state where the dynamics is universal since it depends only on the global symmetry of the system.…”

“…[32] by an explicit evaluation of the moments, turned out to be given by the weight function of the Q-Hermite polynomials with Q ¼ η where ηðN; qÞ is a suppression factor related to the commutation of two products of q Majorana operators. A similar calculation can be carried out for odd q.…”

“…The SYK model in the strong coupling limit has a maximal Lyapunov exponent [20,[23][24][25][26][27], a finite zero temperature entropy [20,21,28,29], a linear specific heat in the low temperature limit [30,31], the density of the low energy excitations grows exponentially [32][33][34][35] and short range spectral correlations are given by random matrix theory [33,36,37]. We note that the observation of a nonzero Lyapunov exponent at the Ehrenfest time and random matrix like level statistics at the Heisenberg time, a much larger time scale of the order of mean level spacing, are both signatures of quantum chaos.…”

The bulk dual of SYK: cubic couplings

“…Like a vector model, SYK has an O(N ) symmetry, with fields in the fundamental representation. Yet, the physical properties of SYK are closer to, for instance, strongly coupled N = 4 Yang-Mills: SYK is maximally chaotic [2] and appears to display features of random matrix theories [6][7][8][9][10][11][12][13][14]. The SYK model is currently in a different state from both the gauge theory and the O(N ) theory examples of AdS/CFT.…”

Approximating the Sachdev-Ye-Kitaev model with Majorana wires

“…As one notable example, Majorana-fermion zero modes [1,2] capture the essence of non-Abelian statistics and topological quantum computation [3,4], and correspondingly now form the centerpiece of a vibrant experimental effort [5][6][7][8][9][10][11][12][13][14][15][16]. More recently, randomly interacting Majorana fermions governed by the "Sachdev-YeKitaev (SYK) model" [17][18][19] were shown to exhibit sharp connections to chaos, quantum-information scrambling, and black holes-naturally igniting broad interdisciplinary activity (see, e.g., [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]). The goal of this Rapid Communication is to exploit hardware components of a Majorana-based topological quantum computer for a tabletop implementation of the SYK model, thus uniting these very different topics.…”