Quantum ergodicity in the SYK model

TL;DR

This work presents a replica-field-theory treatment of the SYK model to understand its ergodicity at long times. It identifies a vast set of nonergodic collective modes, labeled by Clifford-algebra generators, whose densities grow exponentially with mode index while individual decay rates increase with that index, producing a slow approach to a universal ergodic regime described by random matrix theory. The authors derive a key spectral-correlation formula incorporating massive modes, derive the ergodic time scale, and show that nonlinear corrections do not disrupt the main results. The analytical predictions for $R_2(\omega)$ and the spectral form factor $K(\tau)$ align qualitatively and semi-quantitatively with existing numerical data, strengthening the connection between many-body quantum chaos and RMT in the SYK context.

Abstract

We present a replica path integral approach describing the quantum chaotic dynamics of the SYK model at large time scales. The theory leads to the identification of non-ergodic collective modes which relax and eventually give way to an ergodic long time regime (describable by random matrix theory). These modes, which play a role conceptually similar to the diffusion modes of dirty metals, carry quantum numbers which we identify as the generators of the Clifford algebra: each of the $2^N$ different products that can be formed from $N$ Majorana operators defines one effective mode. The competition between a decay rate quickly growing in the order of the product and a density of modes exponentially growing in the same parameter explains the characteristics of the system's approach to the ergodic long time regime. We probe this dynamics through various spectral correlation functions and obtain favorable agreement with existing numerical data.

Figures (5)

• Figure 1: Left: Semiclassical representation of the spectral two-point function through Green function amplitudes. Inset top: microscopic structure of scattering vertex in coordinate (left) and momentum (right) representation. Inset center: abbreviated representation of momentum conserving two particle mode. Inset bottom: spectral correlation function as one loop diagram involving two modes. At low energies higher order loop processes gain importance. Further discussion, see text.
• Figure 2: Left: building blocks of the Majorana relaxation modes. Scattering now is off the operators $X_a\equiv \chi_{i_1}\chi_{i_2}\chi_{i_3}\chi_{i4}$ entering the scattering Hamiltonian. Modes in Fock space are deconfined in that the many body states $|n\rangle$ and $|m\rangle$ correlated by scattering can be very different $|n-m|=\mathcal{O}(N)$. (Here, $|n-m|$ is the Hamming distance between $n$ and $m$, i.e. the number of binary symbols in $n$ that need to be switched to get to $m$.) Right: the conserved quantum numbers of the process are the labels, $\nu$, of the basis states of the Majorana Clifford algebra, as discussed in the text.
• Figure 3: Number variance of systems with $N=22$ and $N=34$, resp., compared to that of the GUE. Discussion, see text.
• Figure 4: Spectral form factor as a function of scaled time $\tau$ (left) and of physical time $t$ (right). Discussion, see text.
• Figure 5: Contraction rules used in the perturbative computation of traces of higher order in the matrices $y_\mu$. Discussion, see text.