Decoherence and Microscopic Diffusion at SYK

TL;DR

The paper analyzes non-equilibrium dynamics in SYK-type models with $N$ fermions and $k$-body random interactions, showing that ensemble-averaged dynamics exhibit perfect decoherence and a Markovian rate equation for microscopic state probabilities. By exploiting exact permutation symmetry, it reduces the problem to $m+1$ variables and derives a closed set of equations for $p_\alpha$, along with a two-tier diffusion approximation that yields a computable kernel matrix. The kernel spectrum $\{\lambda_i^k(m)\}$ is shown to follow a structured pattern, with explicit forms for small $k$ and a late-time scale $t_r\sim \mathcal{O}(N)/\Gamma$, indicating universal long-time decay of correlators. Collectively, these results provide analytic control over decoherence, 1/$N$ effects, and long-time dynamics, offering a concrete framework for the quantum-to-classical transition and potential experimental tests in SYK-inspired systems.

Abstract

Sachdev-Ye-Kitaev (SYK) or embedded random ensembles are models of $N$ fermions with random k-body interactions. They play an important role in understanding black hole dynamics, quantum chaos, and thermalization. We study out of equilibrium scenarios in these systems and show they display perfect decoherence at all times. This peculiar feature makes them very attractive in the context of the quantum-to-classical transition and the emergence of classical general relativity. Based on this feature and unitarity, we propose a rate/continuity equation for the dynamics of the $\mathcal{O}(e^N)$ microstates probabilities. The effective permutation symmetry of the models drastically reduces the number of variables, allowing for compact expressions of n-point correlation functions and entropy of the microscopic distribution. Further assuming a generalized Fermi golden rule allows finding analytic formulas for the kernel spectrum at finite $N$, providing a series of short and long time scales controlling the out of equilibrium dynamics of this model. This approach to chaos, long time scales, and $1/N$ corrections might be tested in future experiments.