Anatomy of information scrambling and decoherence in the integrable Sachdev-Ye-Kitaev model

TL;DR

This paper analytically studies the time evolution of information scrambling in an integrable SYK model coupled to a bath, revealing stages of polynomial growth, linear decrease, and eventual saturation, with implications for quantum information processing.

Contribution

It provides the first complete analytical description of OTOC dynamics in an integrable SYK model, including effects of a Markovian bath and spectral form factors.

Findings

OTOC exhibits polynomial growth, saturation, and linear decrease stages.

Linear decrease in OTOC is governed by the spectral form factor.

Environmental coupling causes exponential decay of OTOC at long times.

Abstract

The growth of information scrambling, captured by out-of-time-order correlation functions (OTOCs), is a central indicator of the nature of many-body quantum dynamics. Here, we compute analytically the complete time dependence of the OTOC for an integrable Sachdev-Ye-Kitaev (SYK) model, $N$ Majoranas with random two-body interactions of infinite range, coupled to a Markovian bath at finite temperature. In the limit of no coupling to the bath, the time evolution of scrambling experiences different stages. For $t \lesssim \sqrt{N}$, after an initial polynomial growth, the OTOC approaches saturation in a power-law fashion with oscillations superimposed. At $t \sim \sqrt{N}$, the OTOC reverses trend and starts to decrease linearly in time. The reason for this linear decrease is that, despite being a subleading $1/N$ effect, the OTOC in this region is governed by the spectral form factor of the antisymmetric couplings of the SYK model. The linear decrease stops at $t \sim 2N$, the Heisenberg time, where saturation occurs. The effect of the environment is an overall exponential decay of the OTOC for times longer than the inverse of the coupling strength to the bath. The oscillations at $t \lesssim \sqrt{N}$ indicate lack of thermalization -- a desired feature for a better performance of quantum information devices.