# Tunable Quantum Chaos in the Sachdev-Ye-Kitaev Model Coupled to a Thermal Bath

## TL;DR

This paper investigates how coupling SYK models to a thermal bath affects quantum chaos indicators like Lyapunov exponent and butterfly velocity, revealing tunable chaos and thermalization behaviors.

## Contribution

It introduces a model of SYK chains coupled to a thermal bath, demonstrating controllable quantum chaos properties and spatially dependent thermalization effects.

## Key findings

- Lyapunov exponent decreases with stronger bath coupling
- Butterfly velocity transitions from 2T to linear T dependence
- Spatial dependence of chaos metrics when only chain ends are coupled

## Abstract

The Sachdev-Ye-Kitaev (SYK) model describes Majorana fermions with random interaction, which displays many interesting properties such as non-Fermi liquid behavior, quantum chaos, emergent conformal symmetry and holographic duality. Here we consider a SYK model or a chain of SYK models with $N$ Majorana fermion modes coupled to another SYK model with $N^2$ Majorana fermion modes, in which the latter has many more degrees of freedom and plays the role as a thermal bath. For a single SYK model coupled to the thermal bath, we show that although the Lyapunov exponent is still proportional to temperature, it monotonically decreases from $2\pi/\beta$ ($\beta=1/(k_BT)$, $T$ is temperature) to zero as the coupling strength to the thermal bath increases. For a chain of SYK models, when they are uniformly coupled to the thermal bath, we show that the butterfly velocity displays a crossover from a $\sqrt{T}$-dependence at relatively high temperature to a linear $T$-dependence at low temperature, with the crossover temperature also controlled by the coupling strength to the thermal bath. If only the end of the SYK chain is coupled to the thermal bath, the model can introduce a spatial dependence of both the Lyapunov exponent and the butterfly velocity. Our models provide canonical examples for the study of thermalization within chaotic models.