Quantum chaos and phase transition in the Yukawa-SYK model

TL;DR

This paper investigates the quantum chaotic behavior of the Yukawa-SYK model across different phases, revealing that chaos remains maximal in the non-Fermi liquid state and exploring the relationship between chaos and phase transitions.

Contribution

It provides a detailed analysis of chaos in the Yukawa-SYK model, including the effect of chemical potential and phase transitions, which was not previously understood.

Findings

OTOCs for fermions and bosons are unaffected by chemical potential in the non-Fermi liquid state.

Lyapunov exponents remain maximal despite the presence of chemical potential.

The boundary of phase stability coincides with regions of maximal chaos.

Abstract

We analyze the quantum chaotic behavior of the Yukawa-SYK model as a function of filling and temperature, which describes random Yukawa interactions between $N$ complex fermions and $M$ bosons in zero spatial dimensions, for both the non-Fermi liquid and insulating states at finite temperature and chemical potential. We solve the ladder equations for the out-of-time-order correlator (OTOC) for both the bosons and fermions. Despite the appearance of the chemical potential in the Hamiltonian, which explicitly introduces an additional energy scale, the OTOCs for the fermions and bosons in the non-Fermi liquid state turn out to be unaffected, and the Lyapunov exponents that diagnose chaos remain maximal. As the chemical potential increases, the system is known to experience a first-order transition from a critical phase to a gapped insulating phase. We postulate that the boundary of the region in parameter space where each phase is (meta)stable coincides with the curve on which the Lyapunov exponent is maximal. By calculating the exponent in the insulating phase and comparing to numerical results on the boundaries of stability, we show that this is plausible.