Quantum chaos and phase transition in the Yukawa-SYK model

We analyze the quantum chaotic behavior of the Yukawa-SYK model, which describes random Yukawa interactions between N complex fermions and M bosons in zero spatial dimensions, in both the non-Fermi liquid and insulating phases at finite temperature and chemical potential. Using Green's functions, we solve the ladder equations for the out-of-time-order correlator (OTOC) for both bosons and fermions. Despite the appearance of the chemical potential in the Hamiltonian, which explicitly introduces an additional energy scale to the problem, the OTOCs for the fermions and bosons in the non-Fermi liquid phase turn out to be unaffected as long as the system remains in the "self-tuned critical" phase, and the Lyapunov exponents that diagnose chaos remain maximal. As the chemical potential increases, the system is known to transition from a critical phase to a gapped insulating phase. We conjecture that the boundary of the region in parameter space where each phase is 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, we show that this is plausible.