Dynamics of Order Parameters of Non-stoquastic Hamiltonians in the Adaptive Quantum Monte Carlo Method

TL;DR

This paper derives deterministic flow equations for order parameters in non-stoquastic Hamiltonians using adaptive quantum Monte Carlo, enabling analysis of phase transitions and stability in complex quantum spin models.

Contribution

It introduces a novel derivation of macroscopic flow equations for non-stoquastic Hamiltonians within the adaptive quantum Monte Carlo framework.

Findings

Flow equations are consistent with mean-field saddle-point solutions.

The method captures effects of anti-ferromagnetic XX interactions as transverse field fluctuations.

Stability analysis of equilibrium solutions is performed.

Abstract

We derive macroscopically deterministic flow equations with regard to the order parameters of the ferromagnetic $p$-spin model with infinite-range interactions. The $p$-spin model has a first-order phase transition for $p>2$. In the case of $p\geq5$ ,the $p$-spin model with anti-ferromagnetic XX interaction has a second-order phase transition in a certain region. In this case, however, the model becomes a non-stoqustic Hamiltonian, resulting in a negative sign problem. To simulate the $p$-spin model with anti-ferromagnetic XX interaction, we utilize the adaptive quantum Monte Carlo method. By using this method, we can regard the effect of the anti-ferromagnetic XX interaction as fluctuations of the transverse magnetic field. A previous study derived deterministic flow equations of the order parameters in the quantum Monte Carlo method. In this study, we derive macroscopically deterministic flow equations for the magnetization and transverse magnetization from the master equation in the adaptive quantum Monte Carlo method. Under the Suzuki-Trotter decomposition, we consider the Glauber-type stochastic process. We solve these differential equations by using the Runge-Kutta method and verify that these results are consistent with the saddle-point solution of mean-field theory. Finally, we analyze the stability of the equilibrium solutions obtained by the differential equations.