Effects of dynamical paths on the energy gap and the corrections to free energy in path integrals of mean-field quantum spin systems

TL;DR

This paper introduces a novel method incorporating dynamical paths into path integrals to compute energy gaps and free energy corrections in mean-field quantum spin systems, addressing challenges in disordered systems.

Contribution

It develops a new approach that extends static approximations by including dynamical effects, enabling more accurate analysis of energy gaps in complex quantum spin models.

Findings

Successfully applied to the infinite-range ferromagnetic Ising model.

Effective in analyzing the Hopfield model with transverse field.

Provides formulae for first excited-state energy calculations.

Abstract

In current studies of mean-field quantum spin systems, much attention is placed on the calculation of the ground-state energy and the excitation gap, especially the latter which plays an important role in quantum annealing. In pure systems, the finite gap can be obtained by various existing methods such as the Holstein-Primakoff transform, while the tunneling splitting at first-order phase transitions has also been studied in detail using instantons in many previous works. In disordered systems, however, it remains challenging to compute the gap of large-size systems with specific realization of disorder. Hitherto, only quantum Monte Carlo techniques are practical for such studies. Recently, Knysh [Nature Comm. \textbf{7}, 12370 (2016)] proposed a method where the exponentially large dimensionality of such systems is condensed onto a random potential of much lower dimension, enabling efficient study of such systems. Here we propose a slightly different approach, building upon the method of static approximation of the partition function widely used for analyzing mean-field models. Quantum effects giving rise to the excitation gap and non-extensive corrections to the free energy are accounted for by incorporating dynamical paths into the path integral. The time-dependence of the trace of the time-ordered exponential of the effective Hamiltonian is calculated by solving a differential equation perturbatively, yielding a finite-size series expansion of the path integral. Formulae for the first excited-state energy are proposed to aid in computing the gap. We illustrate our approach using the infinite-range ferromagnetic Ising model and the Hopfield model, both in the presence of a transverse field.