On the path integral representation for quantum spin models and its application to the quantum cavity method and to Monte Carlo simulations

TL;DR

This paper develops a continuous imaginary time path integral approach for quantum spin models, extending the cavity method and enabling efficient Monte Carlo simulations, with validation on a Bethe lattice ferromagnet.

Contribution

It introduces an analytical procedure for continuous time limit in quantum cavity method and demonstrates its application to quantum Monte Carlo simulations.

Findings

The continuous time quantum cavity method is formulated for a broad class of models.

The method's predictions are validated against quantum Monte Carlo simulations.

The approach simplifies the simulation of quantum spin models on sparse graphs.

Abstract

The cavity method is a well established technique for solving classical spin models on sparse random graphs (mean-field models with finite connectivity). Laumann et al. [arXiv:0706.4391] proposed recently an extension of this method to quantum spin-1/2 models in a transverse field, using a discretized Suzuki-Trotter imaginary time formalism. Here we show how to take analytically the continuous imaginary time limit. Our main technical contribution is an explicit procedure to generate the spin trajectories in a path integral representation of the imaginary time dynamics. As a side result we also show how this procedure can be used in simple heat-bath like Monte Carlo simulations of generic quantum spin models. The replica symmetric continuous time quantum cavity method is formulated for a wide class of models, and applied as a simple example on the Bethe lattice ferromagnet in a transverse field. The results of the methods are confronted with various approximation schemes in this particular case. On this system we performed quantum Monte Carlo simulations that confirm the exactness of the cavity method in the thermodynamic limit.