Quantum Monte Carlo study of the transverse-field quantum Ising model on infinite-dimensional structures

TL;DR

This study uses quantum Monte Carlo simulations to analyze the critical behavior of the transverse-field quantum Ising model on infinite-dimensional networks, confirming mean-field characteristics and proposing a new observable for critical analysis.

Contribution

It demonstrates that quantum Ising models on infinite-dimensional structures exhibit mean-field critical behavior and introduces a practical order parameter for critical analysis.

Findings

Both networks show mean-field critical behavior.

The cumulant method struggles with dynamic critical exponent estimation.

A quantum expectation value-based order parameter is effective for critical analysis.

Abstract

In a number of classical statistical-physical models, there exists a characteristic dimensionality called the upper critical dimension above which one observes the mean-field critical behavior. Instead of constructing high-dimensional lattices, however, one can also consider infinite-dimensional structures, and the question is whether this mean-field character extends to quantum-mechanical cases as well. We therefore investigate the transverse-field quantum Ising model on the globally coupled network and the Watts-Strogatz small-world network by means of quantum Monte Carlo simulations and the finite-size scaling analysis. We confirm that both the structures exhibit critical behavior consistent with the mean-field description. In particular, we show that the existing cumulant method has a difficulty in estimating the correct dynamic critical exponent and suggest that an order parameter based on the quantum-mechanical expectation value can be a practically useful numerical observable to determine critical behavior when there is no well-defined dimensionality.