Detecting Quantum and Classical Phase Transitions via Unsupervised Machine Learning of the Fisher Information Metric

TL;DR

This paper introduces ClassiFIM, an unsupervised machine learning approach that effectively detects quantum and classical phase transitions by estimating the Fisher information metric from limited measurement data, outperforming prior methods in efficiency.

Contribution

The paper presents a novel unsupervised ML method, ClassiFIM, for estimating the Fisher information metric to detect phase transitions, with theoretical proofs and empirical validation on various models.

Findings

ClassiFIM reliably detects topological and dynamical phase transitions.

It requires less training data and only classical measurements.

It outperforms prior unsupervised ML methods in accuracy and resource efficiency.

Abstract

The detection of quantum and classical phase transitions in the absence of an order parameter is possible using the Fisher information metric (FIM), also known as fidelity susceptibility. Here, we propose and investigate an unsupervised machine learning (ML) task: estimating the FIM given limited samples from a multivariate probability distribution of measurements made throughout the phase diagram. We utilize an unsupervised ML method called ClassiFIM (developed in a companion paper) to solve this task and demonstrate its empirical effectiveness in detecting both quantum and classical phase transitions using a variety of spin and fermionic models, for which we generate several publicly available datasets with accompanying ground-truth FIM. We find that ClassiFIM reliably detects both topological (e.g., XXZ chain) and dynamical (e.g., metal-insulator transition in Hubbard model) quantum phase transitions. We perform a detailed quantitative comparison with prior unsupervised ML methods for detecting quantum phase transitions. We demonstrate that ClassiFIM is competitive with these prior methods in terms of appropriate accuracy metrics while requiring significantly less resource-intensive training data compared to the original formulation of the prior methods. In particular, ClassiFIM only requires classical (single-basis) measurements. As part of our methodology development, we prove several theorems connecting the classical and quantum fidelity susceptibilities through equalities or bounds. We also significantly expand the existence conditions of the fidelity susceptibility, e.g., by relaxing standard differentiability conditions. These results may be of independent interest to the mathematical physics community.