From generating functions to the geometric Binder cumulant

TL;DR

This paper explores the use of generating functions and geometric Binder cumulants in quantum systems to detect phase transitions, extending formalism to include degeneracy points and validating with model systems.

Contribution

It introduces a generalized Bargmann invariant as a generating function for cumulants, enabling the use of geometric Binder cumulants to identify quantum phase transitions.

Findings

Geometric Binder cumulants are sensitive to gap closure.

The approach successfully detects metal-insulator transitions.

Validation with model systems confirms the method's effectiveness.

Abstract

We present an overview of the role of generating functions in quantum mechanical contexts, mainly in the modern theory of polarization and in the study of quantum phase transitions. Generating functions enable the derivation of moments and cumulants, quantities which characterize the fluctuations of an underlying probability distribution. In all of the cases we review, the fluctuations are those of a quantum system. We show that the original formalism for geometric phases, in which a quantum system is taken around an adiabatic cycle, can be extended to the case when degeneracy points are encountered along the cycle (quasiadiabatic cycles). The essential tool for this extension is a generalized Bargmann invariant which plays the role of a generating function. From the cumulants generated this way one can form ratios according to the Binder cumulant scheme in statistical mechanics. Such geometric Binder cumulants are sensitive to gap closure, as such, they are useful in identifying metal-insulator transitions, localization, and quantum phase transitions. We present example calculations on simple model systems, whose localization properties are well known, to validate to approach. We also complement our geometric Binder cumulant calculations with results for the fidelity susceptibility, a quantity directly related to the quantum geometry of the parameter space.