Universal critical phase diagram using Gini index

TL;DR

This paper introduces a universal method to reduce multi-parameter critical phase diagrams to a single parameter using the Gini index, demonstrated on various models including Ising and opinion dynamics.

Contribution

The paper presents a novel approach to simplify multi-parameter critical phase surfaces to a single parameter using the Gini index, applicable across different systems.

Findings

Critical phase surfaces can be reduced to a single parameter using the Gini index.

The method is validated on Ising models and opinion dynamics models.

A universal number $g_f$ characterizes the transition in the Gini index.

Abstract

The critical phase surface of a system, in general, can depend on one or more parameters. We show that by calculating the Gini index ($g$) of any suitably defined response function of a system, the critical phase surface can always be reduced to that of a single parameter, starting from $g=0$ and terminating at $g=g_f$, where $g_f$ is a universal number for a chosen response function in a given universality class. We demonstrate the construction with analytical and numerical calculations of mean field transverse field Ising model and site diluted Ising model on the Bethe lattice, respectively. Both models have two parameter critical phase surfaces -- transverse field and temperature for the first case and site dilution and temperature in the second case. Both can be reduced to single parameter transition points in terms of the Gini index. We have additionally demonstrated the validity of the method for a mean field two parameter opinion dynamics model that includes a tri-critical point. The method is generally applicable for any multi-parameter critical transition.