Efficient Computational method using random matrices describing critical thermodynamics

TL;DR

This paper introduces a novel matrix-based spectral method using Wishart-like matrices derived from magnetization data to identify and analyze phase transitions and critical phenomena in the Q-state Potts model, revealing universal eigenvalue behaviors.

Contribution

It presents a new computational approach linking spectral properties of correlation matrices to thermodynamic phase transitions in spin systems, including a didactic toy model for universal behavior.

Findings

Successfully distinguishes second-order and weaker first-order transitions.

Establishes a strong correlation between spectral eigenvalues and thermodynamic properties.

Reveals universal eigenvalue distribution patterns influenced by temperature and correlations.

Abstract

Our research highlights the effectiveness of utilizing matrices akin to Wishart matrices, derived from magnetization time series data under specific dynamics, to elucidate phase transitions and critical phenomena in the Q-state Potts model. By employing appropriate statistical methods, we not only discern second-order transitions but also differentiate weaker first-order transitions through careful analysis of the density of eigenvalues and their fluctuations. Furthermore, we investigate the method's sensitivity to stronger first-order transition points. Importantly, we establish a robust correlation between the system's actual thermodynamics and the spectral thermodynamics encapsulated within the eigenvalues. Our findings are further substantiated by correlation histograms of the time series data, revealing insightful patterns. Expanding upon our core findings, we present a didactic analysis that draws parallels between the spectral properties of criticality in a spin system and matrices intentionally imbued with correlations (a toy model). Within this framework, we observe a universal behavior characterized by the distribution of eigenvalues into two distinct groups, separated by a gap dependent on the level of correlation, influenced by temperature-induced changes in the spin system.