The number of spanning trees as an indicator of critical phenomena: When Kirchhoff meets Ising

TL;DR

This paper demonstrates that the number of spanning trees derived from graphs constructed from Ising model simulations can serve as an indicator of critical phenomena, linking graph topology to physical phase transitions.

Contribution

It introduces a novel approach using the spectral properties of graphs, specifically the count of spanning trees, to detect criticality in physical systems like the Ising model.

Findings

Number of spanning trees correlates with critical points in the Ising model.

Spectral properties of adjacency and Laplacian matrices reveal critical behavior.

Structural entropy from Kirchhoff's theorem encodes system criticality.

Abstract

Visibility graphs are spatial interpretations of time series. When derived from the time evolution of physical systems, the graphs associated with such series may exhibit properties that can reflect aspects such as ergodicity, criticality, or other dynamical behaviors. It is important to describe how the criticality of a system is manifested in the structure of the corresponding graphs or, in a particular way, in the spectra of certain matrices constructed from them. In this paper, we show how the critical behavior of an Ising spin system manifests in the spectra of the adjacency and Laplacian matrices constructed from an ensemble of time evolutions simulated via Monte Carlo (MC) Markov Chains, even for small systems and short MC steps. In particular, we show that the number of spanning trees -- or its logarithm -- , which represents a kind of \emph{structural entropy} or \emph{topological complexity} here obtained from Kirchhoff's theorem, can, in an alternative way, describe the criticality of the spin system. These findings parallel those obtained from the spectra of correlation matrices, which similarly encode signatures of critical and chaotic behavior.