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

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.

Figures (4)

• Figure 1: Time evolution of magnetization for three different random seeds at two temperatures, $T=0.45$ and $T=15.6$. The corresponding visibility graphs, constructed from these series, reflect distinct spectral properties in the random adjacency matrices derived from them.
• Figure 2: Density of eigenvalues of adjacency matrices constructed from visibility graphs of magnetization time series at different temperatures. At high temperatures, the densities resemble those of Gaussian noise visibility graphs, shown with bars for comparison. At lower temperatures, the distributions exhibit heavier tails.
• Figure 3: Eigenvalue spacing distribution. The same temperatures used for the density plots are shown here. Shorter tails are observed at higher temperatures, while longer tails appear at lower temperatures, consistent with the behavior seen in the eigenvalue density distributions.
• Figure 4: Fig. (a) Average of the logarithm of the number of spanning trees (structural entropy) per MC step for different temperatures. Fig. (b) Derivative of the quantity shown in Fig. (a) for the same temperatures. Both quantities appear to signal a phase transition at $T \approx T_{C} = \frac{1}{2} \ln(1 + \sqrt{2})$, indicating that the critical point corresponds to the maximum of the structural entropy.