Analysis of the phase transition in the $2D$ Ising ferromagnet using a Lempel-Ziv string parsing scheme and black-box data-compression utilities

TL;DR

This paper evaluates information-theoretic measures, including data compression-based estimators, to detect the phase transition in a 2D Ising ferromagnet by analyzing spin sequences at various temperatures.

Contribution

It introduces and compares algorithmic entropy estimators, like Lempel-Ziv and black-box compression utilities, for identifying critical points in the 2D Ising model.

Findings

Multi-information peaks at the critical temperature.

Algorithmic entropy estimators effectively detect phase transition.

Black-box compression utilities provide useful approximate measures.

Abstract

In this work we consider information-theoretical observables to analyze short symbolic sequences, comprising time-series that represent the orientation of a single spin in a $2D$ Ising ferromagnet on a square lattice of size $L^2=128^2$, for different system temperatures $T$. The latter were chosen from an interval enclosing the critical point $T_{\rm c}$ of the model. At small temperatures the sequences are thus very regular, at high temperatures they are maximally random. In the vicinity of the critical point, nontrivial, long-range correlations appear. Here, we implement estimators for the entropy rate, excess entropy (i.e. "complexity") and multi-information. First, we implement a Lempel-Ziv string parsing scheme, providing seemingly elaborate entropy rate and multi-information estimates and an approximate estimator for the excess entropy. Furthermore, we apply easy-to-use black-box data compression utilities, providing approximate estimators only. For comparison and to yield results for benchmarking purposes we implement the information-theoretic observables also based on the well-established M-block Shannon entropy, which is more tedious to apply compared to the the first two "algorithmic" entropy estimation procedures. To test how well one can exploit the potential of such data compression techniques, we aim at detecting the critical point of the $2D$ Ising ferromagnet. Among the above observables, the multi-information, which is known to exhibit an isolated peak at the critical point, is very easy to replicate by means of both efficient algorithmic entropy estimation procedures. Finally, we assess how good the various algorithmic entropy estimates compare to the more conventional block entropy estimates and illustrate a simple modification that yields enhanced results.