Information theoretic aspects of the two-dimensional Ising model

TL;DR

This paper investigates the information theoretic properties of the 2D Ising model, revealing how entropy and mutual information diverge logarithmically at criticality, and explores universality across different lattice structures.

Contribution

The study provides precise numerical estimates of entropy differences, mutual information, and their divergence at critical points, advancing understanding of information measures in phase transitions.

Findings

Mutual information diverges logarithmically at the critical point.

Entropy differences scale with system size and boundary conditions.

Logarithmic divergence of excess entropy applies to various subsets of the lattice.

Abstract

We present numerical results for various information theoretic properties of the square lattice Ising model. First, using a bond propagation algorithm, we find the difference $2H_L(w) - H_{2L}(w)$ between entropies on cylinders of finite lengths $L$ and 2L with open end cap boundaries, in the limit $L\to\infty$. This essentially quantifies how the finite length correction for the entropy scales with the cylinder circumference $w$. Secondly, using the transfer matrix, we obtain precise estimates for the information needed to specify the spin state on a ring encircling an infinite long cylinder. Combining both results we obtain the mutual information between the two halves of a cylinder (the "excess entropy" for the cylinder), where we confirm with higher precision but for smaller systems results recently obtained by Wilms et al. -- and we show that the mutual information between the two halves of the ring diverges at the critical point logarithmically with $w$. Finally we use the second result together with Monte Carlo simulations to show that also the excess entropy of a straight line of $n$ spins in an infinite lattice diverges at criticality logarithmically with $n$. We conjecture that such logarithmic divergence happens generically for any one-dimensional subset of sites at any 2-dimensional second order phase transition. Comparing straight lines on square and triangular lattices with square loops and with lines of thickness 2, we discuss questions of universality.