Configurational Information Measures, Phase Transitions, and an Upper Bound on Complexity

TL;DR

The paper addresses how Configurational Entropy $CE$ and Configurational Complexity $CC$ diagnose stability and phase transitions, focusing on the Ising universality class. It derives a model-dependent CC–CE relationship and a model-independent upper bound on $CC$, then tests CIM phenomenology in 2D and 3D Ising lattices near $T_c$ using extensive simulations. Key findings include characteristic spectral reorganization at criticality, a dual magnetization–energy CIM behavior under temperature inversion, and the universal bound $CC/L^d < 1/(e\ln 2)$ bits per node, supporting CIMs as analytical tools for critical phenomena. The work outlines implications for applying CIMs to other universality classes and quantum phase transitions, while highlighting interpretational nuances in defining complexity.

Abstract

Configurational entropy (CE) and configurational complexity (CC) are recently popularized information theoretic measures used to study the stability of solitons. This paper examines their behavior for 2D and 3D lattice Ising Models, where the quasi-stability of fluctuating domains is controlled by proximity to the critical temperature. Scaling analysis lends support to an unproven conjecture that these configurational information measures (CIMs) can detect (in)stability in field theories. The primary results herein are the derivation of a model dependent CC-CE relationship, as well as a model independent upper bound on CC. CIM phenomenology in the Ising universality class reveals multiple avenues for future research.

Figures (3)

• Figure 1: Representative configurations of the 2D Ising model at different temperatures. The left column shows spin configurations, while the right column displays the corresponding energy distributions based on Eq. \ref{['eq: energy density']}. The middle row corresponds to a temperature near the critical point $T_c$, with the top and bottom rows depicting states at higher and lower temperatures, respectively.
• Figure 2: Temperature dependence of configurational information measures (CIMs) for magnetization in the Ising model. The left and right panels correspond to the 2D square and 3D cubic lattices, respectively. The top row displays the configurational complexity per spin, while the bottom row shows the deviation of configurational entropy from its maximal value, $\log L^d$. The shaded regions represent the 80% confidence interval from ensemble fluctuations, with minima marked by circles. The vertical dashed lines indicate the thermodynamic critical temperatures.
• Figure 3: Temperature dependence of configurational information measures (CIMs) for energy fluctuations in the Ising model. The left and right panels correspond to the 2D square and 3D cubic lattices, respectively. The layout and descriptions follow Fig. \ref{['fig: CIMs magnetization']}, demonstrating consistency between magnetization- and energy-based CIMs.