R\'enyi complexity in mean-field disordered systems

TL;DR

This paper introduces a method to compute Rényi complexity in mean-field disordered systems, revealing its relation to free energy differences and its behavior at phase transitions, thus offering a practical approach to characterizing disorder.

Contribution

It provides the first detailed calculation of Rényi complexity for mean-field disordered models, linking it to free energy differences and phase transition phenomena.

Findings

Rényi complexity relates to free energy differences in disordered models.

Rényi complexity vanishes at the Kauzmann transition for models with one-step RSB.

RSB solutions are necessary even in the liquid phase for these models.

Abstract

Configurational entropy, or complexity, plays a critical role in characterizing disordered systems such as glasses, yet its measurement often requires significant computational resources. Recently, R\'enyi entropy, a one-parameter generalization of Shannon entropy, has gained attention across various fields of physics due to its simpler functional form, making it more practical for measurements. In this paper, we compute the R\'enyi version of complexity for prototypical mean-field disordered models, including the random energy model, its generalization, referred to as the random free energy model, and the $p$-spin spherical model. We first demonstrate that the R\'enyi complexity with index $m$ is related to the free energy difference for a generalized annealed Franz-Parisi potential with $m$ clones. Detailed calculations show that for models having one-step replica symmetry breaking (RSB), the R\'enyi complexity vanishes at the Kauzmann transition temperature $T_K$, irrespective of $m>1$, while RSB solutions are required even in the liquid phase. This study strengthens the link between R\'enyi entropy and the physics of disordered systems and provides theoretical insights for its practical measurements.