Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Color symmetry and ferromagnetism in Potts spin glass

Hong-Bin Chen

TL;DR

This work analyzes a augmented Potts spin glass with a ferromagnetic term, establishing that color symmetry breaking and ferromagnetic ordering occur at a single critical point $t_{ ext{c}}$. The authors deploy the Parisi formula with self-overlap correction and a Hopf--Lax representation to link the spin-glass order parameter’s symmetry to the magnetization, proving preserved symmetry for $t\le t_{ ext{c}}$ and spontaneous magnetization for $t>t_{ ext{c}}$. They show $t_{ ext{c}}>0$ and finite by comparing to Curie--Weiss limits, and characterize the maximizers of the Hopf--Lax representation, including differentiability properties of the limiting free energy. The results elucidate a deep connection between color symmetry and ferromagnetic order in Potts spin glasses and provide a rigorous framework for analyzing simultaneous ordering phenomena via variational and Hamilton-Jacobi methods.

Abstract

We consider the Potts spin glass with additional ferromagnetic interaction parametrized by $t$. It has long been observed that the Potts color symmetry breaking for the spin glass order parameter is closely related to the ferromagnetic phase transition. To clarify this, we identify a single critical value $t_\mathrm{c}$, which marks the onset of both color symmetry breaking and the transition to ferromagnetism.

Color symmetry and ferromagnetism in Potts spin glass

TL;DR

This work analyzes a augmented Potts spin glass with a ferromagnetic term, establishing that color symmetry breaking and ferromagnetic ordering occur at a single critical point . The authors deploy the Parisi formula with self-overlap correction and a Hopf--Lax representation to link the spin-glass order parameter’s symmetry to the magnetization, proving preserved symmetry for and spontaneous magnetization for . They show and finite by comparing to Curie--Weiss limits, and characterize the maximizers of the Hopf--Lax representation, including differentiability properties of the limiting free energy. The results elucidate a deep connection between color symmetry and ferromagnetic order in Potts spin glasses and provide a rigorous framework for analyzing simultaneous ordering phenomena via variational and Hamilton-Jacobi methods.

Abstract

We consider the Potts spin glass with additional ferromagnetic interaction parametrized by . It has long been observed that the Potts color symmetry breaking for the spin glass order parameter is closely related to the ferromagnetic phase transition. To clarify this, we identify a single critical value , which marks the onset of both color symmetry breaking and the transition to ferromagnetism.
Paper Structure (12 sections, 9 theorems, 47 equations)

This paper contains 12 sections, 9 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Parisi formulae
  4. Color symmetry
  5. Main result
  6. Related works
  7. Proofs
  8. Properties of the initial condition
  9. Limit of free energy
  10. Critical point
  11. Color symmetry breaking
  12. Ferromagnetism

Key Result

Theorem 1.2

Fix $q \in \mathbb{N}$ and $\beta > 0$. Consider the zero external field case, i.e., $x = 0$. There exists a critical value $t_{\mathrm{c}} > 0$, defined as the first point where the map $t \mapsto \lim_{N \to \infty} F_{\beta,N}(t,0)$ becomes non-linear (see e.t_c=), such that the following phase t For $t > t_{\mathrm{c}}$: The critical value $t_{\mathrm{c}}$ also corresponds to the point where

Theorems & Definitions (23)

  • Definition 1.1
  • Theorem 1.2
  • Remark 2.1: Matching notation in chen2023on
  • Lemma 2.2: Properties of the initial condition
  • proof
  • Proposition 2.3: Hopf--Lax formula for the limit
  • proof
  • Lemma 2.4: Envelop theorem
  • proof
  • Lemma 2.5
  • ...and 13 more