On the continuity of phase transition of three-dimensional square-lattice XY models

TL;DR

The paper investigates the continuity of magnetization at the critical temperature for the three-dimensional XY model on a cubic lattice. It extends switching-lemma techniques to directed graphs arising from the random path representation by introducing edge pairing and finitely-paired graphs, under a finite-loop assumption at criticality. The main result shows $m^*(\beta_c)=0$ (i.e., continuity of the magnetization) under Assumption uniqClus, with supporting results linking plus/free boundary correlations and Gaussian domination arguments. This work broadens the toolkit for analyzing phase-transition continuity in XY models by adapting switching-and-cancellation strategies to directed graphical representations.

Abstract

We study the continuity of magnetization at the phase transition of the ferromagnetic XY model in the three-dimensional square lattice with the nearest neighborhood interaction. We assume that, at the critical temperature, with probability 1, for every edge in the infinite directed graph generated by the random path representation, finitely many edges exist so that they form a finite loop. Then, we prove that the phase transition is continuous at the critical temperature. The main technical contribution is to find a switching lemma to establish a bijection between equally weighted graphs.

Figures (4)

• Figure 1: Two images
• Figure 2: An example for finitely-paired graph
• Figure 3: Before surgical switching
• Figure 4: After surgical switching

Theorems & Definitions (24)

• Lemma 1.1
• Theorem 2.1
• Theorem 2.3
• Proposition 4.1
• Lemma 5.1
• Proposition 5.2
• proof
• Lemma 6.1
• Definition 6.2
• Definition 6.3