Ising-like transitions in the O($n$) loop model on the square lattice

TL;DR

This paper investigates the phase diagram of the O(n) loop model on a square lattice, identifying Ising-like phase transitions and critical points across various n values using transfer-matrix and finite-size scaling methods.

Contribution

It provides new insights into the critical behavior and phase transitions of the O(n) loop model, especially for large n, and explores how the phase diagram topology depends on allowed vertex configurations.

Findings

Identifies Ising-like phase transitions for large n with checkerboard loop ordering.

Determines critical points for -2 ≤ n ≤ 2.

Shows phase diagram topology varies with vertex set choices.

Abstract

We explore the phase diagram of the O($n$) loop model on the square lattice in the $(x,n)$ plane, where $x$ is the weight of a lattice edge covered by a loop. These results are based on transfer-matrix calculations and finite-size scaling. We express the correlation length associated with the staggered loop density in the transfer-matrix eigenvalues. The finite-size data for this correlation length, combined with the scaling formula, reveal the location of critical lines in the diagram. For $n>>2$ we find Ising-like phase transitions associated with the onset of a checkerboard-like ordering of the elementary loops, i.e., the smallest possible loops, with the size of an elementary face, which cover precisely one half of the faces of the square lattice at the maximum loop density. In this respect, the ordered state resembles that of the hard-square lattice gas with nearest-neighbor exclusion, and the finiteness of $n$ represents a softening of its particle-particle potentials. We also determine critical points in the range $-2\leq n\leq 2$. It is found that the topology of the phase diagram depends on the set of allowed vertices of the loop model. Depending on the choice of this set, the $n>2$ transition may continue into the dense phase of the $n \leq 2$ loop model, or continue as a line of $n \leq 2$ O($n$) multicritical points.