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

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Phase Transitions

Background:

• The O(n) loop model is a fundamental model in statistical mechanics used to study phase transitions.
• Understanding its phase diagram is crucial for relating microscopic interactions to macroscopic behavior.
• Previous studies have explored aspects of the O(n) loop model, but a comprehensive phase diagram analysis is ongoing.

Purpose of the Study:

• To determine the phase diagram of the O(n) loop model on a square lattice in the (x,n) plane.
• To identify critical lines and phase transitions, particularly for n>>2 and in the range -2≤n≤2.
• To investigate the influence of allowed vertex configurations on the model's topological phase diagram.

Main Methods:

• Utilized transfer-matrix calculations to analyze the O(n) loop model.
• Employed finite-size scaling techniques to extract critical exponents and locate phase boundaries.
• Expressed correlation lengths in terms of transfer-matrix eigenvalues to identify loop density ordering.

Main Results:

• Identified Ising-like phase transitions for n>>2, characterized by checkerboard ordering of elementary loops.
• This ordered state shows similarities to the hard-square lattice gas model, with 'n' acting as a softening parameter.
• Determined critical points for -2≤n≤2, revealing a dependence of the phase diagram topology on allowed vertex sets.

Conclusions:

• The phase diagram of the O(n) loop model exhibits rich behavior, including Ising-like transitions for large 'n'.
• The model's critical properties are influenced by the choice of allowed vertices, affecting the continuity of phase transitions.
• This work provides a detailed mapping of the O(n) loop model phase diagram, offering insights into lattice gas models and critical phenomena.