Critical line of an n-component cubic model

TL;DR

This paper investigates the critical behavior of an n-component cubic model on a square lattice, determining critical points and universal quantities through transfer matrix methods and finite-size scaling, with results aligning with theoretical predictions for certain n values.

Contribution

It constructs a transfer matrix for the cubic model, performs finite-size scaling to find critical points, and analyzes universal quantities, extending understanding of the model's critical behavior.

Findings

Critical points determined for various n values.

Results agree with O(n) predictions for n<2.

Self-dual plane does not contain critical points for 1<n<2.

Abstract

We consider a special case of the n-component cubic model on the square lattice, for which an expansion exists in Ising-like graphs. We construct a transfer matrix and perform a finite-size-scaling analysis to determine the critical points for several values of n. Furthermore we determine several universal quantities, including three critical exponents. For n<2, these results agree well with the theoretical predictions for the critical O(n) branch. This model is also a special case of the ($N_\alpha,N_\beta$) model of Domany and Riedel. It appears that the self-dual plane of the latter model contains the exactly known critical points of the n=1 and 2 cubic models. For this reason we have checked whether this is also the case for 1<n<2. However, this possibility is excluded by our numerical results.