Critical behavior of two-dimensional cubic and MN models in the five-loop renormalization-group approximation

TL;DR

This study uses five-loop renormalization-group calculations to analyze the critical behavior of two-dimensional cubic and MN models, revealing fixed point structures and confirming some theoretical predictions.

Contribution

The paper provides the first five-loop RG analysis of 2D cubic and MN models, offering detailed insights into fixed points and their stability.

Findings

For N=2, the RG flows match the line of fixed points from XY to Ising models.

For N≥3, the five-loop terms shift the cubic fixed point towards the Ising fixed point.

A new stable fixed point is identified for moderate M and N values.

Abstract

The critical thermodynamics of the two-dimensional N-vector cubic and MN models is studied within the field-theoretical renormalization-group (RG) approach. The beta functions and critical exponents are calculated in the five-loop approximation and the RG series obtained are resummed using the Borel-Leroy transformation combined with the generalized Pad\'e approximant and conformal mapping techniques. For the cubic model, the RG flows for various N are investigated. For N=2 it is found that the continuous line of fixed points running from the XY fixed point to the Ising one is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both beta functions closer to each another. For the cubic model with N\geq 3, the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N>2 is an artifact of the perturbative analysis. For the quenched dilute O(M) models ($MN$ models with N=0) the results are compatible with a stable pure fixed point for M\geq1. For the MN model with M,N\geq2 all the non-perturbative results are reproduced. In addition a new stable fixed point is found for moderate values of M and N.