New Universality Classes for Two-Dimensional $\sigma$-Models

TL;DR

This paper proposes that two-dimensional $O(N)$-invariant lattice $\sigma$-models with mixed actions have a one-parameter family of continuum limits, confirmed by high-precision Monte Carlo simulations, indicating new universality classes beyond traditional models.

Contribution

It introduces a new continuum limit scenario for 2D $\sigma$-models with mixed actions, supported by extensive numerical evidence.

Findings

Existence of a one-parameter family of continuum limits.

Confirmation that $RP^{N-1}$ and $N$-vector models are in different universality classes.

High-precision Monte Carlo data supports the theoretical predictions.

Abstract

We argue that the two-dimensional $O(N)$-invariant lattice $\sigma$-model with mixed isovector/isotensor action has a one-parameter family of nontrivial continuum limits, only one of which is the continuum $\sigma$-model constructed by conventional perturbation theory. We test the proposed scenario with a high-precision Monte Carlo simulation for $N=3,4$ on lattices up to $512 \times 512$, using a Wolff-type embedding algorithm. [CPU time $\approx$ 7 years IBM RS-6000/320H] The finite-size-scaling data confirm the existence of the predicted new family of continuum limits. In particular, the $RP^{N-1}$ and $N$-vector models do not lie in the same universality class.