Critical spin chains and loop models with $PSU(n)$ symmetry

TL;DR

This paper introduces and analyzes two-dimensional critical models with $PSU(n)$ symmetry, describing their spectra and conformal field theories, and relating them to known $O(n)$ and Potts models.

Contribution

It develops new $PSU(n)$ symmetric models in 2D, computes their spectra, and conjectures a relation to $O(n)$ models via orbifolding.

Findings

Models exhibit a $PSU(n)$ symmetric CFT for any complex $n$

Spectra are similar but simpler than $O(n)$ and Potts CFTs

Conjecture: $O(n)$ CFT is a $bZ_2$ orbifold of the $PSU(n)$ CFT

Abstract

Starting with the Ising model, statistical models with global symmetries provide fruitful approaches to interesting physical systems, for example percolation or polymers. These include the $O(n)$ model (symmetry group $O(n)$) and the Potts model (symmetry group $S_Q$). Both models make sense for $n,Q\in \mathbb{C}$ and not just $n,Q\in \mathbb{N}$, and both give rise to a conformal field theory in the critical limit. Here, we study similar models based on the group $PSU(n)$. We focus on the two-dimensional case, where the models can be described either as gases of non-intersecting orientable loops, or as alternating spin chains. This allows us to determine their spectra either by computing a twisted torus partition function, or by studying representations of the walled Brauer algebra. In the critical limit, our models give rise to a CFT that exists for any $n\in\mathbb{C}$ and has a global $PSU(n)$ symmetry. Its spectrum is similar to those of the $O(n)$ and Potts CFTs, but a bit simpler. We conjecture that the $O(n)$ CFT is a $\mathbb{Z}_2$ orbifold of the $PSU(n)$ CFT, where $\mathbb{Z}_2$ acts as complex conjugation.