Truncated Conformal Space Approach for Perturbed Wess-Zumino-Witten $SU(2)_k$ Models

TL;DR

This paper applies the truncated conformal space approach to study perturbations of $SU(2)_k$ Wess-Zumino-Witten models, successfully reproducing spectra and finite size effects in several cases, and comparing results with field theory predictions.

Contribution

It demonstrates the effectiveness of TCSA in analyzing perturbed $SU(2)_k$ models, including cases with marginal perturbations and equivalences to known models.

Findings

TCSA reproduces the sine-Gordon spectrum at specific couplings.

TCSA matches known spectra for $SU(2)_2$ perturbed models.

Discrepancies found between TCSA and Lorentz-invariant field theory calculations for marginal perturbations.

Abstract

We outline the application of the truncated conformal space approach (TCSA) to perturbations of $SU(2)_k$ Wess-Zumino-Witten theories. As examples of this methodology, we consider two distinct perturbations of $SU(2)_1$ and one of $SU(2)_2$. $SU(2)_1$ is first perturbed by its spin-1/2 field, a model which is equivalent to the sine-Gordon model at a particular value of its coupling $\beta$. The sine-Gordon spectrum is correctly reproduced as well as the corresponding finite size corrections. We next study $SU(2)_1$ with a marginal current-current perturbation. The TCSA results can be matched to perturbation theory within an appropriate treatment of the UV divergences. We find however that these results do not match field theoretic computations on the same model performed with a Lorentz invariant regulator. Finally, we consider $SU(2)_2$ perturbed by its spin-1 field, which is equivalent to three decoupled massive Majorana fermions. In this case as well the TCSA reproduces accurately the known spectrum.