# Spectrum of the Wilson-Fisher conformal field theory on the torus

## TL;DR

This paper analyzes the finite-size spectrum of the Wilson-Fisher O(N) conformal field theory on a torus using epsilon expansion and numerical methods, revealing universal spectral features that identify different universality classes.

## Contribution

It derives universal effective Hamiltonians for the zero modes and combines analytical epsilon expansion with numerical diagonalization to study the critical spectrum.

## Key findings

- Effective Hamiltonians are N-dimensional anharmonic oscillators.
- Numerical solutions match analytical predictions.
- Spectral differences serve as fingerprints for universality classes.

## Abstract

We study the finite-size spectrum of the O($N$) symmetric Wilson-Fisher conformal field theory (CFT) on the $d=2$ spatial-dimension torus using the expansion in $\epsilon=3-d$. This is done by deriving a set of universal effective Hamiltonians describing fluctuations of the zero momentum modes. The effective Hamiltonians take the form of $N$-dimensional quantum anharmonic oscillators, which are shown to be strongly coupled at the critical point for small $\epsilon$. The low-energy spectrum is solved numerically for $N = 1,2,3,4$. Using exact diagonalization (ED), we also numerically study explicit lattice models known to be in the O($2$) and O($3$) universality class, obtaining estimates of the low-lying critical spectrum. The analytic and numerical results show excellent agreement and the critical low energy torus spectra are qualitatively different among the studied CFTs, identifying them as a useful fingerprint for detecting the universality class of a quantum critical point.