Spectrum of the Wilson-Fisher Conformal Field Theory on the Torus

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 ε=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 ε. The low-energy spectrum is solved numerically for N=1,2,3,4. Using exact diagonalization, 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.