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

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 is an element of = 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 c. 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.