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 $\epsilon =3-d$. 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.

(v2) 53 pages, 10 figures. Some clarifications, Appendix B expanded

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