Modular Invariance of Finite Size Corrections and a Vortex Critical Phase

Abstract

We find the exact finite size and lattice corrections, to the partition function $Z$, of a continuous spin Gaussian model on a toroidal triangular lattice, with periods ${L}_{0}$ and ${L}_{1}$. The spins carry a representation of the fundamental group of the torus labeled by ${u}_{0}$ and ${u}_{1}$ and a mass $m$. Summing ${Z}^{\ensuremath{-}1/2}$ over ${u}_{i}$ gives the corresponding result for the Ising model. The limits $m\ensuremath{\rightarrow}0$ and ${u}_{i}\ensuremath{\rightarrow}0$ do not commute. With $m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ and ${u}_{i}$ nonzero the model exhibits a vortex critical phase. In the continuum limit, for arbitrary $m$, the finite size corrections to $\ensuremath{-}\mathrm{ln}Z$ are modular invariant. In the limit ${L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$ the ``cylinder charge'' $c({u}_{0}{,m}^{2}{L}_{0}^{2})$ ranges nonmonotonically from $2[{1+6u}_{0}({u}_{0}\ensuremath{-}1)]$ for $m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ to zero for $m\ensuremath{\rightarrow}\ensuremath{\infty}$.