Modular invariance, lattice field theories and finite size corrections

TL;DR

This paper develops a lattice field theory approach to analyze finite size effects and modular invariance in one and two-dimensional quantum field theories, revealing new insights into vortex phases and critical behavior.

Contribution

It introduces a combinatorial lattice model for quantum fields on a circle and torus, providing exact finite size corrections and exploring non-commuting limits and vortex critical phases.

Findings

Exact finite size and lattice corrections to the partition function are computed.

Finite size corrections are modular invariant and expressed via elliptic theta functions.

The cylinder charge varies non-monotonically with mass, linking to the central charge.

Abstract

We give a lattice theory treatment of certain one and two dimensional quantum field theories. In one dimension we construct a combinatorial version of a non-trivial field theory on the circle which is of some independent interest in itself while in two dimensions we consider a field theory on a toroidal triangular lattice. We take a continuous spin Gaussian model on a toroidal triangular lattice with periods $L_0$ and $L_1$ where the spins carry a representation of the fundamental group of the torus labeled by phases $u_0$ and $u_1$. We compute the {\it exact finite size and lattice corrections}, to the partition function $Z$, for arbitrary mass $m$ and phases $u_i$. Summing $Z^{-1/2}$ over a specified set of phases gives the corresponding result for the Ising model on a torus. An interesting property of the model is that the limits $m\rightarrow0$ and $u_i\rightarrow0$ do not commute. Also when $m=0$ the model exhibits a {\it vortex critical phase} when at least one of the $u_i$ is non-zero. In the continuum or scaling limit, for arbitrary $m$, the finite size corrections to $-\ln Z$ are {\it modular invariant} and for the critical phase are given by elliptic theta functions. In the cylinder limit $L_1\rightarrow\infty$ the ``cylinder charge'' $c(u_0,m^2L_0^2)$ is a non-monotonic function of $m$ that ranges from $2(1+6u_0(u_0-1))$ for $m=0$ to zero for $m\rightarrow\infty$ but from which one can determine the central charge $c$. The study of the continuum limit of these field theories provides a kind of quantum theoretic analog of the link between certain combinatorial and analytic topological quantities.