Modular invariance, lattice field theories and finite size corrections

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 $m\to 0$ and $u_i\to 0$ 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 $L_1\to \infty$ the ``cylinder charge'' $c(u_0,m^2{L}_0^2)$ is a non-monotonic function of $m$ that ranges from $2(1+6u_0(u_0-1))$ for $m=0$ to zero for $m\to \infty$ 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.

25 pages Plain TeX

References in corpus (3)

• Determinants of Laplacians, the Ray-Singer Torsion on Lens Spaces and the Riemann zeta function
• Environmentally Friendly Renormalization
• Modular Invariance of Finite Size Corrections and a Vortex Critical Phase

Cited by in corpus (9)

• Universal Finite-size Scaling Functions with Exact Non-universal Metric Factors
• Exact Finite-Size-Scaling Corrections to the Critical Two-Dimensional Ising Model on a Torus. II. Triangular and hexagonal lattices
• Exact partition functions of the Ising model on MxN planar lattices with periodic-aperiodic boundary conditions
• Dimers and the Critical Ising Model on Lattices of genus>1
• Finite size corrections for the Ising model on higher genus triangular lattices
• From Ginzburg-Landau to Hilbert-Einstein via Yamabe
• Ising Model on the Affine Plane
• Dimer geometry, amoebae and a vortex dimer model
• Dimer Models, Free Fermions and Super Quantum Mechanics