1996/06/21 by Charles Nash, C. Nash, Denjoe O' Connor · 1 citation
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat #hep-lat #hep-th
paper · pdf · doi:10.1006/aphy.1998.5868
published as Annals Phys. 273 (1999) 72-98 · 25 pages Plain TeX
arxiv created 1996/06/21 · arxiv updated 2009/11/30
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 L0 and L1 where the spins carry a representation of the fundamental group of the torus labeled by phases u0 and u1. We compute the \it exact finite size and lattice corrections, to the partition function Z, for arbitrary mass m and phases ui. 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→0 and ui→0 do not commute. Also when m=0 the model exhibits a \it vortex critical phase when at least one of the ui 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 L1→∞ the ``cylinder charge'' c(u0,m2L02) is a non-monotonic function of m that ranges from 2(1+6u0(u0-1)) for m=0 to zero for m→∞ 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.