Discrete phase transitions associated with topological lattice field theories in D >= 2 dimensions.

Abstract

We investigate the neighborhood of topological lattice field theories (TLFT's) in the parameter space of general lattice field theories in D\ensuremath{\ge}2 dimensions, and discuss the phase structures associated with them. We first discuss the decomposition of a TLFT into a direct sum of irreducible TLFT's, which cannot be decomposed anymore. Using this decomposed form, we discuss the phase structure and the renormalization group flow near a TLFT in a certain restricted parameter space. We find that a TLFT is on a multiple first-order phase transition point as well as on the fixed point of the flow. The phase structure is controlled by the physical states on a (D-1)-sphere of the TLFT. The flow agrees with the Nienhuis-Nauenberg criterion. We also discuss the neighborhood of a TLFT in general directions by a perturbative method, the so-called cluster expansion. We investigate especially the ${\mathit{Z}}_{\mathit{p}}$ analogue of the Turaev-Viro model, and find that the TLFT is in general on a higher order discrete phase transition point. The phase structures depend on the topology of the base manifold and are controlled by the physical states on topologically nontrivial surfaces. We also discuss the correlation lengths of local fluctuations, and find some long-range modes propagating along topological defects. Thus various discrete phase transitions are associated with TLFT's.