Discrete Phase Transitions Associated to Topological Lattice Field Theories in Dimension $D\ge 2$

Abstract

We investigate the neighborhood of Topological Lattice Field Theories (TLFTs) in the parameter space of general lattice field theories in dimension $D\geq 2$, and discuss the phase structures associated to them. We first define a volume-dependent TLFT, and discuss its decomposition to a direct sum of irreducible TLFTs, which cannot be decomposed anymore. Using this decomposed form, we discuss phase structures and renormalization group flows of volume-dependent TLFTs. We find that TLFTs are on multiple first order phase transition points as well as on fixed points of the flow. The phase structures are controlled by the physical states on $(D-1)$-sphere of TLFTs. The flow agrees with the Nienhuis-Nauenberg criterion. We also discuss the neighborhood of a TLFT in general directions by a perturbative method, so-called cluster expansion. We investigate especially the $Z_p$ 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 non-trivial surfaces. We also discuss the correlation lengths of local fluctuations, and find long-range modes propagating along topological defects. Thus various discrete phase transitions are associated to TLFTs.