Q-colourings of the triangular lattice: Exact exponents and conformal field theory

TL;DR

This paper analyzes Q-colourings of the triangular lattice using integrable models and conformal field theory, deriving exact critical exponents and exploring their implications for phase transitions and related quantum systems.

Contribution

It provides an exact solution for the Q-colouring problem on the triangular lattice and identifies the associated conformal field theory for 2<Q<=4, extending previous understanding.

Findings

Exact critical exponents for Q-colourings derived

Identification of conformal field theory regime IV

Implications for phase diagram of antiferromagnetic Potts model

Abstract

We revisit the problem of Q-colourings of the triangular lattice using a mapping onto an integrable spin-one model, which can be solved exactly using Bethe Ansatz techniques. In particular we focus on the low-energy excitations above the eigenlevel g_2, which was shown by Baxter to dominate the transfer matrix spectrum in the Fortuin-Kasteleyn (chromatic polynomial) representation for Q_0 <= Q <= 4, where Q_0 = 3.819671... We argue that g_2 and its scaling levels define a conformally invariant theory, the so-called regime IV, which provides the actual description of the (analytically continued) colouring problem within a much wider range, namely 2 < Q <= 4. The corresponding conformal field theory is identified and the exact critical exponents are derived. We discuss their implications for the phase diagram of the antiferromagnetic triangular-lattice Potts model at non-zero temperature. Finally, we relate our results to recent observations in the field of spin-one anyonic chains.