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

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) = 3.819 671.... 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 Q is an element of (2, 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.