Jesper Jacobsen

Laboratoire de Physique Théorique, École Normale Supérieure, Paris.

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

We revisit the problem of proper vertex colourings of the triangular lattice, using Q distinct colours. Our approach is based on 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 (in 1986) 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, that we call regime IV and which extends the three previously known regimes for the spin-one model. The new regime IV provides the actual description of the (analytically continued) colouring problem within a much wider range, namely 2 < Q <= 4, although g_2 is only dominant on the smaller range. 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.