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

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.

51 pages (pdflatex). Contains a .bbl file, and 9 figures in pdf format. The tex source uses IOP and tikz macros (not included). v2 contains many changes with respect to v1. Final version published in journal

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• Probing symmetries of quantum many-body systems through gap ratio statistics
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• Conformally invariant boundary conditions in the antiferromagnetic Potts model and the $SL(2,\mathbb{R})/U(1)$ sigma model
• The continuum limit of $a_{N-1}^{(2)}$ spin chains
• Elaborating the phase diagram of spin-1 anyonic chains
• The phase diagram for the bisected-hexagonal-lattice five-state Potts antiferromagnet
• Quantum deformation of cubic string field theory