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Exact Potts model partition functions on strips of the honeycomb lattice. (English) Zbl 0978.82020
Summary: We present exact calculations of the partition function of the \(q\)-state Potts model on (i) open, (ii) cyclic, and (iii) Möbius strips of the honeycomb (brick) lattice of width \(L_y=2\) and arbitrarily great length. In the infinite-length limit the thermodynamic properties are discussed. The continuous locus of singularities of the free energy is determined in the \(q\) plane for fixed temperature and in the complex temperature plane for fixed \(q\) values. We also give exact calculations of the zero-temperature partition function (chromatic polynomial) and \(W(q)\), the exponent of the ground-state entropy, for the Potts antiferromagnet for honeycomb strips of type (iv) \(L_y=3\), cyclic, (v) \(L_y=3\), Möbius, (vi) \(L_y=4\), cylindrical, and (vii) \(L_y=4\), open. In the infinite-length limit we calculate \(W(q)\) and determine the continuous locus of points where it is nonanalytic. We show that our exact calculation of the entropy for the \(L_y=4\) strip with cylindrical boundary conditions provides an extremely accurate approximation, to a few parts in \(10^5\) for moderate \(q\) values, to the entropy for the full 2D honeycomb lattice (where the latter is determined by Monte Carlo measurements since no exact analytic form is known).

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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