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Ground state entropy of the Potts antiferromagnet on triangular lattice strips. (English) Zbl 1046.82505
Summary: The authors present exact calculations of the zero-temperature partition function (chromatic polynomial) \(P\) for the \(q\)-state Potts antiferromagnet on triangular lattice strips of arbitrarily great length \(L_x\) vertices and of width \(L_y\) vertices and, in the \(L_x\to\infty\) limit, the exponent of the ground state entropy \(W= e^{S_0/k_B}\). The strips considered, with their boundary conditions \((BC)\), are (a) \((FBC_y,PBC_x)=\) cyclic for \(L_y= 3,4\), (b) \((FBC_y, TPBC_x)=\) Möbius, \(L_y=3\), (c) \((PBC_y,PBQ_x)=\) toroidal, \(L_y= 3\), (d) \((PBC_y,TPBC_x)=\) Klein bottle, \(L_y=3\), (e) \((PBC_yF BC_x) =\) cylindrical, \(L_y=5,6\), and (f) \((FBC_y,FBC_x)=\) free, \(L_y=5\), where \(F,P\), and \(TP\) denote free, periodic, and twisted periodic. Several interesting features are found, including the presence of terms in \(P\) proportional to \(\cos(2\pi L_x/3)\) for case (c). The continuous locus of points \({\mathcal B}\) where \(W\) is nonanalytic in the \(q\) plane is discussed for each case and a comparative discussion is given of the respective loci \({\mathcal B}\) for families with different boundary conditions. Numerical values of \(W\) are given for infinite-length strips of various widths and are shown to approach values for the 2D lattice rapidly. A remark is also made concerning a zero-free region for chromatic zeros. Some results are given for strips of other lattices.

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