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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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References:
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