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arXiv:cond-mat/9910377 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Ground-state degeneracy of Potts antiferromagnets on two-dimensional lattices: Approach using infinite cyclic strip graphs

Robert E. Shrock, Shan-Ho Tsai

Published Oct 1, 1999• 43 citations
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Abstract

The q-state Potts antiferromagnet on a lattice Lambda exhibits nonzero ground-state entropy S0=kB ln W for sufficiently large q and hence is an exception to the third law of thermodynamics. An outstanding challenge has been the calculation of W(sq,q) on the square (sq) lattice. We present here an exact calculation of W on an infinite-length cyclic strip of the square lattice, which embodies the expected analytic properties of W(sq,q). Similar results are given for the kagomé lattice.

DOI
10.1103/physreve.60.3512

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The q-state Potts antiferromagnet on a lattice Lambda exhibits nonzero ground-state entropy S0=kB ln W for sufficiently large q and hence is an exception to the third law of thermodynamics. An outstanding challenge has been the calculation of W(sq,q) on the square (sq) lattice. We present here an exact calculation of W on an infinite-length cyclic strip of the square lattice, which embodies the expected analytic properties of W(sq,q). Similar results are given for the kagomé lattice.
We study the ground state degeneracy per site (exponent of the ground state entropy) $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{L}_{y},q)$ for the qqq-state Potts antiferromagnet on infinitely long strips with width Ly{L}_{y}Ly​ of 2D lattices \ensuremath{\Lambda} with free and periodic boundary conditions in the yyy direction, denoted FBCy{\mathrm{FBC}}_{y}FBCy​ and PBCy.{\mathrm{PBC}}_{y}.PBCy​. We show that the approach of WWW to its 2D thermodynamic limit as Ly{L}_{y}Ly​ increases is quite rapid; for moderate values of qqq and ${L}_{y}\ensuremath{\simeq}4,$ $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{L}_{y},q)$ is within about 5% and ${O(10}^{\ensuremath{-}3})$ of the 2D value $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{(L}_{y}=\ensuremath{\infty}),q)$ for FBCy{\mathrm{FBC}}_{y}FBCy​ and PBCy,{\mathrm{PBC}}_{y},PBCy​, respectively. The approach of WWW to the 2D thermodynamic limit is proved to be monotonic (nonmonotonic) for FBCy{\mathrm{FBC}}_{y}FBCy​ (PBCy).({\mathrm{PBC}}_{y}).(PBCy​). It is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states.
We present exact calculations of the zero-temperature partition function (chromatic polynomial) and W(q), the exponent of the ground-state entropy, for the q-state Potts antiferromagnet with next-nearest-neighbor spin-spin couplings on square lattice strips, of width L(y)=3 and L(y)=4 vertices and arbitrarily great length Lx vertices, with both free and periodic boundary conditions. The resultant values of W for a range of physical q values are compared with each other and with the values for the full two-dimensional lattice. These results give insight into the effect of such nonnearest-neighbor couplings on the ground-state entropy. We show that the q=2 (Ising) and q=4 Potts antiferromagnets have zero-temperature critical points on the Lx-->infinity limits of the strips that we study. With the generalization of q from Z+ to C, we determine the analytic structure of W(q) in the q plane for the various cases.
We present exact calculations of the zero-temperature partition function, , and ground-state degeneracy (per site), , for the q -state Potts antiferromagnet on a number of families of graphs for which the boundary of regions of analyticity of W in the complex q -plane is noncompact and has the properties that (i) in the -plane, the point z = 0 is a multiple point on and (ii) includes support for . These families are generated by the method of homeomorphic expansion. Our results give further insight into the conditions for the validity of large- q series expansions for the reduced function .
We calculate the large-qqq series expansion for the ground-state degeneracy (= exponent of the ground-state entropy) per site of the qqq-state Potts antiferromagnet on the $(3\ensuremath{\cdot}{12}^{2})$ lattice, to order O(y19){O(y}^{19})O(y19), where $y=1/(q\ensuremath{-}1)$. We note a remarkable agreement, to O(y18){O(y}^{18})O(y18), between this series and a rigorous lower bound derived recently.