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arXiv:cond-mat/0102190 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Exact T=0 partition functions for Potts antiferromagnets on sections of the simple cubic lattice

Jesús Salas, Robert E. Shrock

Published Jun 25, 2001• 20 citations
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Abstract

We present exact solutions for the zero-temperature partition function of the q-state Potts antiferromagnet (equivalently, the chromatic polynomial $P)$ on tube sections of the simple cubic lattice of fixed transverse size ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ and arbitrarily great length ${L}_{z},$ for sizes ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}=2\ifmmode\times\else\texttimes\fi{}3$ and $2\ifmmode\times\else\texttimes\fi{}4$ and boundary conditions (a) $({\mathrm{FBC}}_{x},{\mathrm{FBC}}_{y},{\mathrm{FBC}}_{z})$ and (b) $({\mathrm{PBC}}_{x},{\mathrm{FBC}}_{y},{\mathrm{FBC}}_{z}),$ where FBC (PBC) denote free (periodic) boundary conditions. In the limit of infinite length, ${L}_{z}\ensuremath{\rightarrow}\ensuremath{\infty},$ we calculate the resultant ground-state degeneracy per site W (=exponent of the ground-state entropy). Generalizing q from ${Z}_{+}$ to $C,$ we determine the analytic structure of W and the related singular locus $\mathcal{B}$ which is the continuous accumulation set of zeros of the chromatic polynomial. For the ${L}_{z}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit of a given family of lattice sections, W is analytic for real q down to a value ${q}_{c}.$ We determine the values of ${q}_{c}$ for the lattice sections considered and address the question of the value of ${q}_{c}$ for a d-dimensional Cartesian lattice. Analogous results are presented for a tube of arbitrarily great length whose transverse cross section is formed from the complete bipartite graph ${K}_{m,m}.$

DOI
10.1103/physreve.64.011111

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We present exact solutions for the zero-temperature partition function of the q-state Potts antiferromagnet (equivalently, the chromatic polynomial P)P)P) on tube sections of the simple cubic lattice of fixed transverse size Lx\ifmmode×\else\texttimes\fiLy{L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}Lx​\ifmmode×\else\texttimes\fiLy​ and arbitrarily great length Lz,{L}_{z},Lz​, for sizes Lx\ifmmode×\else\texttimes\fiLy=2\ifmmode×\else\texttimes\fi3{L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}=2\ifmmode\times\else\texttimes\fi{}3Lx​\ifmmode×\else\texttimes\fiLy​=2\ifmmode×\else\texttimes\fi3 and 2\ifmmode×\else\texttimes\fi42\ifmmode\times\else\texttimes\fi{}42\ifmmode×\else\texttimes\fi4 and boundary conditions (a) (FBCx,FBCy,FBCz)({\mathrm{FBC}}_{x},{\mathrm{FBC}}_{y},{\mathrm{FBC}}_{z})(FBCx​,FBCy​,FBCz​) and (b) (PBCx,FBCy,FBCz),({\mathrm{PBC}}_{x},{\mathrm{FBC}}_{y},{\mathrm{FBC}}_{z}),(PBCx​,FBCy​,FBCz​), where FBC (PBC) denote free (periodic) boundary conditions. In the limit of infinite length, Lz\ensuremath→\ensuremath∞,{L}_{z}\ensuremath{\rightarrow}\ensuremath{\infty},Lz​\ensuremath→\ensuremath∞, we calculate the resultant ground-state degeneracy per site W (=exponent of the ground-state entropy). Generalizing q from Z+{Z}_{+}Z+​ to C,C,C, we determine the analytic structure of W and the related singular locus B\mathcal{B}B which is the continuous accumulation set of zeros of the chromatic polynomial. For the Lz\ensuremath→\ensuremath∞{L}_{z}\ensuremath{\rightarrow}\ensuremath{\infty}Lz​\ensuremath→\ensuremath∞ limit of a given family of lattice sections, W is analytic for real q down to a value qc.{q}_{c}.qc​. We determine the values of qc{q}_{c}qc​ for the lattice sections considered and address the question of the value of qc{q}_{c}qc​ for a d-dimensional Cartesian lattice. Analogous results are presented for a tube of arbitrarily great length whose transverse cross section is formed from the complete bipartite graph Km,m.{K}_{m,m}.Km,m​.
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 .
The Q-state Potts model can be extended to noninteger and even complex Q by expressing the partition function in the Fortuin-Kasteleyn (F-K) representation. In the F-K representation the partition function Z(Q,a) is a polynomial in Q and v=a-1 (a=e(betaJ)) and the coefficients of this polynomial, Phi(b,c), are the number of graphs on the lattice consisting of b bonds and c connected clusters. We introduce the random-cluster transfer matrix to compute Phi(b,c) exactly on finite square lattices with several types of boundary conditions. Given the F-K representation of the partition function we begin by studying the critical Potts model Z(CP)=Z(Q,a(c)(Q)), where a(c)(Q)=1+square root[Q]. We find a set of zeros in the complex w=square root[Q] plane that map to (or close to) the Beraha numbers for real positive Q. We also identify Q(c)(L), the value of Q for a lattice of width L above which the locus of zeros in the complex p=v/square root[Q] plane lies on the unit circle. By finite-size scaling we find that 1/Q(c)(L)-->0 as L-->infinity. We then study zeros of the antiferromagnetic (AF) Potts model in the complex Q plane and determine Q(c)(a), the largest value of Q for a fixed value of a below which there is AF order. We find excellent agreement with Baxter's conjecture Q(AF)(c)(a)=(1-a)(a+3). We also investigate the locus of zeros of the ferromagnetic Potts model in the complex Q plane and confirm that Q(FM)(c)(a)=(a-1)(2). We show that the edge singularity in the complex Q plane approaches Q(c) as Q(c)(L) approximately Q(c)+AL(-y(q)), and determine the scaling exponent y(q) for several values of Q. Finally, by finite-size scaling of the Fisher zeros near the antiferromagnetic critical point we determine the thermal exponent y(t) as a function of Q in the range 2</=Q</=3. Using data for lattices of size 3</=L</=8 we find that y(t) is a smooth function of Q and is well fitted by y(t)=(1+Au+Bu2)/(C+Du) where u=-(2/pi)cos(-1)(squareroot[Q]/2). For Q=3 we find y(t) approximately 0.6; however if we include lattices up to L=12 we find y(t) approximately 0.50(8) in rough agreement with a recent result of Ferreira and Sokal [J. Stat. Phys. 96, 461 (1999)].
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 partition function for the q-state Potts model for general q, temperature and magnetic field on strips of the square lattices of width Ly=2L_{y}=2Ly​=2 and arbitrary length $L_x = m $ with periodic longitudinal boundary conditions. A new representation of the transfer matrix for the q-state Potts model is introduced which can be used to calculate the determinant of the transfer matrix for an arbirary m×mm \times mm×m lattice with periodic boundary conditions.
We calculate complex-temperature (CT) zeros of the partition function for the qqq-state Potts model on the honeycomb and kagom\'e lattices for several values of qqq. These give information on the CT phase diagrams. A comparison of results obtained for different boundary conditions and a discussion of some CT singularities are given. Among other results, our findings show that the Potts antiferromagnet with q=4q=4q=4 and q=5q=5q=5 on the kagom\'e lattice has no phase transition at either finite or zero temperature.
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.