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 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
Lz, 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)
(FBCx,FBCy,FBCz) and (b)
(PBCx,FBCy,FBCz), 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
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
qc. We determine the values of
qc for the lattice sections considered and address the question of the value of
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.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)].
The distribution of the zeros of the partition function in the complex temperature plane (Fisher zeros) of the two-dimensional Q-state Potts model is studied for non-integer Q. On L×L self-dual lattices studied (L⩽8), no Fisher zero lies on the unit circle p0=eiθ in the complex p=(eβJ−1)/Q plane for Q<1, while some of the Fisher zeros lie on the unit circle for Q>1 and the number of such zeros increases with increasing Q. The ferromagnetic and antiferromagnetic properties of the Potts model are investigated using the distribution of the Fisher zeros. For the Potts ferromagnet we verify the den Nijs formula for the thermal exponent yt. For the Potts antiferromagnet we also verify the Baxter conjecture for the critical temperature and present new results for the thermal exponents in the range 0<Q<3.
We calculate the large-
q series expansion for the ground-state degeneracy (= exponent of the ground-state entropy) per site of the
q-state Potts antiferromagnet on the $(3\ensuremath{\cdot}{12}^{2})$ lattice, to order
O(y19), where $y=1/(q\ensuremath{-}1)$. We note a remarkable agreement, to
O(y18), between this series and a rigorous lower bound derived recently.
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.