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 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
q-state Potts antiferromagnet on infinitely long strips with width
Ly of 2D lattices \ensuremath{\Lambda} with free and periodic boundary conditions in the
y direction, denoted
FBCy and
PBCy. We show that the approach of
W to its 2D thermodynamic limit as
Ly increases is quite rapid; for moderate values of
q 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 and
PBCy, respectively. The approach of
W to the 2D thermodynamic limit is proved to be monotonic (nonmonotonic) for
FBCy (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 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 distributions of the partition function zeros in the complex a=e2betaJ1 plane of the square-lattice Ising model with nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions are investigated as a function of R=J2/J1. Starting from the well-known two-circle distribution of the zeros a=+/-1+sqrt[2]eitheta for R=0 , finally the partition function zeros lie on the unit circle a=eitheta for R=infinity . Between these two ends, the changes in the zero distributions are described. Using the partition function zeros, the critical point ac(R) and the thermal scaling exponent yt(R) are estimated for the Ising ferromagnet (equivalently, antiferromagnet) and superantiferromagnet. For the special case of R=1/2 , the possible implications of the zero distributions are also discussed.
The phase diagram of the
q=3 state Potts antiferromagnet in a field is studied by exact mappings and Monte Carlo methods, and it is shown that systems with macroscopic ground-state degeneracy may exhibit conventional-type critical ordering. Our Monte Carlo data for the square-lattice case indicate a finite-temperature ordering in a range of field values where the entropy of the ground state is extensive. In a particular limit this phase transition can be shown to be of the Ising type by mapping the Potts model onto a colored, nearest-neighbor exclusion lattice gas. We also discuss the phase diagram of the body-centered-cubic Ising antiferromagnet in a field in order to show that renormalization-group arguments alone are not sufficient to restrict the diversity of possible low-temperature phases occurring in systems with a macroscopically degenerate ground state.
An importance sampling Monte Carlo method has been used to study $L\ifmmode\times\else\texttimes\fi{}L$ Ising square lattices with periodic boundary conditions. The behavior of the thermal and magnetic properties was determined over a wide range of
R, the ratio of next-nearest- to nearest-neighbor coupling. From these data we have extracted the
R dependence of the critical temperature and a range of other critical parameters. Within our experimental error the critical exponents are independent of
R but the critical amplitudes are not. The data do however suggest a change in $\ensuremath{\alpha}$ for
R slightly greater than \textonehalf{}. Variations in critical entropy and internal energy may be as large as those resulting from changes in lattice or spin dimensionalities. Our results are compared and contrasted with those obtained by a range of theoretical methods.