Upper and Lower Bounds for Ground State Entropy of Antiferromagnetic Potts Models

We derive rigorous upper and lower bounds for the ground state entropy of the q-state Potts antiferromagnet on the honeycomb and triangular lattices. These bounds are quite restrictive, especially for large q. ∗email: [email protected] ∗∗email: [email protected] Nonzero ground state disorder and associated entropy, S0 6= 0, is an important subject in statistical mechanics; a physical realization is provided by ice, for which S0 = 0.82± 0.05 cal/(K-mole), i.e., S0/kB = 0.41± 0.03 [1, 2]. Ground state (g.s.) entropy may or may not be associated with frustration. An early example with frustration is the Ising (equivalently, q = 2 Potts) antiferromagnet on the triangular lattice [3]. However, g.s. entropy is also exhibited in the simpler context of models without frustration, such as the q-state Potts antiferromagnet (AF) [4]-[6] on the square (sq) and honeycomb (hc) lattices for (integral) q ≥ 3 and on the triangular (tri) lattice for q ≥ 4. Of these three 2D lattices, S0 has been calculated exactly for the triangular case [7], but, aside from the single value S0(sq, q = 3)/kB = (3/2) ln(4/3) [8], not for the square or honeycomb lattices. Therefore, it is valuable to have rigorous upper and lower bounds on this quantity. Using a “coloring matrix” method, Biggs derived such bounds for the square lattice [9]. Here we shall extend his method to derive analogous bounds for the honeycomb lattice and compare the results with our recent Monte Carlo measurements [10, 11] and with large-q series [12]. We also derive such bounds for the triangular lattice; the interest in this case is that the bounds can be compared with the exact result of Baxter [7]. We make use of the fact that the partition function at T = 0, Z(Λ, q,K = −∞), for the q-state zero-field Potts AF on a lattice Λ (where K = βJ , β = 1/(kBT ), and J < 0 denotes the spin-spin coupling) is equal to the chromatic polynomial P (Λ, q). Here, P (G, q) is the number of ways of coloring the graph G with q colors such that no adjacent vertices (sites) have the same color [13]. Define the reduced, per site free energy for the Potts AF in the thermodynamic limit as f(Λ, q,K) = limN→∞N −1 lnZ(Λ, q,K). From the general relation between the entropy S, the internal energy U , and the reduced free energy, S = βU + f (henceforth, we use units such that kB = 1), together with the property that limK→−∞ βU(β) = 0, as is true of the q-state Potts AF models considered here, it follows that S0(Λ, q) = f(Λ, q,K = −∞) = lnW (Λ, q), where W (Λ, q) is the asymptotic limit W (Λ, q) = lim N→∞ P (Λ, q) (1) Given this connection, we shall express our bounds on the g.s. entropy S0(Λ, q) in terms of the equivalent function W (Λ, q). As we have discussed earlier [11], the formal eq. (1) is not, in general, adequate to define W (Λ, q) because of a noncommutativity of limits lim N→∞ lim q→qs P (Λ, q) 6= lim q→qs lim N→∞ P (Λ, q) (2) at certain special points qs. We denote the definitions based on the first and second orders of limits in (2) as W (Λ, q)Dnq and W (Λ, q)Dqn, respectively. This noncommutativity can occur for q < qc(Λ), where qc(Λ) denotes the maximal (finite) real value of q where W (Λ, q) is