Antiferromagnet on the Honeycomb Lattice

We study the q-state Potts antiferromagnet with q = 3 on the honeycomb lattice. Using an analytic argument together with a Monte Carlo simulation, we conclude that this model is disordered for all T ≥ 0. We also calculate the ground state entropy to be S0/kB = 0.507(10) and discuss this result. ∗email: [email protected] ∗∗email: [email protected] The effect of ground state disorder and associated nonzero ground state entropy S0 has been a subject of longstanding interest. A physical example is ice, for which S0 = 0.82±0.05 cal/(K-mole), i.e., S0/kB = 0.41 ± 0.03 [1, 2]. Among spin models, an example is the Ising antiferromagnet (AF) on the triangular lattice. In the context of this model, Wannier argued that a nonzero ground state (g.s.) entropy implies the absence of long-range order, viz., staggered magnetization Mst for T ≥ 0 [3]. Another example is the Ising AF on the kagomé lattice [4, 5]. In both of these Ising models, the nonzero g.s. entropy has the effect of removing a phase transition at finite temperature. The Ising AF on the triangular lattice is critical at T = 0 [6], while on the kagomé lattice, with a larger value of S0, it is disordered even at T = 0 [5]. In these two cases, the nonzero g.s. entropy is associated with frustration. However, there are also spin models, such as the antiferromagnetic q-state Potts model [7]-[9] on the square (sq) and honeycomb (hc) lattice, which exhibit g.s. entropy without frustration. Because of the absence of frustration, these models constitute ideally simple cases where one can study the effects of ground state entropy on the thermodynamics of a statistical mechanical model. In contrast to the ferromagnetic (FM) Potts model, which has a finite-temperature phase transition for dimensionality d > 1, the question of whether the q-state Potts AF has a phase transition at finite (or zero) temperature is more delicate and depends on both the value of q and the type of lattice. The q = 3 Potts AF on the square lattice has been well studied; an exact result of Baxter showed that it is critical at T = 0 [11], in agreement with a renormalization group argument [12], and several Monte Carlo simulations have been performed on it [13, 14]. However, to our knowledge, the behavior of the q = 3 Potts AF on the honeycomb lattice has not been definitely established. We report here the results of a study of this model. The (isotropic, nearest-neighbor, zero-field) q-state Potts model on a lattice Λ is defined by the partition function Z = ∑ {σn} e −βH with the Hamiltonian H = −J ∑