The Critical Line of an Ising Antiferromagnet on Square and Honeycomb Lattices

Abstract

We show that the singularity of the free energy of Ising models in the absence of a magnetic field on the triangular, square, and honeycomb lattices is related to zeros of the pseudopartition function on an elementary cycle. Using the Griffiths' smoothness postulate, we extend these results to the case in a magnetic field and derive a formula of the critical line of an Ising antiferromagnet, which is in good agreement with the numerical results.[S0031-9007 (96)02173-4] PACS numbers: 05.50.+ q, 64.60.Cn, 75.10.Hk, 75.40.CxSince Onsager's famous solution of the square lattice Ising model in the absence of a magnetic field [1], the Ising model became a standard model for testing the scaling and universality hypotheses [2,3]. However, the Ising model in a magnetic field has not been solved exactly so far, although some exact results are known. Of particular interest is to determine the critical line in the ͑h, T ͒ plane, along which the free energy becomes singular. Yang and Lee [4] proved that for the Ising ferromagnet, the critical line is located at h 0. For the Ising antiferromagnet, the series expansion method was used to obtain related information [5]. Müller-Hartmann and Zittartz obtained the critical line by considering the interfacial tension [6,7]. Wu and coworkers [8,9] formulated the Ising model on the honeycomb lattice as an 8-vertex model and identified the critical line as a locus invariant under a generalized weak-graph transformation. In this paper we introduce a new approach by considering zeros of the partition function on an elementary cycle and using Griffiths' smoothness postulate [10]. A closed-form formula of the critical line of an Ising antiferromagnet is obtained, which is in good agreement with the numerical results [8,11,12].The partition function of the Ising model in the presence of a magnetic field is given bywhere S i 61 and K ij is the interaction strength. The sum over ͗ij͘ runs over the pairs of nearest neighbors on the lattices. Let us consider the ferromagnetic case, K ij . 0. The Ising partition function on an elementary cycle of the triangular, square, and honeycomb lattices (see Fig. 1) can be written, respectively, as z t 2͓e b͑K

Real space renormalization group study of the two-dimensional super-antiferromagnetic Ising model

“…Linear chain approximation (LCA), a mix of exact results and MF, shows a second-order line transition between the SAF and FIF states and an evidence of reentrant behavior at low temperatures with the presence of two critical temperatures for the same value of the magnetic field [25]. These later results were also obtained by two others groups: Wang and Kim, who have introduced a new approach considering zeros of the Ising partition function on an elementary cycle of the square lattice [20] and also Neto et al by considering an effective-field theory (EFT) and Bethe-Peierls (BP) approximation [21]. On the other hand, Rottman [23], by using a generalization of the interface method [26] reported a phase diagram similar to the isotropic antiferromagnetic Ising model with applied magnetic field [27] without any reentrant behavior at low temperatures.…”

“…2 the MF results [24], the curve found by Wang and Kim in Ref. [20], and that obtained via interface method [23]. The MF theory (curve b) predicts the existence of a tricritical point at coordinates (2.667, 1.756) while in the results found by Wang and Kim, the critical line presents a reentrant behavior with the presence of two critical temperatures for the same value of magnetic field above H/ J = 2 (see curve c).…”

“…The same procedure is carried out in the original cell and the matching of these quantities in the two graphs yields the following recursion relations [24], curve (c) was found by Wang and Kim in Ref. [20], while curve (d) denotes the interface method results [23].…”

“…Among the recently studied systems is the SAF Ising model [20,21]. In two dimensions, the system is represented by the Ising model with ferromagnetic couplings between first neighbors along the x direction and antiferromagnetic couplings in the y direction, giving rise to a ground state with a horizontal stripe or SAF structure.…”

Monte Carlo study of the Ising antiferromagnetic with a longitudinal field on the anisotropic square lattice

“…In recent years, the effect of a longitudinal field in the Ising antiferromagnetic on an anisotropic square lattice has been discussed [1][2][3][4][5][6][7]. The phase diagram in the temperature longitudinal field plane was obtained by using several approximative methods (for example, effective field theory (EFT), Bethe-Peierls approximation (BP), mean field approximation (MFA), etc.…”