Study of the Antiferromagnetic Blume-Capel Model on kagomé Lattice

We study the anti-ferromagnetic (AF) Ising model and the AF Blume-Capel (BC) model on the kagomé lattice. Using the Wang-Landau sampling method, we estimate the joint density functions for both models on the lattice, and we obtain the exact critical magnetic fields at zero temperature by using the micro-canonical analysis. We also show the patterns of critical lines for the models from micro-canonical analysis.


Introduction
Monte Carlo simulations currently play a major role in statistical physics to study phase transitions and critical properties. They are well-known in the case of anti-ferromagnetic (AF) Ising on square lattice. However, the studies of frustrated systems such as triangular [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] and kagomé lattices [19,20] are also highlighted by the difficult task to understand the various phenomena. It has been reported that AF Ising models have three types of conjectured patterns of critical lines [21]. The Blume-Capel (BC) model [22,23,24,25,26,27] has been studied to understand systems such as metamagents, ternary alloys, and multicomponent fluids. This spin-1 Ising model shows first-order transitions and second-order transition. In this paper, we study the critical lines and properties of the AF BC model as well as those of the AF Ising model on the kagomé lattice using the Wang-Landau sampling method [28,29,30,31] and microcanonical analysis [1].

Model and Simulation
The Hamiltonian of the AF Ising model defined on L × L square lattice in two dimensions is given by where E = i,j σ i σ j and M = i σ i , and σ i = ±1. J < 0, and h is an external magnetic field. The Hamiltonian of the AF BC model is as follows: where D is the crystal field (also called the single-spin anisotropy parameter or the spin impurity chemical potential), M 2 = i σ i 2 , and σ i = −1, 0, or 1. Here, a negative coupling constant (J < 0) defines the AF BC model. The physical origin of the crystal field D arises from the non-central potentials for the metal atoms coordinated with various ligands in the crystal.
Using the WL algorithm, the 3d random walks are performed in the joint space [32, 33] A = (E, M ) for the AF Ising model and A = E, M, M 2 for the AF BC model by randomly changing the states of spins A, where the order parameter M is i S i and the square of order parameter M 2 is i S i 2 , but the state A associated with each spin configuration is only accepted with a probability proportional to the reciprocal of the joint density of states g (A) [32,33,34,35].
Therefore, the transition probability from state A to A is   We can observe that at zero temperature the term with the minimum E t is dominant, where E t = (E − hM ) is the total energy. The total energy E t can be interpreted as the intersection on the E axis of the linear line E = hM + E t in the ME diagram. Note that the slope of the line connecting the top right vertex and the bottom right vertex is six which is the critical magnetic field h c , where h c = 4 for AF Ising and AF BC models on kagomé lattice. Figure 3 shows the estimated critical lines of the AF Ising and AF BC models on kagomé lattice. Although the patterns of critical lines for both models are the same, the transitions in the vicinity of T ∼ 0 are different from each other. As shown in Figure 3, the critical line of AF BC model at T ∼ 0 and h = 4 indicates the first order phase transition, whereas that of AF Ising at T ∼ 0 and h = 4 indicates the second order phase transition.

Conclusions
In this preliminary study by using the small kagomé lattices, we study the AF BC and the AF Ising model on kagomé lattices using the Wang-Landau sampling method and micro-canonical anaysis. We find critical lines of both models and find the critical magnetic field h c = 4 at zero temperature. We also find that the critical line of AF BC model at T ∼ 0 indicates the first order phase transition and that of AF Ising indicates the second order phase transition. Thus, our future intention is to extend the present investigations to the critical properties and phenomena of the AF BC model on the kagomé lattice with nonzero D and non-zero h field.