Phase transitions and thermodynamic properties of the antiferromagnetic Potts model on a face-centered cubic lattice

Using the Monte Carlo method we investigate the phase transitions and thermodynamic properties of magnetic structures with noncollinear directions of magnetic moments corresponded to antiferromagnetic q=4 Potts model on a face-centered cubic lattice. Monte Carlo simulations are performed on lattices with linear sizes L=2044. Thermodynamic parameters: the order parameter mAF, susceptibility , internal energy U, and specific heat С are evaluated for all studied systems. By employing the fourth order Binder cumulant method, a first order transition is shown to be occurred in the model.


Introduction
There exists at present a rich body of the experimental material on magnetic substances with crystalline structure and properties of AnX-type compounds with the NaCl structure.These are so called monopnictides and monochalcogenides, where An is a rear-earth element or actinide, X is V or VI groups` element (UN, UAs, USb).They display most interesting properties.The evolution of the magnetic neuronography increases a number of experiments in determination of different compound magnetic structures.Rossat-Mignod et.al. [1] proved that a single wave vector component is insufficient to describe the experimental data for UP, UAs, and USb compounds.Therefore, multi-k structures with wave vectors comprised more than one component (2-k structures for two-component wave vector and 3-k structures for three-component vector) were proposed.The interpretation of USb neuronography is in a better agreement with experimental data of a 3-k structure than a collinear model.We notice that this 3-k structure corresponds precisely to the q=4 standard Potts model illustrated in Fig. 1.
The investigation of phase transitions (PTs) and thermodynamic properties of the 4-state antiferromagnetic (AF) Potts model is the main goal of this study.

Model and Method
The Hamiltonian of the antiferromagnetic q=4 standard Potts model on a face-centered cubic (FCC) lattice (Fig. 1 ) reads where J is the parameter of antiferromagnetic exchange interaction (J<0),  i,j is the angle between interacting spins S i -S j .As is evident from Fig. 1, in the model spins occupying lattice sites can orient in hypertetrahedron q-symmetric directions in the dimensionality space q-1 retaining equal angles between any two different directions.
The cubic ferromagnetic q=4 Potts model with nearest neighbor ferromagnetic interactions were studied in works [2,3].Authors reported that a second order transition occurred in the models considered.In our study we use the Metropolis algorithm along with the Wolff cluster algorithm [4].We performe the Monte Carlo simulations for systems of linear sizes 4(LLL) with periodic boundary conditions, where L=28-40.Initial configurations were given so that every spin inside each of sublattices is ordered.To bring the system to equilibrium state the relaxation time  0 for all systems with L is computed.Next, the averaging is performed along the Markov chain of length =150 0 and additional averaging is done by ten different initial spin configurations.

Simulation Results
The temperature behavior of the specific heat and susceptibility is evaluated from fluctuation correlations [5] ) )( ( where K=J/k B T, N=4L 3 is the number of sites, U is the intrinsic energy, m is the order parameter, angle brackets denotes the ensemble average.The order parameter for the antiferromagnetic Potts model is defined as Figures 2, 3, 4 and 5 depict the temperature dependences of the specific heat C, order parameter m AF , internal energy E and Binder cumulants V L for 3D 4-state antiferromagnetic Potts model on a FCC lattice for systems with L=28; 36; 40.From now on an error of data in all figures doesn`t exceed symbol sizes used at plotting the diagrams.As is evident from Figs. 2, 3, and 4, the temperature dependences of the specific heat C, order parameter m AF , internal energy E for all studied systems exhibit a behavior featured to a first order transition (a jump occurs at PT point).Beyond that, we investigate the dependence of the Binder's cumulants V L (T) on temperature T for this model (Fig. 5).A clearly marked jump at PT point indicates a first order transition.The fourth-order Binder cumulants method [7] has proved to be the most efficient method for the analysis of the character of the phase transition at T= T t (L).0.6 0.9 ) , ( 3) , ( 1) where Е is the internal energy and т is the order parameter of the system with L. This technique has proved to be successful for the detection of a PT order, a detail description of that was provided in [8][9][10][11][12].The averaged value V L (T) tends to a certain non-trivial value V * at the first order transition 7) at L and T=T t (L), where V * is distinct from 2/3 at L, which is shown in the insert in Fig. 5 for the 3D q=4 antiferromagnetic Potts model on a FCC lattice.
Thus, in this study we evaluate within the unified technique the phase transitions and thermodynamic properties of magnetic structures with non-collinear directions of magnetic moments corresponded to the antiferromagnetic q=4 Potts model on a FCC lattice.The temperature dependences of thermodynamic parameters for the specific heat C, susceptibility , order parameter m AF , and Binder cumulants V L (T) demonstrate the first order transition.The investigation of the antiferromagnetic q=4 Potts model on a FCC lattice permits to elucidate magnetic and thermodynamic properties of real crystals with multi-k structures.

Fig. 1 .
Fig. 1.Identification of the USb diffraction pattern in the 3-k structure (a) and in the collinear one (b).4-State Potts model (c).

Fig. 2 .Fig. 3 .
Fig. 2. The temperature dependence of the specific heat C for the 3D 4-state Potts model on the FCC lattice.

Fig. 4 .Fig. 5 .
Fig. 4. The temperature dependence of the average energy for the 3D 4-state Potts model on the FCC lattice.