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The rst order phase transition line in the vicinity of the tricritical Ising point region is studied in the 3D standard AshkinTeller model on a cubic lattice. The large-scale Monte Carlo computer experiments using the Binder-and Challa-like cumulants, the latter modied by Musial, are proposed and performed. Specic behavior of the ChallaMusial cumulants for weak rst order phase transitions is discovered and its interpretation is proposed. The paper proves the arbitrarily weak rst order character of phase transitions when approaching to the Ising point. [2] in which this model is expressed in terms of two Ising models put on the same lattice with spins s i and σ i at each lattice site, respectively. As in standard Ising model, only twospin interactions of a constant magnitude J 2 between the nearest neighbors are considered. Fan extended these independent Ising models to the AshkinTeller (AT) one by introducing the four-spin interaction of a constant magnitude J 4 also only between couples of nearest-neighboring spins. Thus, we have the Hamiltonian H:where [i, j] denotes summation over nearest-neighboring lattice sites, K i = −J i /k B T , with i = 2 or 4, and T is the temperature of the system. We consider the standard AT model in 3D put on the cubic lattice. The research done for this model and its applications can be found in many papers, e.g. [36]. The K 2 (K 4 ) phase diagram of the AT model is very diversied as three components of the order parameter can order independently: s , σ and sσ where the symbol . . . denotes the thermal average.The aim of our paper is to verify the existence of arbitrarily weak temperature-driven rst order phase transitions when approaching the Ising point situated at K 4 = 0 where obviously there is continuous Ising phase transition, as from Eq. (1) follows that here we have the pure Ising model with ferromagnetic interactions between the nearest neighbors. It has been signalized [3,4] that for K 4 > 0 up to the Potts point situated at K 4 = K 2 ≈ 0.157154 the phase transitions * corresponding author; e-mail: gmusial@amu.edu.pl are of the rst order but Arnold and Zhang suggested also that they are arbitrarily weak when approaching an Ising point [3].For this purpose in the next section we propose the large-scale Monte Carlo (MC) computer experiments to measure precisely the latent heat for the phase transitions under consideration. The Monte Carlo experimentWe consider a nite-size cubic samples of the standard AT model to go next to the thermodynamic limit and in consequence to obtain the results suitable for a macroscopic system. The number of degrees of freedom in our system is too large to be able to take into account all the states of the system. Therefore, we have to use the tools of statistical mechanics including MC method with importance sampling of states.In contrast to simple MC simulations, we propose the MC computer experiment in which one not only calculates the thermodynamic quantities but also precisely determines their uncertainties. Such a way of obtaining of the...

The method of computation of the latent heat based on the energy distribution histogram is applied to the standard 3D Ashkin-Teller (AT) model. Similarly as in the original method for the q-state Potts model for strong first order phase transitions, the characteristic histogram with two peaks in the critical region have been observed. Positions of two minima of negative logarithm of internal energy probability for samples of finite size show good linear scalability to the thermodynamic limit. The applicability of this method has been confirmed by proving that the latent heat values are consistent with the ones obtained by us using the analysis of the behavior of the cumulants of the type of Challa and of Lee-Kosterlitz. The presented method is far more efficient than the one based on those cumulants. [2] in terms of two Ising models put on the same lattice with spins s i and σ i at each lattice site i. In consequence, only two-spin interactions of a constant magnitude J 2 between the nearest neighbors are considered. These Ising models are extended to the AT one by the four-spin interaction of a constant magnitude J 4 also only between couples of nearest-neighboring spins. Thus, the effective Hamiltonian H is of the formwhere K i ¡J i {k B T , with i 2 or 4, ri, js denotes summation over nearest-neighboring lattice sites, and T is the temperature of the system.We consider this standard AT model in 3D put on the cubic lattice. It should be called the standard one as there are many extensions of the AT model (see e.g. Its K 2 pK 4 q phase diagram high complexity is the consequence of the fact that three components of the order parameter can order independently: not only xsy and xσy, but also xsσy where x. . .y denotes the thermal average.The aim of our paper is to present the method of computation of latent heat based on the energy distribution histogram, originally proposed for the q-state Potts * corresponding author; e-mail: djeziorek@wp.pl model [11], applied by us to the 3D AT model. To confirm the applicability of this method, we compare results of our analysis with the ones obtained by us using the analysis of the behavior of the cumulants of the type of Challa V L [12] and Lee-Kosterlitz U L [11], modified by us [7,10]. The latter for the first time applied by us to the AT model what is explained in Section 2. Similarly as in the original method for the q-state Potts model [11] for strong first order phase transitions, in Section 3 we demonstrate the characteristic histogram with two peaks in the critical region also in the 3D AT model and we compute the latent heat on this basis. The method of computationJust like in the q-state Potts model with q equivalent ordered states and one disordered, we observe characteristic histogram of two peaks in the critical area also in the 3D AT model. As shown on the right in Fig. 1 for spins s, maxima of these peaks appear at the energy value E ¡,L for the ordered state and at E ,L for the disordered one, and they are separated by a minimum of E m,L . In this paper we ...

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