Computation of Latent Heat based on the Energy Distribution Histogram in the 3D Ashkin-Teller Model

Abstract

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 ...

Computation of latent heat in the system of multi-component order parameter: 3D Ashkin-Teller model

“…8 shows the results of our analyzes for the energy E α,−,L (lower lines) and E min α,+,L (upper lines) of the whole Hamiltonian (1), for energy of interaction of degrees of freedom s (the same result is for σ) and the product sσ separately, explained in the legend box, for systems with different sizes 16 ≤ L ≤ 28 at the point K 4 = 0.18 and K 2,c = 0.145189 (12). The values E α,−,L and E α,+,L in the L −2 function scale linearly to the respective bulk values E α,− and E α,+ [3,4,12]. Therefore, the individual lines in Fig.…”

“…For sufficiently strong first order phase transitions, a characteristic histogram of the internal energy E distribution with two peaks in the close critical region can be observed [3,4]. For samples of finite-size L, the maxima of these peaks appear at the energy value E −,L for the ordered state and at E +,L for the unordered one and they are separated by the minimum of E m,L value.…”

“…In Eqs 3and 4E n α L is the n-th moment of the interaction energy of α-degrees of freedom (α = s, σ or their product sσ) in Hamiltonian (1) separately, which is averaged over an ensemble of independent samples of the size L×L×L. The Challa and Lee-Kosterlitz cumulants have been adapted [4,10] by taking not only the whole Hamiltonian (1) for energy E as originally proposed by Challa et al [2] and by Lee and Kosterlitz [3], but also the Hamiltonian individual terms to be able to compute the latent heat for each component of the order parameter separately [10,12,19]. Of course, we make sure that the latent heat l H computed on the basis of the whole Hamiltonian (1) is equal to the sum of the latent heats l α (α = s, σ or their product sσ) coming from α degrees of freedom within their error bars.…”

“…In this paper, we propose precise determination of the value of latent heat in the computer experiment based on various cumulants and a histogram of energy distribution. We exploit Binder [1], Challa [2] and Lee-Kosterlitz [3] cumulants as well as the internal energy distribution histogram method [3,4] which were introduced for systems with one independent order parameter, such as the Ising like models. The non-trivial generalization of the widely exploited Ising model of current interest is the Ashkin-Teller (AT) model [5] which is one of the most important models in statistical physics and every year a dozen works are devoted to it (see e.g.…”

“…Simultaneously, we independently use Challa and Lee-Kosterlitz cumulants, which require adaptation [10,12] to enable locating the phase transition point for the individual order parameter components and their contribution to the latent heat. Knowing the phase transition point position accurately enough, for strong phase transitions we can determine the latent heat value with a higher precision using the internal energy distribution histogram method [3] adapted to the considered system [4]. A detailed description of the method proposed in this paper and how to systematically validate the results obtained are clarified in Section II.…”