A new effective correlation mean-field theory for the ferromagnetic spin-1 Blume–Capel model in a transverse crystal field

Abstract

A new approximating technique is developed so as to study the quantum ferromagnetic spin-1 Blume-Capel model in the presence of a transverse crystal field in the square lattice. Our proposal consists of approaching the spin system by considering islands of finite clusters whose frontiers are surrounded by non-interacting spins that are treated by the effective-field theory. The resulting phase diagram is qualitatively correct, in contrast to most effective-field treatments, in which the first-order line exhibits spurious behavior by not being perpendicular to the anisotropy axis at low temperatures. The effect of the transverse anisotropy is also verified by the presence of quantum phase transitions. The possibility of using larger sizes constitutes an advantage to other approaches where the implementation of larger sizes is costly computationally.PACS numbers: 64.60.Ak; 64.60.Fr; 68.35.Rh I. INTRODUCTIONIn general, many-body systems with interactions are very difficult to solve exactly. A way to overcome this difficulty is by approaching the many-body problem by a one-body problem, in which a mean-field replaces the interactions affecting the body. This idea is applied to the ferromagnetic Ising Model (see reference [1]). In the most simple mean-field approach, the nearest-neighbor interactions affecting each spin S i are replaced in such a way that S i now interacts with an effective field given by zJ S i , where z is the coordination number, J the exchange constant, and S i is the thermal average of the spin i. This is the so called "Weiss mean-field approach" [2]. Nevertheless, it neglects the spin correlations, and it leads the transition temperature T c as well as the values of the critical exponents away from the exact values (T c = zJ/k B , for all dimensions). However, for the one-dimensional case, the Ising model lacks of a phase transition at finite temperature, but Weiss' approach wrongly predicts that T c = 2J/k B . A further step for improving the solution of this problem, is to use the proposal of Hans Bethe, which consists in considering that a central spin should interact with all its nearest-neighbour spins forming a cluster [3]. Then, that cluster would interact to an effective field that approaches the next-nearest-neighbor spins surrounding the cluster. Thus, this improvement gives T c = 2J/k B ln(z/(z − 2)), which not only betters the approximation of the critical temperatures, but leads correctly to T c = 0, for the one-dimensional case. In this way, the correlations between the spins has been included to some degree by considering a cluster of spins interacting with its nearest-neighbors.A further step in approaching the many-body problem in spin systems is the effective-field approach. It is used in spin models with finite-size clusters based on the following Hamiltonian splitting: