The spin-S Blume-Capel RG flow diagram

Abstract

Using the Finite Size Scaling Renormalisation Group we obtain the two-dimensional flow diagram of the Blume-Capel model, for S = 1 and S = 3/2. In the first case our results are similar to those of Mean Field Theory, which predicts the existence of first and second order transitions whith a tricritical point.In the second case, however, our results are different. While we obtain, in the S = 1 case, a phase diagram presenting a multicritical point, the Mean Field approach predicts only a second order transition and a critical end point.Work partially supported by Brazilian agencies FINEP, CNPq, CAPES and FAPEMIG.

Critical behavior of the spin-32Blume-Capel model in two dimensions

“…Despite of that, we have carefully swept paths of constant temperatures for t ≥ 0.81 and constant d for d ≥ 1.92 and we have not found any indication of the first-order line, separating the low temperature ordered phases, terminating in the second order transition line. This is in disagreement with previous renormalization group calculations [12,18] and reproduce the qualitative behavior predicted by mean field like calculations [15], and are in agreement with our results of finite-sizes calculations on strips (see Fig. 2).…”

“…For spin S = 3/2 it gives a second order phase transition with no tricritical point and a separated first-order transition line which ends up in an isolated multicritical point. In contradiction with these results a calculation [12] based on a renormalization group introduced in [17] give us for the two dimensional model a unique first-order transition line at low temperature which ends up in the second-order transition line at a tetracritical point. Similar results are also obtained by using a Migdal-Kadanoff real space renormalization group approach [18].…”

“…In this scenario a multicritical point will exist as an isolated end point of a first-order transition line. On the other hand approximate calculations based on renormalization-group methods in real space [11,12] give us (scenario b) a multicritical point along the disorder-order transition line. This point being the end point of the first-order transition line.…”

“…In the case where S = 1 this Hamiltonian was proposed originally by Blume and Capel [1] for treating magnetic systems and has also been used in describing 3 He-4 He mixtures [2]. This S = 1 model was studied by a variety of methods such as mean field [1], two-spin cluster [3], variational methods [4], constant coupling approximation [5], Monte Carlo simulations [6,7], finite-size-scaling on its transfer matrix [8] or quantum Hamiltonian [9,10] and renormalization group methods [11,12]. It is well established that for dimension d ≥ 2 the S = 1 model presents a phase diagram with ordered ferromagnetic and disordered paramagnetic phases separated by a transition line which changes from a second-order character (Ising type) to a first-order one at a tricritical point.…”

Critical behavior of the spin-3/2 Blume-Capel model on a random two-dimensional lattice

“…The second sum is taken over the N spins on a D-dimensional lattice. The case where S = 1 has been extensively studied by several approximate techniques in two-and three-dimensions and its phase diagram is well established [29][30][31][32][33][34][35]. The case S > 1 has also been investigated according to several procedures [36][37][38][39][40][41][42].…”