First-order transition and tricritical behavior of the transverse crystal field spin-1 Ising model Emanuel Costabile a , J. Roberto Viana a , J. Ricardo de Sousa a,b,n , Alberto S. de Arruda c a Departamento de Física, Universidade Federal do Amazonas 3000, Japiim, 69077-000 Manaus-AM, Brazil b National Institute of Science and Technology for Complex Systems, Universidade Federal do Amazonas, 3000 Japiim, 69077-000 Manaus-AM, Brazil c Instituto de Física, Universidade Federal de Mato Grosso, 78060-900 Cuiabá-MT, Brazil article info Article history: Received 28 July 2014 Received in revised form 19 February 2015 Accepted 23 February 2015 by C. Lacroix Available online 3 April 2015 abstract The phase diagram of the spin-1 Ising model in the presence of a transverse crystal-field anisotropy (D x ) is studied within the framework of an effective-field theory with correlation. The effect of the coordination number (z) on the phase diagram in the T –D x plane is investigated. We observe only second-order transitions for coordination number z o7, while that for z Z7 we have first- and second- order transitions, with the presence of two tricritical points. The lower tricritical temperature (T t ) decreases monotonically with the increasing value of z, and in the limit of z-1 we found T t ¼ 0, corresponding to the mean-field solution [Ricardo de Sousa and Branco, Phys. Rev. E 77 (2008) 012104] with a single tricritical point in the phase diagram. & 2015 Elsevier Ltd. All rights reserved. 1. Introduction Phase transitions in spin systems are among the most inter- esting phenomena that occurs in nature. Theoretically, many investigations have been devoted for decades to study the phase transition of the spin-1 Ising model with longitudinal crystal-field [1–15], which is described by the following Hamiltonian: H ¼ J X 〈i;j〉 S z i S z j þ D X i ðS z i Þ 2 ; ð1Þ where the first sum is over pairs of nearest-neighbor spins, J is the exchange interaction, D represents the longitudinal crystal-field, and S z i is the z-component of a spin-1 operator at site i. The Hamiltonian (1) represents the spin-1 Blume–Capel (BC) model [1] and has been applied to explain the tricritical behavior in ternary fluid mixtures [16], isotropic mixtures of helium 3 He– 4 He [1] as well as magnetic and crystallographic phase transitions in compounds such as DyVO 4 [17]. The behavior of the thermodynamic properties and phase transitions of the model (1) are well established in the literature. These are, for example: (i) The isotropic D ¼ 0 limit is reduced to a spin-1 classical Ising model and undergoes a second-order phase transition at T ¼ T c ð0Þ. (ii) Moreover in the limit D-1 the most energetic favorable state is the one with full occupation on the greatest spin component, i.e., S z i ¼ 71, meaning that we have a simple Ising model with components 71 instead of 71/2. (iii) Further in the opposite large limit ðD-1Þ, the favorable state is the one with full occupation on the smallest spin component, i.e., S z i ¼ 0. At zero temperature, there is a first-order transition point at D c =J ¼ z=2(z being the coordination number of the lattice). Thus, for D 4D c the spin configuration of the ground state is S z i ¼ 0 (disordered state with the magnetization per spin defined by m ¼ð1=NÞ P N i ¼ 1 S z i  ¼ 0). However, for D oD c the ground state is degenerate with S z i ¼ 71 (ordered state with m71). (iv) At finite dimensions,2 rd o1, extant results indicate the exi- stence of a phase transition from the high-temperature para- magnetic (P) phase (m¼ 0) to low-temperature ferromagnetic (F) phase (m a0) at a transition temperature T c that is dependent on the crystal field D values, i.e., T c (D). By increasing D, T c ðDÞ transition line decreases, and changes from a second-order character to a first-order type at a tric- ritical point (TCP), where three phases become identical (i.e., m¼ 0 [P phase] and 7m 0 [F phase]). (v) We recall here that the first method used to treat the spin-1 BC model was the mean-field approximation (MFA) [8], where the TCP found is located at D t =J ¼ 2zlnð2Þ=3 and k B T t =J ¼ z=3. (vi) In the case of infinite dimensions, d-1 (or z-1), the classical model (1) is exactly solvable when J is replaced by J/N (N is the total number of sites) and the first sum is performed over all pairs. The exact solution to this model with long-range interaction is equivalent to (have the same Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ssc Solid State Communications http://dx.doi.org/10.1016/j.ssc.2015.02.018 0038-1098/& 2015 Elsevier Ltd. All rights reserved. n Corresponding author. Solid State Communications 212 (2015) 30–34