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Journal of the Physical Society of Japan

Phase Diagrams of Spin-3/2 Ising Model on a Square Lattice in Terms of Corner Transfer Matrix Renormalization Group Method

Norihiro Tsushima, Tsuyoshi Horiguchi

Published May 15, 1998• 25 citations
Publisher

Abstract

We obtain phase diagrams of a spin-3/2 Ising model on a square lattice by using a corner transfer matrix renormalization group method. We find that three phases appear in the phase diagrams; they are the ferromagnetic(FM) phase, the paramagnetic(PM) phase and a ferrimagnetic(FI) phase. It turns out that a tricritical point appears in a region of space spanned by parameters of coupling constants and the temperature. We estimate values of critical exponent ν and α at a transition temperature between the FM phase and the PM phase, at that between the FI phase and the FM phase, and also at that between the FI phase and the PM phase, by using the finite size scaling analysis.

DOI
10.1143/jpsj.67.1574

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We obtain phase diagrams of a spin-3/2 Ising model on a square lattice by using a corner transfer matrix renormalization group method. We find that three phases appear in the phase diagrams; they are the ferromagnetic(FM) phase, the paramagnetic(PM) phase and a ferrimagnetic(FI) phase. It turns out that a tricritical point appears in a region of space spanned by parameters of coupling constants and the temperature. We estimate values of critical exponent ν and α at a transition temperature between the FM phase and the PM phase, at that between the FI phase and the FM phase, and also at that between the FI phase and the PM phase, by using the finite size scaling analysis.
We investigate a spin-3/2 Ising model on a square lattice by using Monte Carlo (MC) simulations and a density matrix renormalization group (DMRG) method. We obtain phase diagrams of the ground state. We carried out the MC simulations in order to calculate the critical temperature and the critical exponent η. We estimate the critical exponents ν, α and γ by using a finite size scaling analysis. We also calculate the critical temperature, the central charge c and the critical exponent ν by using the DMRG method. From the results obtained by the MC simulations and by the DMRG method, we discuss a universality class of the system. It is found that the values of critical exponents are determined by means of the degeneracy of the ground state. This result suggests that the universality does not hold for the spin-3/2 Ising model. Furthermore we find a possibility of a breakdown of the hyperscaling law. We also confirm that the ferrimagnetic phase exists in a parameter region where the ground state has a fourfold degeneracy, and that a transition between the ferrimagnetic phase and the ferromagnetic phase exists in addition to that between the ferromagnetic phase and the paramagnetic phase.
We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.
The mixed spin 3–spin 3/2 ferrimagnetic Ising model was simulated using cooling algorithm on cellular automaton (CA). The simulations were carried out in the intervals −4 ≤ DA/J ≤ 8 and −4 ≤ DB/J ≤ 8 for the square lattices with periodic boundary conditions. The ground-state phase diagram of the model has different types of ferrimagnetic phases. Although only the antiferromagnetic nearest-neighbor interaction was contained in the Hamiltonian, the compensation points emerged through DA/J = 2 at kT/J = 0. The values of the critical exponents (ν, α , β and γ) were estimated within the framework of the finite-size scaling theory and power-law relations for the selected DA/J values (−2, 0, 1, 2, and 4). The estimated critical exponent values were in good agreement with the universal values of the two-dimensional Ising model (ν = 1, α = α′ = 0, β = 0.125, β′ = 0.875 and γ = γ′ = 1.75).
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (JJJ) and biquadratic (KKK) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in JJJ, KKK, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (HHH) and another odd interaction (LLL) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.
The ferromagnetic-paramagnetic phase transition of the two-dimensional frustrated Ising model on a hyperbolic lattice is investigated by use of the corner transfer matrix renormalization group method. The model contains a ferromagnetic nearest-neighbor interaction J1 and a competing antiferromagnetic interaction J2 . A mean-field-like second-order phase transition is observed when the ratio kappa=J_{2}J_{1} is less than 0.203. In the region 0.203<kappa<14 , the spontaneous magnetization is discontinuous at the transition temperature. Such tricritical behavior suggests that the phase transitions on hyperbolic lattices need not always be mean-field-like.
We study a square-lattice three-state Potts antiferromagnet with a staggered polarization field at finite temperature. Numerically treating the transfer matrices, we determine two phase boundaries separating the model-parameter space into three parts. We confirm that one of them belongs to the ferromagnetic three-state Potts criticality, which is in accord with a recent prediction, and another to the Ising-type; these are both corresponding to the massless renormalization-group flows stemming from the Gaussian fixed points. We also discuss a field theory to describe the latter Ising transition.