The global phase diagram of a vectorial generalization of the Blume-Emery-Griffiths model is obtained using Migdal’s approximate renormalization procedure. Classical two-component spins and non-magnetic impurities populate a triangular lattice, with nearest-neighbor interactions. The phase diagram in thermodynamic field space is divided into magnetic and impurity-rich phases by a first-order surface of discontinuous impurity concentrations, terminating in an Ising-type critical line. The magnetic region is further divided into a high-temperature paramagnetic phase and a low-temperature Kosterlitz-Thouless phase. The exponent η=1/4 of the pure system is preserved at the higher-order surface separating these two phases. This surface terminates in a line of critical end-points on the first-order surface, and, consequently, no tricritical point occurs for any values of the model parameters. However, the Ising critical line and the line of critical end-points approach each other in a certain limit, yielding an effective tricritical phase diagram. Within the Kosterlitz-Thouless phase, lines of constant η bunch together as the effective tricritical point is approached, in apparent agreement with tricritical scaling. This is also a model for superfluidity and phase separation in helium films.
The global phase diagram of a vectorial generalization of the Blume-Emery-Griffiths model is obtained using Migdal’s approximate renormalization procedure. Classical two-component spins and non-magnetic impurities populate a triangular lattice, with nearest-neighbor interactions. The phase diagram in thermodynamic field space is divided into magnetic and impurity-rich phases by a first-order surface of discontinuous impurity concentrations, terminating in an Ising-type critical line. The magnetic region is further divided into a high-temperature paramagnetic phase and a low-temperature Kosterlitz-Thouless phase. The exponent η=1/4 of the pure system is preserved at the higher-order surface separating these two phases. This surface terminates in a line of critical end-points on the first-order surface, and, consequently, no tricritical point occurs for any values of the model parameters. However, the Ising critical line and the line of critical end-points approach each other in a certain limit, yielding an effective tricritical phase diagram. Within the Kosterlitz-Thouless phase, lines of constant η bunch together as the effective tricritical point is approached, in apparent agreement with tricritical scaling. This is also a model for superfluidity and phase separation in helium films.
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (J) and biquadratic (K) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in J, K, $\ensuremath{\Delta}$ 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 (H) and another odd interaction (L) 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.
We study the S=1 Ising model, equivalent to the three-state lattice-gas model, with nearest-neighbor, pairwise interactions on a two-dimensional, triangular lattice. We pay particular attention to the antiferromagnetic phase diagrams. We show its relation to other well-studied models (S=(1/2 Ising, Blume-Capel, Blume-Emery-Griffiths), classify the ground states, and calculate finite-temperature phase diagrams using transfer matrices and finite-size scaling for infinite strips of three and six sites width. The phase diagrams are quite complicated, with surfaces of first- and second-order transitions that intersect along lines of multicritical points of various kinds, providing a rich laboratory for studying a number of first-order phase transitions, critical and multicritical phenomena within the framework of one single model.
We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.
The Blume-Caple model on the triangular lattice is considered in this paper. Using the (perturbative) real-space renormalization group (RSRG) technique we analytically obtain the running of the parameters of the model, i.e., the exchange integral J and the crystal field Δ. The observations show that the first-order calculations are insufficient to capture the tricritical phenomena, and the RG analysis based on second-order expansion of the partition function show qualitative agreement with the previous works. This reveals that the second-order perturbation works appropriately, to obtain the qualitative features of the phase diagram of the model, e.g., a tricritical point which is unstable towards the Ising critical point from one side (which forms the line of second order, separating the ferromagnetic and paramagnetic phases) and the critical point at and from the other side (which forms the line of first-order transition separating ferromagnetic and zero state phases). This analysis shows also that there is another unstable critical point at , with the RG eigenvalue in the Δ-direction (towards S = 0 zero state and paramagnetic phases).
The spin-3/2 Ising model, with nearest-neighbor interactions only, is the prototypical system with two different ordering species, with concentrations regulated by a chemical potential. Its global phase diagram, obtained in d=3 by renormalization-group theory in the Migdal-Kadanoff approximation or equivalently as an exact solution of a d=3 hierarchical lattice, with flows subtended by 40 different fixed points, presents a very rich structure containing eight different ordered and disordered phases, with more than 14 different types of phase diagrams in temperature and chemical potential. It exhibits phases with orientational and/or positional order. It also exhibits quintuple phase transition reentrances. Universality of critical exponents is conserved across different renormalization-group flow basins via redundant fixed points. One of the phase diagrams contains a plastic crystal sequence, with positional and orientational ordering encountered consecutively as temperature is lowered. The global phase diagram also contains double critical points, first-order and critical lines between two ordered phases, critical end points, usual and unusual (inverted) bicritical points, tricritical points, multiple tetracritical points, and zero-temperature criticality and bicriticality. The four-state Potts permutation-symmetric subspace is contained in this model.
We use the higher-order tensor renormalization group method to study the two-dimensional generalized XY model that admits integer and half-integer vortices. This model is the deformation of the classical XY model and has a rich phase structure consisting of nematic, ferromagnetic, and disordered phases and three transition lines belonging to the Berezinskii-Kosterlitz-Thouless and Ising class. We explore the model for a wide range of temperatures, T , and the deformation parameter, Δ , and compute specific heat along with integer and half-integer magnetic susceptibility, finding both Berezinskii-Kosterlitz-Thouless-like and Ising-like transitions and the region where they meet. Published by the American Physical Society 2024