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Physica Scripta

Renormalization group analysis of Blume–Capel ferromagnet on two-dimensional triangular lattice

M. N. Najafi

Published Jul 9, 2015• 0 citations
Publisher

Abstract

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).

DOI
10.1088/0031-8949/90/8/085206

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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).
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 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.
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The phase diagram of the two-dimensional Blume-Capel model with a random crystal field is investigated within the framework of a real-space renormalization-group approximation. Our results suggest that, for any amount of randomness, the model exhibits a line of Ising-like continuous transitions, as in the pure model, but no first-order transition. At zero temperature the transition is also continuous, but not in the same universality class as the Ising model. In this limit, the attractor (in the renormalization-group sense) is the percolation fixed point of the site diluted spin-1/2 Ising model. The results we found are in qualitative agreement with general predictions made by Berker and Hui on the critical behavior of random models.
A new formulation of the renormalization-group in real space, suitable for quantum spin systems is proposed. The method is applied to the two-dimensional spin-12\frac{1}{2}21​ $X\ensuremath{-}Y$ model on a triangular lattice, the renormalization-group transformation being evaluated up to second order in an appropriate cumulant expansion. To first order an unstable fixed point of the transformation is found, corresponding to a critical temperature and critical indices in satisfactory qualitative agreement with present high-temperature series expansion estimates. In the second-order calculation, however, this fixed point disappears, thus throwing some doubt on the conventional picture of criticality as furnished by high-temperature series. The free energy of the model is also computed. For relatively small values of the nearest-neighbor coupling it is in good agreement with that found by high-temperature series analysis.
The spin-5/2 Blume–Capel model was studied using the mean-field approximation and the Migdal–Kadanoff renormalization group method for two- and three-dimensional systems. We determined the phase diagrams in the (crystal field, temperature) plane where the system exhibited first- and second-order phase transitions as well as isolated critical, bicritical and triple points. In order to show first-order transitions at low temperature, we presented the total magnetization per site and the derivative of the free energy as a function of the crystal field. Moreover, the critical exponents of the system were calculated by linearizing the renormalization transformation at the vicinity of the second-order fixed points.