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Physical review. B, Solid state

Renormalization-group results for the Blume-Capel model in two and three dimensions

Theodore W. Burkhardt, H. J. F. Knops

Published Feb 1, 1977• 64 citations
Publisher

Abstract

A Kadanoff lower-bound renormalization transformation with three variational parameters has been applied to the Blume-Capel model by Burkhardt. We report some additional results obtained in a particular one-parameter subspace of the three variational parameters. In this subspace the transformation treats the three possible Blume-Capel ground-state configurations with all the spins in the same state on an equal footing and preserves the three-state permutational symmetry of the Potts model. There are fixed points associated with first-order, critical, and tricritical transitions. The variational parameters maximizing the free energy at the critical and tricritical fixed points differ by only a few percent. The critical exponents associated with the different transitions and various points on the critical surfaces are calculated for $d=2,3$ dimensions.

DOI
10.1103/physrevb.15.1602

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World Journal of Condensed Matter Physics•Jan 1, 2016•10 citations
A Kadanoff lower-bound renormalization transformation with three variational parameters has been applied to the Blume-Capel model by Burkhardt. We report some additional results obtained in a particular one-parameter subspace of the three variational parameters. In this subspace the transformation treats the three possible Blume-Capel ground-state configurations with all the spins in the same state on an equal footing and preserves the three-state permutational symmetry of the Potts model. There are fixed points associated with first-order, critical, and tricritical transitions. The variational parameters maximizing the free energy at the critical and tricritical fixed points differ by only a few percent. The critical exponents associated with the different transitions and various points on the critical surfaces are calculated for d=2,3d=2,3d=2,3 dimensions.
The critical behavior of the Blume-Capel model for d=2,3d=2,3d=2,3 is analyzed using a renormalization transformation based on Kadanoff's lower-bound transformation. There are fixed points associated with first-order, second-order, and tricritical phase transitions. The discontinuity fixed point onto which the line of first-order phase transition is mapped has two relevant eigenoperators with eigenvalue 2d{2}^{d}2d, corresponding to the discontinuities in $〈\ensuremath{\sigma}〉$ and $〈{\ensuremath{\sigma}}^{2}〉$. Quite different values of the variational parameters in the renormalization transformation optimize the free energy at each fixed point, making it difficult to calculate crossover behavior with the present approach.
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.
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.
Purpose The authors investigate the magnetic properties of a mixed spin-3/2 and spin-2 Blume-Capel model on square and cubic lattices with two different single-ion anisotropies. Design/methodology/approach To study the critical behavior of a mixed spin-3/2 and spin-2 system, the authors have used a real space renormalization group approximation and specifically the Migdal-Kadanoff technique. The authors give the phase diagrams for two different cases: (1) on the (Δ/|J|, 1/|J|) plane with ΔA = ΔB = Δ, and (2) on the (ΔA/|J|, 1/|J|) and (ΔB/|J|, 1/|J|) planes for selected values of ΔB/|J| and ΔA/|J|, respectively. Findings The phase diagrams obtained show that the system exhibits both second- and first-order phase transitions as well as tricritical points for some values of the anisotropies. Moreover, using the variation of the free energy and its derivative at low temperatures, the authors have seen the appearance of first-order transitions at very low temperatures. Originality/value Few investigations of mixed spin-3/2 and spin-2 systems with crystal field have been realized. For this reason, the authors use the renormalization group approach to complete the work done on these systems. In absence of an exact solution, this contributes to the synthesis of the approximation results on mixed spins models.
The fixed-point topology of Kadanoff's lower-bound renormalization transformation is discussed for the d=2d=2d=2 square-lattice Ising model in the three-dimensional space of coupling constants for the nearest-neighbor, next-nearest neighbor, and four-spin interactions. An obstacle one encounters in trying to calculate a critical surface is pointed out. In the three-dimensional space of coupling constants Kadanoff's fixed point has two relevant vectors and can only be reached from a critical line, not from a critical surface. Another fixed point is reported which does have only one relevant vector and can be reached from a critical surface. Another fixed point is reported which does have only one relevant vector and can be reached from a critical surface. The critical exponents of this fixed point are close to the exact values, though not nearly so close as for Kadanoff's fixed point. The properties of both fixed points as a function of the variational parameter ppp are described. Critical values of the nearest-neighbor couplings leading to the fixed points are compared.
We study the mixed spin-1 and spin-3/2 Blume-Capel model under crystal field in the tridimensional semi-infinite case. This has been done by using the real-space renormalization group approximation and specifically the Migdal-Kadanoff technique. As a function of the ratio R of bulk and surface interactions and the ratios R1 and R2 of bulk and surface crystals fields on the spin-1 and spin-3/2 respectively, we have determined various types of phase diagrams. Besides second- order transition lines, first-order phase transition lines terminating at tricritical points are obtained. We found that there existed nine main types of phase diagram showing a variety of phase transitions associated with the surface, including ordinary, extraordinary, surface and special phase transitions.