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International Journal of Modern Physics B

Multicritical Points from Mean Field Renormalization Group

D. Peña Lara, J. A. Plascak, J. Ricardo de Souza

Published Jul 20, 1998• 2 citations
Publisher

Abstract

The mean field renormalization group method is extended to study tri-critical phenomena. By taking into account the third-order terms in the expansion of the magnetizations of the different clusters with respect to the symmetry breaking fields, together with the scaling assumption of the magnetization at a first-order fixed point, one additional equation is obtained which locates, in an unique manner, the tricritical point. This approach is applied in the study of the classical spin-σ Blume–Capel model and the quantum spin-1/2 anisotropic Heisenberg model in the presence of a transverse field as well as with Dzyaloshinsky–Moriya interactions.

DOI
10.1142/s0217979298001046

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The mean field renormalization group method is extended to study tri-critical phenomena. By taking into account the third-order terms in the expansion of the magnetizations of the different clusters with respect to the symmetry breaking fields, together with the scaling assumption of the magnetization at a first-order fixed point, one additional equation is obtained which locates, in an unique manner, the tricritical point. This approach is applied in the study of the classical spin-σ Blume–Capel model and the quantum spin-1/2 anisotropic Heisenberg model in the presence of a transverse field as well as with Dzyaloshinsky–Moriya interactions.
The global phase diagram for a three-component lattice gas or spin-one Ising model with general single-site and nearest-neighbor "ferromagnetic" interactions is worked out for two-dimensional lattices using a Migdal-Kadanoff recursion relation. It differs in important qualitative respects from the corresponding mean-field phase diagram. The set of fixed points and flows provides the characteristic phase diagrams of the three-state Potts multicritical point and the ordinary (n=1n=1n=1) tricritical point in a complete set of symmetry-breaking fields. The latter is associated, in this renormalization-group scheme, with seven distinct critical fixed points, a number which is surprisingly large.
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.
This article concludes a series of papers [Folk, Holovatch, and Moser, Phys. Rev. E 78, 041124 (2008); 78, 041125 (2008); 79, 031109 (2009)] where the tools of the field theoretical renormalization group were employed to explain and quantitatively describe different types of static and dynamic behavior in the vicinity of multicritical points. Here we give the complete two-loop calculation and analysis of the dynamic renormalization-group flow equations at the multicritical point in anisotropic antiferromagnets in an external magnetic field. We find that the time scales of the order parameters characterizing the parallel and perpendicular ordering with respect to the external field scale in the same way. This holds independent whether the Heisenberg fixed point or the biconical fixed point in statics is the stable one. The nonasymptotic analysis of the dynamic flow equations shows that due to cancellation effects the critical behavior is described, in distances from the critical point accessible to experiments, by the critical behavior qualitatively found in one-loop order. Although one may conclude from the effective dynamic exponents (taking almost their one-loop values) that weak scaling for the order parameter components is valid, the flow of the time-scale ratios is quite different, and they do not reach their asymptotic values.
Using a Monte Carlo renormalization-group (MCRG) method we study tricritical behavior in two very different two-dimensional spin models: the Blume-Capel ferromagnet and the next-nearest-neighbor Ising antiferromagnet. The MCRG method is used to locate accurately tricritical points independent of the convergence of eigenvalue estimates. Despite the different symmetries and the different renormalization transformations used for the two models, in both cases we find four relevant tricritical eigenvalues which are essentially identical for both models. For the antiferromagnet, there is no indication of any decomposition of the tricritical point as predicted by mean-field theory, for ratios of intrasublattice to intersublattice coupling as small as (1/8).
We investigate the phase diagram for the spin-3/23/23/2 ferromagnetic Blume-Capel model in a transverse crystal field using the standard mean-field approximation within the framework of Bogoliubov inequality for free energy. We draw a very rich phase diagram with first- and second-order transition lines; tricritical and tetracritical points; critical endpoint of order 2 and double critical endpoint. Additionally, the behaviour of magnetisation as a function of temperature over a wide range of values of both longitudinal and transverse crystal fields is also analysed. To the best of our knowledge, this quantum spin model has only been studied employing an effective field theory, which in turn was not able to characterise completely the multicritical phenomena in its phase diagram, because that procedure is not based on free energy. Thus, our findings on the phase diagram for the present model are novel as they have been not previously reported.
The conventional renormalization-group analysis of the tricritical behavior of a metamagnet is completed by a description of the first-order transition below the tricritical temperature, and the conditions for the discontinuity in the magnetization are explicitly verified in an Ising spin model. The singularity structure of the crossover scaling function is derived from relations between the scaling fields for the critical and tricritical fixed points. Numerical calculations illustrating various aspects of the theory are given for a square-lattice Ising model in a four-cell cluster approximation.