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International Journal of Modern Physics B

SUPERIORITY OF GENERALIZED BOND-MOVING RENORMALIZATION IN STUDYING PHASE TRANSITION OF COMPLEX SPIN SYSTEMS

Chun-Yang Wang, Wen-Xian Yang, Hong Du

Published Oct 2, 2013• 0 citations
Publisher

Abstract

The superiority of the generalized Migdal–Kadanoff bond-moving renormalization (GMKBR) method in studying phase transition of complex spin systems is revealed by a sampled application on the classical S4model. In three different cases of nearest-neighbor, next-nearest neighbor and external field, we obtain the critical points and further calculate the critical exponents according to the scaling theory. In all the cases, it is shown that the method is of great convenience in obtaining results which are in good conformity with the previous ones.

DOI
10.1142/s0217979213501877

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The superiority of the generalized Migdal–Kadanoff bond-moving renormalization (GMKBR) method in studying phase transition of complex spin systems is revealed by a sampled application on the classical S4model. In three different cases of nearest-neighbor, next-nearest neighbor and external field, we obtain the critical points and further calculate the critical exponents according to the scaling theory. In all the cases, it is shown that the method is of great convenience in obtaining results which are in good conformity with the previous ones.
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We study the critical properties of the spin-continuous S4S^{4}S4 system on the typical translational invariant triangular lattices by combining the recently-developed generalized Migdal-Kadanoff bond-moving recursion procedures with the cumulative expansion technique. In three different cases of nearest-neighbor, next nearest neighbor and external field we obtain the critical points and further calculate the critical exponents according to the scaling theory. In all case it is found that there exists three fixed points. The correlation length critical exponents obtained near the Wilson-Fisher fixed points are found getting smaller and smaller with the increasing of the system complexity. Others are found similar to the results of the classical Gaussian model and different from those of the Ising system.
Considering in symmetrical half-length bond operations, we present in this paper two types of newly-developed generalizations of the remarkable Migdal-Kadanoff bond-moving renormalization group transformation recursion procedures. The predominance in application of these generalized procedures are illustrated by making use of them to study the critical behavior of the spin-continuous Gaussian model constructed on the typical translational invariant lattices and fractals respectively. Results such as the correlation length critical exponents obtained by these means are found to be in good conformity with the classical results from other previous studies.
We report in a series of papers two types of generalized Migdal-Kadanoff bond-moving renormalization group transformation recursion procedures. In this first part the symmetrical operation of half length bonds on translational invariant lattices are considered. As an illustration of their predominance in application, the procedures are used to study the critical behavior of the spin-continuous Gaussian model constructed on the triangular lattices. Results such as the correlation length critical exponents obtained by this means are found to be in good conformity with the classical results from other studies.
In Part I the global phase diagram of the general spin-1, nearest-neighbor, ferromagnetic Ising model was studied for d=2d=2d=2 dimensions using an approximate renormalization-group approach of the Migdal-Kadanoff type. Here general ddd is considered with emphasis on $\ensuremath{\epsilon}=d\ensuremath{-}1$ expansions and on the variation of the tricritical exponents with d< 4d<~4d< 4. A dilution field is introduced which can be chosen to yield tricritical exponents for d=2d=2d=2, in better agreement with more reliable estimates and conjectured exact results. The three-phase monohedra, which describe the densities of coexisting phases near a tricritical point, are calculated quantitatively for d=2d=2d=2, and their shape contrasted with that predicted by classical theory: they are found, in particular, to be significantly flattened.
Classical and quantum spin systems are widely used models in both experimental and theoretical condensed matter physics. In many materials, the electronic interactions can be difficult to model exactly. However, in some insulators (Mott insulators), the magnetic (spin) interactions can be captured well with spin-only models. Several models are studied in this thesis. First, I report the solution of a long-standing issue in a classical frustrated spin model (i.e., where the quantum effects are neglected and the complexity is due to competing interactions): the nature of the thermal phase transition to a stripe state in the two dimensional (2D) J1-J2 model. Here J1 and J2 are nearest- and next-nearest-neighbor couplings. Monte Carlo simulations with single-spin updates are used for the calculations, and an extended-ensemble method, a generalization of the Wang-Landau algorithm, is also developed and tested. I focus on the study of "weak universality" behavior (continuously varying critical exponents, with one of the exponents staying fixed), for which I show a correspondence with a known class of conformal field theories with charge c = 1. Next, moving to quantum spins, to shed light on magnetic systems studied experimentally and to investigate new types of quantum states of interest in developing theories of quantum magnetism, I study the S = 1/2 Heisenberg model on the 2D square lattice with added six-spin interactions (the so-called J-Q3 model) as well as a set of 3D quantum antiferromagnets on dimerized lattices. Here I use the stochastic series expansion quantum Monte Carlo method. In the study of the J-Q3 model, I report on a similar weak-universality behavior as in the classical J1-J2 model, but with a mapping to a different known class of c = 1 conformal field theories. The critical behavior of the system again shows continuously changing exponents, in a way which corresponds to a gradual weakening of Z4 symmetry-breaking to an emergent U(1) symmetry. In the study of dimerized antiferromagnets, I report on a universal behavior of the Néel temperature TN, which can be related to ground state parameters independently of the microscopic interaction details in several different models.
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.