Migdal-Kadanoff Recursion Relations Research Articles

Renormalization-group approach to surface critical behavior in the semi-infinite spin-32Blume-Emery-Griffiths model

Using a renormalization-group transformation in position space based on the Migdal-Kadanoff recursion relations, we investigate the three-dimensional semi-infinite cubic spin-$\frac{3}{2}$ Blume-Emery-Griffiths model with single-site uniaxial anisotropy and nearest-neighbor pair interactions, both bilinear and biquadratic. The fixed-point topologies and corresponding phase diagrams predicted for this model are presented. Attention is focused on the renormalization-group description of the surface critical behaviors, in comparison with the mean-field approximation results. Three main classes of system phase diagrams are determined, featuring a rich variety of phase transitions associated with the surface with type being according to the values of bulk and surface interaction parameters. Surface critical, tricritical, bicritical, and multicritical points are also found.

Phase diagrams of the three-dimensional semi-infinite Blume-Emery-Griffiths model

Within a real-space renormalization-group framework based on the Migdal-Kadanoff recursion relations, we investigate the three-dimensional semi-infinite Blume-Emery-Griffiths model with nearest-neighbor interactions, both bilinear and biquadratic, and with a crystal-field interaction. According to the values of the interactions on the surface and in the bulk of the system, we determine the various generic types of phase diagrams in the case of repulsive biquadratic interactions. Our analysis has led to a classification scheme with eleven fundamental types of phase diagrams describing a large variety of phase transitions associated with the surface and multicritical topologies.

Phase transitions in the spin- [formula omitted] Blume-Emery-Griffiths model

The spin- 3 2 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions (both bilinear ( J) and biquadratic ( K)) and crystal-field interaction (Δ) is studied using a renormalization-group transformation in position-space based on the Migdal-Kadanoff recursion relations. The global phase diagram in ( J, K, Δ) space (with J, K ⩾ 0) is found to have two surfaces of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary trictical line, an isolated critical line of end points, and a line of multicritical points. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined.

Phase diagrams of the Blume-Emery-Griffiths model: real-space renormalization group investigation and finite size scaling analysis

Abstract An approximate real-space renormalization group scheme based on the Migdal-Kadanoff recursion relations and the finite size scaling method are used to study the phase diagrams of the two-dimensional spin-1 Ising model with nearest-neighbor interactions, both bilinear and biquadratic, and with a crystal-field interaction. Contrary to previous studies, we clarify the behavior of the model at low temperature including the region J + K

On the uniqueness of the solutions to the Migdal-Kadanoff recursion relations when perturbative corrections are included

We show that solutions to the Migdal-Kadanoff recursion relations via Martinelli-Parisi-Roberts are unique. We use Z 4 theory as a specific example to illustrate this.

Origin of nonuniversality in micellar solutions.

An anomalous variation of critical indices has recently been observed in a homologous series of micellar binary solutions. Examination of existing theories of micellar structure shows that a generalization of the Ising model is needed to describe correctly the critical behavior in micellar solutions. It implies the existence of a term that is marginal under renormalization-group transformations. Modified Migdal-Kadanoff recursion relations for this model exhibit a nonuniversal line of fixed points, in qualitative agreement with the experimental behavior.

Convergence of Migdal-Kadanoff iterations in non-abelian lattice gauge models

We study the Migdal-Kadanoff recursion relations for lattice gauge models with gauge groups SU(N) or U(N) in dimensionsd<4. It is shown that the Gibbs factor of a plaquette with Wilson action is driven to 1 for all values of the “temperature” (coupling constant). For models recently proposed by K. R. Ito, where Migdal's and Kadanoff's recursion relations hold exactly, a lower bound on the string tension is derived. The results obtained by us extend those derived by Ito for U(1). Our method is based on analytic continuation of the Gibbs factors, which are class functions, in the central angles.

Analytic determination of theSU 2 lattice gauge theory stable under Migdal-Kadanoff renormalization

The Migdal-Kadanoff recursion relations are applied to the most general plaquette action. The stable action, determined by a weak-coupling expansion, lies between the Manton and the heat kernel ones, in agreement with previous numerical investigations. Moreover, it is shown that certain contributions to the renormalization of the coupling constant exactly coincide with the perturbative ones; this happens for those terms of pure group-theoretical origin, which are the same in the perturbative expansions of the 4D gauge theory and of the corresponding 2D chiral model.

Migdal-Kadanoff approach to superfluid film formation near a wall in3He-4He mixtures

The semi-infinite generalised Blume-Emery-Griffiths model of 3He-4He mixtures with a vector order parameter is analysed by means of Migdal-Kadanoff recursion relations. Superfluid film formation near the wall is put into evidence as a Kosterlitz-Thouless type transition. At higher 3H concentrations one observes the separation of a 4He rich normal film near the wall.

A success of the Martinelli-Parisi expansion: the crossover to first-order transition in the 2D Potts model

The authors show that an instability for q>qc of the Migdal-Kadanoff recursion relations for the two-dimensional Potts model is related to the appearance of a discontinuity fixed point in the Martinelli-Parisi expansion (1981). Consequently, at first order in this expansion, they can evidence a crossover to discontinuous transition, without explicit introduction of vacancies. Through numerical derivation of the free energy, they simulate remarkably well the discontinuous jump in the internal energies.

Perturbative corrections to Migdal-Kadanoff recursion relations for Z 4 theory

For the Z 4 lattice gauge theory in four dimensions or the associated spin system in two dimensions we examine the critical structure by making a correction to the bond-shifting operator. This is done for the most general single plaquette or nearest neighbor action. The two-dimensional problem in coupling space becomes a four-dimensional problem when the correction is made to the bond-shifting operators. The four-dimensional problem is decomposed into two two-dimensional problems which allows for the effective application of the Migdal-Kadanoff recursion relations in differential form. The solutions to the two-dimensional problems are then recomposed into the solution of the original two-dimensional problem in coupling space. Of particular interest is the region β 2 < 0, β 1 ⩾ 0, where β 1 and β 2 are the coupling constants in the fundamental and adjoint representations respectively. We find a critical structure, including a second crossover, which leads to a splitting of the phase structure in this region which is in agreement with Monte Carlo predictions.

Migdal-Kadanoff study of Z 4 symmetric systems with generalized action

For the Z 4 lattice gauge theory in four dimensions or the related spin system in two dimensions, we consider real space renormalization group transformations on the most general single plaquette or nearest neighbor action. The Migdal-Kadanoff recursion relations in differential form have several fixed points and predict a rich phase structure.

Migdal-Kadanoff recursion relations in SU(2) and SU(3) gauge theories

We study the Migdal recursion relations and the reformulation due to Kadanoff for SU(2) and SU(3) lattice gauge theory, using analytic approximations for large and small couplings and numerical methods for all couplings. In SU(2) we obtain the beta function and the expectation value of the plaquette, which is compared with recent Monte Carlo results. In analogy to U(1), we find that a Villain form (periodic gaussian) for the exponential of the plaquette action is a good approximation to the result of the Migdal renormalization transformation. We also perform some calculations in SU(3) and find that its behavior is similar to SU(2).