Migdal Renormalization Group Research Articles

Analytic solution inSU(N) lattice-gauge model of Migdal

Migdal's renormalization group transformation for theSU (N) pure lattice-gauge model is solved in the weak coupling regime by a perturbative expansion in the coupling constant. The heat-kernel model is shown to be a low-order perturbative solution of the Migdal recursion relation. The parameters of the β-function are fixed within theSU (N) Migdal model. The critical coupling constant of the deconfining phase transition is approximately calculated and compared with constants obtained by the heat-kernel model and numerical calculations. The improved actions for theSU (2) andSU (3) are presented for realistic lattice-gauge methods such as the Monte Carlo simulation.

SU ( N) lattice-gauge theory in the Migdal renormalization group model

The renormalization group model of Migdal for SU( N) pure lattice-gauge theory is shown to give 15/2 π 2(≈0.76) of the Stefan-Boltzmann value for internal energy and the relevant lattice Coulomb potential. The similarities between the Migdal model and the one-loop perturbative calculations in the N- and the lattice-size-dependences are shown.

Scaling behavior in lattice gauge theory

By means of Migdal's renormalization group transformation for the SU(2) pure lattice-gauge model we calculate the scaling behavior of the lattice spacing as a function of 1 g 2 ( g=coupling constant). The immediate observation of scaling behavior shows that the physical quantity should be evaluated in the scaling regime from g which is defined on the unique renormalized trajectory of the parameter space.

Crossover in the XY model from three to two dimensions.

We study the crossover between two- and three-dimensional behaviors of the Villain form of the XY spin model on a three-dimensional lattice with anisotropic couplings ${\mathit{J}}_{1}$=${\mathit{J}}_{2}$=J, ${\mathit{J}}_{3}$=\ensuremath{\alpha}J (0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}1). The \ensuremath{\alpha} dependence of various quantities is examined by means of duality transformations, Migdal renormalization group, and Monte Carlo simulations. For the specific heat, a crossover into the Kosterlitz-Thouless behavior takes place around \ensuremath{\alpha}\ensuremath{\simeq}0.2, which corresponds to ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{cos}}$\ensuremath{\simeq}0.015 in the cosine form of the XY model. Some implications for models of high-${\mathit{T}}_{\mathit{c}}$ superconductivity are discussed.

Migdal Scheme for the Effective Couplings of Polyakov Loops

The Migdal renormalization group (MRG) procedure is applied to the finite temperature lattice gauge theory. With its effective action form in terms of the Polyakov loop we analyse the strong and weak coupling approximations of the evolution of renormalized couplings. The distribution of renormalization group trajectory is obtained and the critical coupling is measured.

SU(3) Lattice Gauge Theory in Migdal Renormalization Group Approach

The SU(3) lattice gauge theory at zero and finite temperatures is investigated in Migdal renor­ malization group approach. At zero temperature, the coupling constant renormalization is studied by unique renormalized trajectory method. For bare theory starting from the Wilson action, crossover behavior is obtained and for bare theory starting from the mixed action with large /3a (adjoint representation coupling), sharp stepwise transition is observed. At finite temperatures, the renormalization flow and the internal energy show a clear deconfinement phase transition. The internal energy at high temperatures shows T4 behavior reminiscent of the Stefan-Boltzmann law. The deconfining temperature exhibits renormalization group scaling law. The scaling behavior obtained by coupling constant renormalization at T=O and that by deconfinement transition is compared with each other and found to be consistent.

Migdal renormalization group approach to deconfinement phase transition

The Migdal renormalization group approach is applied to a finite temperature lattice gauge theory. Imposing the periodic boundary condition in the timelike orientation, the phase structure of the finite temperature lattice gauge system with a gauge groupG in (d+1)-dimensional space is determined by two kinds of recursion equations, describing spacelike and timelike correlations, respectively. One is the recursion equation for ad-dimensional gauge system with the gauge groupG, and the other corresponds to ad-dimensional spin system for which the effective theory is described by the nearest neighbor interaction of the Wilson lines. Detailed phase structure is investigated for theSU(2) gauge theory in (3+1)-dimensional space. Deconfinement phase transition is obtained. Using the recursion equation for the three dimensional spin system of the Wilson lines, it is shown that the flow of the renormalization group trajectories leads to a phase transition of the three dimensional Ising model.

Analytic study of the Migdal-Kadanoff recursion formula

After proposing lattice gauge field models in which the Migdal renormalization group recursion formulas are exact, we study the recursion formulas analytically. IfD is less than 4, it is shown that the effective actions ofD-dimensionalU(1) lattice gauge models are uniformly driven to the high temperature region no matter how low the initial temperature is. If the initial temperature is large enough, this holds for anyD and gauge groupG. These are also the cases for the recursion formulas of Kadanoff type. It turns out, however, that the string tension forD=3 obtained by these methods is rather big compared with the one already obtained by Mack, Göpfert and by the present author. The reason is clarified.

Unique Trajectory Method in Migdal Renormalization Group Approach and SU(2) Lattice Gauge Theory

In a preceding paper (Ref. 1)), we presented unique trajectory and applied it to the lattice gauge theory (LGT)2) of the icosahedral subgroup [(120)3) of 5U(2). The unique trajectory method, which is a combined method of Migdal renormalization group approach )-6) and Wilson-Kogut topological argument,7) leads to the relation between the bare coupling constants and the lattice constant by fixing the renormalized coupling constant at some value. The obtained results are interesting because we observed not only the ordinary second order phase transition but also the delicate phenomena like the crossover structureS) and the stepwise transition like a first order phase transition is observed_ These observations suggest that it is valuable to investigate the phase structure of other LGT's like 5U(2), 5U(3), etc. by this method. In this paper, we apply the unique trajectory method to the continuous group 5U(2) case and investigate the detailed p!lase structure. The unique trajectory method is briefly summarized as follows. By the Migdal renormalization group transformation, we obtain trajectories in the space of renormalized coupling constants for various bare theories. All of these trajectories starting from different bare coupling constants converge to a single trajectory by the successive transformations. This is a unique trajectory or a unique renormalized trajectory.7) The existence of this unique trajectory suggests that we can derive a relation between various bare theories with a different bare coupling constant /:Jb and a different lattice constant a by fixing the renormalized coupling constant. Namely, we fix a renormalized coupling constant /:Jc at any point on the unique trajectory, which we call gate, and calculate the

Unique Trajectory Method in Migdal Renormalization Group Approach and Crossover Phenomena

Migdal renormalization group approach, combined with Wilson· Kogut topological argu· ment, is applied to four dimensional lattice gauge theory of finite subgroup [(120) of 5U(2). i) A slight (compared with the Monte Carlo results) but clear crossover from strong coupling regime to weak coupling regime is observed for the Wilson action. ii) For mixed action, of the fundamental and the adjoint representation, a clearer stepwise transition, suggesting first order phase transition, is found at 1:S/:Iab:S3 (where /:lab denotes the bare inverse coupling constant of the adjoint representation). This stepwise transition changes into crossover for smaller /:lab iii) There are four critical lines in (/:1/, /:I:) plane starting from a quadruple point (/:1/ ~ 0.75, /:I: ~ 3.2) where /:1/ denotes the bare inverse coupling constant of the fundamental representation; 1) 50(3) critical line, 2) Z(2) critical line, 3) a critical line due to the discreteness of [(120), 4) a critical line related to crossover. In this investigation, the unique trajectory of renormal· ization group is very important and plays a powerful role in finding crossover and stepwise transition.

Migdal renormalization group approach to quantum spin systems

The Migdal RG approximation is extended to quantum spin systems such as the Heisenberg and XY-models. This yields the non-existence of phase transition in the two-dimensional Heisenberg model. The phase transition of the two-dimensional XY-model is also studied.