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Physical Review Letters

Absence of the Kosterlitz-Thouless Fixed Points in the Migdal-Kadanoff Recursion Formulas

Keiichi R. Ito

Published Jun 3, 1985• 12 citations
Publisher

Abstract

It is shown that the Migdal approximate-renormalization recursion formulas always bring the initial state into the high-temperature region for the U(1)-invariant models at the critical dimensions. This means that the Migdal recursion formulas fail to exhibit the Kosterlitz-Thouless transitions. This is also the case for the recursion formulas of Kadanoff type. The method developed here is extended to non-Abelian systems by some additional tricks.

DOI
10.1103/physrevlett.54.2383

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It is shown that the Migdal approximate-renormalization recursion formulas always bring the initial state into the high-temperature region for the U(1)-invariant models at the critical dimensions. This means that the Migdal recursion formulas fail to exhibit the Kosterlitz-Thouless transitions. This is also the case for the recursion formulas of Kadanoff type. The method developed here is extended to non-Abelian systems by some additional tricks.
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.
Recent Monte Carlo simulations suggest that changing the shape of the nearest-neighbor XY\mathrm{XY}XY interaction might turn the infinite-order Kosterlitz-Thouless transition into a first-order one. In this Communication, an alternative explanation for the Monte Carlo data is proposed which is based on the renormalization pattern that results for this model in a Migdal-Kadanoff approximation.
We make use of the calculation of Achiam and Kosterlitz and investigate the generalization of the Migdal-type renormalization-group calculation to critical dynamics, by looking at the one- and two-dimensional kinetic Ising model with no conserved magnetization. In the two-dimensional case the dynamical critical index ZM(magneticperturbation)=2.064{Z}_{M}(\mathrm{magnetic}\mathrm{perturbation})=2.064ZM​(magneticperturbation)=2.064 and ZE(energylikeperturbation)=1.819{Z}_{E}(\mathrm{energylike}\mathrm{perturbation})=1.819ZE​(energylikeperturbation)=1.819 for scale factor $\ensuremath{\lambda}=1$. ZM{Z}_{M}ZM​ involves the static $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ exponent, while ZE{Z}_{E}ZE​ involves $\frac{1}{\ensuremath{\nu}}$, and therefore, it is not surprising that ZM{Z}_{M}ZM​ is closer to the high-temperature expansion results, since $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ in the Migdal approximation for statics is much closer to the exact value than $\frac{1}{\ensuremath{\nu}}$. In one dimension we obtain ZM=ZE=2{Z}_{M}={Z}_{E}=2ZM​=ZE​=2, which is the exact result of Glauber.
Using analytical and discretized Migdal-Kadanoff renormalization-group methods it is shown that similar to the standard XY model the two-component uniform multipolar systems undergo a Kosterlitz-Thouless transition in two dimensions. The critical behavior of the multipole-multipole correlation function is governed by a set of critical exponents that depend on the rank t of multipole moments. At the Kosterlitz-Thouless transition they become universal for all systems. An exactly renormalizable multipolar model is introduced. The behavior of this model on fractal and hierarchical lattices is studied. The T=0 scaling exponent of the two-component multipolar systems on a Sierpinski carpet is shown to be negative, so no Kosterlitz-Thouless transition can be expected to take place in this case.
The quantum spin 1/2 X-Y model in dimensions d = 1, 2 and 3 is studied by means of the extended Migdal approximation for the scale factor of the transformation b = 2. It is found, with the imposed conditions to get the recursion equations, that an unstable fixed point between a T = 0 fixed point and that of T = ∞ exists for d > 1. However, alternate imposed conditions lead to an unusual behavior that the T = 0 fixed point does not appear in any dimension and that a stable fixed point appears at a lower temperature than that of an unstable one for d > 2. The same model on the simple cubic and face-centered cubic lattices is also treated by the two-step decimation transformation.