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Progress of Theoretical Physics

Study of Quantum Spin 1/2 X-Y Model by Means of Extended Migdal Approximation

T. Tatsumi

Published Feb 1, 1981• 11 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.65.451

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•May 7, 1992•0 citations
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.
The spin 1/2 anisotropic Heisenberg model is studied by means of the real space renormalization transformations.Thermodynamic properties of one-dimensional systems are calculated by using an approximate decimation method.These results agree well with those obtained by Bonner and Fisher and by Katsura.Two-and three-dimensional systems are studied by generalizing the Migdal-Kadanoff renormalization transformations to quantum spin systems.For the Ising and Heisenberg models in two and three dimensions and for the X-V model in three dimensions, the critical properties thus obtained are in qualitative agreement with those obtained from high temperature series expansions.For the two-dimensional X-V model, our results suggest that this model undergoes some kind of phase transition and that the critical behaviour of this model is similar to that expected for the classical X-V and planar models.Contents I. Hiroshi Takano 2-3.Three-dimensional systems III.Decimation method for one-dimensional systems 3-1.Transformation function and approximation 3-2.Renormalization equations for scale factor Q= 2 3-3.Free energy IV.The Migdal-Kadanoff transformations for two-and three-dimensional systems 4-1.Fixed points and critical lines 4-2.Critical properties 4-3.Thermodynamic properties V. Summary and conclusions Appendix A. Real space renormalization transformations A-I.Renormalization transformations A-2.Critical properties A-3.Free energy
The spin-(1/2 anisotropic Heisenberg model is studied by a modified Migdal-Kadanoff method, called the cluster-decimation approximation. The free energy for a small cluster of spins with an appropriate boundary condition is assumed to be preserved in the bond-moving step and the approximate decimation of Suzuki and Takano is adopted in the site-decimation step. Critical temperatures obtained by the present method are in general much better than those determined by preserving the high-temperature series expansion of the free energy. In particular the XY fixed point and the isotropic Heisenberg fixed point for the square lattice are not found in the present calculation.
The existence of a massless phase in several different two-dimensional O(2)-symmetric spin models is investigated within the approximate Migdal/Kadanoff renormalisation transformation. The models investigated include the planar rotor model, the truncated model used by Nienhuis (1982) to derive exact results for O(2)-symmetric models, and the step model of Guttmann and Joyce (1973). The truncated model is found to exhibit a 'fixed' line similar to that seen in the planar rotor model, but only for values of the coupling for which the model is unphysical. The step model is found to be always disordered, but a variant exhibits a fixed line. The significance of the results beyond the Migdal approximation is assessed.
The spin wave theory of Kondo and Yamaji is applied to the spin 1/2 XY model in one to three dimensions. In one dimension, the value of the gap Δ which appears in the spectrum of x -component spin remains finite up to 0 K, and no phase transition occurs. The value of the nearest neighbour correlation function is in good agreement with the exact values. In three dimensions, Δ vanishes at a finite temperature, and the second order phase transition occurs. In two dimensions, as T →0, Δ becomes very small for square lattice and tends to zero as exp (- T 0 / T ) for triangular lattice, which means the non-existence of the phase transition. The susceptibility, however, becomes anomalously large for low temperatures, and the extrapolation of Curie-Weiss law gives a “fictitious” transition temperature whose value agrees with those given by the high temperature expansion and the real space renormalization group theory.
Abstract In Chapter 8 we gave an introduction to the formalism of the renormalization group. A renormalization transformation changes the scale of a system by a factor b. Systems at criticality flow to fixed points under the transformation and their critical exponents and scaling behaviour are determined by linearizing the recursion equations about the fixed point. The aim of this chapter is two-fold: firstly to illustrate these ideas by giving several examples of renormalization group transformations and secondly to demonstrate the ways in which they can be used to calculate critical properties. Our examples are divided into three classes. In Section 9.1, we shall consider a one-dimensional model where the recursion relations can be written down exactly. The drawback of this as a canonical example is that any critical behaviour is at zero temperature. The decimation transformation presented in one dimension is an example of a real-space transformation and the second task, in Section 9.2, is to show how these may be used approximately to treat models in higher dimensions. Thirdly, in Section 9.5, we shall mention the E-expansion where the renormalization takes place in momentum space. This enables an understanding of the crossover to mean-field behaviour. However, the formalism is somewhat involved and the details are beyond the scope of this book.