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Study of Quantum Spin Systems using Real Space Renormalization Transformations

Hiroshi Takano

Published Aug 20, 1979• 0 citations

Abstract

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

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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.

A real-space effective-field renormalization-group method (ERFG) recently derived for computing critical properties of Ising spins is extended to treat the quantum spin-1/2 anisotropic Heisenberg model. The formalism is based on a generalized but approximate Callen–Suzuki spin relation and utilizes a convenient differential operator expansion technique. The method is illustrated in several lattice structures by employing its simplest approximation version in which clusters with one (N′=1) and two (N=2) spins are used. The results are compared with those obtained from the standard mean-field (MFRG) and Migdal–Kadanoff (MKRG) renormalization-group treatments and it is shown that this technique leads to rather accurate results. It is shown that, in contrast with the MFRG and MKRG predictions, the EFRG, besides correctly distinguishing the geometries of different lattice structures, also provides a vanishing critical temperature for all two-dimensional lattices in the isotropic Heisenberg limit. For the simple cubic lattice, the dependence of the transition temperature Tc with the exchange anisotropy parameter Δ [i.e., Tc(Δ)], and the resulting value for the critical thermal crossover exponent φ [i.e., Tc≂Tc(0)+AΔ1/φ ] are in quite good agreement with results available in the literature in which more sophisticated treatments are used.

A new formulation of the renormalization-group in real space, suitable for quantum spin systems is proposed. The method is applied to the two-dimensional spin-

\frac{1}{2}

$X\ensuremath{-}Y$ model on a triangular lattice, the renormalization-group transformation being evaluated up to second order in an appropriate cumulant expansion. To first order an unstable fixed point of the transformation is found, corresponding to a critical temperature and critical indices in satisfactory qualitative agreement with present high-temperature series expansion estimates. In the second-order calculation, however, this fixed point disappears, thus throwing some doubt on the conventional picture of criticality as furnished by high-temperature series. The free energy of the model is also computed. For relatively small values of the nearest-neighbor coupling it is in good agreement with that found by high-temperature series analysis.

A simple modification of the real-space renormalisation group method is used to discuss critical behaviour of site diluted spin- 1 / 2 Ising and Heisenberg ferromagnets in two and three dimensions. The critical parameters and critical exponents are evaluated, and are shown to be in good agreement with the exact results and series calculations for site-diluted Ising ferromagnets on the square and triangular lattices. Good values of the critical parameters have also been obtained for the site-diluted Heisenberg and Ising models in two and three dimensions using the Migdal-Kadanoff approximation (1976).

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.