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Progress of Theoretical Physics

Study on the Critical Properties of the One-Dimensional Quantum XXZ Model with General Spin S Using the Quantum Renormalization Group Method

M. Inoue, Masuo Suzuki

Published May 1, 1986• 1 citation
Publisher

Abstract

For the isotropic Heisenberg model with an arbitrary spin 5, the correlation exponent l/ defined by ~ ~ T-v near the critical point Tc=O is evaluated as l/=1 with the use of the real space renormalization group method.This yields y=2 for the susceptibility exponent y defined by Xo~ T-r , using the scaling relation y=l + (2-1]) l/= 1 + l/.Our general result y=2 for all values of spin 5 includes Fisher's result in the classical limit 5-->00.For the anisotropic Heisenberg model (XXZ model) with a spin 5=1/2, an anisotropy scaling law is derived.

DOI
10.1143/ptp.75.1066

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For the isotropic Heisenberg model with an arbitrary spin 5, the correlation exponent l/ defined by ~ ~ T-v near the critical point Tc=O is evaluated as l/=1 with the use of the real space renormalization group method.This yields y=2 for the susceptibility exponent y defined by Xo~ T-r , using the scaling relation y=l + (2-1]) l/= 1 + l/.Our general result y=2 for all values of spin 5 includes Fisher's result in the classical limit 5-->00.For the anisotropic Heisenberg model (XXZ model) with a spin 5=1/2, an anisotropy scaling law is derived.
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.
Abstract Using the quantum renormalization group approach, we have investigated the entanglement and critical behavior of an anisotropic XXZ Heisenberg spin chain. The resolution of the quantum renormalization equations yields stable and unstable fixed points, corresponding respectively to the phases and critical point of the system. It is shown that a quantum phase transition exists at a critical anisotropy parameter , separating the Néel phase from the spin-fluid phase. Then, the concurrence between two blocks of sites and the quantum coherence have been calculated and presented for various renormalization group iterations. In the thermodynamic limit, i.e. , for higher renormalization steps, the concurrence and coherence exhibit two different saturated values in the two regions separated by Δ c . The first derivative of concurrence diverges at the critical point as the size of the chain increases, showing nonanalytic behavior. Moreover, the scaling behavior has been examined near the critical point with some universal exponential scaling laws.
We have applied our recent approach [M. Kargarian et al., Phys. Rev. A 76, 060304(R) (2007)] to study the quantum-information properties of the anisotropic s=1/2s=1/2s=1/2 Heisenberg chain. We have investigated the underlying quantum-information properties such as the evolution of concurrence, entanglement entropy, nonanalytic behaviors, and the scaling close to the quantum critical point of the model. Both the concurrence and the entanglement entropy develop two saturated values after enough iterations of the renormalization of coupling constants. These values are associated with the two different phases, i.e., N\'eel and spin liquid phases. The nonanalytic behavior comes from the divergence of the first derivative of both measures of entanglement as the size of the system becomes large. The renormalization scheme demonstrates how the minimum value of the first derivative and its position scales with an exponent of the system size. It is shown that this exponent is directly related to the critical properties of the model, i.e., the exponent governing the divergence of the correlation length close to the quantum critical point. We also use a renormalization method based on the quantum group concept in order to get more insight about the critical properties of the model and the renormalization of entanglement.
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
A real-space renormalization-group theory is proposed for the anisotropic Heisenberg chain which yields physically reasonable results for all values of the temperature and of the anisotropy. The method is based on a four-spin cluster and a transformation which preserves the sublattice structure of the antiferromagnet. At low temperatures the transformation maps the four lowest energy levels of the four-spin cluster onto the four levels of two spins, giving ground-state energies in excellent agreement with exact results. The specific-heat exponent for the isotropic antiferromagnet is given correctly, but the exponent for larger transverse coupling is somewhat too small. A scaling form is proposed for the free energy near the isotropic antiferromagnetic fixed point, and the crossover to Ising-like behavior is discussed.
The anisotropic Heisenberg model has been investigated by interpolating anisotropy between the Ising and isotropic Heisenberg limits via general spin high temperature series (HTS) for the zero-field susceptibility of the ferromagnets for the square lattice. This is done by locating the critical temperature ( T c (2) ) and the susceptibility exponent (γ) from the analysis of the series by Pade approximant and ratio methods. The critical temperature is found to be spin dependent. It decreases as anisotropy changes from the Ising limit towards the isotropic limit and remains finite (lower than that of Stanley and Kaplan) for all the spins except for S =1/2 at the isotropic limit. Exponent γ is almost Ising like for the anisotropic system and anomalous jump in it occurs for the isotropic case. In addition, expressions for the magnetization are derived by using spin-wave theory. The limit of complete isotropy has been found not to be a special case for the susceptibility, in contrast to magnetization.