Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

physica status solidi (b)

Spin Anisotropy and Critical Exponents for a Heisenberg System A Renormalization Group Approach

S. Fakioǧlu

Published Mar 1, 1985• 4 citations
Publisher

Abstract

Abstract Firstly the renormalization group technique is applied to an isotropic Heisenberg spin system with long‐range interaction and a set of critical exponents are calculated. These exponents are expressed in terms of a potential parameter s̀. All critical exponents except β, the magnetization exponent, depend on s̀. Secondly, the anisotropy is considered and the related crossover exponent φ is evaluated. This later parameter turns out to be equal to γ, the susceptibility exponent, as predicted by some authors.

DOI
10.1002/pssb.2221280122

Related papers

Renormalization Group and Critical Exponents for Classical Heisenberg Ferromagnet with Dipole–Dipole Interaction

S. Fakioǧlu

Abstract The renormalization group technique is applied to the three‐dimensional classical Heisenberg spin system with modified dipole–dipole interaction. For this system, the values and dependence of several critical exponents are investigated. These exponents are: α, the specific heat exponent; β and δ, the magnetization exponents; γ, the susceptibility exponent; η, the correlation function exponent; and v, the correlation length exponent. All the critical exponents except β depend upon the potential range σ. When the potential range σ = 3/2, the modified dipole–dipole results agree with mean‐field values of the critical exponents.

physica status solidi (b)•Mar 1, 1980•8 citations

Crossover Exponent of the Spin Anisotropic n-Vector Model with Short-Range Interaction in 1/n Expansion

S. Hikami, R. Abe

The critical behavior of a spin anisotropic n-vector model is studied in 1/n expansion. The critical exponent for the anisotropy (the crossover exponent) is derived in powers of 1/n to first order by considering the shift of critical temperature and by studying the derivatives of the susceptibility with respect to the anisotropic parameter. It is shown that obtained crossover exponent φ= 2/(d-2) + 4γ1/3n + O(n-2) is consistent with the result of Wilson's ε expansion.

Progress of Theoretical Physics•Aug 1, 1974•24 citations

On the Phase Transition and Critical Behaviour in the Two-Dimensional Anisotropic Heisenberg Model

R. Shanker, R. A. Singh

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.

Journal of the Physical Society of Japan•Mar 15, 1976•2 citations

Critical behavior of the anisotropic Heisenberg model by effective-field renormalization group

J. Ricardo de Sousa, I. P. Fittipaldi

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.

Journal of Applied Physics•May 15, 1994•15 citations

Crossover behaviour and effective critical exponents in isotropic and anisotropic Heisenberg systems

Ann Bruce, D. J. Wallace

Renormalisation group methods are used to determine, to second order in epsilon =4-d, the scaling function describing the crossover from Gaussian to Heisenberg behaviour in the susceptibility of an isotropic n-component spin system. The results are used in conjunction with an earlier Feynman graph calculation to obtain an O( epsilon 2 ) representation of the n-to-m-component susceptibility crossover function, and the corresponding effective exponents, for an anisotropic n-component system.

Journal of Physics A Mathematical and General•Jul 1, 1976•74 citations

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

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.

Progress of Theoretical Physics•May 1, 1986•1 citation
Abstract Firstly the renormalization group technique is applied to an isotropic Heisenberg spin system with long‐range interaction and a set of critical exponents are calculated. These exponents are expressed in terms of a potential parameter s̀. All critical exponents except β, the magnetization exponent, depend on s̀. Secondly, the anisotropy is considered and the related crossover exponent φ is evaluated. This later parameter turns out to be equal to γ, the susceptibility exponent, as predicted by some authors.
Abstract The renormalization group technique is applied to the three‐dimensional classical Heisenberg spin system with modified dipole–dipole interaction. For this system, the values and dependence of several critical exponents are investigated. These exponents are: α, the specific heat exponent; β and δ, the magnetization exponents; γ, the susceptibility exponent; η, the correlation function exponent; and v, the correlation length exponent. All the critical exponents except β depend upon the potential range σ. When the potential range σ = 3/2, the modified dipole–dipole results agree with mean‐field values of the critical exponents.
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.
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.
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.