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

Progress of Theoretical Physics

Expansion of a Critical Exponent in Inverse Powers of Spin Dimensionality

Ryuzo Abe

Published Jan 1, 1973• 84 citations
Publisher

Abstract

A systematic expansion method is developed for studying a critical exponent. The expansion parameter in the present theory is 1/n where n is the spin dimensionality. Since the limit n →∞ corresponds to the spherical model, the present method means that the deviation of critical exponent from its spherical model value is expanded in powers of 1/n. The susceptibility exponent γ is discussed and the terms up to 1/n are exactly derived for the d-dimensional simple hypercubical system with nearest-neighbor interaction (d is assumed to take a continuous value). The result is proved to be consistent with recent Wilson's ε expansion.

DOI
10.1143/ptp.49.113

Related papers

Critical Exponent of the Ising Model in the High Density Limit

Ryuzo Abe

The system under consideration is the d-dimensional simple hypercubical Ising ferromagnet with nearest-neighbor interaction. A critical exponent γ for the susceptibility is discussed in the high density limit (d →∞), and formal expressions for γ in powers of 1/d are derived. It is shown that the expansion for γ is not determined uniquely, contrary to the case of Curie point. It is suggested that the deviation of γ from mean field result is proportional to exp(-Bd) in the high density limit, and B is estimated by an extrapolation method.

Progress of Theoretical Physics•Jan 1, 1972•20 citations

Critical Exponent of the Ising Model in the High Density Limit. II

Ryuzo Abe

The system under consideration is the d-dimensional simple hypercubical Ising ferromagnet with nearest-neighbor interaction. A critical exponent γ for the susceptibility is discussed in the high density limit (d →∞) by means of ratio method. It is suggested that the mean field exponent is valid if d exceeds a critical dimensionality dc. By a numerical analysis it is estimated that dc ≃4.

Progress of Theoretical Physics•Apr 1, 1972•1 citation

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

Critical Indices for a System of Spins of Arbitrary Dimensionality Situated on a Lattice of Arbitrary Dimensionality

Eugene Stanley

A model Hamiltonian H=−JΣ(ij)Si·Sj is considered, where Si are isotropically interacting D-dimensional classical spins. Thus the S=½ Ising, classical planar, and classical Heisenberg models are obtained if D=1, 2, and 3, respectively; moreover, as D→∞, the free energy corresponding to H approaches the free energy of the exactly-soluble spherical model of Berlin and Kac. H can also be solved exactly providing the lattice dimensionality d=1. For lattices of higher dimensionality, we must resort to approximation techniques; hence we calculate high-temperature expansions for arbitrary D and d of the susceptibility, internal energy (specific heat), and the second moment. (With a few exceptions, such expansions have been available in the past only for D=1, 3 and d=1–3.) Somewhat smoother series to facilitate more reliable extrapolation are then obtained by re-expanding all of the series in terms of the nearest-neighbor spin correlation function for a one-dimensional lattice. We then study the dependence on spin and lattice dimensionality of the critical temperatures Tc(D, d) and the critical exponents γ(D, d) [(χ∼(T-Tc)−γ], α(D, d) [C∼(T-Tc)−α], ν(D, d) [inverse correlation range κ∼(T-Tc)−v]. These ``critical properties'' are all found to vary monotonically (and in most cases, smoothly) with D and d. The assumptions behind making critical phenomena predictions from a dozen or so terms (of an infinite series!) are given increased plausibility by the fact that we have extended the various expansions for the spherical model to 80–100 terms and find that none of the predictions based upon ∼10 terms is changed [e.g., Tc(∞, 2)=0, γ(∞, 3)=2, etc.].

Journal of Applied Physics•Mar 1, 1969•34 citations

Series studies of critical exponents in continuous dimensions

John P. VAN DYKE, William J. Camp

Using results we have previously derived for the high‐temperature susceptibility expansion of classical models as a closed‐form function of lattice dimension, we study the dimensional dependence of the critical exponent γ and critical temperature, as well as the correction‐to‐scaling exponent Δ1, for Ising‐like (n=1) models. The numerical results are obtained by extrapolation of 10‐th order series on loose‐packed hypercubical lattices, and 8‐th order series on close‐packed hypertriangular lattices. The critical exponent γ increases monotonically with decreasing dimension, d, for d<4, and apparently tends to infinity at d=1; while the critical temperature decreases monotonically and smoothly to zero at d=1. Detailed contact is made with the ε‐expansion estimates for critical exponents obtained in the context of renormalization group theory.

AIP conference proceedings•Jan 1, 1976•4 citations

Scaling with a Parameter in Spin Systems near the Critical Point. I

Masuo Suzuki

The Hamiltonian of the form H=H0+λH1 is discussed, where λ is a parameter to change symmetry, dimensionality, or potential range. Scaling with the parameter λ is studied for thermodynamic quantities such as the free energy. By assuming the scaled form F(ε,λ)=ε2-αF(λ/εφ) for the singular part of the free energy near the critical point Tc(ε=(T-Tc)/Tc), the expression (or explicit value) of the critical exponent φ is obtained in each case of change of symmetry, dimensionality and potential range. In particular, the universal relation φ=γ is found for change of dimensionality, where γ is the critical exponent of the susceptibility in the unperturbed Hamiltonian H0. An extension to dynamical critical phenomena is also discussed briefly, particularly in connection with the critical slowing down. A possibility is suggested to derive this generalized scaling with the parameter λ by applying the usual scaling law to each term of the perturbational expansion with respect to the parameter λ.

Progress of Theoretical Physics•Oct 1, 1971•61 citations
A systematic expansion method is developed for studying a critical exponent. The expansion parameter in the present theory is 1/n where n is the spin dimensionality. Since the limit n →∞ corresponds to the spherical model, the present method means that the deviation of critical exponent from its spherical model value is expanded in powers of 1/n. The susceptibility exponent γ is discussed and the terms up to 1/n are exactly derived for the d-dimensional simple hypercubical system with nearest-neighbor interaction (d is assumed to take a continuous value). The result is proved to be consistent with recent Wilson's ε expansion.
The system under consideration is the d-dimensional simple hypercubical Ising ferromagnet with nearest-neighbor interaction. A critical exponent γ for the susceptibility is discussed in the high density limit (d →∞), and formal expressions for γ in powers of 1/d are derived. It is shown that the expansion for γ is not determined uniquely, contrary to the case of Curie point. It is suggested that the deviation of γ from mean field result is proportional to exp(-Bd) in the high density limit, and B is estimated by an extrapolation method.
A model Hamiltonian H=−JΣ(ij)Si·Sj is considered, where Si are isotropically interacting D-dimensional classical spins. Thus the S=½ Ising, classical planar, and classical Heisenberg models are obtained if D=1, 2, and 3, respectively; moreover, as D→∞, the free energy corresponding to H approaches the free energy of the exactly-soluble spherical model of Berlin and Kac. H can also be solved exactly providing the lattice dimensionality d=1. For lattices of higher dimensionality, we must resort to approximation techniques; hence we calculate high-temperature expansions for arbitrary D and d of the susceptibility, internal energy (specific heat), and the second moment. (With a few exceptions, such expansions have been available in the past only for D=1, 3 and d=1–3.) Somewhat smoother series to facilitate more reliable extrapolation are then obtained by re-expanding all of the series in terms of the nearest-neighbor spin correlation function for a one-dimensional lattice. We then study the dependence on spin and lattice dimensionality of the critical temperatures Tc(D, d) and the critical exponents γ(D, d) [(χ∼(T-Tc)−γ], α(D, d) [C∼(T-Tc)−α], ν(D, d) [inverse correlation range κ∼(T-Tc)−v]. These ``critical properties'' are all found to vary monotonically (and in most cases, smoothly) with D and d. The assumptions behind making critical phenomena predictions from a dozen or so terms (of an infinite series!) are given increased plausibility by the fact that we have extended the various expansions for the spherical model to 80–100 terms and find that none of the predictions based upon ∼10 terms is changed [e.g., Tc(∞, 2)=0, γ(∞, 3)=2, etc.].
Using results we have previously derived for the high‐temperature susceptibility expansion of classical models as a closed‐form function of lattice dimension, we study the dimensional dependence of the critical exponent γ and critical temperature, as well as the correction‐to‐scaling exponent Δ1, for Ising‐like (n=1) models. The numerical results are obtained by extrapolation of 10‐th order series on loose‐packed hypercubical lattices, and 8‐th order series on close‐packed hypertriangular lattices. The critical exponent γ increases monotonically with decreasing dimension, d, for d<4, and apparently tends to infinity at d=1; while the critical temperature decreases monotonically and smoothly to zero at d=1. Detailed contact is made with the ε‐expansion estimates for critical exponents obtained in the context of renormalization group theory.
The Hamiltonian of the form H=H0+λH1 is discussed, where λ is a parameter to change symmetry, dimensionality, or potential range. Scaling with the parameter λ is studied for thermodynamic quantities such as the free energy. By assuming the scaled form F(ε,λ)=ε2-αF(λ/εφ) for the singular part of the free energy near the critical point Tc(ε=(T-Tc)/Tc), the expression (or explicit value) of the critical exponent φ is obtained in each case of change of symmetry, dimensionality and potential range. In particular, the universal relation φ=γ is found for change of dimensionality, where γ is the critical exponent of the susceptibility in the unperturbed Hamiltonian H0. An extension to dynamical critical phenomena is also discussed briefly, particularly in connection with the critical slowing down. A possibility is suggested to derive this generalized scaling with the parameter λ by applying the usual scaling law to each term of the perturbational expansion with respect to the parameter λ.