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

Physical review. B, Solid state

S=12XY model on cubic lattices. II. Longitudinal susceptibilities

M. Howard Lee

Published Jul 1, 1975• 3 citations
Publisher

Abstract

The longitudinal susceptibility for the $S=\frac{1}{2}\mathrm{XY}$ model on cubic lattices is discussed on the basis of its relationship to the free energy. A series expansion for the susceptibility is developed and its behavior is studied at an asymptotic limit. The asymptotic behavior of the series expansion shows that the longitudinal susceptibility is nondivergent at the critical point ${T}_{c}$, its "singular" part behaving as ${(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\alpha}+1}$, where $\ensuremath{\alpha}$ is the specific-heat exponent. A related quantity, the partial longitudinal susceptibility, which is important in the study of spin dynamics, is also shown to have a similar nondivergent critical behavior.

DOI
10.1103/physrevb.12.276

Related papers

High-temperature series for the susceptibility of the spin- S Ising model

John P. VAN DYKE, William J. Camp

We have extended the series for the zero-field susceptibility of the spin-$S$ Ising model to eighth order in the reduced temperature $K$, on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series ${h}_{n}(S)$ are expressed as simple polynomials in $X=S(S+1)$. For the face-centered-cubic lattice, an accurate polynomial fit to the critical point ${K}_{c}(S)$ is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all $S$. However, the large-$S$ series seems to exhibit more rapid apparent convergence.

Physical review. B, Solid state•Apr 1, 1974•42 citations

N -vector spin models on the simple-cubic and the body-centered-cubic lattices: A study of the critical behavior of the susceptibility and of the correlation length by high-temperature series extended to order β21

P. Butera, M. Comi

High-temperature expansions for the free energy, the susceptibility, and the second correlation moment of the classical $N$-vector model [also known as the $O(N)$ symmetric classical spin-Heisenberg model or as the lattice $O(N)$ nonlinear $\ensuremath{\sigma}$ model] on the simple-cubic and the body-centered-cubic lattices are extended to order ${\ensuremath{\beta}}^{21}$ for arbitrary $N$. The series for the second field derivative of the susceptibility is extended to order ${\ensuremath{\beta}}^{17}$. We report here on the analysis of the computed series for the susceptibility and the (second moment) correlation length which yields updated estimates of the critical parameters for various values of the spin dimensionality $N$, including $N=0$ (the self-avoiding walk model), $N=1$ (the Ising spin-1/2 model), $N=2$ (the $\mathrm{XY}$ model), and $N=3$ (the classical Heisenberg model). For all values of $N$ we confirm a good agreement with the present renormalization-group estimates. A study of the series for the other observables will appear in a forthcoming paper.

arXiv:hep-lat/9703018•Physical review. B, Condensed matter•Oct 1, 1997•139 citations

High temperature series for the susceptibility of the Ising model. II. Three dimensional lattices

M F Sykes, David S. Gaunt, P D Roberts +1 more

For pt. I see ibid., vol. 5, no. 5, 624. Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K for the simple cubic lattice to v 17 , the body-centred cubic lattice to v 15 , and the face-centred cubic lattice to v 12 , inclusive. The coefficients are analysed by the ratio method and it is estimated that, subject to the implicit assumptions thereof, the critical temperatures correspond to v f =0.21813+or-0.00001 (SC), v f =0.15612+or-0.00003 (BCC) and v f =0.10174+or-0.00001 (FCC). The symptotic behaviour of the coefficients is studied in detail; for the loose-packed lattices the decay of the even-odd oscillation in the ratios is found to be consistent with the assumption that the critical index of the high temperature specific heat is very close to 1/8.

Journal of physics. A, Proceedings of the Physical Society. General•May 1, 1972•139 citations

High-temperature longitudinal susceptibility for the S=12 XY model: Some numerical evidences

I. M. Kim, M. Howard Lee

The longitudinal component of the $\mathrm{XY}$-model susceptibility is nondivergent at $T={T}_{c}$. In an earlier paper it was shown that as $T\ensuremath{\rightarrow}{T}_{c}$+ the longitudinal susceptibility is determined essentially by nearest-neighbor correlations and it behaves as the derivative of the free energy with respect to inverse termperature. We provide here numerical evidences in support of the earlier conclusions.

Physical review. B, Condensed matter•Jun 1, 1979•2 citations

Extension to order β23 of the high-temperature expansions for the spin- 12 Ising model on simple cubic and body-centered cubic lattices

P. Butera, M. Comi

Using a renormalized linked-cluster-expansion method, we have extended to order ${\ensuremath{\beta}}^{23}$ the high-temperature series for the susceptibility $\ensuremath{\chi}$ and the second-moment correlation length $\ensuremath{\xi}$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}.$ Our best estimates for the inverse critical temperatures are ${\ensuremath{\beta}}_{c}^{\mathrm{sc}}=0.221654(1)$ and ${\ensuremath{\beta}}_{c}^{\mathrm{bcc}}=0.1573725(6).$ For the susceptibility exponent we get $\ensuremath{\gamma}=1.2375(6)$ and for the correlation length exponent $\ensuremath{\nu}=0.6302(4).$ The ratio of the critical amplitudes of $\ensuremath{\chi}$ above and below the critical temperature is estimated to be ${C}_{+}{/C}_{\ensuremath{-}}=4.762(8).$ The analogous ratio for $\ensuremath{\xi}$ is estimated to be ${f}_{+}{/f}_{\ensuremath{-}}=1.963(8).$ For the correction-to-scaling amplitude ratio we obtain ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$

arXiv:hep-lat/0006009•Physical review. B, Condensed matter•Dec 1, 2000•40 citations

The spin model. II. Analysis of high temperature series expansions of some thermodynamic quantities in three dimensions

Jos Rogiers, Donald D. Betts, T. Lookman

In this paper we reanalyze the series for the fluctuation of the order parameter, the fourth order fluctuation in the order parameter, and the specific heat for the spin [Formula: see text] model on three dimensional lattices. We find for the exponents, respectively, γ = 1.333 ± 0.001, γ2 = 4.725 ± 0.01, and α = 0.00 ± 0.04. We find no evidence for a correction to scaling term with exponent [Formula: see text] for the specific heat or for the susceptibility. Precise estimates for critical entropy and critical energy are obtained. From the critical amplitudes we obtain further support for the lattice–lattice scaling hypothesis and the one scale factor universality hypothesis.

Canadian Journal of Physics•Apr 1, 1978•15 citations
The longitudinal susceptibility for the S=12XYS=\frac{1}{2}\mathrm{XY}S=21​XY model on cubic lattices is discussed on the basis of its relationship to the free energy. A series expansion for the susceptibility is developed and its behavior is studied at an asymptotic limit. The asymptotic behavior of the series expansion shows that the longitudinal susceptibility is nondivergent at the critical point Tc{T}_{c}Tc​, its "singular" part behaving as (T\ensuremath−Tc)\ensuremath−\ensuremathα+1{(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\alpha}+1}(T\ensuremath−Tc​)\ensuremath−\ensuremathα+1, where \ensuremathα\ensuremath{\alpha}\ensuremathα is the specific-heat exponent. A related quantity, the partial longitudinal susceptibility, which is important in the study of spin dynamics, is also shown to have a similar nondivergent critical behavior.
We have extended the series for the zero-field susceptibility of the spin-SSS Ising model to eighth order in the reduced temperature KKK, on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series hn(S){h}_{n}(S)hn​(S) are expressed as simple polynomials in X=S(S+1)X=S(S+1)X=S(S+1). For the face-centered-cubic lattice, an accurate polynomial fit to the critical point Kc(S){K}_{c}(S)Kc​(S) is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all SSS. However, the large-SSS series seems to exhibit more rapid apparent convergence.
High-temperature expansions for the free energy, the susceptibility, and the second correlation moment of the classical NNN-vector model [also known as the O(N)O(N)O(N) symmetric classical spin-Heisenberg model or as the lattice O(N)O(N)O(N) nonlinear $\ensuremath{\sigma}$ model] on the simple-cubic and the body-centered-cubic lattices are extended to order ${\ensuremath{\beta}}^{21}$ for arbitrary NNN. The series for the second field derivative of the susceptibility is extended to order ${\ensuremath{\beta}}^{17}$. We report here on the analysis of the computed series for the susceptibility and the (second moment) correlation length which yields updated estimates of the critical parameters for various values of the spin dimensionality NNN, including N=0N=0N=0 (the self-avoiding walk model), N=1N=1N=1 (the Ising spin-1/2 model), N=2N=2N=2 (the XY\mathrm{XY}XY model), and N=3N=3N=3 (the classical Heisenberg model). For all values of NNN we confirm a good agreement with the present renormalization-group estimates. A study of the series for the other observables will appear in a forthcoming paper.
For pt. I see ibid., vol. 5, no. 5, 624. Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K for the simple cubic lattice to v 17 , the body-centred cubic lattice to v 15 , and the face-centred cubic lattice to v 12 , inclusive. The coefficients are analysed by the ratio method and it is estimated that, subject to the implicit assumptions thereof, the critical temperatures correspond to v f =0.21813+or-0.00001 (SC), v f =0.15612+or-0.00003 (BCC) and v f =0.10174+or-0.00001 (FCC). The symptotic behaviour of the coefficients is studied in detail; for the loose-packed lattices the decay of the even-odd oscillation in the ratios is found to be consistent with the assumption that the critical index of the high temperature specific heat is very close to 1/8.
The longitudinal component of the XY\mathrm{XY}XY-model susceptibility is nondivergent at T=TcT={T}_{c}T=Tc​. In an earlier paper it was shown that as $T\ensuremath{\rightarrow}{T}_{c}$+ the longitudinal susceptibility is determined essentially by nearest-neighbor correlations and it behaves as the derivative of the free energy with respect to inverse termperature. We provide here numerical evidences in support of the earlier conclusions.
Using a renormalized linked-cluster-expansion method, we have extended to order ${\ensuremath{\beta}}^{23}$ the high-temperature series for the susceptibility $\ensuremath{\chi}$ and the second-moment correlation length $\ensuremath{\xi}$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}.$ Our best estimates for the inverse critical temperatures are ${\ensuremath{\beta}}_{c}^{\mathrm{sc}}=0.221654(1)$ and ${\ensuremath{\beta}}_{c}^{\mathrm{bcc}}=0.1573725(6).$ For the susceptibility exponent we get $\ensuremath{\gamma}=1.2375(6)$ and for the correlation length exponent $\ensuremath{\nu}=0.6302(4).$ The ratio of the critical amplitudes of $\ensuremath{\chi}$ above and below the critical temperature is estimated to be ${C}_{+}{/C}_{\ensuremath{-}}=4.762(8).$ The analogous ratio for $\ensuremath{\xi}$ is estimated to be ${f}_{+}{/f}_{\ensuremath{-}}=1.963(8).$ For the correction-to-scaling amplitude ratio we obtain ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$
In this paper we reanalyze the series for the fluctuation of the order parameter, the fourth order fluctuation in the order parameter, and the specific heat for the spin [Formula: see text] model on three dimensional lattices. We find for the exponents, respectively, γ = 1.333 ± 0.001, γ2 = 4.725 ± 0.01, and α = 0.00 ± 0.04. We find no evidence for a correction to scaling term with exponent [Formula: see text] for the specific heat or for the susceptibility. Precise estimates for critical entropy and critical energy are obtained. From the critical amplitudes we obtain further support for the lattice–lattice scaling hypothesis and the one scale factor universality hypothesis.