Journal of Applied Physics

Transverse susceptibility of the Ising model

Yung‐Li Wang, Christopher D. Wentworth

Published Apr 15, 1985• 9 citations

Abstract

A high-temperature series for the transverse susceptibility of the Ising model is obtained to the tenth order in J/kBT for the plane square lattice and the fcc lattice. The temperature behavior of the three-dimensional lattice is studied and compared with the temperature behavior of the two-dimensional lattice.

DOI: 10.1063/1.335085

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Series expansions for the zero-temperature transverse Ising model

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Journal of Applied Physics

Transverse susceptibility of the Ising model

Yung‐Li Wang, Christopher D. Wentworth

Published Apr 15, 1985• 9 citations

Publisher

Abstract

DOI: 10.1063/1.335085

Related papers

Series expansions for the zero-temperature transverse Ising model

L.G. Marland

Series expansions are presented for the magnetisation, susceptibility, magnetic field derivatives of the susceptibility and 'specific heat' of the zero-temperature transverse Ising model. Coefficients in these series have been calculated to tenth order in lambda 2 (where lambda is the transverse field) for the linear chain, to eighth order for the square lattice and to seventh order for the triangular lattice. These series yield estimates of the low-temperature critical exponents alpha ', beta ', gamma ' and Delta ' of the two- and three-dimensional Ising models. They provide good evidence for the symmetry of exponents above and below the critical point, e.g. gamma '= gamma .Series expansions are presented for the magnetisation, susceptibility, magnetic field derivatives of the susceptibility and 'specific heat' of the zero-temperature transverse Ising model. Coefficients in these series have been calculated to tenth order in lambda 2 (where lambda is the transverse field) for the linear chain, to eighth order for the square lattice and to seventh order for the triangular lattice. These series yield estimates of the low-temperature critical exponents alpha ', beta ', gamma ' and Delta ' of the two- and three-dimensional Ising models. They provide good evidence for the symmetry of exponents above and below the critical point, e.g. gamma '= gamma .

Journal of Physics A Mathematical and GeneralAug 1, 198150 citations

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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.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.

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Journal of physics. A, Proceedings of the Physical Society. GeneralMay 1, 197290 citations

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Takuma Ishikawa, Takehiko Oguchi

The susceptibility of the restricted-dimensional (one- or two-dimensional-like) Ising model is analyzed by means of the high temperature series expansion. The exchange interaction in a chain (or plane), and inter chains (or planes) are J and λ J , respectively. The first eight terms in the susceptibility series are explicitly obtained. The critical temperature T c and the critical exponent γ are obtained as a function of λ. It is found that the susceptibility shows restricted-dimensional behavior at high temperature ( T ≫ T c ), and shows three-dimensional behavior in the very vicinity of the critical temperature ( $T{\gtrsim}T_{\text{c}}$ ) for small λ.The susceptibility of the restricted-dimensional (one- or two-dimensional-like) Ising model is analyzed by means of the high temperature series expansion. The exchange interaction in a chain (or plane), and inter chains (or planes) are J and λ J , respectively. The first eight terms in the susceptibility series are explicitly obtained. The critical temperature T c and the critical exponent γ are obtained as a function of λ. It is found that the susceptibility shows restricted-dimensional behavior at high temperature ( T ≫ T c ), and shows three-dimensional behavior in the very vicinity of the critical temperature ($T{\gtrsim}T_{\text{c}}$) for small λ.

Journal of the Physical Society of JapanOct 1, 19721 citation

General- R High-Temperature Series for the Susceptibility, Second Moment, and Specific Heat of sc and fcc Ising Models with Lattice Anisotropy

Fredric Harbus, Eugene Stanley

Of particular current interest is the critical behavior of functions on crossing over from one lattice dimensionality to another. To this end, we report high-temperature series for an Ising model with lattice anisotropy-i.e., with different exchange constants for different lattice directions. The Hamiltonian is ${\mathcal{H}}_{l,\mathrm{anis}}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j}\ensuremath{\equiv}\ensuremath{-}{J}_{\mathrm{xy}}(\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}+R\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j})$ where ${s}_{i}=\ifmmode\pm\else\textpm\fi{}1$, the first summation is over all nearest-neighbor pairs in the $\mathrm{xy}$ plane, and the second sum is over pairs coupled in the $z$ direction. The susceptibility, second moment, and specific-heat series are explicitly presented for arbitrary ${J}_{\mathrm{xy}}$ and ${J}_{z}$ for the simple cubic (sc) and face-centered cubic (fcc) lattices to tenth order in inverse temperature. The general- $R$ series are essential if one wishes to study the Riedel-Wegner crossover exponent appropriate to changing lattice dimensionality, since for $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}=0$, both the sc and fcc lattices reduce to two-dimensional square lattices, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc reduces to noninteracting linear chains.Of particular current interest is the critical behavior of functions on crossing over from one lattice dimensionality to another. To this end, we report high-temperature series for an Ising model with lattice anisotropy-i.e., with different exchange constants for different lattice directions. The Hamiltonian is ${\mathcal{H}}_{l,\mathrm{anis}}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j}\ensuremath{\equiv}\ensuremath{-}{J}_{\mathrm{xy}}(\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}+R\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j})$ where ${s}_{i}=\ifmmode\pm\else\textpm\fi{}1$, the first summation is over all nearest-neighbor pairs in the $\mathrm{xy}$ plane, and the second sum is over pairs coupled in the $z$ direction. The susceptibility, second moment, and specific-heat series are explicitly presented for arbitrary ${J}_{\mathrm{xy}}$ and ${J}_{z}$ for the simple cubic (sc) and face-centered cubic (fcc) lattices to tenth order in inverse temperature. The general-$R$ series are essential if one wishes to study the Riedel-Wegner crossover exponent appropriate to changing lattice dimensionality, since for $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}=0$, both the sc and fcc lattices reduce to two-dimensional square lattices, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc reduces to noninteracting linear chains.

Physical review. B, Solid stateJan 1, 197333 citations

Susceptibility scaling functions for ferromagnetic Ising films

T. Weston Capehart, Michael Ellis Fisher

The susceptibility of simple-cubic-lattice ferromagnetic Ising films of $n$ two-dimensional lattice layers is studied by extrapolation of high-temperature series expansions (to eleventh and twelfth orders) for $n=3,4,\dots{},14$ layers with periodic boundary conditions, and $n=3,4,\dots{},10$ layers with free-surface conditions. The corresponding surface susceptibility series for sc, bcc, and fcc lattices are analyzed in the light of universality hypotheses. On the basis of finite-size scaling theory explicit scaling functions are constructed describing the crossover from two-dimensional to three-dimensional critical behavior in terms of the ratio $\frac{n}{\ensuremath{\xi}(T)}$, where $\ensuremath{\xi}\ensuremath{\sim}\ensuremath{\Delta}{T}^{\ensuremath{-}\ensuremath{\nu}}$ is the bulk correlation length.The susceptibility of simple-cubic-lattice ferromagnetic Ising films of $n$ two-dimensional lattice layers is studied by extrapolation of high-temperature series expansions (to eleventh and twelfth orders) for $n=3,4,\dots{},14$ layers with periodic boundary conditions, and $n=3,4,\dots{},10$ layers with free-surface conditions. The corresponding surface susceptibility series for sc, bcc, and fcc lattices are analyzed in the light of universality hypotheses. On the basis of finite-size scaling theory explicit scaling functions are constructed describing the crossover from two-dimensional to three-dimensional critical behavior in terms of the ratio $\frac{n}{\ensuremath{\xi}(T)}$, where $\ensuremath{\xi}\ensuremath{\sim}\ensuremath{\Delta}{T}^{\ensuremath{-}\ensuremath{\nu}}$ is the bulk correlation length.

Physical review. B, Solid stateJun 1, 1976123 citations

T{\gtrsim}T_{\text{c}}

) for small λ.

\mathrm{xy}

plane, and the second sum is over pairs coupled in the

z

direction. The susceptibility, second moment, and specific-heat series are explicitly presented for arbitrary

{J}_{\mathrm{xy}}

and

{J}_{z}

for the simple cubic (sc) and face-centered cubic (fcc) lattices to tenth order in inverse temperature. The general-

R

series are essential if one wishes to study the Riedel-Wegner crossover exponent appropriate to changing lattice dimensionality, since for $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}=0$, both the sc and fcc lattices reduce to two-dimensional square lattices, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc reduces to noninteracting linear chains.

The susceptibility of simple-cubic-lattice ferromagnetic Ising films of

n

two-dimensional lattice layers is studied by extrapolation of high-temperature series expansions (to eleventh and twelfth orders) for

n=3,4,\dots{},14

layers with periodic boundary conditions, and

n=3,4,\dots{},10

layers with free-surface conditions. The corresponding surface susceptibility series for sc, bcc, and fcc lattices are analyzed in the light of universality hypotheses. On the basis of finite-size scaling theory explicit scaling functions are constructed describing the crossover from two-dimensional to three-dimensional critical behavior in terms of the ratio $\frac{n}{\ensuremath{\xi}(T)}$, where $\ensuremath{\xi}\ensuremath{\sim}\ensuremath{\Delta}{T}^{\ensuremath{-}\ensuremath{\nu}}$ is the bulk correlation length.