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

Critical Properties of the Two-Dimensional Spin-1/2 XY Ferromagnet

T. Onogi, Seiji Miyashita, Masuo Suzuki

Springer series in solid-state sciencesJan 1, 19875 citations

Spherical Model with Long-Range Ferromagnetic Interactions

G S Joyce

The behavior of the spherical model of a ferromagnet with an interaction energy between the magnetic spins which varies with distance as $\frac{1}{{r}^{d+\ensuremath{\sigma}}}$ (where $d$ is the dimensionality of the lattice and $\ensuremath{\sigma}>0$) is analyzed. It is shown that the model exhibits a ferromagnetic transition in one and two dimensions, providing $0<\ensuremath{\sigma}<d$. (The usual spherical model with nearest-neighbor interactions does not have a transition in one and two dimensions.) The critical-point behavior is investigated. It is found that the singularities in the specific heat and susceptibility are dependent on $\ensuremath{\sigma}$ and $d$ , but the behavior of the magnetization is independent of $\ensuremath{\sigma}$ and $d$ . In three dimensions the susceptibility diverges as ${(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}$, where $\ensuremath{\gamma}=1$ for $0<\ensuremath{\sigma}<\frac{3}{2}$, $\ensuremath{\gamma}=\frac{\ensuremath{\sigma}}{(3\ensuremath{-}\ensuremath{\sigma})}$ for $\frac{3}{2}<\ensuremath{\sigma}<2$ and $\ensuremath{\gamma}=2$ for $\ensuremath{\sigma}>2$. The asymptotic form of the spin-spin correlation function $\ensuremath{\Gamma}(\mathrm{r})$ is studied in the neighborhood of the critical temperature ${T}_{c}$ . At $T={T}_{c}$ , $\ensuremath{\Gamma}(\mathrm{r})$ decays for large $r$ as $\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}$. Several two-dimensional models with long-range interactions falling off as $\frac{1}{{r}^{2}}$ in certain directions only are also investigated.The behavior of the spherical model of a ferromagnet with an interaction energy between the magnetic spins which varies with distance as $\frac{1}{{r}^{d+\ensuremath{\sigma}}}$ (where $d$ is the dimensionality of the lattice and $\ensuremath{\sigma}>0$) is analyzed. It is shown that the model exhibits a ferromagnetic transition in one and two dimensions, providing $0<\ensuremath{\sigma}<d$. (The usual spherical model with nearest-neighbor interactions does not have a transition in one and two dimensions.) The critical-point behavior is investigated. It is found that the singularities in the specific heat and susceptibility are dependent on $\ensuremath{\sigma}$ and $d$, but the behavior of the magnetization is independent of $\ensuremath{\sigma}$ and $d$. In three dimensions the susceptibility diverges as ${(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}$, where $\ensuremath{\gamma}=1$ for $0<\ensuremath{\sigma}<\frac{3}{2}$, $\ensuremath{\gamma}=\frac{\ensuremath{\sigma}}{(3\ensuremath{-}\ensuremath{\sigma})}$ for $\frac{3}{2}<\ensuremath{\sigma}<2$ and $\ensuremath{\gamma}=2$ for $\ensuremath{\sigma}>2$. The asymptotic form of the spin-spin correlation function $\ensuremath{\Gamma}(\mathrm{r})$ is studied in the neighborhood of the critical temperature ${T}_{c}$. At $T={T}_{c}$, $\ensuremath{\Gamma}(\mathrm{r})$ decays for large $r$ as $\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}$. Several two-dimensional models with long-range interactions falling off as $\frac{1}{{r}^{2}}$ in certain directions only are also investigated.

Physical ReviewJun 3, 1966189 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

Critical properties of a spin- Heisenberg ferromagnet

K. Moorjani, T. Tanaka

Physics Letters AMay 1, 19690 citations

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

) for small λ.

The behavior of the spherical model of a ferromagnet with an interaction energy between the magnetic spins which varies with distance as $\frac{1}{{r}^{d+\ensuremath{\sigma}}}$ (where

d

is the dimensionality of the lattice and $\ensuremath{\sigma}>0$) is analyzed. It is shown that the model exhibits a ferromagnetic transition in one and two dimensions, providing $0<\ensuremath{\sigma}<d$. (The usual spherical model with nearest-neighbor interactions does not have a transition in one and two dimensions.) The critical-point behavior is investigated. It is found that the singularities in the specific heat and susceptibility are dependent on $\ensuremath{\sigma}$ and

d

, but the behavior of the magnetization is independent of $\ensuremath{\sigma}$ and

d

. In three dimensions the susceptibility diverges as ${(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}$, where $\ensuremath{\gamma}=1$ for $0<\ensuremath{\sigma}<\frac{3}{2}$, $\ensuremath{\gamma}=\frac{\ensuremath{\sigma}}{(3\ensuremath{-}\ensuremath{\sigma})}$ for $\frac{3}{2}<\ensuremath{\sigma}<2$ and $\ensuremath{\gamma}=2$ for $\ensuremath{\sigma}>2$. The asymptotic form of the spin-spin correlation function $\ensuremath{\Gamma}(\mathrm{r})$ is studied in the neighborhood of the critical temperature

{T}_{c}

. At

T={T}_{c}

, $\ensuremath{\Gamma}(\mathrm{r})$ decays for large

r

as $\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}$. Several two-dimensional models with long-range interactions falling off as

\frac{1}{{r}^{2}}

in certain directions only are also investigated.

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.