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

Dynamical critical behavior of long-range spin models

Paraj Titum, Mohammad F. Maghrebi

Bulletin of the American Physical SocietyMar 5, 20190 citations

Kinetic Gaussian Model with Long-Range Interactions

Xiang-Mu Kong, Yang Zhan-Ru

In this paper dynamical critical phenomena of the Gaussian model with long-range interactions decaying as on d-dimensional hypercubic lattices (d = 1, 2, and 3) are studied. First, the critical points are exactly calculated, and it is found that the critical points depend on the value of δ and the range of interactions. Then the critical dynamics is considered. We calculate the time evolutions of the local magnetizations and the spin-spin correlation functions, and further the dynamic critical exponents are obtained. For one-, two- and three-dimensional lattices, it is found that the dynamic critical exponents are all which agrees with the result when only considering nearest neighboring interactions, and that they are all δ if It shows that the dynamic critical exponents are independent of the spatial dimensionality but depend on the value of δ.In this paper dynamical critical phenomena of the Gaussian model with long-range interactions decaying as on d-dimensional hypercubic lattices (d = 1, 2, and 3) are studied. First, the critical points are exactly calculated, and it is found that the critical points depend on the value of δ and the range of interactions. Then the critical dynamics is considered. We calculate the time evolutions of the local magnetizations and the spin-spin correlation functions, and further the dynamic critical exponents are obtained. For one-, two- and three-dimensional lattices, it is found that the dynamic critical exponents are all which agrees with the result when only considering nearest neighboring interactions, and that they are all δ if It shows that the dynamic critical exponents are independent of the spatial dimensionality but depend on the value of δ.

Communications in Theoretical PhysicsDec 15, 20040 citations

Non-perturbative effects in two-dimensional lattice O(N) models

Michael C. Ogilvie

Nuclear Physics BDec 1, 19814 citations

Kinetic Gaussian Model with Long-Range Interactions

KONGXiang-Mu, Xiang-Mu, Yang +1 more

In this paper dynamical critical phenomena of the Gaussian model with long-range interactions decaying as 1/rd+δ (δ＞ 0) on d-dimensional hypercubic lattices (d = 1, 2, and 3) are studied. First, the critical points are exactly calculated, and it is found that the critical points depend on the value of δ and the range of interactions. Then the critical dynamics is considered. We calculate the time evolutions of the local magnetizations and the spin-spin correlation functions, and further the dynamic critical exponents are obtained. For one-, two- and three-dimensional lattices, it is found that the dynamic critical exponents are all z = 2 if δ＞ 2, which agrees with the result when only considering nearest neighboring interactions, and that they are all δ if 0 ＜δ＜ 2. It shows that the dynamic critical exponents are independent of the spatial dimensionality but depend on the value of δ.In this paper dynamical critical phenomena of the Gaussian model with long-range interactions decaying as 1/rd+δ (δ＞ 0) on d-dimensional hypercubic lattices (d = 1, 2, and 3) are studied. First, the critical points are exactly calculated, and it is found that the critical points depend on the value of δ and the range of interactions. Then the critical dynamics is considered. We calculate the time evolutions of the local magnetizations and the spin-spin correlation functions, and further the dynamic critical exponents are obtained. For one-, two- and three-dimensional lattices, it is found that the dynamic critical exponents are all z = 2 if δ＞ 2, which agrees with the result when only considering nearest neighboring interactions, and that they are all δ if 0 ＜δ＜ 2. It shows that the dynamic critical exponents are independent of the spatial dimensionality but depend on the value of δ.

理论物理通讯：英文版Jan 1, 20040 citations

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.