Physical review. B, Condensed matter

Critical exponents for ferromagnetic systems with Ruderman-Kittel-Kasuya-Yosida interactions

R. Beutler, M. Fähnle

Published Dec 1, 1989• 3 citations

Abstract

The asymptotic critical exponents of a ferromagnetic spherical model on a simple cubic lattice with Ruderman-Kittel-Kasuya-Yosida interactions are calculated. These interactions are of long range but oscillating in sign and therefore lead to the same asymptotic critical behavior as the corresponding nearest-neighbor model.

DOI: 10.1103/physrevb.40.11417

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Spherical Model with Long-Range Ferromagnetic Interactions

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Physical review. B, Condensed matter

Critical exponents for ferromagnetic systems with Ruderman-Kittel-Kasuya-Yosida interactions

R. Beutler, M. Fähnle

Published Dec 1, 1989• 3 citations

Publisher

Abstract

DOI: 10.1103/physrevb.40.11417

Related papers

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.

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Ferromagnets with the number of spin components $n>2$ and with power-law interactions ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ are studied in dimensions $d=2+\ensuremath{\epsilon}$ for small $\ensuremath{\epsilon}$. The following theorem is proved: Let ${\ensuremath{\eta}}_{\mathrm{SR}}$ be the value of the critical exponent $\ensuremath{\eta}$ for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality $2\ensuremath{-}\ensuremath{\sigma}<{\ensuremath{\eta}}_{\mathrm{SR}}$ is satisfied. In this case the critical exponents depend on $d$ and $n$ but not on $\ensuremath{\sigma}$. In the opposite case, the value of $\ensuremath{\eta}$ is ${\ensuremath{\eta}}_{\mathrm{LR}}=2\ensuremath{-}\ensuremath{\sigma}$ and the critical exponent $\ensuremath{\nu}$ depends on $\ensuremath{\sigma}$ in addition to $d$ and $n$ .Ferromagnets with the number of spin components $n>2$ and with power-law interactions ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ are studied in dimensions $d=2+\ensuremath{\epsilon}$ for small $\ensuremath{\epsilon}$. The following theorem is proved: Let ${\ensuremath{\eta}}_{\mathrm{SR}}$ be the value of the critical exponent $\ensuremath{\eta}$ for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality $2\ensuremath{-}\ensuremath{\sigma}<{\ensuremath{\eta}}_{\mathrm{SR}}$ is satisfied. In this case the critical exponents depend on $d$ and $n$ but not on $\ensuremath{\sigma}$. In the opposite case, the value of $\ensuremath{\eta}$ is ${\ensuremath{\eta}}_{\mathrm{LR}}=2\ensuremath{-}\ensuremath{\sigma}$ and the critical exponent $\ensuremath{\nu}$ depends on $\ensuremath{\sigma}$ in addition to $d$ and $n$.

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P. H. Lundow, Ian A. Campbell

Three-dimensional Ising model ferromagnets on different lattices with nearest-neighbor interactions, and on simple-cubic lattices with equivalent interactions out to further neighbors, are studied numerically. The susceptibility data for all these systems are analyzed using the critical renormalization-group theory formalism over the entire temperature range above ${T}_{c}$ with an appropriate choice of scaling variable and scaling expressions. Representative experimental data on a metallic ferromagnet (Ni) and an elementary fluid (Xe) are interpreted in the same manner so as to estimate effective coordination numbers.Three-dimensional Ising model ferromagnets on different lattices with nearest-neighbor interactions, and on simple-cubic lattices with equivalent interactions out to further neighbors, are studied numerically. The susceptibility data for all these systems are analyzed using the critical renormalization-group theory formalism over the entire temperature range above ${T}_{c}$ with an appropriate choice of scaling variable and scaling expressions. Representative experimental data on a metallic ferromagnet (Ni) and an elementary fluid (Xe) are interpreted in the same manner so as to estimate effective coordination numbers.

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Computer simulation of the critical behavior of magnetic systems with competition between the short- and long-range interactions

С. В. Белим, И. Б. Ларионов, R. V. Soloneckiy

Critical behavior of Ising ferromagnetic systems with the competition between short- and long-range interactions near the point of a second-order phase transition has been investigated. The effects of long-range interaction lead to the need to calculate the influence of next-nearest spins, which lie inside the sphere of a specific radius. The study was carried out for a power law of decreaseing the exchange integral with distance for a three-dimensional Ising model with a cubic lattice. A comparison with the results of a field-theoretical approach and data of real experiments has been carried out. It has been shown that the critical temperature grows linearly with an increase in the parameters of long-range interaction. Critical exponents have been calculated at different parameters of long-range interaction. It is shown that at small dimensions of a system it is the short-range interaction that exerts the dominant effect on the critical behavior of the system. With an increase in the linear dimension of the system, long-range forces begin to dominate.Critical behavior of Ising ferromagnetic systems with the competition between short- and long-range interactions near the point of a second-order phase transition has been investigated. The effects of long-range interaction lead to the need to calculate the influence of next-nearest spins, which lie inside the sphere of a specific radius. The study was carried out for a power law of decreaseing the exchange integral with distance for a three-dimensional Ising model with a cubic lattice. A comparison with the results of a field-theoretical approach and data of real experiments has been carried out. It has been shown that the critical temperature grows linearly with an increase in the parameters of long-range interaction. Critical exponents have been calculated at different parameters of long-range interaction. It is shown that at small dimensions of a system it is the short-range interaction that exerts the dominant effect on the critical behavior of the system. With an increase in the linear dimension of the system, long-range forces begin to dominate.

The Physics of Metals and MetallographyNov 1, 20165 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.

Ferromagnets with the number of spin components

n>2

and with power-law interactions ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ are studied in dimensions $d=2+\ensuremath{\epsilon}$ for small $\ensuremath{\epsilon}$. The following theorem is proved: Let ${\ensuremath{\eta}}_{\mathrm{SR}}$ be the value of the critical exponent $\ensuremath{\eta}$ for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality $2\ensuremath{-}\ensuremath{\sigma}<{\ensuremath{\eta}}_{\mathrm{SR}}$ is satisfied. In this case the critical exponents depend on

d

and

n

but not on $\ensuremath{\sigma}$. In the opposite case, the value of $\ensuremath{\eta}$ is ${\ensuremath{\eta}}_{\mathrm{LR}}=2\ensuremath{-}\ensuremath{\sigma}$ and the critical exponent $\ensuremath{\nu}$ depends on $\ensuremath{\sigma}$ in addition to

d

and

n

{T}_{c}

with an appropriate choice of scaling variable and scaling expressions. Representative experimental data on a metallic ferromagnet (Ni) and an elementary fluid (Xe) are interpreted in the same manner so as to estimate effective coordination numbers.