The crossover behavior of the susceptibilities ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}$ of spin systems with anisotropic exchange coupling is discussed on the basis of an "extended" scaling hypothesis which gives ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}(T,g)\ensuremath{\approx}A{t}^{\ensuremath{-}\ensuremath{\gamma}}{X}_{\ensuremath{\alpha}}(\frac{Bg}{{t}^{\ensuremath{\varphi}}})$, where $g$ is the anisotropy parameter, $t=\frac{(T\ensuremath{-}{T}_{c0})}{{T}_{c0}}$, with ${T}_{c0}$ and $\ensuremath{\gamma}$ being the iso tropic critical temperature, and susceptibility exponent, while $A$ and $B$ are model-dependent amplitudes. Analysis of high-temperature series expansions for ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}(T,g)$ as polynomials in $g$ for rhombic and axial anisotropy in the fcc, bcc, and simple cubic classical Heisenberg ( $n=3$ ) and classical $XY$ and planar-spin models ( $N=2$ ), verifies the scaling with crossover exponents $\ensuremath{\varphi}=1.25\ifmmode\pm\else\textpm\fi{}0.015$ ( $n=3$ ) and $\ensuremath{\varphi}=1.175\ifmmode\pm\else\textpm\fi{}0.015$ ( $n=2$ ). The universality of the scaling functions ${X}_{\ensuremath{\alpha}}(x)$ is demonstrated both for small $x$ and in the anisotropic limit $T\ensuremath{\rightarrow}{T}_{c}(g)$, where $x\ensuremath{\rightarrow}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}$; for $n=3$ , accurate representations are constructed in the form ${X}_{\ensuremath{\alpha}}(x)\ensuremath{\simeq}{P}_{\ensuremath{\alpha}}(\frac{x}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}}){(1\ensuremath{-}\frac{x}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}})}^{\ensuremath{-}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}}$, where $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}$ is the anisotropic susceptibility exponent, while ${P}_{\ensuremath{\alpha}}(z)$ varies smoothly from ${P}_{\ensuremath{\alpha}}(0)=1$ to ${P}_{\ensuremath{\alpha}}(1)\ensuremath{\simeq}1.10$.The crossover behavior of the susceptibilities ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}$ of spin systems with anisotropic exchange coupling is discussed on the basis of an "extended" scaling hypothesis which gives ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}(T,g)\ensuremath{\approx}A{t}^{\ensuremath{-}\ensuremath{\gamma}}{X}_{\ensuremath{\alpha}}(\frac{Bg}{{t}^{\ensuremath{\varphi}}})$, where $g$ is the anisotropy parameter, $t=\frac{(T\ensuremath{-}{T}_{c0})}{{T}_{c0}}$, with ${T}_{c0}$ and $\ensuremath{\gamma}$ being the iso tropic critical temperature, and susceptibility exponent, while $A$ and $B$ are model-dependent amplitudes. Analysis of high-temperature series expansions for ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}(T,g)$ as polynomials in $g$ for rhombic and axial anisotropy in the fcc, bcc, and simple cubic classical Heisenberg ($n=3$) and classical $XY$ and planar-spin models ($N=2$), verifies the scaling with crossover exponents $\ensuremath{\varphi}=1.25\ifmmode\pm\else\textpm\fi{}0.015$ ($n=3$) and $\ensuremath{\varphi}=1.175\ifmmode\pm\else\textpm\fi{}0.015$ ($n=2$). The universality of the scaling functions ${X}_{\ensuremath{\alpha}}(x)$ is demonstrated both for small $x$ and in the anisotropic limit $T\ensuremath{\rightarrow}{T}_{c}(g)$, where $x\ensuremath{\rightarrow}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}$; for $n=3$, accurate representations are constructed in the form ${X}_{\ensuremath{\alpha}}(x)\ensuremath{\simeq}{P}_{\ensuremath{\alpha}}(\frac{x}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}}){(1\ensuremath{-}\frac{x}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}})}^{\ensuremath{-}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}}$, where $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}$ is the anisotropic susceptibility exponent, while ${P}_{\ensuremath{\alpha}}(z)$ varies smoothly from ${P}_{\ensuremath{\alpha}}(0)=1$ to ${P}_{\ensuremath{\alpha}}(1)\ensuremath{\simeq}1.10$.

Physical review. B, Solid stateSep 1, 1974256 citations

Crossover exponent of the anisotropic n-vector model

R. Oppermann

Physics Letters AApr 1, 197415 citations

Quasi-long-range order in the random anisotropy Heisenberg model

Dima E. Feldman

The random field and random anisotropy N -vector models are studied with the functional renormalization group in 4−ε dimensions. The random anisotropy Heisenberg ( N =3) model has a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law 〈 m ( r 1 ) m ( r 2 )〉∼¦ r 1 − r 2 ¦ − 0.62ε . The magnetic susceptibility diverges at low fields as χ∼ H −1+0.15ε . In the random field N -vector model the correlation length is finite at arbitrarily weak disorder for any N >3.The random field and random anisotropy N -vector models are studied with the functional renormalization group in 4−ε dimensions. The random anisotropy Heisenberg ( N =3) model has a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law 〈 m ( r 1 ) m ( r 2 )〉∼¦ r 1 − r 2 ¦ − 0.62ε . The magnetic susceptibility diverges at low fields as χ∼ H −1+0.15ε . In the random field N -vector model the correlation length is finite at arbitrarily weak disorder for any N >3.

arXiv:cond-mat/9812280Journal of Experimental and Theoretical Physics LettersJul 1, 19996 citations

Strongly Anisotropic Scaling

Malte Henkel

Springer eBooksJan 1, 19990 citations

Role of collision anisotropy in two-photon processes in a magnetic field. Phenomenological analysis

G. N. Nikolaev, Sergei G Rautian

It is shown that the nonlinear susceptibility tensor of an isotropic medium is axially symmetric because of the anisotropy of collisions characterized by a symmetry axis along the field wave vectors. In the collinear geometry a two-photon process is described by six functions of the frequencies and wave vectors of the fields, whereas in the case of a mirror-symmetric medium it is described by three such functions. A new difference method is proposed for separating just the collision anisotropy contribution to the signal of a two-photon process in a magnetic field. The method makes it possible to obtain several types of difference signal. In a probe field variant the number of possible signals is four and eight for mirror-symmetric and mirror-asymmetric media, respectively. In a variant involving observation of fluorescence at right-angles to the magnetic field and also parallel to the direction of propagation of the exciting light, the number of possible difference signals is eight and sixteen for mirror-symmetric and mirror-asymmetric media, respectively, whereas in the case of observations along a magnetic field there is only one type of signal. An analysis is made of the parity of the difference resonances relative to reversal of the magnetic field direction.It is shown that the nonlinear susceptibility tensor of an isotropic medium is axially symmetric because of the anisotropy of collisions characterized by a symmetry axis along the field wave vectors. In the collinear geometry a two-photon process is described by six functions of the frequencies and wave vectors of the fields, whereas in the case of a mirror-symmetric medium it is described by three such functions. A new difference method is proposed for separating just the collision anisotropy contribution to the signal of a two-photon process in a magnetic field. The method makes it possible to obtain several types of difference signal. In a probe field variant the number of possible signals is four and eight for mirror-symmetric and mirror-asymmetric media, respectively. In a variant involving observation of fluorescence at right-angles to the magnetic field and also parallel to the direction of propagation of the exciting light, the number of possible difference signals is eight and sixteen for mirror-symmetric and mirror-asymmetric media, respectively, whereas in the case of observations along a magnetic field there is only one type of signal. An analysis is made of the parity of the difference resonances relative to reversal of the magnetic field direction.

Soviet Journal of Quantum ElectronicsMay 31, 19840 citations

g

is the anisotropy parameter, $t=\frac{(T\ensuremath{-}{T}_{c0})}{{T}_{c0}}$, with

{T}_{c0}

and $\ensuremath{\gamma}$ being the iso tropic critical temperature, and susceptibility exponent, while

A

and

B

are model-dependent amplitudes. Analysis of high-temperature series expansions for ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}(T,g)$ as polynomials in

g

for rhombic and axial anisotropy in the fcc, bcc, and simple cubic classical Heisenberg (

n=3

) and classical

XY

and planar-spin models (

N=2

), verifies the scaling with crossover exponents $\ensuremath{\varphi}=1.25\ifmmode\pm\else\textpm\fi{}0.015$ (

n=3

) and $\ensuremath{\varphi}=1.175\ifmmode\pm\else\textpm\fi{}0.015$ (

n=2

). The universality of the scaling functions ${X}_{\ensuremath{\alpha}}(x)$ is demonstrated both for small

x

and in the anisotropic limit $T\ensuremath{\rightarrow}{T}_{c}(g)$, where $x\ensuremath{\rightarrow}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}$; for

n=3

, accurate representations are constructed in the form ${X}_{\ensuremath{\alpha}}(x)\ensuremath{\simeq}{P}_{\ensuremath{\alpha}}(\frac{x}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}}){(1\ensuremath{-}\frac{x}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}})}^{\ensuremath{-}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}}$, where $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}$ is the anisotropic susceptibility exponent, while ${P}_{\ensuremath{\alpha}}(z)$ varies smoothly from ${P}_{\ensuremath{\alpha}}(0)=1$ to ${P}_{\ensuremath{\alpha}}(1)\ensuremath{\simeq}1.10$.