Physical Review Letters

Critical Behavior of Anisotropic Cubic Systems in the Limit of Infinite Spin Dimensionality

Amnon Aharony A

Published Dec 17, 1973• 52 citations

Abstract

The zero-field critical behavior above ${T}_{c}$ of ferromagnets (or ferroelectrics) with a Hamiltonian of cubic symmetry is studied, in $d$ space dimensions, in the limit of infinite spin dimensionality, $n\ensuremath{\rightarrow}\ensuremath{\infty}$, with the cubic term of order unity, but the usual ${S}^{4}$ term of order ${n}^{\ensuremath{-}1}$. A diagrammatic expansion about an Ising model (instead of the normal Gaussian model) is used. Characteristic cubic behavior is discovered, with renormalized Ising-model exponents (in the Fisher sense).

DOI: 10.1103/physrevlett.31.1494

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Critical Behavior of Anisotropic Cubic Systems

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Short-time critical behavior of anisotropic cubic systems

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Physical Review Letters

Critical Behavior of Anisotropic Cubic Systems in the Limit of Infinite Spin Dimensionality

Amnon Aharony A

Published Dec 17, 1973• 52 citations

Publisher

Abstract

The zero-field critical behavior above

{T}_{c}

of ferromagnets (or ferroelectrics) with a Hamiltonian of cubic symmetry is studied, in

d

space dimensions, in the limit of infinite spin dimensionality,

n\ensuremath{\rightarrow}\ensuremath{\infty}

, with the cubic term of order unity, but the usual

{S}^{4}

term of order

{n}^{\ensuremath{-}1}

. A diagrammatic expansion about an Ising model (instead of the normal Gaussian model) is used. Characteristic cubic behavior is discovered, with renormalized Ising-model exponents (in the Fisher sense).

DOI: 10.1103/physrevlett.31.1494

Related papers

Critical Behavior of Anisotropic Cubic Systems

Amnon Aharony A

The critical behavior in zero field above ${T}_{c}$ of ferromagnets or ferroelectrics with a Hamiltonian of cubic symmetry is studied, to order ${\ensuremath{\epsilon}}^{2}$, by exact renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with $n$ -component spins. For $\ensuremath{\epsilon}=1$, $n\ensuremath{\ge}3$, a crossover from isotropic (Heisenberg) to characteristic cubic behavior occurs, with the new value $2{v}^{C}=1+[\frac{(n\ensuremath{-}1)}{3n}]\ensuremath{\epsilon}+[\frac{(n\ensuremath{-}1)}{324{n}^{3}}](17{n}^{2}+290n\ensuremath{-}424){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, and cubic symmetry appearing in the four-spin correlation function. Experiments on structural phase transitions are considered briefly.The critical behavior in zero field above ${T}_{c}$ of ferromagnets or ferroelectrics with a Hamiltonian of cubic symmetry is studied, to order ${\ensuremath{\epsilon}}^{2}$, by exact renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with $n$-component spins. For $\ensuremath{\epsilon}=1$, $n\ensuremath{\ge}3$, a crossover from isotropic (Heisenberg) to characteristic cubic behavior occurs, with the new value $2{v}^{C}=1+[\frac{(n\ensuremath{-}1)}{3n}]\ensuremath{\epsilon}+[\frac{(n\ensuremath{-}1)}{324{n}^{3}}](17{n}^{2}+290n\ensuremath{-}424){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, and cubic symmetry appearing in the four-spin correlation function. Experiments on structural phase transitions are considered briefly.

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Investigation of surface critical behavior of semi-infinite systems with cubic anisotropy

Z. Usatenko

The critical behavior at the special surface transition and crossover bevavior from special to ordinary surface transition in semi-infinite n-component anisotropic cubic models are investigated by applying the field theoretic approach directly in d=3 dimensions up to the two-loop approximation. The crossover behavior for random semi-infinite Ising-like system, which is the nontrivial particular case of the cubic model in the limit $n\to 0$ , is also investigated. The numerical estimates of the resulting two-loop series expansions for the critical exponents of the special surface transition, surface crossover critical exponent $Φ$ and the surface critical exponents of the layer, $α_{1}$ , and local specific heats, $α_{11}$ , are computed by means of Pade and Pade-Borel resummation techniques. For $nn_{c}$ the cubic fixed point is stable, where $n_{c}$ is the marginal spin dimensionality of the cubic model. The obtained results indicate that the surface critical behavior of semi-infinite systems with cubic anisotropy is characterized by new set of surface critical exponents for $n>n_{c}$ .The critical behavior at the special surface transition and crossover bevavior from special to ordinary surface transition in semi-infinite n-component anisotropic cubic models are investigated by applying the field theoretic approach directly in d=3 dimensions up to the two-loop approximation. The crossover behavior for random semi-infinite Ising-like system, which is the nontrivial particular case of the cubic model in the limit $n\to 0$, is also investigated. The numerical estimates of the resulting two-loop series expansions for the critical exponents of the special surface transition, surface crossover critical exponent $Φ$ and the surface critical exponents of the layer, $α_{1}$, and local specific heats, $α_{11}$, are computed by means of Pade and Pade-Borel resummation techniques. For $nn_{c}$ the cubic fixed point is stable, where $n_{c}$ is the marginal spin dimensionality of the cubic model. The obtained results indicate that the surface critical behavior of semi-infinite systems with cubic anisotropy is characterized by new set of surface critical exponents for $n>n_{c}$.

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The renormalization group approach is applied to the study of the short-time critical behavior of the d-dimensional n-component spin systems with cubic anisotropy. First, the system is quenched from high temperature to the critical temperature and then relaxes to equilibrium within model A dynamics. The asymptotic scaling laws and the initial slip exponents ${\ensuremath{\theta}}^{\ensuremath{'}}$ and $\ensuremath{\theta}$ of the order parameter and the response function, respectively, are calculated to the second order in $\ensuremath{\epsilon}=4\ensuremath{-}d.$ Logarithmic corrections to short-time behaviors of both the autocorrelation and the order parameter are found in $d=4$ dimension.The renormalization group approach is applied to the study of the short-time critical behavior of the d-dimensional n-component spin systems with cubic anisotropy. First, the system is quenched from high temperature to the critical temperature and then relaxes to equilibrium within model A dynamics. The asymptotic scaling laws and the initial slip exponents ${\ensuremath{\theta}}^{\ensuremath{'}}$ and $\ensuremath{\theta}$ of the order parameter and the response function, respectively, are calculated to the second order in $\ensuremath{\epsilon}=4\ensuremath{-}d.$ Logarithmic corrections to short-time behaviors of both the autocorrelation and the order parameter are found in $d=4$ dimension.

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Critical Behavior of Magnets with Dipolar Interactions. V. Uniaxial Magnets in d Dimensions

Amnon Aharony A

The exact renormalization-group approach is used to study the critical behavior for $T>{T}_{c}$ , $H=0$ of a uniaxial ferromagnetic (or ferroelectric) system in $d$ dimensions, with exchange and dipolar interactions between the (single-component) spins. Normal Ising-like behavior is retained for $t=\frac{T}{{T}_{c}}\ensuremath{-}1\ensuremath{\gg}\stackrel{^}{g}=\frac{{(g{\ensuremath{\mu}}_{B})}^{2}}{J{a}^{d}}$, where $J$ is the exchange parameter, $g{\ensuremath{\mu}}_{B}$ is the magnetic moment per spin, and $a$ is the lattice spacing. Crossover to a characteristic dipolar behavior occurs when ${t}^{\ensuremath{\varphi}}\ensuremath{\approx}\stackrel{^}{g}$, where $\ensuremath{\varphi}=1+\frac{\ensuremath{\epsilon}}{6}$ (to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$). For $t\ensuremath{\ll}\stackrel{^}{g}$, the leading temperature singularity in the Fourier transform of the spin-spin correlation function $\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ becomes ${\ensuremath{\xi}}^{2}\ifmmode\times\else\texttimes\fi{}{[1+{(\ensuremath{\xi}q)}^{2}\ensuremath{-}{h}_{0}{(\ensuremath{\xi}{q}^{z})}^{2}+{g}_{0}{(\frac{{q}^{z}}{q})}^{2}]}^{\ensuremath{-}1}$, where ${h}_{0}$ and ${g}_{0}$ are of order $\stackrel{^}{g}$ and $\ensuremath{\xi}(t)$ varies as ${t}^{\ensuremath{-}\frac{1}{2}}$ for $d>3$ , as ${t}^{\ensuremath{-}\frac{1}{2}}{|\mathrm{ln}t|}^{\frac{1}{6}}$ for $d=3$ , and as ${t}^{\ensuremath{-}\ensuremath{\nu}}$ with $\frac{1}{2\ensuremath{\nu}}=1\ensuremath{-}\frac{(3\ensuremath{-}d)}{6}+O({(3\ensuremath{-}d)}^{2})$ for $d<3$ . The susceptibility displays the expected demagnetization effects, namely, $({\ensuremath{\chi}}^{\ensuremath{-}1}\ensuremath{-}{g}_{0}\mathcal{D})\ensuremath{\propto}{\ensuremath{\xi}}^{\ensuremath{-}2}$. The experimental situation is mentioned briefly.The exact renormalization-group approach is used to study the critical behavior for $T>{T}_{c}$, $H=0$ of a uniaxial ferromagnetic (or ferroelectric) system in $d$ dimensions, with exchange and dipolar interactions between the (single-component) spins. Normal Ising-like behavior is retained for $t=\frac{T}{{T}_{c}}\ensuremath{-}1\ensuremath{\gg}\stackrel{^}{g}=\frac{{(g{\ensuremath{\mu}}_{B})}^{2}}{J{a}^{d}}$, where $J$ is the exchange parameter, $g{\ensuremath{\mu}}_{B}$ is the magnetic moment per spin, and $a$ is the lattice spacing. Crossover to a characteristic dipolar behavior occurs when ${t}^{\ensuremath{\varphi}}\ensuremath{\approx}\stackrel{^}{g}$, where $\ensuremath{\varphi}=1+\frac{\ensuremath{\epsilon}}{6}$ (to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$). For $t\ensuremath{\ll}\stackrel{^}{g}$, the leading temperature singularity in the Fourier transform of the spin-spin correlation function $\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ becomes ${\ensuremath{\xi}}^{2}\ifmmode\times\else\texttimes\fi{}{[1+{(\ensuremath{\xi}q)}^{2}\ensuremath{-}{h}_{0}{(\ensuremath{\xi}{q}^{z})}^{2}+{g}_{0}{(\frac{{q}^{z}}{q})}^{2}]}^{\ensuremath{-}1}$, where ${h}_{0}$ and ${g}_{0}$ are of order $\stackrel{^}{g}$ and $\ensuremath{\xi}(t)$ varies as ${t}^{\ensuremath{-}\frac{1}{2}}$ for $d>3$, as ${t}^{\ensuremath{-}\frac{1}{2}}{|\mathrm{ln}t|}^{\frac{1}{6}}$ for $d=3$, and as ${t}^{\ensuremath{-}\ensuremath{\nu}}$ with $\frac{1}{2\ensuremath{\nu}}=1\ensuremath{-}\frac{(3\ensuremath{-}d)}{6}+O({(3\ensuremath{-}d)}^{2})$ for $d<3$. The susceptibility displays the expected demagnetization effects, namely, $({\ensuremath{\chi}}^{\ensuremath{-}1}\ensuremath{-}{g}_{0}\mathcal{D})\ensuremath{\propto}{\ensuremath{\xi}}^{\ensuremath{-}2}$. The experimental situation is mentioned briefly.

Physical review. B, Solid stateOct 1, 1973228 citations

Surface critical behaviour of semi-infinite systems with cubic anisotropy at the ordinary transition

Z Usatenko, Józef Spałek

The critical behaviour at the ordinary transition in semi-infinite n -component anisotropic cubic models is investigated by applying the field theoretic approach in d = 3 dimensions up to the two-loop approximation. Numerical estimates of the resulting two-loop series expansions for the critical exponents of the ordinary transition are computed by means of Padé resummation techniques. For n < n c the system belongs to the universality class of the isotropic n -component model, while for n > n c the cubic fixed point becomes stable, where n c < 3 is the marginal spin dimensionality of the cubic model. The obtained results indicate that the surface critical behaviour of the semi-infinite systems with cubic anisotropy is characterized by a new set of surface critical exponents for n > n c .The critical behaviour at the ordinary transition in semi-infinite n -component anisotropic cubic models is investigated by applying the field theoretic approach in d = 3 dimensions up to the two-loop approximation. Numerical estimates of the resulting two-loop series expansions for the critical exponents of the ordinary transition are computed by means of Padé resummation techniques. For n < n c the system belongs to the universality class of the isotropic n -component model, while for n > n c the cubic fixed point becomes stable, where n c < 3 is the marginal spin dimensionality of the cubic model. The obtained results indicate that the surface critical behaviour of the semi-infinite systems with cubic anisotropy is characterized by a new set of surface critical exponents for n > n c .

arXiv:cond-mat/0312320Journal of Physics A Mathematical and GeneralJul 1, 20043 citations

Phase transition in the anisotropic cubic model

Yurij V. Holovatch

Ferroelectrics Letters SectionDec 1, 19872 citations

The zero-field critical behavior above

{T}_{c}

of ferromagnets (or ferroelectrics) with a Hamiltonian of cubic symmetry is studied, in

d

space dimensions, in the limit of infinite spin dimensionality,

n\ensuremath{\rightarrow}\ensuremath{\infty}

, with the cubic term of order unity, but the usual

{S}^{4}

term of order

{n}^{\ensuremath{-}1}

The critical behavior in zero field above

{T}_{c}

of ferromagnets or ferroelectrics with a Hamiltonian of cubic symmetry is studied, to order ${\ensuremath{\epsilon}}^{2}$, by exact renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with

n

-component spins. For $\ensuremath{\epsilon}=1$, $n\ensuremath{\ge}3$, a crossover from isotropic (Heisenberg) to characteristic cubic behavior occurs, with the new value $2{v}^{C}=1+[\frac{(n\ensuremath{-}1)}{3n}]\ensuremath{\epsilon}+[\frac{(n\ensuremath{-}1)}{324{n}^{3}}](17{n}^{2}+290n\ensuremath{-}424){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, and cubic symmetry appearing in the four-spin correlation function. Experiments on structural phase transitions are considered briefly.

n\to 0

, is also investigated. The numerical estimates of the resulting two-loop series expansions for the critical exponents of the special surface transition, surface crossover critical exponent

Φ

and the surface critical exponents of the layer,

α_{1}

, and local specific heats,

α_{11}

, are computed by means of Pade and Pade-Borel resummation techniques. For

nn_{c}

the cubic fixed point is stable, where

n_{c}

is the marginal spin dimensionality of the cubic model. The obtained results indicate that the surface critical behavior of semi-infinite systems with cubic anisotropy is characterized by new set of surface critical exponents for

n>n_{c}

d=4

dimension.

The exact renormalization-group approach is used to study the critical behavior for

T>{T}_{c}

H=0

of a uniaxial ferromagnetic (or ferroelectric) system in

d

dimensions, with exchange and dipolar interactions between the (single-component) spins. Normal Ising-like behavior is retained for $t=\frac{T}{{T}_{c}}\ensuremath{-}1\ensuremath{\gg}\stackrel{^}{g}=\frac{{(g{\ensuremath{\mu}}_{B})}^{2}}{J{a}^{d}}$, where

J

is the exchange parameter, $g{\ensuremath{\mu}}_{B}$ is the magnetic moment per spin, and

a

is the lattice spacing. Crossover to a characteristic dipolar behavior occurs when ${t}^{\ensuremath{\varphi}}\ensuremath{\approx}\stackrel{^}{g}$, where $\ensuremath{\varphi}=1+\frac{\ensuremath{\epsilon}}{6}$ (to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$). For $t\ensuremath{\ll}\stackrel{^}{g}$, the leading temperature singularity in the Fourier transform of the spin-spin correlation function $\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ becomes ${\ensuremath{\xi}}^{2}\ifmmode\times\else\texttimes\fi{}{[1+{(\ensuremath{\xi}q)}^{2}\ensuremath{-}{h}_{0}{(\ensuremath{\xi}{q}^{z})}^{2}+{g}_{0}{(\frac{{q}^{z}}{q})}^{2}]}^{\ensuremath{-}1}$, where

{h}_{0}

and

{g}_{0}

are of order $\stackrel{^}{g}$ and $\ensuremath{\xi}(t)$ varies as ${t}^{\ensuremath{-}\frac{1}{2}}$ for

d>3

, as ${t}^{\ensuremath{-}\frac{1}{2}}{|\mathrm{ln}t|}^{\frac{1}{6}}$ for

d=3

, and as ${t}^{\ensuremath{-}\ensuremath{\nu}}$ with $\frac{1}{2\ensuremath{\nu}}=1\ensuremath{-}\frac{(3\ensuremath{-}d)}{6}+O({(3\ensuremath{-}d)}^{2})$ for

d<3

. The susceptibility displays the expected demagnetization effects, namely, $({\ensuremath{\chi}}^{\ensuremath{-}1}\ensuremath{-}{g}_{0}\mathcal{D})\ensuremath{\propto}{\ensuremath{\xi}}^{\ensuremath{-}2}$. The experimental situation is mentioned briefly.