Renormalisation group recursion relations are obtained to two-loop order for an n-component Landau-Ginzburg-Wilson model containing both hypercubic anisotropy and quenched random impurities. The fixed points are enumerated and their stability properties analysed, using the epsilon expansion about four spatial dimensions to second order. The existence of a new fixed point corresponding to an anisotropic impure system is reported. It is found that both isotropic and anisotropic pure systems are unstable to the addition of random impurities for n<n R identical to 4-4 epsilon +O( epsilon 2 ), while both pure and random isotropic systems are unstable to anisotropy for n>n C identical to 4-2 epsilon +O( epsilon 2 ). Singularities in the epsilon expansions for the exponents of the random fixed points are shown to be associated with the onset of focal behaviour (the acquisition of complex eigenvalues). These, together with the slow convergence of the expansions leads one to doubt the reliability of extrapolations to three dimensions.Renormalisation group recursion relations are obtained to two-loop order for an n-component Landau-Ginzburg-Wilson model containing both hypercubic anisotropy and quenched random impurities. The fixed points are enumerated and their stability properties analysed, using the epsilon expansion about four spatial dimensions to second order. The existence of a new fixed point corresponding to an anisotropic impure system is reported. It is found that both isotropic and anisotropic pure systems are unstable to the addition of random impurities for n<n R identical to 4-4 epsilon +O( epsilon 2 ), while both pure and random isotropic systems are unstable to anisotropy for n>n C identical to 4-2 epsilon +O( epsilon 2 ). Singularities in the epsilon expansions for the exponents of the random fixed points are shown to be associated with the onset of focal behaviour (the acquisition of complex eigenvalues). These, together with the slow convergence of the expansions leads one to doubt the reliability of extrapolations to three dimensions.

Journal of Physics A Mathematical and GeneralApr 21, 198721 citations

Critical behaviour of two-component spin systems with quenched random nonmagnetic impurities in cubic anisotropic crystals

Y. Yamazaki, M. Ochiai, Yoshiichi Fukuda

By means of the renormalisation-group approach, static and dynamic critical behaviours are studied for the XY spin system with cubic anisotropy and with quenched random nonmagnetic impurities. The stability of the quenched-random-impurity fixed point in d ( identical to 4- epsilon ) dimensions, the flow diagram of the interactions and the critical exponents are investigated, within the two-loop approximation, in terms of the epsilon 1/2 expansion. It is shown that the transition that appears is a first-order transition.By means of the renormalisation-group approach, static and dynamic critical behaviours are studied for the XY spin system with cubic anisotropy and with quenched random nonmagnetic impurities. The stability of the quenched-random-impurity fixed point in d ( identical to 4- epsilon ) dimensions, the flow diagram of the interactions and the critical exponents are investigated, within the two-loop approximation, in terms of the epsilon 1/2 expansion. It is shown that the transition that appears is a first-order transition.

Journal of Physics C Solid State PhysicsSep 10, 19856 citations

Static and dynamic properties of systems with extended defects: two-loop approximation

Ian D. Lawrie, Vladimir V. Prudnikov

Static and dynamic critical behaviour is studied for spin systems with quenched impurities that are correlated along an epsilon d -dimensional 'line' and randomly distributed in d- epsilon d dimensions. These impurities make the system anisotropic and modify its critical properties. Critical exponents are calculated at the second order of an expansion in epsilon and epsilon d (with epsilon =4-d). Although renormalisation-group functions depend on the ratio epsilon d /( epsilon + epsilon d ), critical exponents are found not to involve this ratio. Pade-like approximants are used to give numerical estimates for the exponents.Static and dynamic critical behaviour is studied for spin systems with quenched impurities that are correlated along an epsilon d -dimensional 'line' and randomly distributed in d- epsilon d dimensions. These impurities make the system anisotropic and modify its critical properties. Critical exponents are calculated at the second order of an expansion in epsilon and epsilon d (with epsilon =4-d). Although renormalisation-group functions depend on the ratio epsilon d /( epsilon + epsilon d ), critical exponents are found not to involve this ratio. Pade-like approximants are used to give numerical estimates for the exponents.

Journal of Physics C Solid State PhysicsApr 10, 198484 citations

Critical behavior of the two-dimensionalN-component Landau-Ginzburg Hamiltonian with cubic anisotropy

Pasquale Calabrese, Alessio Celi

We study the two-dimensional N-component Landau-Ginzburg Hamiltonian with cubic anisotropy. We compute and analyze the fixed-dimension perturbative expansion of the renormalization-group functions to four loops. The relations of these models with N-color Ashkin-Teller models, discrete cubic models, the planar model with fourth-order anisotropy, and the structural phase transition in adsorbed monolayers are discussed. Our results for $N=2$ $(XY$ model with cubic anisotropy) are compatible with the existence of a line of fixed points joining the Ising and the $O(2)$ fixed points. Along this line the exponent $\ensuremath{\eta}$ has the constant value $1/4,$ while the exponent $\ensuremath{\nu}$ runs in a continuous and monotonic way from 1 to $\ensuremath{\infty}$ [from Ising to $O(2)].$ In the four-loop approximation, for $N>~3$ we find a cubic fixed point in the region $u,v>~0.$ We study the two-dimensional N-component Landau-Ginzburg Hamiltonian with cubic anisotropy. We compute and analyze the fixed-dimension perturbative expansion of the renormalization-group functions to four loops. The relations of these models with N-color Ashkin-Teller models, discrete cubic models, the planar model with fourth-order anisotropy, and the structural phase transition in adsorbed monolayers are discussed. Our results for $N=2$ $(XY$ model with cubic anisotropy) are compatible with the existence of a line of fixed points joining the Ising and the $O(2)$ fixed points. Along this line the exponent $\ensuremath{\eta}$ has the constant value $1/4,$ while the exponent $\ensuremath{\nu}$ runs in a continuous and monotonic way from 1 to $\ensuremath{\infty}$ [from Ising to $O(2)].$ In the four-loop approximation, for $N>~3$ we find a cubic fixed point in the region $u,v>~0.$

arXiv:cond-mat/0111118Physical review. B, Condensed matterNov 11, 200217 citations

Critical behaviour of thin films with quenched impurities

L. Craco, L. De Cesare, Ileana Rabuffo +2 more

The critical behaviour of thin films containing quenched random impurities and inhomogeneities is investigated by the renormalization-group method to the one-loop order within the framework of the n-component φ4-model. The finite-size crossover in impure films has been considered on the basis of the fundamental relationship between the effective dimensionality Deff and the characteristic lengths of the system. The fixed points, their stability properties and the critical exponents are obtained and analyzed, using an ε̃=(4−Deff)-expansion near the effective spatial dimensionality Deff of the fluctuation modes in D-dimensional hyperslabs with two types of quenched impurities: point-like impurities with short-range random correlations and extended (linear) impurities with infinite-range random correlations along the small-size spatial direction. The difference between the critical properties of infinite systems and films is demonstrated and investigated. A new critical exponent, describing the scaling properties of the thickness of films with extended impurities has been derived and calculated. A special attention is paid to the critical behaviour of real impure films (D=3).The critical behaviour of thin films containing quenched random impurities and inhomogeneities is investigated by the renormalization-group method to the one-loop order within the framework of the n-component φ4-model. The finite-size crossover in impure films has been considered on the basis of the fundamental relationship between the effective dimensionality Deff and the characteristic lengths of the system. The fixed points, their stability properties and the critical exponents are obtained and analyzed, using an ε̃=(4−Deff)-expansion near the effective spatial dimensionality Deff of the fluctuation modes in D-dimensional hyperslabs with two types of quenched impurities: point-like impurities with short-range random correlations and extended (linear) impurities with infinite-range random correlations along the small-size spatial direction. The difference between the critical properties of infinite systems and films is demonstrated and investigated. A new critical exponent, describing the scaling properties of the thickness of films with extended impurities has been derived and calculated. A special attention is paid to the critical behaviour of real impure films (D=3).

Jan 1, 19997 citations

Randomly dilute spin models with cubic symmetry

Pasquale Calabrese, Andrea Pelissetto C, Ettore Vicari

We study the combined effect of cubic anisotropy and quenched uncorrelated impurities on multicomponent spin models. For this purpose, we consider the field-theoretical approach based on the Ginzburg-Landau-Wilson ${\ensuremath{\varphi}}^{4}$ Hamiltonian with cubic-symmetric quartic interactions and quenched randomness coupled to the local energy density. We compute the renormalization-group functions to six loops in the fixed-dimension $(d=3)$ perturbative scheme. The analysis of such high-order series provides an accurate description of the renormalization-group flow. The results are also used to determine the critical behavior of three-dimensional antiferromagnetic three- and four-state Potts models in the presence of quenched impurities.We study the combined effect of cubic anisotropy and quenched uncorrelated impurities on multicomponent spin models. For this purpose, we consider the field-theoretical approach based on the Ginzburg-Landau-Wilson ${\ensuremath{\varphi}}^{4}$ Hamiltonian with cubic-symmetric quartic interactions and quenched randomness coupled to the local energy density. We compute the renormalization-group functions to six loops in the fixed-dimension $(d=3)$ perturbative scheme. The analysis of such high-order series provides an accurate description of the renormalization-group flow. The results are also used to determine the critical behavior of three-dimensional antiferromagnetic three- and four-state Potts models in the presence of quenched impurities.

arXiv:cond-mat/0202292Physical review. B, Condensed matterJan 22, 200327 citations

N=2

(XY

model with cubic anisotropy) are compatible with the existence of a line of fixed points joining the Ising and the

O(2)

fixed points. Along this line the exponent $\ensuremath{\eta}$ has the constant value

1/4,

while the exponent $\ensuremath{\nu}$ runs in a continuous and monotonic way from 1 to $\ensuremath{\infty}$ [from Ising to

O(2)].

In the four-loop approximation, for

N>~3

we find a cubic fixed point in the region

u,v>~0.

(d=3)

perturbative scheme. The analysis of such high-order series provides an accurate description of the renormalization-group flow. The results are also used to determine the critical behavior of three-dimensional antiferromagnetic three- and four-state Potts models in the presence of quenched impurities.