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Journal of Physical Studies

Classical to quantum crossover of the critical behaviour of impure systems

I. P. Takov

Published Jan 1, 1999• 2 citations
Publisher

Abstract

The classical{to{quantum dimensional crossover in a wide class of systems with quenched impurities and inhomogeneities is investigated within the framework of the renormalization group.The one{loop renormalization group equations are derived with the help of a double (; Æ){expansion, where the small parameter Æ describes the quantum eects and = 2 d is the usual expansion parameter (0 < 2; d is the spatial dimensionality).The classical{to{quantum crossover phenomenon is established and the critical exponents are calculated to rst order in and Æ.The analogy between this approach and that of double (; Æ){expansion in treating the extended impurities problem is discussed.

DOI
10.30970/jps.03.422

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The classical{to{quantum dimensional crossover in a wide class of systems with quenched impurities and inhomogeneities is investigated within the framework of the renormalization group.The one{loop renormalization group equations are derived with the help of a double (; Æ){expansion, where the small parameter Æ describes the quantum eects and = 2 d is the usual expansion parameter (0 < 2; d is the spatial dimensionality).The classical{to{quantum crossover phenomenon is established and the critical exponents are calculated to rst order in and Æ.The analogy between this approach and that of double (; Æ){expansion in treating the extended impurities problem is discussed.
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 theoretic renormalization-group approach is applied to the study of short-time critical behavior of the ddd-dimensional spin systems (model A) in the presence of quenched impurities with a long-range correlations decaying as ${r}^{\ensuremath{-}(d\ensuremath{-}\ensuremath{\rho})}$. The asymptotic scaling laws are studied in the frame of a double expansion in $ϵ=4\ensuremath{-}d$ and $\ensuremath{\rho}$ with $\ensuremath{\rho}$ of order ϵϵϵ. In d<4d<4d<4, the initial slip exponents ${\ensuremath{\theta}}^{\ensuremath{'}}$ of the magnetization and $\ensuremath{\theta}$ of the response function, are calculated up to two-loop order. The crossover between fixed points is obtained. The long-time limit of the fluctuation-dissipation ratio is found in the aging regime, and its connection to equilibrium quantities is discussed. The comparison of our results with those of other systems without long-range correlated quenched impurities is also investigated.
The theoretical renormalization-group approach is applied to the study of the short-time critical behavior of the Ginzburg-Landau model with long-range random impurities which have a power-like correlation r—(d—ρ. The system initially at a high temperature is firstly quenched to the critical temperature Tc and then released to an evolution with a model A dynamics. The asymptotic scaling laws are studied in the frame of a double expansion in ϵ = 4 — d and δ = ϵ + ρ with δ of the order of ϵ. The initial slip exponent θ′ for the order parameter is calculated up to the first order in ϵ = 4 — d or in ϵ1/2 corresponding to different fixed points, respectively. For d > 4, the short-time behavior of the order parameter is investigated in one loop. Two different logarithmic and exponential-logarithmic corrections to short-time behaviors of both the autocorrelation and the order parameter are also solved in d = 4 dimension. Crossover between the nonrandom behavior and quenched behavior is found for n = 4 (where n is the spin dimensionality) in d = 4 dimension.
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.
The critical behaviour of a system undergoing a dimensional crossover is investigated. A renormalization group equation is obtained that interpolates between different dimensions. The susceptibility of an Ising-like system is treated in some detail, in particular and effective critical exponent for the susceptibility is computed, which for the particular crossover examined interpolates between the susceptibility exponents appropriate for a (4- in )-dimensional and a (3- in )-dimensional Ising model.