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Journal of Physics A Mathematical and General

Influence of long-range correlated impurities upon the phase transition in model superconductors

Elka Korutcheva, YONKO T. MILLEV

Published Jun 21, 1984• 2 citations
Publisher

Abstract

Direct renormalisation group epsilon -analysis yields a fixed point which is stable over a domain in a specific parameter connected with the long-range character of the correlations. The critical exponents are calculated to first order in epsilon . The existence of a stable fixed point is interpreted as signalling a second-order phase transition near four dimensions of space.

DOI
10.1088/0305-4470/17/9/011

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Direct renormalisation group epsilon -analysis yields a fixed point which is stable over a domain in a specific parameter connected with the long-range character of the correlations. The critical exponents are calculated to first order in epsilon . The existence of a stable fixed point is interpreted as signalling a second-order phase transition near four dimensions of space.
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The application of renormalisation group recursion relations to the calculation of critical exponents in d dimensions to second order in epsilon =4-d is discussed. It is shown that in the limit of a large scale factor b, the fixed point value, u* of the four-spin coupling constant is correctly given to second order by a renormalisation group equation in which all irrelevant variables, mass renormalisation graphs, and mass terms in the propagators are ignored. Related approximations in the calculation of the critical exponents are justified. The unexpected equality to order epsilon 2 of u* and u 0c , the Feynman graph coupling constant, is discussed.
The critical properties of a random Ising model with long-range isotropic interactions decaying as R -(d+ sigma ) are analysed by using renormalisation group methods in an expansion in epsilon '=2 sigma -d, where d is dimensionality. For epsilon '>O the critical behaviour is described by a stable fixed point O( epsilon 1 2/') in the physical region of parameter space. The crossover to short-range behaviour is analysed and takes place when sigma =2+ epsilon /106, epsilon =4-d>O. There is then a small region O< sigma -2<O( epsilon ) where the system is still dominated by long-range behaviour. The results for d=1 are compared with those obtained previously for the random hierarchical model and it is found that the two models do not show the same critical behaviour in the epsilon ' expansion.
The numerical renormalization group method is used to investigate zero-temperature phase transitions in quantum impurity systems, in particular in the particle–hole symmetric soft-gap Anderson model. The model displays two stable phases whose fixed points can be built up of non-interacting single-particle states. In contrast, the quantum phase transitions turn out to be described by interacting fixed points, and their excitations cannot be described in terms of free particles. We show that the structure of the many-body spectrum of these critical fixed points can be understood using renormalized perturbation theory close to certain values of the bath exponents which play the role of critical dimensions. Contact is made with perturbative renormalization group calculations for the soft-gap Anderson and Kondo models. A complete description of the quantum critical many-particle spectra is achieved using suitable marginal operators; technically this can be understood as epsilon-expansion for full many-body spectra.
Tricritical scaling is studied in a single-component Landau-Ginzburg-Wilson model of spatial dimensionality d=4- epsilon . At first order in epsilon , a single expression for the order-parameter susceptibility is exhibited, which behaves correctly in all limits and is uniformly applicable throughout the tricritical region. This is achieved by means of a renormalised group scheme in which couplings depend explicitly on the order parameter and all three loci of ordinary critical points are controlled by a single fixed point.
We study the critical behavior of the q-state Potts model with long-range interactions decaying asymptotically as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d+\ensuremath{\sigma})}$ in the presence of random-Tc{T}_{c}Tc​ impurities correlated over large distances such that the correlations fall off as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}a)}$, a<d. We find that the renormalization-group scaling equations have a new fixed point in the appropriate double \ensuremath{\epsilon}, x expansion, where \ensuremath{\epsilon}=3\ensuremath{\sigma}-d and x=a-\ensuremath{\sigma}. This fixed point, however, is never both stable and physical.