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Journal of Physics A Mathematical and General

On the critical behaviour of bosonised models in a long-range correlated random field

Gaetano Busiello, L. De Cesare, Ileana Rabuffo

Published Jul 11, 1985• 2 citations
Publisher

Abstract

The renormalisation group approach is used for exploring the critical behaviour of bosonised models in the presence of a long-range correlated random field. The zero-temperature regime is studied in some detail for different types of random field correlation functions. As a result, the random critical exponents are not derivable from the pure ones with appropriate dimensional shifts.

DOI
10.1088/0305-4470/18/10/032

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The renormalisation group approach is used for exploring the critical behaviour of bosonised models in the presence of a long-range correlated random field. The zero-temperature regime is studied in some detail for different types of random field correlation functions. As a result, the random critical exponents are not derivable from the pure ones with appropriate dimensional shifts.
The critical properties of a ddd-dimensional Bose model in the presence of a random quenched field are investigated by renormalization-group techniques via the replica trick. To first order in $\ensuremath{\epsilon}=6\ensuremath{-}d$ it is found that both at nonzero- and zero-temperature regimes the grand-canonical critical exponents are the same as those of the corresponding classical model so that thermal and quantum fluctuations appear to be irrelevant. Furthermore, at zero temperature, the scaling laws already established for the pure model are retrieved with the replacement of ddd with $d\ensuremath{-}4$, but the random critical exponents are not derivable from the pure ones with the dimensional shift $d\ensuremath{\rightarrow}d\ensuremath{-}4$.
We study the critical behaviour of the 1D random field Ising model (RFIM) with long-range interactions (∝ r −( d + σ ) ) by the nonperturbative functional renormalization group. We find two distinct regimes of critical behaviour as a function of σ , separated by a critical value σ c . What distinguishes these two regimes is the presence or not of a cusp-like nonanalyticity in the functional dependence of the renormalized cumulants of the random field at the fixed point. This change of behaviour can be associated with the characteristics of the large-scale avalanches present in the system at zero temperature. We propose some ways to check these predictions through lattice simulations. We also discuss the differences with the RFIM on the Dyson hierarchical lattice.
Abstract The effect of quenched random fields on the zero‐temperature critical behaviour is studied by the means of the Wilson‐Fisher renormalisation‐group recursion relations to first order in the ϵ‐expansion. A number of quantum models is considered. The simultaneous influence of both, the random‐field and the classical‐to‐quantum dimensional crossover phenomena on the quantum critical behaviour is investigated. A special attention is paid to the properties of some reasonable particular forms of Gaussian random correlations which might be of relevance to the description of natural systems.
We study the long-distance behavior of the O(N)O(N)O(N) model in the presence of random fields and random anisotropies correlated as $\ensuremath{\sim}1∕{x}^{d\ensuremath{-}\ensuremath{\sigma}}$ for large separation xxx using the functional renormalization group. We compute the fixed points and analyze their regions of stability within a double $\ensuremath{\epsilon}=d\ensuremath{-}4$ and $\ensuremath{\sigma}$ expansion. We find that the long-range disorder correlator remains analytic but generates short-range disorder whose correlator develops the usual cusp. This allows us to obtain the phase diagrams in $(d,\ensuremath{\sigma},N)$ parameter space and compute the critical exponents to first order in $\ensuremath{\epsilon}$ and $\ensuremath{\sigma}$. We show that the standard renormalization group methods with a finite number of couplings used in previous studies of systems with long-range correlated random fields fail to capture all critical properties. We argue that our results may be relevant to the behavior of $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{A}$ in aerogel.
A variety of phenomena is observed experimentally in random-field (RF) systems realized by the application of an external field to diluted antiferromagnets. At low temperatures, infinitely long hysteretic effects are manifested by the history dependence of the final states: long-range order is observed if the field is applied after cooling, while domain states are reached when field cooled. While no indications for critical fluctuations are detected in 2-D systems, scaling behavior, for both the correlation length and the specific heat, is observed in 3-D systems over an intermediate range of temperatures. The related critical properties seem to be well described by the corresponding ones in the 2-D pure Ising model. The renormalization-group approach, which yields for the equilibrium critical exponents their values of the pure model in d−2 dimensions, is reviewed. A generalization of the dimensional-reduction approach, which accounts self-consistently for renormalized responses of the RF system, is presented. The dynamical effects are implicitly incorporated through the variation in the critical response between the local and the global regimes, associated with short- and long-time scales, respectively. In both regimes the lower critical dimension is found to be d=2 in accordance with stability arguments. The short-time critical behavior indicates a dimensional reduction by one for the 3-D thermal exponents, in agreement with the experimental results.
We study the critical behavior of the q-state Potts model in the presence of quenched random fields, which locally prefer ordering of any one of the q states, and are long-range correlated such that [H(0)H(R${)]}_{\mathrm{av}\mathrm{\ensuremath{\sim}}{R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}\ensuremath{\theta})}}$. We find a stable critical behavior controlled by a new random fixed point, and obtain a set of critical exponents within the \ensuremath{\epsilon}\ifmmode \tilde{}\else \~{}\fi{} expansion (\ensuremath{\epsilon}\ifmmode \tilde{}\else \~{}\fi{}=8+\ensuremath{\theta}-d), which cannot be related to those of a pure Potts model of lower dimensionality.