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arXiv:0803.1587 · International Conference on Path Integrals

A FUNCTIONAL RENORMALIZATION GROUP APPROACH TO SYSTEMS WITH LONG-RANGE CORRELATED DISORDER

Andrei A. Fedorenko

Published Nov 1, 2008• 0 citations
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Abstract

We studied the statics and dynamics of elastic manifolds in disordered media with long-range correlated disorder using functional renormalization group (FRG). We identified different universality classes and computed the critical exponents and universal amplitudes describing geometric and velocity-force characteristics. In contrast to uncorrelated disorder, the statistical tilt symmetry is broken resulting in a nontrivial response to a transverse tilting force. For instance, the vortex lattice in disordered superconductors shows a new glass phase whose properties interpolate between those of the Bragg and Bose glasses formed by pointlike and columnar disorder, respectively. Whereas there is no response in the Bose glass phase (transverse Meissner effect), the standard linear response expected in the Bragg-glass gets modified to a power law response in the presence of disorder correlations. We also studied the long distance properties of the O(N) spin system with random fields and random anisotropies correlated as 1/x^{d-sigma}. Using FRG we obtained the phase diagram in (d,sigma,N)-parameter space and computed the corresponding critical exponents. We found that below the lower critical dimension 4+sigma, there can exist two different types of quasi-long-range-order with zero order-parameter but infinite correlation length.

DOI
10.1142/9789812837271_0070

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We studied the statics and dynamics of elastic manifolds in disordered media with long-range correlated disorder using functional renormalization group (FRG). We identified different universality classes and computed the critical exponents and universal amplitudes describing geometric and velocity-force characteristics. In contrast to uncorrelated disorder, the statistical tilt symmetry is broken resulting in a nontrivial response to a transverse tilting force. For instance, the vortex lattice in disordered superconductors shows a new glass phase whose properties interpolate between those of the Bragg and Bose glasses formed by pointlike and columnar disorder, respectively. Whereas there is no response in the Bose glass phase (transverse Meissner effect), the standard linear response expected in the Bragg-glass gets modified to a power law response in the presence of disorder correlations. We also studied the long distance properties of the O(N) spin system with random fields and random anisotropies correlated as 1/x^{d-sigma}. Using FRG we obtained the phase diagram in (d,sigma,N)-parameter space and computed the corresponding critical exponents. We found that below the lower critical dimension 4+sigma, there can exist two different types of quasi-long-range-order with zero order-parameter but infinite correlation length.
Some physical systems, although dierent by their microscopic nature, in the vicinity of critical points can have similar thermodynamic and structural properties governed by universal power laws.Such systems are grouped into the so-called universality classes.The latter are determined by such global characteristics as the type of interaction, space dimensionality, symmetry, and the number of components of the order parameter.Therefore, it is interesting to investigate how the mutual inuence of these characteristics can aect the critical features of the system.The goal of our research is to analyze the changes in the critical behavior of a many-particle magnetic system under the inuence of two competing factors: long-range interaction and weak structural disorder.Using the example of an n-vector model in a d-dimensional space, we will investigate ferromagnetic ordering in a structurally disordered magnet with the long-range interaction decaying with the distance x as J(x) ∼ x -d-σ , with the control parameter σ.The analysis is carried out using the eld-theoretical renormalization group methods, which makes it possible to identify the universality classes of the system under consideration, their change with global parameters, and to determine the universal characteristics of critical behavior (critical exponents and marginal dimensions).It will be shown that there exists such a region of parameters (d, n, σ), where the long-range interaction and structural disorder lead to a synergistic eect and the appearance of a new, random long-range universality class.We obtain the renormalization group functions in the three-loop approximation, and calculate the correlation length critical exponent ν(ϵ ′ , n) as a perturbation theory series in ϵ ′ = 2σ -d.Quantitative estimates are obtained using asymptotic series resummation methods.
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.
We apply the Gaussian variational method (GVM) to study the equilibrium statistical mechanics of two related systems: (i) classical elastic manifolds, such as flux lattices, in the presence of columnar disorder correlated along the \ensuremath{\tau} direction, and (ii) interacting quantum particles in a static random potential. We find localization by disorder, the localized phase being described by a replica-symmetry-broken solution confined to the mode \ensuremath{\omega}=0. For classical systems we compute the correlation function of relative displacements. In d=2+1, in the absence of dislocations, the GVM allows one to describe the Bose glass phase. Along the columns the displacements saturate at a length ${\mathit{l}}_{\mathrm{\ensuremath{\perp}}}$, indicating flux-line localization. Perpendicularly to the columns long-range order is destroyed. We find a divergent tilt modulus c44{\mathit{c}}_{44}c44​=\ensuremath{\infty} and a x\ensuremath{\sim}${\mathrm{\ensuremath{\tau}}}^{1/2}$ scaling. Quantum systems are studied using the analytic continuation from imaginary to real time \ensuremath{\tau}\ensuremath{\rightarrow}it. We compute the conductivity and find that it behaves at small frequency as \ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\approxeq}${\mathrm{\ensuremath{\omega}}}^{2}$ in all dimensions (d4) for which disorder is relevant. We compute the quantum localization length \ensuremath{\xi}. In d=1, where the model also describes interacting fermions in a static random potential, we find a delocalization transition and obtain analytically both the low- and high-frequency behavior of the conductivity for any value of the interaction. We show that the marginality condition appears as the condition to obtain the correct physical behavior. Agreement with renormalization group results is found whenever it can be compared. \textcopyright{} 1996 The American Physical Society.
A renormalization-group treatment is used for investigating the critical properties of d-dimensional systems with anisotropic quenched disorder totally correlated in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{d}}$ dimensions and randomly distributed in the remaining d-${\mathrm{\ensuremath{\epsilon}}}_{\mathit{d}}$ space directions (d=4-\ensuremath{\epsilon}) with long-range correlation which decays with the distance as a power law involving a characteristic parameter \ensuremath{\sigma}. Within a triple (\ensuremath{\epsilon},${\mathrm{\ensuremath{\epsilon}}}_{\mathit{d}}$,\ensuremath{\sigma}) expansion it is shown that the recursion relations possess a novel focus-type random fixed point leading to new anisotropic critical properties. Under certain conditions, different critical regimes and crossovers are found to occur, consistent with the modified version of the Harris criterion obtained for the random problem under study.
The large-distance behaviors of the random field and random anisotropy O(N)O(N)O(N) models are studied with the functional renormalization group in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. The random anisotropy Heisenberg (N=3)(N=3)(N=3) model is found to have a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law $〈\mathbf{m}({\mathbf{r}}_{1})\mathbf{m}({\mathbf{r}}_{2})〉\ensuremath{\sim}|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}{|}^{\ensuremath{-}0.62\ensuremath{\epsilon}}.$ The magnetic susceptibility diverges at low fields as $\ensuremath{\chi}\ensuremath{\sim}{H}^{\ensuremath{-}1+0.15\ensuremath{\epsilon}}.$ In the random field O(N)O(N)O(N) model the correlation length is found to be finite at the arbitrarily weak disorder for any N>3.N>3.N>3. The random field case is studied with a simple method, based on a rigorous inequality. This approach allows one to avoid the integration of the functional renormalization-group equations.
Abstract Criticality in the class of disordered systems comprising the random-field Ising model (RFIM) and elastic manifolds in a random environment is controlled by zero-temperature fixed points that must be treated through a functional renormalization group (RG). We apply the nonperturbative functional RG approach that we have previously used to describe the RFIM in and out of equilibrium (Balog et al 2018 Phys. Rev . B 97 094204) to the simpler and by now well-studied case of the random elastic manifold model. We recover the main known properties, critical exponents and scaling functions, of both the pinned phase of the manifold at equilibrium and the depinning threshold in the athermally and quasi-statically driven case for any dimension . This successful benchmarking of our theoretical approach gives strong support to the results that we have previously obtained for the RFIM, in particular concerning the distinct universality classes of the equilibrium and out-of-equilibrium (hysteresis) critical points below a critical dimension .