Journal ArticleDOI

# Disordered elastic systems and one-dimensional interfaces

01 Jun 2012-Physica B-condensed Matter (North-Holland)-Vol. 407, Iss: 11, pp 1725-1733

AbstractWe briefly introduce the generic framework of disordered elastic systems (DES), giving a short ‘recipe’ of a DES modeling and presenting the quantities of interest in order to probe the static and dynamical disorder-induced properties of such systems. We then focus on a particular low-dimensional DES, namely the one-dimensional interface in short-ranged elasticity and short-ranged quenched disorder. Illustrating different elements given in the introductory sections, we discuss specifically the consequences of the interplay between a finite temperature T > 0 and a finite interface width ξ > 0 on the static geometrical fluctuations at different lengthscales, and the implications on the quasistatic dynamics.

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### Cites background from "Disordered elastic systems and one-..."

• ...(a) Interface velocity v as a function of the driving force F , adapted from [29]....

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• ...In the case of more general disorder, it may also be necessary to consider all the n-th order moments of the probability distribution function (PDF) of relative displacements ∆u(r), which reflect the characteristic scaling properties of the system [29]:...

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• ...(d) The roughness B(r) of the interface grows with the length scale r, adapted from [29]....

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• ...(a) Relative displacement ∆u(r) for r = z2 − z1 of a one-dimensional interface from its flat, elastically optimal configuration under the influence of weak collective pinning in a disorder potential, adapted from [29] (b) Random bond disorder preserves the symmetry of the double-well potential, while locally changing its depth....

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##### References
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Book ChapterDOI

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01 Jan 2012

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### "Disordered elastic systems and one-..." refers background in this paper

• ...the imaging of the imbibition line of a fluid on a disordered substrate [26], of a crack front along an heterogeneous weak plane [27] by ultra-fast CCD camera, or of avalanches in ferromagnetic thin films [5]....

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Abstract: With the high-temperature superconductors a qualitatively new regime in the phenomenology of type-II superconductivity can be accessed. The key elements governing the statistical mechanics and the dynamics of the vortex system are (dynamic) thermal and quantum fluctuations and (static) quenched disorder. The importance of these three sources of disorder can be quantified by the Ginzburg number $Gi=\frac{{(\frac{{T}_{c}}{{H}_{c}^{2}}\ensuremath{\varepsilon}{\ensuremath{\xi}}^{3})}^{2}}{2}$, the quantum resistance $Qu=(\frac{{e}^{2}}{\ensuremath{\hbar}})(\frac{{\ensuremath{\rho}}_{n}}{\ensuremath{\varepsilon}\ensuremath{\xi}})$, and the critical current-density ratio $\frac{{j}_{c}}{{j}_{o}}$, with ${j}_{c}$ and ${j}_{o}$ denoting the depinning and depairing current densities, respectively (${\ensuremath{\rho}}_{n}$ is the normal-state resistivity and ${\ensuremath{\varepsilon}}^{2}=\frac{m}{M}l1$ denotes the anisotropy parameter). The material parameters of the oxides conspire to produce a large Ginzburg number $\mathrm{Gi}\ensuremath{\sim}{10}^{\ensuremath{-}2}$ and a large quantum resistance $\mathrm{Qu}\ensuremath{\sim}{10}^{\ensuremath{-}1}$, values which are by orders of magnitude larger than in conventional superconductors, leading to interesting effects such as the melting of the vortex lattice, the creation of new vortex-liquid phases, and the appearance of macroscopic quantum phenomena. Introducing quenched disorder into the system turns the Abrikosov lattice into a vortex glass, whereas the vortex liquid remains a liquid. The terms "glass" and "liquid" are defined in a dynamic sense, with a sublinear response $\ensuremath{\rho}={\frac{\ensuremath{\partial}E}{\ensuremath{\partial}j}|}_{j\ensuremath{\rightarrow}0}$ characterizing the truly superconducting vortex glass and a finite resistivity $\ensuremath{\rho}(j\ensuremath{\rightarrow}0)g0$ being the signature of the liquid phase. The smallness of $\frac{{j}_{c}}{{j}_{o}}$ allows one to discuss the influence of quenched disorder in terms of the weak collective pinning theory. Supplementing the traditional theory of weak collective pinning to take into account thermal and quantum fluctuations, as well as the new scaling concepts for elastic media subject to a random potential, this modern version of the weak collective pinning theory consistently accounts for a large number of novel phenomena, such as the broad resistive transition, thermally assisted flux flow, giant and quantum creep, and the glassiness of the solid state. The strong layering of the oxides introduces additional new features into the thermodynamic phase diagram, such as a layer decoupling transition, and modifies the mechanism of pinning and creep in various ways. The presence of strong (correlated) disorder in the form of twin boundaries or columnar defects not only is technologically relevant but also provides the framework for the physical realization of novel thermodynamic phases such as the Bose glass. On a macroscopic scale the vortex system exhibits self-organized criticality, with both the spatial and the temporal scale accessible to experimental investigations.

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Abstract: This book contains a detailed and self-contained presentation of the replica theory of infinite range spin glasses. The authors also explain recent theoretical developments, paying particular attention to new applications in the study of optimization theory and neural networks. About two-thirds of the book are a collection of the most interesting and pedagogical articles on the subject.

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### "Disordered elastic systems and one-..." refers methods in this paper

• ...We first used the so-called ‘replica trick’, well-known in the study of spin glasses [39], in order to average first over the disorder and so to transform the random part Hdis in the full Hamiltonian of one interface (u1), into an effective non-random coupling between n copies of the interface (u1⁄4 fu1, ....

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• ...The two main analytical tools used on DES are on one hand the Functional Renormalization Group (FRG) where the whole disorder correlator (3) evolves under the renormalization procedure [17,34–37], and on the other hand the Gaussian Variational Method (GVM) as introduced by Mézard and Parisi on DES, and involving Replica to treat the disorder [38,39]....

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Abstract: In many materials with a highly anisotropic band structure, electron-phonon interactions lead to a novel type of ground state called the charge-density wave. The condensate is pinned to the underlying lattice by impurities and by boundary effects, but can, even for small electric fields, carry current in a fashion originally envisioned by Fr\"ohlich. This review discusses some of the underlying theories and the main experimental observations on this new collective transport phenomenon. The frequency- and electric-field-dependent conductivity, current oscillations, electric-field-dependent transport coefficients and elastic properties, together with nuclear-magnetic-resonance experiments, provide clear evidence for a translational motion of the condensate. Various theories, involving classical and quantum-mechanical concepts, are able to account for a broad variety of experimental findings, which were also made in the presence of combined dc and ac fields.

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### "Disordered elastic systems and one-..." refers background in this paper

• ...ferroelectric [1, 2, 3] or ferromagnetic [4, 5, 6] domain walls, contact line in wetting experiments [7] or propagating cracks in paper and thin materials [8]) or periodic systems (typically vortex lattices in type-II superconductors [9], classical [10] or quantum [11] Wigner crystals, or electronic crystals displaying charge or spin density waves [12, 13])....

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