Disordered elastic systems and one-dimensional interfaces
TL;DR: In this paper, the authors 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.
Abstract: We 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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TL;DR: In this article, the stationary state fluctuations of a growing one-dimensional interface described by the Kardar-Parisi-Zhang (KPZ) dynamics with a noise featuring smooth spatial correlations of characteristic range ξ were investigated.
Abstract: We investigate the stationary-state fluctuations of a growing one-dimensional interface described by the Kardar-Parisi-Zhang (KPZ) dynamics with a noise featuring smooth spatial correlations of characteristic range ξ. We employ nonperturbative functional renormalization group methods to resolve the properties of the system at all scales. We show that the physics of the standard (uncorrelated) KPZ equation emerges on large scales independently of ξ. Moreover, the renormalization group flow is followed from the initial condition to the fixed point, that is, from the microscopic dynamics to the large-distance properties. This provides access to the small-scale features (and their dependence on the details of the noise correlations) as well as to the universal large-scale physics. In particular, we compute the kinetic energy spectrum of the stationary state as well as its nonuniversal amplitude. The latter is experimentally accessible by measurements at large scales and retains a signature of the microscopic noise correlations. Our results are compared to previous analytical and numerical results from independent approaches. They are in agreement with direct numerical simulations for the kinetic energy spectrum as well as with the prediction, obtained with the replica trick by Gaussian variational method, of a crossover in ξ of the nonuniversal amplitude of this spectrum.
25 citations
TL;DR: In the fully connected model, it is shown that the weak-strong pinning transition coincides with a peculiar localization transition of the ground state, and the soft modes of the dynamical matrix at the depinning transition are characterized.
Abstract: We characterize the soft modes of the dynamical matrix at the depinning transition, and compare the matrix with the properties of the Anderson model (and long-range generalizations). The density of states at the edge of the spectrum displays a universal linear tail, different from the Lifshitz tails. The eigenvectors are instead very similar in the two matrix ensembles. We focus on the ground state (soft mode), which represents the epicenter of avalanche instabilities. We expect it to be localized in all finite dimensions, and make a clear connection between its localization length and the Larkin length of the depinning model. In the fully connected model, we show that the weak-strong pinning transition coincides with a peculiar localization transition of the ground state.
24 citations
TL;DR: It is shown that the region of validity of models of thermal activation on mesoscopically rough surfaces typically corresponds to velocities of less than 1 mm/s, and a model based on independent defects is developed and used to show deviations from the purely exponential law.
Abstract: From simple models of thermally activated contact line dynamics far below the depinning transition, one expects the velocity to depend exponentially on the applied force and the activation area to be the size of the defects on the surface. We study contact line motion on evaporated gold films and find that the dynamics are activated, but the activation area is not straightforwardly linked to the surface roughness. Surprisingly, the activation area can be significantly smaller than any features on the surface. Furthermore, it depends strongly on the liquid. We show that this indicates that the line is close to the depinning threshold at experimentally accessible velocities. A model based on independent defects is developed and used to show deviations from the purely exponential law. The dynamics are written entirely in terms of properties of the surface and partially wetting liquid. In addition, we are able to show that the region of validity of models of thermal activation on mesoscopically rough surfaces...
23 citations
TL;DR: In this article, the evolution of magnetic domain walls under the application of alternating magnetic fields within the creep regime, well beyond a small fluctuation limit of the domain wall position, was studied.
Abstract: The domain wall response under constant external magnetic fields reveals a complex behavior where sample disorder plays a key role. Furthermore, the response to alternating magnetic fields has only been explored in limited cases and analyzed in terms of the constant field solution. Here we unveil phenomena in the evolution of magnetic domain walls under the application of alternating magnetic fields within the creep regime, well beyond a small fluctuation limit of the domain wall position. Magnetic field pulses were applied in ultrathin ferromagnetic films with perpendicular anisotropy, and the resulting domain wall evolution was characterized by polar magneto-optical Kerr effect microscopy. Whereas the dc characterization is well predicted by the elastic interface model, striking unexpected features are observed under the application of alternating square pulses: Magneto-optical images show that after a characteristic number of cycles, domain walls evolve toward strongly distorted shapes concomitantly with a modification of domain area. The morphology of domain walls is characterized with a roughness exponent when possible and contrasted with alternative observables which are more suitable for the characterization of this transient evolution. The final stationary convergence as well as the underlying physics is discussed.
15 citations
TL;DR: This work addresses numerically the time and temperature dependence of the roughness B(t), which quantifies the DP end point transverse fluctuations, and shows how the amplitude D[over ̃](∞)(T,ξ) controls the different regimes experienced by B( t)-in agreement with the analytical predictions of a DP toy model approach.
Abstract: We study numerically the geometrical and free-energy fluctuations of a static one-dimensional (1D) interface with a short-range elasticity, submitted to a quenched random-bond Gaussian disorder of finite correlation length $\ensuremath{\xi}g0$ and at finite temperature $T$. Using the exact mapping from the static 1D interface to the 1+1 directed polymer (DP) growing in a continuous space, we focus our analysis on the disorder free energy of the DP end point, a quantity which is strictly zero in the absence of disorder and whose sample-to-sample fluctuations at a fixed growing time $t$ inherit the statistical translation invariance of the microscopic disorder explored by the DP. Constructing a new numerical scheme for the integration of the Kardar-Parisi-Zhang evolution equation obeyed by the free energy, we address numerically the time and temperature dependence of the disorder free-energy fluctuations at fixed finite $\ensuremath{\xi}$. We examine, on one hand, the amplitude ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}}_{t}$ and effective correlation length ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\xi}}}_{t}$ of the free-energy fluctuations and, on the other hand, the imprint of the specific microscopic disorder correlator on the large-time shape of the free-energy two-point correlator. We observe numerically the crossover to a low-temperature regime below a finite characteristic temperature ${T}_{c}(\ensuremath{\xi})$, as previously predicted by Gaussian variational method computations and scaling arguments and extensively investigated analytically in [Phys. Rev. E 87, 042406 (2013)]. Finally, we address numerically the time and temperature dependence of the roughness $B(t)$, which quantifies the DP end point transverse fluctuations, and we show how the amplitude ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}}_{\ensuremath{\infty}}(T,\ensuremath{\xi})$ controls the different regimes experienced by $B(t)$---in agreement with the analytical predictions of a DP toy model approach.
15 citations
References
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01 Jan 2012
139,059 citations
"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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TL;DR: The Ginzburg number as discussed by the authors was introduced to account for thermal and quantum fluctuations and quenched disorder in high-temperature superconductors, leading to interesting effects such as melting of the vortex lattice, the creation of new vortex-liquid phases, and the appearance of macroscopic quantum phenomena.
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.
4,502 citations
TL;DR: In this paper, a detailed and self-contained presentation of the replica theory of infinite range spin glasses is presented, paying particular attention to new applications in the study of optimization theory and neural networks.
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.
3,846 citations
"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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TL;DR: 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 as mentioned in this paper, which can, even for small electric fields, carry current in a fashion originally envisioned by Frohlich.
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.
1,308 citations
"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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TL;DR: Kinetic interfaces form the basis of a fascinating, interdisciplinary branch of statistical mechanics as mentioned in this paper, which can be unified via an intriguing nonlinear stochastic partial differential equation whose consequences and generalizations have mobilized a sizeable community of physicists concerned with a statistical description of kinetically roughened surfaces.
1,015 citations