Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles

Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar-Parisi-Zhang (KPZ) terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random-field magnets. Emphasis is given to numerical and experimental tests of the theory.

Physical aging and rejuvenation in the vortex matter of superconducting Nb50Zr50

We report the experimental evidence of physical aging and rejuvenation in the vortex matter of a conventional low-T_Csuperconducting Nb₅₀Zr₅₀alloy. The underlying naturally formed microstructure indicates a landscape of pinning potential for the flux lines, on the basis of which the pinning properties are explained. Magnetic relaxation measurements were used to construct the two-time auto correlation function which is a function of the measuring time 't' and waiting time 't_w' after the vortex state is prepared. The main characteristic features of the phenomenon of physical aging, which are the breaking of time-translation invariance and dynamical scaling are seen. Successive aging of the vortex matter after following different histories in the (H, T) phase space is non-cumulative in nature, which is also known as the phenomena of rejuvenation. These experimental observations of relaxation dynamics along with the features of microstructure of our sample seem to agree with the theoretical models of aging phenomenon in a system of elastic lines pinned by random quenched disorder that leads to hierarchical modes of relaxation.

Chaos in the Bose-glass phase of a one-dimensional disordered Bose fluid

We show that the Bose-glass phase of a one-dimensional disordered Bose fluid exhibits a chaotic behavior, i.e., an extreme sensitivity to external parameters. Using bosonization, the replica formalism and the nonperturbative functional renormalization group, we find that the ground state is unstable to any modification of the disorder configuration ("disorder" chaos) or variation of the Luttinger parameter ("quantum" chaos, analog to the "temperature" chaos in classical disordered systems). This result is obtained by considering two copies of the system, with slightly different disorder configurations or Luttinger parameters, and showing that intercopy statistical correlations are suppressed at length scales larger than an overlap length ξ_{ov}∼|ε|^{-1/α} (|ε|≪1 is a measure of the difference between the disorder distributions or Luttinger parameters of the two copies). The chaos exponent α can be obtained by computing ξ_{ov} or by studying the instability of the Bose-glass fixed point for the two-copy system when ε≠0. The renormalized, functional, intercopy disorder correlator departs from its fixed-point value-characterized by "cuspy" singularities-via a chaos boundary layer, in the same way as it approaches the Bose-glass fixed point when ε=0 through a quantum boundary layer. Performing a linear analysis of perturbations about the Bose-glass fixed point, we find α=1.

Mott-Glass Phase of a One-Dimensional Quantum Fluid with Long-Range Interactions

We investigate the ground-state properties of quantum particles interacting via a long-range repulsive potential V_{σ}(x)∼1/|x|^{1+σ} (-1<σ) or V_{σ}(x)∼-|x|^{-1-σ} (-2≤σ<-1) that interpolates between the Coulomb potential V_{0}(x) and the linearly confining potential V_{-2}(x) of the Schwinger model. In the absence of disorder the ground state is a Wigner crystal when σ≤0. Using bosonization and the nonperturbative functional renormalization group we show that any amount of disorder suppresses the Wigner crystallization when -3/2<σ≤0; the ground state is then a Mott glass, i.e., a state that has a vanishing compressibility and a gapless optical conductivity. For σ<-3/2 the ground state remains a Wigner crystal.

Dimensional reduction breakdown and correction to scaling in the random-field Ising model

We provide a theoretical analysis by means of the nonperturbative functional renormalization group (NP-FRG) of the corrections to scaling in the critical behavior of the random-field Ising model (RFIM) near the dimension d_{DR}≈5.1 that separates a region where the renormalized theory at the fixed point is supersymmetric and critical scaling satisfies the d→d-2 dimensional reduction property (d>d_{DR}) from a region where both supersymmetry and dimensional reduction break down at criticality (d<d_{DR}). We show that the NP-FRG results are in very good agreement with recent large-scale lattice simulations of the RFIM in d=5 and we detail the consequences for the leading correction-to-scaling exponent of the peculiar boundary-layer mechanism by which the dimensional-reduction fixed point disappears and the dimensional-reduction-broken fixed point emerges in d_{DR}.

Bose-glass phase of a one-dimensional disordered Bose fluid: Metastable states, quantum tunneling, and droplets

We study a one-dimensional disordered Bose fluid using bosonization, the replica method, and a nonperturbative functional renormalization-group approach. We find that the Bose-glass phase is described by a fully attractive strong-disorder fixed point characterized by a singular disorder correlator whose functional dependence assumes a cuspy form that is related to the existence of metastable states. At nonzero momentum scale k, quantum tunneling between the ground state and low-lying metastable states leads to a rounding of the cusp singularity into a quantum boundary layer (QBL). The width of the QBL depends on an effective Luttinger parameter K_{k}∼k^{θ} that vanishes with an exponent θ=z-1 related to the dynamical critical exponent z. The QBL encodes the existence of rare "superfluid" regions, controls the low-energy dynamics, and yields a (dissipative) conductivity vanishing as ω^{2} in the low-frequency limit. These results reveal the glassy properties (pinning, "shocks," or static avalanches) of the Bose-glass phase and can be understood within the "droplet" picture put forward for the description of glassy (classical) systems.

Manifolds in a high-dimensional random landscape: Complexity of stationary points and depinning

We obtain explicit expressions for the annealed complexities associated, respectively, with the total number of (i) stationary points and (ii) local minima of the energy landscape for an elastic manifold with internal dimension d<4 embedded in a random medium of dimension N≫1 and confined by a parabolic potential with the curvature parameter μ. These complexities are found to both vanish at the critical value μ_{c} identified as the Larkin mass. For μ<μ_{c} the system is in complex phase corresponding to the replica symmetry breaking in its T=0 thermodynamics. The complexities vanish, respectively, quadratically (stationary points) and cubically (minima) at μ_{c}^{-}. For d≥1 they admit a finite "massless" limit μ=0 which is used to provide an upper bound for the depinning threshold under an applied force.

Glassy properties of the Bose-glass phase of a one-dimensional disordered Bose fluid

We study a one-dimensional disordered Bose fluid using bosonization, the replica method, and a nonperturbative functional renormalization-group approach. The Bose-glass phase is described by a fully attractive strong-disorder fixed point characterized by a singular disorder correlator whose functional dependence assumes a cuspy form that is related to the existence of metastable states. At nonzero momentum scale, quantum tunneling between these metastable states leads to a rounding of the nonanalyticity in a quantum boundary layer that encodes the existence of rare superfluid regions responsible for the ω^{2} behavior of the (dissipative) conductivity in the low-frequency limit. These results can be understood within the "droplet" picture put forward for the description of glassy (classical) systems.

Glassy Properties of Anderson Localization: Pinning, Avalanches, and Chaos

I present the results of extensive numerical simulations, which reveal the glassy properties of Anderson localization in dimension two at zero temperature: pinning, avalanches, and chaos. I first show that strong localization confines quantum transport along paths that are pinned by disorder but can change abruptly and suddenly (avalanches) when the energy is varied. I determine the roughness exponent ζ characterizing the transverse fluctuations of these paths and find that its value ζ=2/3 is the same as for the directed polymer problem. Finally, I characterize the chaos property, namely, the fragility of the conductance with respect to small perturbations in the disorder configuration. It is linked to interference effects and universal conductance fluctuations at weak disorder and more spin-glass-like behavior at strong disorder.

Universality and Specificity in Protein Fluctuation Dynamics

We investigate the universal scaling of protein fluctuation dynamics with a site-specific diffusive model of protein motion, which predicts an initial subdiffusive regime in the configurational relaxation. The long-time dynamics of proteins is controlled by an activated regime. We argue that the hierarchical free energy barriers set the time scales of biological processes and establish an upper limit to the size of single protein domains. We find it compelling that the scaling behavior for the protein dynamics is in close agreement with the Kardar-Parisi-Zhang scaling exponents.

Avalanches and dimensional reduction breakdown in the critical behavior of disordered systems

We investigate the connection between a formal property of the critical behavior of several disordered systems, known as "dimensional reduction," and the presence in these systems at zero temperature of collective events known as "avalanches." Avalanches generically produce nonanalyticities in the functional dependence of the cumulants of the renormalized disorder. We show that this leads to a breakdown of the dimensional reduction predictions if and only if the fractal dimension characterizing the scaling properties of the avalanches is exactly equal to the difference between the dimension of space and the scaling dimension of the primary field. This is proven by combining scaling theory and the functional renormalization group. We therefore clarify the puzzle of why dimensional reduction remains valid in random field systems above a nontrivial dimension (but fails below), always applies to the statistics of branched polymer, and is always wrong in elastic models of interfaces in a random environment.

First-principles derivation of static avalanche-size distributions

We study the energy minimization problem for an elastic interface in a random potential plus a quadratic well. As the position of the well is varied, the ground state undergoes jumps, called shocks or static avalanches. We introduce an efficient and systematic method to compute the statistics of avalanche sizes and manifold displacements. The tree-level calculation, i.e., mean-field limit, is obtained by solving a saddle-point equation. Graphically, it can be interpreted as the sum of all tree graphs. The 1-loop corrections are computed using results from the functional renormalization group. At the upper critical dimension the shock statistics is described by the Brownian force model (BFM), the static version of the so-called Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model in the nonequilibrium context of depinning. This model can itself be treated exactly in any dimension and its shock statistics is that of a Lévy process. Contact is made with classical results in probability theory on the Burgers equation with Brownian initial conditions. In particular we obtain a functional extension of an evolution equation introduced by Carraro and Duchon, which recursively constructs the tree diagrams in the field theory.

Interference in disordered systems: a particle in a complex random landscape

We consider a particle in one dimension submitted to amplitude and phase disorder. It can be mapped onto the complex Burgers equation, and provides a toy model for problems with interplay of interferences and disorder, such as the Nguyen-Spivak-Shklovskii model of hopping conductivity in disordered insulators and the Chalker-Coddington model for the (spin) quantum Hall effect. We also propose a direct realization in an experiment with cold atoms. The model has three distinct phases: (I) a high-temperature or weak disorder phase, (II) a pinned phase for strong amplitude disorder, and (III) a diffusive phase for strong phase disorder, but weak amplitude disorder. We compute analytically the renormalized disorder correlator, equivalent to the Burgers velocity-velocity correlator at long times. In phase III, it assumes a universal form. For strong phase disorder, interference leads to a logarithmic singularity, related to zeros of the partition sum, or poles of the complex Burgers velocity field. These results are valuable in the search for the adequate field theory for higher-dimensional systems.

Spontaneous versus explicit replica symmetry breaking in the theory of disordered systems

We investigate the relation between spontaneous and explicit replica symmetry breaking in the theory of disordered systems. On general ground, we prove the equivalence between the replicon operator associated with the stability of the replica-symmetric solution in the standard replica scheme and the operator signaling a breakdown of the solution with analytic field dependence in a scheme in which replica symmetry is explicitly broken by applied sources. This opens the possibility to study, via the recently developed functional renormalization group, unresolved questions related to spontaneous replica symmetry breaking and spin-glass behavior in finite-dimensional disordered systems.

Cusps, self-organization, and absorbing states

Elastic interfaces embedded in (quenched) random media exhibit metastability and stick-slip dynamics. These nontrivial dynamical features have been shown to be associated with cusp singularities of the coarse-grained disorder correlator. Here we show that annealed systems with many absorbing states and a conservation law but no quenched disorder exhibit identical cusps. On the other hand, similar nonconserved systems in the directed percolation class are also shown to exhibit cusps but of a different type. These results are obtained both by a recent method to explicitly measure disorder correlators and by defining an alternative new protocol inspired by self-organized criticality, which opens the door to easily accessible experimental realizations.

Measuring functional renormalization group fixed-point functions for pinned manifolds

Exact numerical minimization of interface energies is used to test the functional renormalization group analysis for interfaces pinned by quenched disorder. The fixed-point function R(u) (the correlator of the coarse-grained disorder) is computed. In dimensions D=d+1, a linear cusp in R''(u) is confirmed for random bond (d=1, 2, 3), random field (d=0, 2, 3), and periodic (d=2, 3) disorders. The functional shocks that lead to this cusp are seen. Small, but significant, deviations from the 1-loop calculation are compared to 2-loop corrections and chaos is measured.

Nonperturbative functional renormalization group for random-field models: the way out of dimensional reduction

We develop a nonperturbative functional renormalization group approach for the random-field O(N) model that allows us to investigate the ordering transition in any dimension and for any value of N including the Ising case. We show that the failure of dimensional reduction and standard perturbation theory is due to the nonanalytic nature of the zero-temperature fixed point controlling the critical behavior, nonanalyticity, which is associated with the existence of many metastable states. We find that this nonanalyticity leads to critical exponents differing from the dimensional reduction prediction only below a critical dimension dc(N)<6, with dc(N=1)>3.

Functional renormalization group at large N for disordered systems

We introduce a method, based on an exact calculation of the effective action at large N, to bridge the gap between mean-field theory and renormalization in complex systems. We apply it to a d-dimensional manifold in a random potential for large embedding space dimension N. This yields a functional renormalization group equation valid for any d, which contains both the O(epsilon=4-d) results of Balents-Fisher and some of the nontrivial results of the Mezard-Parisi solution, thus shedding light on both. Corrections are computed at order O(1/N). Applications to the Kardar-Parisi-Zhang growth model, random field, and mode coupling in glasses are mentioned.