Interface depinning versus absorbing-state phase transitions

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.

Directed percolation with a conserved field and the depinning transition

Conserved directed percolation (C-DP) and the depinning transition of a disordered elastic interface belong to the same universality class, as has been proven very recently by Le Doussal and Wiese [Phys. Rev. Lett. 114, 110601 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.110601] through a mapping of the field theory for C-DP onto that of the quenched Edwards-Wilkinson model. Here, we present an alternative derivation of the C-DP field theoretic functional, starting with the coherent-state path integral formulation of the C-DP and then applying the Grassberger transformation, which avoids the disadvantages of the so-called Doi shift. We revisit the aforementioned mapping with focus on a specific term in the field theoretic functional that has been problematic in the past when it came to assessing its relevance. We show that this term is redundant in the sense of the renormalization group.

Exact mapping of the stochastic field theory for Manna sandpiles to interfaces in random media

We show that the stochastic field theory for directed percolation in the presence of an additional conservation law [the conserved directed-percolation (C-DP) class] can be mapped exactly to the continuum theory for the depinning of an elastic interface in short-range correlated quenched disorder. Along one line of the parameters commonly studied, this mapping leads to the simplest overdamped dynamics. Away from this line, an additional memory term arises in the interface dynamics; we argue that this does not change the universality class. Since C-DP is believed to describe the Manna class of self-organized criticality, this shows that Manna stochastic sandpiles and disordered elastic interfaces (i.e., the quenched Edwards-Wilkinson model) share the same universal large-scale behavior.

Abelian Manna model in three dimensions and below

The Abelian Manna model of self-organized criticality is studied on various three-dimensional and fractal lattices. The exponents for avalanche size, duration, and area distribution of the model are obtained by using a high-accuracy moment analysis. Together with earlier results on lower-dimensional lattices, the present results reinforce the notion of universality below the upper critical dimension and allow us to determine the coefficients of an ε expansion. By rescaling the critical exponents by the lattice dimension and incorporating the random walker dimension, a remarkable relation is observed, satisfied by both regular and fractal lattices.

Flooding transition in the topography of toppling surfaces of stochastic and rotational sandpile models

A continuous phase transition occurs in the topography of toppling surfaces of stochastic and rotational sandpile models when they are flooded with liquid, say water. The toppling surfaces are extracted from the sandpile avalanches that appear due to sudden burst of toppling activity in the steady state of these sandpile models. Though a wide distribution of critical flooding heights exists, a critical point is defined by merging the flooding thresholds of all the toppling surfaces. The criticality of the transition is characterized by power-law distribution of island area in the critical regime. A finite size scaling theory is developed and verified by calculating several new critical exponents. The flooding transition is found to be an interesting phase transition and does not belong to the percolation universality class. The universality class of this transition is found to depend on the degree of self-affinity of the toppling surfaces characterized by the Hurst exponent H and the fractal dimension D(f) of critical spanning islands. The toppling surfaces of different stochastic sandpile models are found to have a single Hurst exponent, whereas those of different rotational sandpile models have another Hurst exponent. As a consequence, the universality class of different sandpile models remains preserved within the same symmetry of the models.

Statistics of static avalanches in a random pinning landscape

We study the minimum-energy configuration of a d -dimensional elastic interface in a random potential tied to a harmonic spring. As a function of the spring position, the center of mass of the interface changes in discrete jumps, also called shocks or "static avalanches." We obtain analytically the distribution of avalanche sizes and its cumulants within an =4-d expansion from a tree and one-loop resummation using functional renormalization. This is compared with exact numerical minimizations of interface energies for random-field disorder in d=2,3 . Connections to dynamic avalanches are mentioned.

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.

Confirming and extending the hypothesis of universality in sandpiles

Stochastic sandpiles self-organize to an absorbing-state critical point with scaling behavior different from directed percolation (DP) and characterized by the presence of an additional conservation law. This is usually called the C-DP or Manna universality class. There remains, however, an exception to this universality principle: a sandpile automaton introduced by Maslov and Zhang, which was claimed to be in the DP class despite the existence of a conservation law. We show, by means of careful numerical simulations as well as by constructing and analyzing a field theory, that (contrarily to what was previously thought) this sandpile is also in the C-DP or Manna class. This confirms the hypothesis of universality for stochastic sandpiles and gives rise to a fully coherent picture of self-organized criticality in systems with conservation. In passing, we obtain a number of results for the C-DP class and introduce a strategy to easily discriminate between DP and C-DP scaling.

Absorbing states and elastic interfaces in random media: two equivalent descriptions of self-organized criticality

We elucidate a long-standing puzzle about the nonequilibrium universality classes describing self-organized criticality in sandpile models. We show that depinning transitions of linear interfaces in random media and absorbing phase transitions (with a conserved nondiffusive field) are two equivalent languages to describe sandpile criticality. This is so despite the fact that local roughening properties can be radically different in the two pictures, as explained here. Experimental implications of our work as well as promising paths for future theoretical investigations are also discussed.

Fluctuations and correlations in sandpiles and interfaces with boundary pinning

Interfaces are studied in an inhomogeneous critical state where boundary pinning is compensated with a ramped force. Sandpiles driven off the self-organized critical point provide an example of this ensemble in the Edwards-Wilkinson (EW) model of kinetic roughening. A crossover from quenched to thermal noise violates spatial and temporal translational invariances. The bulk temporal correlation functions have the effective exponents beta(1D) approximately 0.88+/-0.03 and beta(2D) approximately 0.52+/-0.05, while at the boundaries beta(b,1D/2D) approximately 0.47+/-0.05. The bulk beta(1D) is shown to be reproduced in a randomly kicked thermal EW model.