Oslo model, hyperuniformity, and the quenched Edwards-Wilkinson model

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.

Density relaxation in conserved Manna sandpiles

We study relaxation of long-wavelength density perturbations in a one-dimensional conserved Manna sandpile. Far from criticality where correlation length ξ is finite, relaxation of density profiles having wave numbers k→0 is diffusive, with relaxation time τ_{R}∼k^{-2}/D with D being the density-dependent bulk-diffusion coefficient. Near criticality with kξ≳1, the bulk diffusivity diverges and the transport becomes anomalous; accordingly, the relaxation time varies as τ_{R}∼k^{-z}, with the dynamical exponent z=2-(1-β)/ν_{⊥}<2, where β is the critical order-parameter exponent and ν_{⊥} is the critical correlation-length exponent. Relaxation of initially localized density profiles on an infinite critical background exhibits a self-similar structure. In this case, the asymptotic scaling form of the time-dependent density profile is analytically calculated: we find that, at long times t, the width σ of the density perturbation grows anomalously, σ∼t^{w}, with the growth exponent ω=1/(1+β)>1/2. In all cases, theoretical predictions are in reasonably good agreement with simulations.

Hydrodynamics, density fluctuations, and universality in conserved stochastic sandpiles

We study conserved stochastic sandpiles (CSSs), which exhibit an active-absorbing phase transition upon tuning density ρ. We demonstrate that a broad class of CSSs possesses a remarkable hydrodynamic structure: There is an Einstein relation σ^{2}(ρ)=χ(ρ)/D(ρ), which connects bulk-diffusion coefficient D(ρ), conductivity χ(ρ), and mass fluctuation, or scaled variance of subsystem mass, σ^{2}(ρ). Consequently, density large-deviations are governed by an equilibrium-like chemical potential μ(ρ)∼lna(ρ), where a(ρ) is the activity in the system. By using the above hydrodynamics, we derive two scaling relations: As Δ=(ρ-ρ_{c})→0^{+}, ρ_{c} being the critical density, (i) the mass fluctuation σ^{2}(ρ)∼Δ^{1-δ} with δ=0 and (ii) the dynamical exponent z=2+(β-1)/ν_{⊥}, expressed in terms of two static exponents β and ν_{⊥} for activity a(ρ)∼Δ^{β} and correlation length ξ∼Δ^{-ν_{⊥}}, respectively. Our results imply that conserved Manna sandpile, a well studied variant of the CSS, belongs to a distinct universality-not that of directed percolation (DP), which, without any conservation law as such, does not obey scaling relation (ii).

Kardar-Parisi-Zhang modes in d-dimensional directed polymers

We define a stochastic lattice model for a fluctuating directed polymer in d≥2 dimensions. This model can be alternatively interpreted as a fluctuating random path in two dimensions, or a one-dimensional asymmetric simple exclusion process with d-1 conserved species of particles. The deterministic large dynamics of the directed polymer are shown to be given by a system of coupled Kardar-Parisi-Zhang (KPZ) equations and diffusion equations. Using nonlinear fluctuating hydrodynamics and mode coupling theory we argue that stationary fluctuations in any dimension d can only be of KPZ type or diffusive. The modes are pure in the sense that there are only subleading couplings to other modes, thus excluding the occurrence of modified KPZ-fluctuations or Lévy-type fluctuations, which are common for more than one conservation law. The mode-coupling matrices are shown to satisfy the so-called trilinear condition.

Multichain models of conserved lattice gas

Conserved lattice-gas models in one dimension exhibit absorbing state phase transition (APT) with simple integer exponents β=1=ν=η, whereas the same on a ladder belong to directed percolation (DP) universality. We conjecture that additional stochasticity in particle transfer is a relevant perturbation and its presence on a ladder forces the APT to be in the DP class. To substantiate this we introduce a class of restricted conserved lattice-gas models on a multichain system (M×L square lattice with periodic boundary condition in both directions), where particles which have exactly one vacant neighbor are active and they move deterministically to the neighboring vacant site. We show that for odd number of chains, in the thermodynamic limit L→∞, these models exhibit APT at ρ_{c}=1/2(1+1/M) with β=1. On the other hand, for even-chain systems transition occurs at ρ_{c}=1/2 with β=1,2 for M=2,4, respectively, and β=3 for M≥6. We illustrate this unusual critical behavior analytically using a transfer-matrix method.

Multicritical absorbing phase transition in a class of exactly solvable models

We study diffusion of hard-core particles on a one-dimensional periodic lattice subjected to a constraint that the separation between any two consecutive particles does not increase beyond a fixed value n+1; an initial separation larger than n+1 can however decrease. These models undergo an absorbing state phase transition when the conserved particle density of the system falls below a critical threshold ρ_{c}=1/(n+1). We find that the ϕ_{k}, the density of 0-clusters (0 representing vacancies) of size 0≤k<n, vanish at the transition point along with activity density ρ_{a}. The steady state of these models can be written in matrix product form to obtain analytically the static exponents β_{k}=n-k and ν=1=η corresponding to each ϕ_{k}. We also show from numerical simulations that, starting from a natural condition, ϕ_{k}(t)s decay as t^{-α_{k}} with α_{k}=(n-k)/2 even though other dynamic exponents ν_{t}=2=z are independent of k; this ensures the validity of scaling laws β=αν_{t} and ν_{t}=zν.