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.

中文翻译：

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无序弹性流形、脱钉、雪崩和沙堆的理论和实验

磁体中的畴壁、超导体中的涡旋晶格、脱钉时的接触线以及许多其他系统都可以建模为受淬灭无序影响的弹性系统。随之而来的场论在其上限临界维数周围具有良好控制的微扰扩展。与标准场论相反，重整化群 (RG) 流涉及一个函数，无序相关因子 Δ(w), 因此被称为函数 RG。Δ(w) 是一个物理可观测值，弹性流形质心的自相关函数。在这篇评论中，我们对其现象学和技术进行了教学介绍。这使我们能够同时处理平衡（静态）和脱钉（动态）。在这些技术的基础上，可以访问雪崩观测值：大小、持续时间和速度的分布，以及空间和时间形状。无序弹性流形和沙堆模型之间存在各种等价关系：在一点驱动的弹性弦和奥斯陆模型；无序的弹性流形和甘露沙堆；电荷密度波和阿贝尔沙堆或循环擦除随机游走。这些系统之间的每个映射都需要我们开发的特定技术，包括通过粗粒度随机运动方程、超对称技术和元胞自动机对离散随机系统进行建模。比二次最近邻相互作用更强的相互作用会导致定向渗流，以及带有额外 Kardar–Parisi–Zhang (KPZ) 项的非线性表面增长。另一方面，无序的 KPZ 可以映射回无序的弹性流形，无论是在定向聚合物上的稳态，还是单个粒子的衰变。涵盖的其他主题包括功能 RG 与复制对称性破缺之间的关系，以及随机场磁铁。重点是理论的数值和实验测试。和具有额外 Kardar-Parisi-Zhang (KPZ) 项的非线性表面生长。另一方面，无序的 KPZ 可以映射回无序的弹性流形，无论是在定向聚合物上的稳态，还是单个粒子的衰变。涵盖的其他主题包括功能 RG 与复制对称性破缺之间的关系，以及随机场磁铁。重点是理论的数值和实验测试。和具有额外 Kardar-Parisi-Zhang (KPZ) 项的非线性表面生长。另一方面，无序的 KPZ 可以映射回无序的弹性流形，无论是在定向聚合物上的稳态，还是单个粒子的衰变。涵盖的其他主题包括功能 RG 与复制对称性破缺之间的关系，以及随机场磁铁。重点是理论的数值和实验测试。