Depinning transition of charge-density waves: mapping onto $O(n)$ symmetric ${\varphi }^4$ theory with $n\to -2$ and loop-erased random walks

Abstract

Driven periodic elastic systems such as charge-density waves (CDWs) pinned by impurities show a non-trivial, glassy dynamical critical behavior. Their proper theoretical description requires the functional renormalization group. We show that their critical behavior close to the depinning transition is related to a much simpler model, $O(n)$-symmetric ${\varphi }^4$ theory in the unusual limit of $n\to -2$. We demonstrate that both theories yield identical results to 4-loop order and give both a perturbative and a non-perturbative proof of their equivalence. As we show, both theories can be used to describe loop-erased random walks (LERWs), the trace of a random walk where loops are erased as soon as they are formed. Remarkably, two famous models of non-self-intersecting random walks, self-avoiding walks (SAWs) and LERWs, can both be mapped onto ${\varphi }^4$ theory taken, with formally $n=0$ and $n\to -2$ components. This mapping allows us to compute the dynamic critical exponent of CDWs at the depinning transition and the fractal dimension of LERWs in $d=3$ with unprecedented accuracy, $z(d=3)=1.6243\pm 0.001$, in excellent agreement with the estimate $z=1.62400\pm 0.00005$ of numerical simulations.

Final version as published in PRL. For details see arXiv:1802.08830

References in corpus (3)

• An exact mapping of the stochastic field theory for Manna sandpiles to interfaces in random media
• Random-Manifold to Random-Periodic Depinning of an Elastic Interface
• Field theory conjecture for loop-erased random walks

Cited by in corpus (12)

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• An exact mapping between loop-erased random walks and an interacting field theory with two fermions and one boson
• Conformal invariance constraints in the $O(N)$ models: a first study within the nonperturbative renormalization group
• Roughness and critical force for depinning at 3-loop order
• Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories
• Large Orders and Strong-Coupling Limit in Functional Renormalization
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