Depinning transition of charge density waves: Mapping onto O(n) symmetric φ4 theory with n → −2 and loop-erased random walks

Depinning transition of charge density waves: Mapping onto O(n) symmetric φ⁴ theory with n → −2 and loop-erased random walks

Kay Jörg Wiese¹, Andrei A. Fedorenko²
¹ CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, PSL Research University, Sorbonne Universités, UPMC, 24 rue Lhomond, 75005 Paris, France.
² Univ. Lyon, ENS de Lyon, Univ. Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France

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 $\phi^4$ 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 $\phi^4$ 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 $z = 1.624 00 \pm 0.00005$ of numerical simulations.

arXiv:1908.11721 [pdf]
Phys. Rev. Lett. 123 (2019) 197601 [pdf]