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

TL;DR

This paper reveals a surprising connection between the depinning transition of charge-density waves, a complex physical phenomenon, and simpler mathematical models like $O(n)$ $\,\phi^4$ theory with $n\to -2$, enabling precise calculations of critical exponents.

Contribution

It establishes an exact mapping between the depinning transition of charge-density waves and $O(n)$ $\,\phi^4$ theory at $n\to -2$, providing new analytical tools for these phenomena.

Findings

Both theories produce identical results to 4-loop order.

The dynamic critical exponent for CDWs in 3D is computed as 1.6243 ± 0.001.

The fractal dimension of loop-erased random walks in 3D matches numerical estimates.

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.