Depinning Transition of Charge-Density Waves: Mapping onto $O(n)$ Symmetric ${\varphi }^4$ Theory with $n\to -2$ and Loop-Erased Random Walks

Driven periodic elastic systems such as charge-density waves (CDWs) pinned by impurities show a nontrivial, 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 four-loop order and give both a perturbative and a nonperturbative 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 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.

Figure 1

Trace of a LERW in blue, with the erased loops in red, on a 2D honeycomb lattice. (Inset) Nesting of the different field theories for LERWs.