Laboratoire de Physique de l'Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, 24 rue Lhomond, 75005 Paris, France.
Université de Lyon, ENS de Lyon, Université Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France.
Phys Rev Lett. 2019 Nov 8;123(19):197601. doi: 10.1103/PhysRevLett.123.197601.
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 ϕ^{4} theory in the unusual limit of n→-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 ϕ^{4} theory, taken with formally n=0 and n→-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±0.001, in excellent agreement with the estimate z=1.62400±0.00005 of numerical simulations.
受杂质钉扎的驱动周期弹性系统，如电荷密度波（CDW），表现出非平凡的玻璃态动力学临界行为。它们的理论描述需要功能重整化群。我们表明，它们在去钉扎转变附近的临界行为与一个简单得多的模型有关，即在 n→-2 的不寻常极限下 O(n)对称的 ϕ^{4}理论。我们证明了这两个理论在四阶环下都得到了相同的结果，并给出了它们等价性的微扰和非微扰证明。正如我们所展示的，这两个理论都可以用来描述环消去随机行走（LERW），即随机行走的轨迹，其中一旦形成环就会被消去。值得注意的是，两个著名的非自相交随机行走模型，自回避行走和 LERW，可以都被映射到 ϕ^{4}理论，其中正式的 n=0 和 n→-2 分量。这种映射允许我们计算 CDW 在去钉扎转变处的动力学临界指数和 LERW 在 d=3 中的分形维数，精度前所未有，z(d=3)=1.6243±0.001，与数值模拟的估计 z=1.62400±0.00005 非常吻合。
