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arXiv:2310.12801 · Physical review. B./Physical review. B

Roughness and critical force for depinning at 3-loop order

Mikhail N. Semeikin, Kay Jörg Wiese

Published Apr 18, 2024• 4 citations
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Abstract

A $d$-dimensional elastic manifold at depinning is described by a renormalized field theory, based on the functional renormalization group (FRG). Here, we analyze this theory to 3-loop order, equivalent to third order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where $d$ is the internal dimension. The critical exponent reads $\ensuremath{\zeta}=\frac{\ensuremath{\epsilon}}{3}+0.04777{\ensuremath{\epsilon}}^{2}\ensuremath{-}0.068354{\ensuremath{\epsilon}}^{3}+O({\ensuremath{\epsilon}}^{4})$. Using that $\ensuremath{\zeta}(d=0)={2}^{\ensuremath{-}}$, we estimate $\ensuremath{\zeta}(d=1)=1.266(20)$, $\ensuremath{\zeta}(d=2)=0.752(1)$, and $\ensuremath{\zeta}(d=3)=0.357(1)$. For Gaussian disorder, the pinning force per site is estimated as ${f}_{\mathrm{c}}=\mathcal{B}{m}^{2}{\ensuremath{\rho}}_{m}+{f}_{\mathrm{c}}^{0}$, where ${m}^{2}$ is the strength of the confining potential, $\mathcal{B}$ a universal amplitude, ${\ensuremath{\rho}}_{m}$ the correlation length of the disorder, and ${f}_{\mathrm{c}}^{0}$ a nonuniversal lattice-dependent term. For charge-density waves, we find a mapping to the standard ${\ensuremath{\phi}}^{4}$ theory with $O(n)$ symmetry in the limit of $n\ensuremath{\rightarrow}\ensuremath{-}2$. This gives ${f}_{\mathrm{c}}=\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{A}}(d){m}^{2}ln(m)+{f}_{\mathrm{c}}^{0}$, with $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{A}}(d)=\ensuremath{-}{\ensuremath{\partial}}_{n}{[\ensuremath{\nu}{(d,n)}^{\ensuremath{-}1}+\ensuremath{\eta}(d,n)]}_{n=\ensuremath{-}2}$, reminiscent of logarithmic conformal field theories.

DOI
10.1103/physrevb.109.134203

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Large-scale numerical simulations are used to study the elastic dynamics of two-dimensional vortex lattices driven on a disordered medium in the case of weak disorder. We investigate the so-called elastic depinning transition by decreasing the driving force from the elastic dynamical regime to the state pinned by the quenched disorder. Similarly to the plastic depinning transition, we find results compatible with a second-order phase transition, although both depinning transitions are very different from many viewpoints. We evaluate three critical exponents of the elastic depinning transition. β = 0.29 ± 0.03 is found for the velocity exponent at zero temperature, and from the velocity–temperature curves we extract the critical exponent δ−1 = 0.28 ± 0.05. Furthermore, in contrast with charge-density waves, a finite-size scaling analysis suggests the existence of a unique diverging length at the depinning threshold with an exponent ν = 1.04 ± 0.04, which controls the critical force distribution, the finite-size crossover force distribution and the intrinsic correlation length. Finally, a scaling relation is found between velocity and temperature with the β and δ critical exponents both independent with regard to pinning strength and disorder realizations.
Large-scale numerical simulations are used to study the elastic dynamics of two-dimensional vortex lattices driven on a disordered medium in the case of weak disorder. We investigate the so-called elastic depinning transition by decreasing the driving force from the elastic dynamical regime to the state pinned by the quenched disorder. Similarly to the plastic depinning transition, we find results compatible with a second-order phase transition, although both depinning transitions are very different from many viewpoints. We evaluate three critical exponents of the elastic depinning transition. β = 0.29 ± 0.03 is found for the velocity exponent at zero temperature, and from the velocity–temperature curves we extract the critical exponent δ −1 = 0.28 ± 0.05. Furthermore, in contrast with charge-density waves, a finite-size scaling analysis suggests the existence of a unique diverging length at the depinning threshold with an exponent ν = 1.04 ± 0.04, which controls the critical force distribution, the finite-size crossover force distribution and the intrinsic correlation length. Finally, a scaling relation is found between velocity and temperature with the β and δ critical exponents both independent with regard to pinning strength and disorder realizations.
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We study both analytically, using the renormalization group (RG) to two loop order, and numerically, using an exact polynomial algorithm, the disorder-induced glass phase of the two-dimensional XYXYXY model with quenched random symmetry-breaking fields and without vortices. In the super-rough glassy phase, i.e., below the critical temperature Tc{T}_{c}Tc​, the disorder and thermally averaged correlation function B(r)B(r)B(r) of the phase field $\ensuremath{\theta}(\mathbf{x})$, $B(r)=\overline{⟨[\ensuremath{\theta}(\mathbf{x})\ensuremath{-}\ensuremath{\theta}(\mathbf{x}+\mathbf{r}){]}^{2}⟩}$ behaves, for $r\ensuremath{\gg}a$, as $B(r)\ensuremath{\simeq}A(\ensuremath{\tau}){ln}^{2}(r/a)$ where r=∣r∣r=|\mathbf{r}|r=∣r∣ and aaa is a microscopic length scale. We derive the RG equations up to cubic order in $\ensuremath{\tau}=({T}_{c}\ensuremath{-}T)/{T}_{c}$ and predict the universal amplitude $A(\ensuremath{\tau})=2{\ensuremath{\tau}}^{2}\ensuremath{-}2{\ensuremath{\tau}}^{3}+\mathcal{O}({\ensuremath{\tau}}^{4})$. The universality of $A(\ensuremath{\tau})$ results from nontrivial cancellations between nonuniversal constants of RG equations. Using an exact polynomial algorithm on an equivalent dimer version of the model we compute $A(\ensuremath{\tau})$ numerically and obtain a remarkable agreement with our analytical prediction, up to $\ensuremath{\tau}\ensuremath{\approx}0.5$.
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.
The critical behavior of three-dimensional weakly diluted quenched Ising model is examined on the base of six-loop renormalization group expansions obtained within the minimal subtraction scheme in 4-ɛ space dimensions. For this purpose the ϕ^{4} field theory with cubic symmetry was analyzed in the replica limit n→0. Along with renormalization group expansions in terms of renormalized couplings the sqrt[ɛ] expansions of critical exponents are presented. Corresponding numerical estimates for the physical, three-dimensional system are obtained by means of different resummation procedures applied both to the sqrt[ɛ] series and to initial renormalization group expansions. The results given by the latter approach are in a good agreement with their counterparts obtained experimentally and within the Monte Carlo simulations, while resumming of sqrt[ɛ] series themselves turned out to be disappointing.