Elastic theory of pinned flux lattices.

The pinning of flux lattices by weak impurity disorder is studied in the absence of free dislocations using both the gaussian variational method and, to O(ǫ = 4 − d), the functional renormalization group. We find universal logarithmic growth of displacements for 2 < d < 4: 〈u(x) − u(0)〉2 ∼ Ad log |x| and persistence of algebraic quasi-long range translational order. When the two methods can be compared they agree within 10% on the value of Ad. We compute the function describing the crossover between the “randommanifold” regime and the logarithmic regime. This crossover should be observable in present decoration experiments. 74.60.Ge, 05.20.-y Typeset using REVTEX 1 It has been argued on the basis of various elastic models for vortex lattices, such as Larkin’s model of independent random forces acting on each vortex, that arbitrarily weak disorder destroys translational order below four dimensions [3]. There is considerable disagreement, however, on the exact behaviour of the density-density correlation function CK(r) = 〈ρK(0)ρK(r)〉 even in the simpler case where dislocations are excluded. While direct extensions of Larkin’s model predict exponential decay in d = 3 [4], it has been pointed out that beyond the Larkin-Ovchinikov pinning length Lc the lattice behaves collectively as an elastic manifold in a random potential with many metastable states, leading to a different power-law roughening of the lattice and to stretched exponential decay of CK(r) [5–7]. On the other hand, Flory-type arguments were proposed making explicit use of the periodicity of the lattice leading to logarithmic roughening [8]. The role of dislocations at weak disorder above two dimensions is presently unsettled, but Bitter decoration experiments [9] show remarkably large regions free of dislocations and provide a strong motivation for a better understanding of the elastic model. Other related pinned elastic systems such as charge density waves, magnetic bubbles, Wigner crystal, are under current active experimental study [10,11]. In this Letter we take into account both the existence of many metastable states and the periodicity of the lattice. We are primarily interested in the triangular Abrikosov lattice (d = 2+1). We also mention the case of d = 2+0 (point vortices in thin films) or d = 1+1 (lines in a plane). We show that in the absence of dislocations, the translational correlation function has a slow algebraic decay in dimension larger than two, and thus quasi-long range order persists. Two important length scales control the crossover towards this asymptotic decay. i) When the mean square of the relative displacement B̃(x) = 1 2 〈[u(x)− u(0)]2〉 of two lines as a function of their separation x is shorter than the square of the Lindemann length l T = 〈u 〉, the thermal wandering of the lines averages enough over the random potential and the model becomes equivalent to the random force Larkin model for which B̃(x) ∼ |x|. At low enough temperature, lT is replaced by the superconducting coherence length ξ0 (i.e. the correlation length of the random potential [5–7]). ii) For l T ≪ B̃(x) ≤ a , B̃(x) ∼ x : 2 this is the random manifold regime where each line sees effectively an independent random potential. iii) For x > ξ, where ξ corresponds to a relative displacement of the order of the lattice spacing a, B̃(x = ξ) ∼ a, the periodicity of the lattice becomes important. We find B̃(x) ∼ Ad log |x| where Ad is a universal amplitude depending on dimension only and isotropy is recovered at large distances. This leads to quasi long range order CK0(r) ∼ (1/r)d. We have computed the full crossover function in d = 3 (Fig. 1). It suggests that all the above regimes could be observed by analysis of the dislocation-free decoration samples. These results are obtained using the Mezard-Parisi variational method [12] first applied in this context by Bouchaud, Mezard and Yedidia (BMY) [6,7]. Our results are at variance with BMY, for reasons detailed below. In addition, we perform an ǫ = 4−d expansion using the functional renormalization group. The amplitudes Ad obtained by these two rather different methods agree at order ǫ within 10%. In d = 2, thermal fluctuations are important (lT = ∞) and the random manifold regime is much reduced. We find a modified Larkin regime with T-dependent exponents B̃(x) ∼ |x| ) and a long distance logarithmic regime. Details can be found in [13]. We denote by Ri the equilibrium position of the lines labeled by an integer i, in the x−y plane, and by u(Ri, z) their in-plane displacements. z denotes the coordinate perpendicular to the planes. For weak disorder a/ξ ≪ 1 it is legitimate to assume that u(Ri, z) is slowly varying on the scale of the lattice and to use a continuum elastic energy, as a function of the continuous variable u(x, z). Impurity disorder is modeled by a gaussian random potential V (x, z) with correlations: V (x, z)V (x′, z′) = ∆(x− x)δ(z− z) where ∆(x) is a short range function of range ξ0 and Fourier transform ∆q. The total energy is: Hel = 1 2 ∫ dxdz[(c11 − c66)(∂αuα) 2 + c66(∂αuβ) 2 + c44(∂zuα) ] + ∫ dxdzV (x, z)ρ(x, z) (1) where α, β denote in-plane coordinates and the density is ρ(x, z) = ∑ i δ(x−Ri − u(Ri, z)). Although we have also performed the calculations directly on the Hamiltonian (1) [13] it is more enlightening to use the following decomposition of the density that keeps track of the discreteness of the lines. In the absence of dislocations, generalizing [14], one intro3 duces the slowly varying field φ(x, z) = x − u(φ(x, z), z). The density can be rewritten as ρ(x, z) = ρ0det[∂αφβ] ∑ K e iK·φ(x,z) ≃ ρ0(1 − ∂αuα(φ(x, z), z) + ∑ K 6=0 e ρK(x)), where ρK(x) = e −iK·u(φ(x,z),z) is the usual translational order parameter defined in terms of the reciprocal lattice vectors K, and ρ0 is the average density. Using the replica trick on (1) the disorder term gives −1/(2T ) ∑ a,b ∫ dxdxdz∆(x − x)ρ(x, z)ρ(x, z). The above decomposition for the density leads to our starting model: Heff = ∫ dqdqz 2(2π)3 ∑ a G 0,αβu a α(q, qz)u a β(q, qz) (2)