Interacting Arrays of Lines and Steps in Random Media

Abstract

The phase diagram of two interacting planar arrays of directed lines in random media is obtained by a renormalization group analysis. The results are discussed in the contexts of the roughening of reconstructed crystal surfaces, and the pinning of flux line arrays in layered superconductors. Among the findings are a glassy flat phase with disordered domain structures, a novel second-order phase transition with continuously varying critical exponents, and the generic disappearance of the glassy "super-rough" phases found previously for a single array. Two kinds of (3x1) microfacets on a (2x1) reconstructed crystal surface. The background (2x1) facets can be on four possible sublattices (marked "a"-"d"). Each (3x1) facet shifts the phase by one sublattice.In this article, we study the effect of point disorders on two interacting species of lines in planar arrays. This problem arises in the study of anisotropically (2x1) reconstructed gold (110) surfaces [12], where two kinds of (3x1) microfacets can be treated as two species of interacting lines [ Fig. 1]. Previous studies of the pure system have revealed a rich phase diagram with a variety of possible phases as a function of the interaction parameters [12,13]. The inclusion of point disorders, say crystalline defects originating from a disordered underlying substrate, induces deformations in the trajectories of the microfacets. Similar issues arise in two layers of magnetically interacting Josephson vortex lines. Performing a renormalization group (RG) analysis in replica space, we are able to obtain a complete picture of the RG-flow. The result is applied to discuss the rich phase diagrams of the anisotropically (2x1) reconstructed crystal surfaces. We also discuss the structure of the glass phases obtained for two planar vortex arrays, and comment on the relevance of our results to the issue of replica-symmetry breaking in a single vortex array.A single species of directed lines confined in a plane containing quenched randomness can be described by the continuum Hamiltonian [2,3] on length scales exceeding the line spacing l. The first part of (1) gives the elastic energy of the line array in terms of a displacement-like scalar field φ(r) (a displacement by l corresponds to a shift of 2π in φ), characterized by an (isotropized) elastic constant K [14]. The second term describes density variations ρ(φ, r) induced by a random potential V (r). The density field has the form ρ(φ(r), r), where ρ 0 = 1/l is the average line density, and r = (x, z) with z along the line direction. The random potential is taken to have zero mean with short-range correlations V (r)V (r ′ ) = gδ 2 (r − r ′ ) of (bare) strength g. An interaction between two such species of lines in the form of r1,r2 V int (r 1 − r 2 )ρ(φ 1 , r 1 )ρ(φ 2 , r 2 ) with a shortranged potential V int [15] leads towith µ = r V int (r) and K µ = µ/8π 2 [14]. We assume the disorder potential V i acting on species i to be statistically identical, with the cross-correlations V 1 (r)V 2 (r ′ ) = g µ δ 2 (r − r ′ ) t...

Topological order in the phase diagram for high-temperature superconductors with point defects

“…For this geometry a Lindemann-like criterion has been derived, which is given below in (1) and relates the stability of the Bragg glass phase to the ratio of the positional correlation length and the FL spacing. These findings are supported by a more rigorous RG analysis [26,27] for a simplified model with only two layers of FLs in a parallel magnetic field. The usual experimental situation with the magnetic field perpendicular to the CuO-layers, which will be considered in the present work, is theoretically less understood mainly because dis-…”

“…There it was found, that unbound dislocations proliferate indeed on the shortest scale at the topological transition described by (1), i.e., in between every layer and thus with a distance l. This suggests that the use of the conventional phenomenological (27) between u * and l, it becomes clear that (1) is the analog of the Lindemann criterion (34) formulated in terms of the underlying transversal scales rather than the corresponding displacements. Using (27), the criterion (1) for the stability of the Bragg glass can be written as…”

Topological order in the vortex-glass phase of high-temperature superconductors

“…Hence the Bragg glass is stable to the spontaneous formation and proliferation of large dislocation loops. This conclusion is further supported by a systematic renormalization-group analysis, the details of which will be given elsewhere [30]. The possibility of a marginally stable Bragg glass for weakly-disorder sample was first suggested in Ref.…”

“…This is evaluated using a Gaussian approximation which can be justified [30] in a controlled fashion and should be reasonably accurate. φφ H contains contributions from (i) the quasi 2D VG regime φφ 2D which dominates for µ ≈ 0; (ii) the 3D VG regime φφ 3D ; and (iii) thermal fluctuations on scales smaller than the correlation length ℓ for large µ.…”

“…For µ < µ c , the system "melts" into a stack of decoupled 2D VG's, distinguished from the Bragg glass by a vanishing shear modulus µ = 0 [33]. The transition at µ = µ c is first order, with a jump in µ [30]. Our variational calculation yields a prefactor c ≈ 5 which depends very weakly on temperatures, as long as we are away from the melting temperature of the pure system [30].…”