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arXiv:cond-mat/9905058 · Physical review. B, Condensed matter

Simulation of the zero-temperature behavior of a three-dimensional elastic medium

David McNamara, A. Alan Middleton, Chen Zeng

Published Oct 1, 1999• 18 citations
arXivHTMLPDF

Abstract

We have performed numerical simulation of a three-dimensional elastic medium, with scalar displacements, subject to quenched disorder. In the absence of topological defects this system is equivalent to a $(3+1)$-dimensional interface subject to a periodic pinning potential. We have applied an efficient combinatorial optimization algorithm to generate exact ground states for this interface representation. Our results indicate that this Bragg glass is characterized by power law divergences in the structure factor $S(k)\ensuremath{\sim}{\mathrm{Ak}}^{\ensuremath{-}3}.$ We have found numerically consistent values of the coefficient A for two lattice discretizations of the medium, supporting universality for A in the isotropic systems considered here. We also examine the response of the ground state to the change in boundary conditions that corresponds to introducing a single dislocation loop encircling the system. The rearrangement of the ground state caused by this change is equivalent to the domain wall of elastic deformations which span the dislocation loop. Our results indicate that these domain walls are highly convoluted, with a fractal dimension ${d}_{f}=2.60(5).$ We also discuss the implications of the domain wall energetics for the stability of the Bragg glass phase. Elastic excitations similar to these domain walls arise when the pinning potential is slightly perturbed. As in other disordered systems, perturbations of relative strength \ensuremath{\delta} introduce a new length scale ${L}^{*}\ensuremath{\sim}{\ensuremath{\delta}}^{\ensuremath{-}1/\ensuremath{\zeta}}$ beyond which the perturbed ground state becomes uncorrelated with the reference (unperturbed) ground state. We have performed a scaling analysis of the response of the ground state to the perturbations and obtain $\ensuremath{\zeta}=0.385(40).$ This value is consistent with the scaling relation $\ensuremath{\zeta}{=d}_{f}/2\ensuremath{-}\ensuremath{\theta},$ where \ensuremath{\theta} characterizes the scaling of the energy fluctuations of low energy excitations.

DOI
10.1103/physrevb.60.10062

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We have performed numerical simulation of a three-dimensional elastic medium, with scalar displacements, subject to quenched disorder. In the absence of topological defects this system is equivalent to a (3+1)(3+1)(3+1)-dimensional interface subject to a periodic pinning potential. We have applied an efficient combinatorial optimization algorithm to generate exact ground states for this interface representation. Our results indicate that this Bragg glass is characterized by power law divergences in the structure factor S(k)\ensuremath∼Ak\ensuremath−3.S(k)\ensuremath{\sim}{\mathrm{Ak}}^{\ensuremath{-}3}.S(k)\ensuremath∼Ak\ensuremath−3. We have found numerically consistent values of the coefficient A for two lattice discretizations of the medium, supporting universality for A in the isotropic systems considered here. We also examine the response of the ground state to the change in boundary conditions that corresponds to introducing a single dislocation loop encircling the system. The rearrangement of the ground state caused by this change is equivalent to the domain wall of elastic deformations which span the dislocation loop. Our results indicate that these domain walls are highly convoluted, with a fractal dimension df=2.60(5).{d}_{f}=2.60(5).df​=2.60(5). We also discuss the implications of the domain wall energetics for the stability of the Bragg glass phase. Elastic excitations similar to these domain walls arise when the pinning potential is slightly perturbed. As in other disordered systems, perturbations of relative strength \ensuremath{\delta} introduce a new length scale L∗\ensuremath∼\ensuremathδ\ensuremath−1/\ensuremathζ{L}^{*}\ensuremath{\sim}{\ensuremath{\delta}}^{\ensuremath{-}1/\ensuremath{\zeta}}L∗\ensuremath∼\ensuremathδ\ensuremath−1/\ensuremathζ beyond which the perturbed ground state becomes uncorrelated with the reference (unperturbed) ground state. We have performed a scaling analysis of the response of the ground state to the perturbations and obtain \ensuremathζ=0.385(40).\ensuremath{\zeta}=0.385(40).\ensuremathζ=0.385(40). This value is consistent with the scaling relation \ensuremathζ=df/2\ensuremath−\ensuremathθ,\ensuremath{\zeta}{=d}_{f}/2\ensuremath{-}\ensuremath{\theta},\ensuremathζ=df​/2\ensuremath−\ensuremathθ, where \ensuremath{\theta} characterizes the scaling of the energy fluctuations of low energy excitations.
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The XY\mathrm{XY}XY model with quenched random disorder is studied by a zero-temperature domain wall renormalization group method in two dimensions (2D) and three dimensions (3D). Instead of the usual phase representation we use the charge (vortex) representation to compute the domain wall, or defect, energy. For the gauge glass corresponding to the maximum disorder we reconfirm earlier predictions that there is no ordered phase in 2D but an ordered phase can exist in 3D at low temperature. However, our simulations yield spin stiffness exponents ${\ensuremath{\theta}}_{s}\ensuremath{\approx}\ensuremath{-}0.36$ in 2D and ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.31$ in 3D, which are considerably larger than previous estimates and strongly suggest that the lower critical dimension is less than three. For the $\ifmmode\pm\else\textpm\fi{}J \mathrm{XY}$ spin glass in 3D, we obtain a spin stiffness exponent ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.10$ which supports the existence of spin glass order at finite temperature in contrast with previous estimates which obtain ${\ensuremath{\theta}}_{s}<0.$ Our method also allows us to study renormalization group flows of both the coupling constant and the disorder strength with a length scale L. Our results are consistent with recent analytic and numerical studies suggesting the absence of a reentrant transition in 2D at low temperature. Some possible consequences and connections with real vortex systems are discussed.
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We study large-scale, low-energy excitations in the Ising spin glass with Gaussian interactions in two dimensions at zero temperature, using an optimization algorithm to determine exact ground states. Periodic boundary conditions are applied. Our results for the fractal dimension of the surface, ds,{d}_{s},ds​, and stiffness exponent, ${\ensuremath{\theta}}^{\ensuremath{'}},$ for ``droplet'' excitations, are in reasonable agreement with estimates from ``domain wall'' calculations, and so support the predictions of the ``droplet theory.'' Restricting our analysis to small lattices, we do not find an effective value of ${\ensuremath{\theta}}^{\ensuremath{'}}$ close to $\ensuremath{-}0.47$ as has been recently proposed. The effects of averaging over droplets of different sizes are studied and are also found to be too small to give ${\ensuremath{\theta}}^{\ensuremath{'}}\ensuremath{\approx}\ensuremath{-}0.47$ for smaller sizes. Larger corrections to finite-size scaling would be needed in three and four dimensions in order for the numerical data used to support the ``TNT'' scenario to be compatible with the droplet theory prediction that the stiffness exponent is the same for droplets and domain walls.
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.