Critical slowing down in polynomial time algorithms

Combinatorial optimization algorithms which compute exact ground state configurations in disordered magnets are seen to exhibit critical slowing down at zero temperature phase transitions. Using arguments based on the physical picture of the model, including vanishing stiffness on scales beyond the correlation length and the ground state degeneracy, the number of operations carried out by one such algorithm, the push-relabel algorithm for the random field Ising model, can be estimated. Some scaling can also be predicted for the 2D spin glass.Comment: 4 pp., 3 fig

Simulation of the Zero Temperature Behavior of a 3-Dimensional Elastic Medium

We have performed numerical simulation of a 3-dimensional elastic medium, with scalar displacements, subject to quenched disorder. We applied an efficient combinatorial optimization algorithm to generate exact ground states for an interface representation. Our results indicate that this Bragg glass is characterized by power law divergences in the structure factor $S(k)\sim A k^{-3}$. 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. Our results indicate that the domain walls formed by this change 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. As in other disordered systems, perturbations of relative strength $\delta$ introduce a new length scale $L^* \sim \delta^{-1/\zeta}$ beyond which the perturbed ground state becomes uncorrelated with the reference (unperturbed) ground state. We have performed scaling analysis of the response of the ground state to the perturbations and obtain $\zeta = 0.385(40)$. This value is consistent with the scaling relation $\zeta=d_f/2- \theta$, where $\theta$ characterizes the scaling of the energy fluctuations of low energy excitations.Comment: 20 pages, 13 figure

Computational Complexity of Determining the Barriers to Interface Motion in Random Systems

The low-temperature driven or thermally activated motion of several condensed matter systems is often modeled by the dynamics of interfaces (co-dimension-1 elastic manifolds) subject to a random potential. Two characteristic quantitative features of the energy landscape of such a many-degree-of-freedom system are the ground-state energy and the magnitude of the energy barriers between given configurations. While the numerical determination of the former can be accomplished in time polynomial in the system size, it is shown here that the problem of determining the latter quantity is NP-complete. Exact computation of barriers is therefore (almost certainly) much more difficult than determining the exact ground states of interfaces.Comment: 8 pages, figures included, to appear in Phys. Rev.

Degeneracy Algorithm for Random Magnets

It has been known for a long time that the ground state problem of random magnets, e.g. random field Ising model (RFIM), can be mapped onto the max-flow/min-cut problem of transportation networks. I build on this approach, relying on the concept of residual graph, and design an algorithm that I prove to be exact for finding all the minimum cuts, i.e. the ground state degeneracy of these systems. I demonstrate that this algorithm is also relevant for the study of the ground state properties of the dilute Ising antiferromagnet in a constant field (DAFF) and interfaces in random bond magnets.Comment: 17 pages(Revtex), 8 Postscript figures(5color) to appear in Phys. Rev. E 58, December 1st (1998

Breakdown of Simple Scaling in Abelian Sandpile Models in One Dimension

We study the abelian sandpile model on decorated one dimensional chains. We determine the structure and the asymptotic form of distribution of avalanche-sizes in these models, and show that these differ qualitatively from the behavior on a simple linear chain. We find that the probability distribution of the total number of topplings $s$ on a finite system of size $L$ is not described by a simple finite size scaling form, but by a linear combination of two simple scaling forms $Prob_L(s) = 1/L f_1(s/L) + 1/L^2 f_2(s/L^2)$, for large $L$, where $f_1$ and $f_2$ are some scaling functions of one argument.Comment: 10 pages, revtex, figures include

Ab-initio self-consistent Gorkov-Green's function calculations of semi-magic nuclei - I. Formalism at second order with a two-nucleon interaction

An ab-initio calculation scheme for finite nuclei based on self-consistent Green's functions in the Gorkov formalism is developed. It aims at describing properties of doubly-magic and semi-magic nuclei employing state-of-the-art microscopic nuclear interactions and explicitly treating pairing correlations through the breaking of U(1) symmetry associated with particle number conservation. The present paper introduces the formalism, necessary to undertake applications at (self-consistent) second-order using two-nucleon interactions, in a detailed and self-contained fashion. First applications of such a scheme will be reported soon in a forthcoming publication. Future works will extend the present scheme to include three-nucleon interactions and implement more advanced truncation schemes.Comment: 38 page

Static and Dynamic Properties of Inhomogeneous Elastic Media on Disordered Substrate

The pinning of an inhomogeneous elastic medium by a disordered substrate is studied analytically and numerically. The static and dynamic properties of a $D$-dimensional system are shown to be equivalent to those of the well known problem of a $D$-dimensional random manifold embedded in $(D+D)$-dimensions. The analogy is found to be very robust, applicable to a wide range of elastic media, including those which are amorphous or nearly-periodic, with local or nonlocal elasticity. Also demonstrated explicitly is the equivalence between the dynamic depinning transition obtained at a constant driving force, and the self-organized, near-critical behavior obtained by a (small) constant velocity drive.Comment: 20 pages, RevTeX. Related (p)reprints also available at http://matisse.ucsd.edu/~hwa/pub.htm

Moving Wigner Glasses and Smectics: Dynamics of Disordered Wigner Crystals

We examine the dynamics of driven classical Wigner solids interacting with quenched disorder from charged impurities. For strong disorder, the initial motion is plastic -- in the form of crossing winding channels. For increasing drive, the disordered Wigner glass can reorder to a moving Wigner smectic -- with the electrons moving in non-crossing 1D channels. These different dynamic phases can be related to the conduction noise and I(V) curves. For strong disorder, we show criticality in the voltage onset just above depinning. We also obtain the dynamic phase diagram for driven Wigner solids and prove that there is a finite threshold for transverse sliding, recently found experimentally.Comment: 4 pages, 4 postscript figure

String-inspired cosmology: Late time transition from scaling matter era to dark energy universe caused by a Gauss-Bonnet coupling

The Gauss-Bonnet (GB) curvature invariant coupled to a scalar field $\phi$ can lead to an exit from a scaling matter-dominated epoch to a late-time accelerated expansion, which is attractive to alleviate the coincident problem of dark energy. We derive the condition for the existence of cosmological scaling solutions in the presence of the GB coupling for a general scalar-field Lagrangian density $p(\phi, X)$, where $X=-(1/2)(\nabla \phi)^2$ is a kinematic term of the scalar field. The GB coupling and the Lagrangian density are restricted to be in the form $f(\phi) \propto e^{\lambda \phi}$ and $p=Xg (Xe^{\lambda \phi})$, respectively, where $\lambda$ is a constant and $g$ is an arbitrary function. We also derive fixed points for such a scaling Lagrangian with a GB coupling $f(\phi) \propto e^{\mu \phi}$ and clarify the conditions under which the scaling matter era is followed by a de-Sitter solution which can appear in the presence of the GB coupling. Among scaling models proposed in the current literature, we find that the models which allow such a cosmological evolution are an ordinary scalar field with an exponential potential and a tachyon field with an inverse square potential, although the latter requires a coupling between dark energy and dark matter.Comment: 18 pages, 4 figures, version to appear in JCA

Interface Motion in Random Media at Finite Temperature

We have studied numerically the dynamics of a driven elastic interface in a random medium, focusing on the thermal rounding of the depinning transition and on the behavior in the $T=0$ pinned phase. Thermal effects are quantitatively more important than expected from simple dimensional estimates. For sufficient low temperature the creep velocity at a driving force equal to the $T=0$ depinning force exhibits a power-law dependence on $T$, in agreement with earlier theoretical and numerical predictions for CDW's. We have also examined the dynamics in the $T=0$ pinned phase resulting from slowly increasing the driving force towards threshold. The distribution of avalanche sizes $S_\|$ decays as $S_\|^{-1-\kappa}$, with $\kappa = 0.05\pm 0.05$, in agreement with recent theoretical predictions.Comment: harvmac.tex, 30 pages, including 9 figures, available upon request. SU-rm-94073