Finite size corrections to disordered systems on Erd\"{o}s-R\'enyi random graphs

We study the finite size corrections to the free energy density in disorder spin systems on sparse random graphs, using both replica theory and cavity method. We derive an analytical expressions for the $O(1/N)$ corrections in the replica symmetric phase as a linear combination of the free energies of open and closed chains. We perform a numerical check of the formulae on the Random Field Ising Model at zero temperature, by computing finite size corrections to the ground state energy density.Comment: Submitted to PR

A New Phenomenology for the Disordered Mixed Phase

A universal phase diagram for type-II superconductors with weak point pinning disorder is proposed. In this phase diagram, two thermodynamic phase transitions generically separate a ``Bragg glass'' from the disordered liquid. Translational correlations in the intervening ``multi-domain glass'' phase are argued to exhibit a significant degree of short-range order. This phase diagram differs significantly from the currently accepted one but provides a more accurate description of experimental data on high and low-T$_c$ materials, simulations and current theoretical understanding.Comment: 15 pages including 2 postscript figures, minor changes in published versio

Stability of Elastic Glass Phases in Random Field XY Magnets and Vortex Lattices in Type II Superconductors

A description of a dislocation-free elastic glass phase in terms of domain walls is developed and used as the basis of a renormalization group analysis of the energetics of dislocation loops added to the system. It is found that even after optimizing over possible paths of large dislocation loops, their energy is still very likely to be positive when the dislocation core energy is large. This implies the existence of an equilibrium elastic glass phase in three dimensional random field X-Y magnets, and a dislocation free, bond-orientationally ordered ``Bragg glass'' phase of vortices in dirty Type II superconductors.Comment: 12 pages, Revtex, no figures, submitted to Phys Rev Letter

The 3-d Random Field Ising Model at zero temperature

We study numerically the zero temperature Random Field Ising Model on cubic lattices of various linear sizes $L$ in three dimensions. For each random field configuration we vary the ferromagnetic coupling strength $J$. We find that in the infinite volume limit the magnetization is discontinuous in $J$. The energy and its first $J$ derivative are continuous. The approch to the thermodynamic limit is slow, behaving like $L^{-p}$ with $p \sim .8$ for the gaussian distribution of the random field. We also study the bimodal distribution $h_{i} = \pm h$, and we find similar results for the magnetization but with a different value of the exponent $p \sim .6$. This raises the question of the validity of universality for the random field problem.Comment: 8 pages, 3 PostScript Figure

A cluster Monte Carlo algorithm with a conserved order parameter

We propose a cluster simulation algorithm for statistical ensembles with fixed order parameter. We use the tethered ensemble, which features Helmholtz's effective potential rather than Gibbs's free energy, and in which canonical averages are recovered with arbitrary accuracy. For the D = 2,3 Ising model our method's critical slowing down is comparable to that of canonical cluster algorithms. Yet, we can do more than merely reproduce canonical values. As an example, we obtain a competitive value for the 3D Ising anomalous dimension from the maxima of the effective potential.Comment: 4 pages, 2 color figures. Minor improvements and update of table

Glassy Motion of Elastic Manifolds

We discuss the low-temperature dynamics of an elastic manifold driven through a random medium. For driving forces well below the $T=0$ depinning force, the medium advances via thermally activated hops over the energy barriers separating favorable metastable states. We show that the distribution of waiting times for these hopping processes scales as a power-law. This power-law distribution naturally yields a nonlinear glassy response for the driven medium, $v\sim\exp(-{\rm const}\times F^{-\mu})$.Comment: 4pages, revte

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

Mott Transition vs Multicritical Phenomenon of Superconductivity and Antiferromagnetism -- Application to κ\kappa-(BEDT-TTF)2_2X --

Interplay between the Mott transition and the multicritical phenomenon of d-wave superconductivity (SC) and antiferromagnetism (AF) is studied theoretically. We describe the Mott transition, which is analogous to a liquid-gas phase transition, in terms of an Ising-type order parameter $\eta$. We reveal possible mean-field phase diagrams produced by this interplay. Renormalization group analysis up to one-loop order gives flows of coupling constants, which in most cases lead to fluctuation-induced first-order phase transitions even when the SO(5) symmetry exists betwen the SC and AF. Behaviors of various physical quantities around the Mott critical point are predicted. Experiments in $\kappa$-(BEDT-TTF)$_2$X are discussed from this viewpoint.Comment: 4 pages, 9 figures, to appear in J. Phys. Soc. Jp

Phase Transitions in the Two-Dimensional XY Model with Random Phases: a Monte Carlo Study

We study the two-dimensional XY model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, $\sigma_{c}$, the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to T=0 in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder $T_{KT}\rightarrow 0$ and for $\sigma > \sigma_{c}$ there is no quasi-ordered phase. At zero temperature there is a phase transition between two different glassy states at $\sigma_{c}$. The functional dependence of the correlation length on $\sigma$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.Comment: LaTex file and 18 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.