Specific Heat of Quantum Elastic Systems Pinned by Disorder

We present the detailed study of the thermodynamics of vibrational modes in disordered elastic systems such as the Bragg glass phase of lattices pinned by quenched impurities. Our study and our results are valid within the (mean field) replica Gaussian variational method. We obtain an expression for the internal energy in the quantum regime as a function of the saddle point solution, which is then expanded in powers of $\hbar$ at low temperature $T$. In the calculation of the specific heat $C_v$ a non trivial cancellation of the term linear in $T$ occurs, explicitly checked to second order in $\hbar$. The final result is $C_v \propto T^3$ at low temperatures in dimension three and two. The prefactor is controlled by the pinning length. This result is discussed in connection with other analytical or numerical studies.Comment: 14 page

Specific heat of the quantum Bragg Glass

We study the thermodynamics of the vibrational modes of a lattice pinned by impurity disorder in the absence of topological defects (Bragg glass phase). Using a replica variational method we compute the specific heat $C_v$ in the quantum regime and find $C_v \propto T^3$ at low temperatures in dimension three and two. The prefactor is controlled by the pinning length. The non trivial cancellation of the linear term in $C_v$ arises from the so-called marginality condition and has important consequences for other mean field models.Comment: 5 pages, RevTex, strongly revised versio

Sinai model in presence of dilute absorbers

We study the Sinai model for the diffusion of a particle in a one dimension random potential in presence of a small concentration $\rho$ of perfect absorbers using the asymptotically exact real space renormalization method. We compute the survival probability, the averaged diffusion front and return probability, the two particle meeting probability, the distribution of total distance traveled before absorption and the averaged Green's function of the associated Schrodinger operator. Our work confirms some recent results of Texier and Hagendorf obtained by Dyson-Schmidt methods, and extends them to other observables and in presence of a drift. In particular the power law density of states is found to hold in all cases. Irrespective of the drift, the asymptotic rescaled diffusion front of surviving particles is found to be a symmetric step distribution, uniform for $|x| < {1/2} \xi(t)$, where $\xi(t)$ is a new, survival length scale ($\xi(t)=T \ln t/\sqrt{\rho}$ in the absence of drift). Survival outside this sharp region is found to decay with a larger exponent, continuously varying with the rescaled distance $x/\xi(t)$. A simple physical picture based on a saddle point is given, and universality is discussed.Comment: 21 pages, 2 figure

Random elastic networks : strong disorder renormalization approach

For arbitrary networks of random masses connected by random springs, we define a general strong disorder real-space renormalization (RG) approach that generalizes the procedures introduced previously by Hastings [Phys. Rev. Lett. 90, 148702 (2003)] and by Amir, Oreg and Imry [Phys. Rev. Lett. 105, 070601 (2010)] respectively. The principle is to eliminate iteratively the elementary oscillating mode of highest frequency associated with either a mass or a spring constant. To explain the accuracy of the strong disorder RG rules, we compare with the Aoki RG rules that are exact at fixed frequency.Comment: 8 pages, v2=final versio

A Crash Course on Aging

In these lecture notes I describe some of the main theoretical ideas emerged to explain the aging dynamics. This is meant to be a very short introduction to aging dynamics and no previous knowledge is assumed. I will go through simple examples that allow one to grasp the main results and predictions.Comment: Lecture Notes (22 pages) given at "Unifying Concepts in Glass Physics III", Bangalore (2004); to be published in JSTA

Functional Renormalization Group and the Field Theory of Disordered Elastic Systems

We study elastic systems such as interfaces or lattices, pinned by quenched disorder. To escape triviality as a result of ``dimensional reduction'', we use the functional renormalization group. Difficulties arise in the calculation of the renormalization group functions beyond 1-loop order. Even worse, observables such as the 2-point correlation function exhibit the same problem already at 1-loop order. These difficulties are due to the non-analyticity of the renormalized disorder correlator at zero temperature, which is inherent to the physics beyond the Larkin length, characterized by many metastable states. As a result, 2-loop diagrams, which involve derivatives of the disorder correlator at the non-analytic point, are naively "ambiguous''. We examine several routes out of this dilemma, which lead to a unique renormalizable field-theory at 2-loop order. It is also the only theory consistent with the potentiality of the problem. The beta-function differs from previous work and the one at depinning by novel "anomalous terms''. For interfaces and random bond disorder we find a roughness exponent zeta = 0.20829804 epsilon + 0.006858 epsilon^2, epsilon = 4-d. For random field disorder we find zeta = epsilon/3 and compute universal amplitudes to order epsilon^2. For periodic systems we evaluate the universal amplitude of the 2-point function. We also clarify the dependence of universal amplitudes on the boundary conditions at large scale. All predictions are in good agreement with numerical and exact results, and an improvement over one loop. Finally we calculate higher correlation functions, which turn out to be equivalent to those at depinning to leading order in epsilon.Comment: 42 pages, 41 figure

Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks

We use the Real Space Renormalization Group (RSRG) method to study extreme value statistics for a variety of Brownian motions, free or constrained such as the Brownian bridge, excursion, meander and reflected bridge, recovering some standard results, and extending others. We apply the same method to compute the distribution of extrema of Bessel processes. We briefly show how the continuous time random walk (CTRW) corresponds to a non standard fixed point of the RSRG transformation.Comment: 24 pages, 5 figure

Exact multilocal renormalization on the effective action : application to the random sine Gordon model statics and non-equilibrium dynamics

We extend the exact multilocal renormalization group (RG) method to study the flow of the effective action functional. This important physical quantity satisfies an exact RG equation which is then expanded in multilocal components. Integrating the nonlocal parts yields a closed exact RG equation for the local part, to a given order in the local part. The method is illustrated on the O(N) model by straightforwardly recovering the $\eta$ exponent and scaling functions. Then it is applied to study the glass phase of the Cardy-Ostlund, random phase sine Gordon model near the glass transition temperature. The static correlations and equilibrium dynamical exponent $z$ are recovered and several new results are obtained. The equilibrium two-point scaling functions are obtained. The nonequilibrium, finite momentum, two-time $t,t'$ response and correlations are computed. They are shown to exhibit scaling forms, characterized by novel exponents $\lambda_R \neq \lambda_C$, as well as universal scaling functions that we compute. The fluctuation dissipation ratio is found to be non trivial and of the form $X(q^z (t-t'), t/t')$. Analogies and differences with pure critical models are discussed.Comment: 33 pages, RevTe

Specific heat of classical disordered elastic systems

We study the thermodynamics of disordered elastic systems, applied to vortex lattices in the Bragg glass phase. Using the replica variational method we compute the specific heat of pinned vortons in the classical limit. We find that the contribution of disorder is positive, linear at low temperature, and exhibits a maximum. It is found to be important compared to other contributions, e.g. core electrons, mean field and non linear elasticity that we evaluate. The contribution of droplets is subdominant at weak disorder in $d=3$.Comment: 4 pages, RevTe

On the definition of a unique effective temperature for non-equilibrium critical systems

We consider the problem of the definition of an effective temperature via the long-time limit of the fluctuation-dissipation ratio (FDR) after a quench from the disordered state to the critical point of an O(N) model with dissipative dynamics. The scaling forms of the response and correlation functions of a generic observable are derived from the solutions of the corresponding Renormalization Group equations. We show that within the Gaussian approximation all the local observables have the same FDR, allowing for a definition of a unique effective temperature. This is no longer the case when fluctuations are taken into account beyond that approximation, as shown by a computation up to the first order in the epsilon-expansion for two quadratic observables. This implies that, contrarily to what often conjectured, a unique effective temperature can not be defined for this class of models.Comment: 32 pages, 5 figures. Minor changes, published versio