arXiv subjects

Pierre Le Doussal

Publications and source records attributed to Pierre Le Doussal.

At least 235 records · Page 13Linked to original sources

arXiv2004

Disorder induced transitions in layered Coulomb gases and application to flux lattices in superconductors

A layered system of charges with logarithmic interaction parallel to the layers and random dipoles in each layer is studied via a variational method and an energy rationale. These methods reproduce the known phase diagram for a single layer where charges unbind by increasing either temperature or disorder, as well as a freezing first order transition within the ordered phase. Increasing interlayer coupling leads to successive transitions in which charge rods correlated in N>1 neighboring layers are unbounded by weaker disorder. Increasing disorder leads to transitions between different N phases. The method is applied to flux lattices in layered superconductors in the limit of vanishing Josephson coupling. The unbinding charges are point defects in the flux lattice, i.e. vacancies or interstitials. We show that short range disorder generates random dipoles for these defects. We predict and accurately locate a disorder-induced defect-unbinding transition with loss of superconducting order, upon increase of disorder. While N=1 charges dominate for most system parameters, we propose that in multi-layer superconductors defect rods can be realized.

cond-mat.dis-nn↗

arXiv2004

Aging in the glass phase of a 2D random periodic elastic system

Using RG we investigate the non-equilibrium relaxation of the (Cardy-Ostlund) 2D random Sine-Gordon model, which describes pinned arrays of lines. Its statics exhibits a marginal ($θ=0$) glass phase for $T<T_g$ described by a line of fixed points. We obtain the universal scaling functions for two-time dynamical response and correlations near $T_g$ for various initial conditions, as well as the autocorrelation exponent. The fluctuation dissipation ratio is found to be non-trivial and continuously dependent on $T$.

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arXiv2004

Thermal fluctuations in pinned elastic systems: field theory of rare events and droplets

Using the functional renormalization group (FRG) we study the thermal fluctuations of elastic objects, described by a displacement field u and internal dimension d, pinned by a random potential at low temperature T, as prototypes for glasses. A challenge is how the field theory can describe both typical (minimum energy T=0) configurations, as well as thermal averages which, at any non-zero T as in the phenomenological droplet picture, are dominated by rare degeneracies between low lying minima. We show that this occurs through an essentially non-perturbative *thermal boundary layer* (TBL) in the (running) effective action Gamma[u] at T>0 for which we find a consistent scaling ansatz to all orders. The TBL resolves the singularities of the T=0 theory and contains rare droplet physics. The formal structure of this TBL is explored around d=4 using a one loop Wilson RG. A more systematic Exact RG (ERG) method is employed and tested on d=0 models. There we obtain precise relations between TBL quantities and droplet probabilities which are checked against exact results. We illustrate how the TBL scaling remains consistent to all orders in higher d using the ERG and how droplet picture results can be retrieved. Finally, we solve for d=0,N=1 the formidable "matching problem" of how this T>0 TBL recovers a critical T=0 field theory. We thereby obtain the beta-function at T=0, *all ambiguities removed*, displayed here up to four loops. A discussion of d>4 case and an exact solution at large d are also provided.

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arXiv2004

Low-temperature properties of some disordered systems from the statistical properties of nearly degenerate two-level excitations

The thermal fluctuations that exist at very low temperature in disordered systems are often attributed to the existence of some two-level excitations. In this paper, we revisit this question via the explicit studies of the following 1D models (i) a particle in 1D random potentials (ii) the random field Ising chain with continuous disorder distribution. In both cases, we define precisely the `two-level' excitations and their statistical properties, and we show that their contributions to various observables are in full agreement at low temperature with the the rigorous results obtained independently. The statistical properties of these two-level excitations moreover yield simple identities at order $T$ in temperature for some generating functions of thermal cumulants. For the random-field Ising chain, in the regime where the Imry-Ma length is large, we obtain that the specific heat is dominated by small non-universal excitations, that depend on the details of the disorder distribution, whereas the magnetic susceptibility and the Edwards-Anderson order parameter are dominated by universal large excitations, whose statistical properties only depend on the variance of the initial disorder via the Imry-Ma length.

cond-mat.dis-nn↗

arXiv2004

Derivation of the Functional Renormalization Group Beta-Function at order 1/N for Manifolds Pinned by Disorder

In an earlier publication, we have introduced a method to obtain, at large N, the effective action for d-dimensional manifolds in a N-dimensional disordered environment. This allowed to obtain the Functional Renormalization Group (FRG) equation for N=infinity and was shown to reproduce, with no need for ultrametric replica symmetry breaking, the predictions of the Mezard-Parisi solution. Here we compute the corrections at order 1/N. We introduce two novel complementary methods, a diagrammatic and an algebraic one, to perform the complicated resummation of an infinite number of loops, and derive the beta-function of the theory to order 1/N. We present both the effective action and the corresponding functional renormalization group equations. The aim is to explain the conceptual basis and give a detailed account of the novel aspects of such calculations. The analysis of the FRG flow, comparison with other studies, and applications, e.g. to the strong-coupling phase of the Kardar-Parisi-Zhang equation are examined in a subsequent publication.

cond-mat.dis-nn↗

arXiv2003

Broad relaxation spectrum and the field theory of glassy dynamics for pinned elastic systems

We study thermally activated, low temperature equilibrium dynamics of elastic systems pinned by disorder using one loop functional renormalization group (FRG). Through a series of increasingly complete approximations, we investigate how the field theory reveals the glassy nature of the dynamics, in particular divergent barriers and barrier distributions controling the spectrum of relaxation times. A naive single relaxation time approximation for each wavevector is found to be unsatisfactory. A second approximation based on a random friction model, yields a size (L) dependent log-normal distribution of relaxation times (mean barriers ~L^θand variance ~ L^{θ/2}) and a procedure to estimate dynamical scaling functions. Finally, we study the full structure of the running dynamical effective action within the field theory. We find that relaxation time distributions are non-trivial (broad but not log-normal) and encoded in a closed hierarchy of FRG equations. A thermal boundary layer ansatz (TBLA) appears as a consistent solution. It extends the one discovered in the statics which was shown to embody droplet thermal fluctuations. Although perturbative control remains a challenge, the structure of the dynamical TBLA which encodes barrier distributions opens the way for deeper understanding of the field theory approach to glasses.

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arXiv2003

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 $d=3$.

cond-mat↗

arXiv2003

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.

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arXiv2003

Functional Renormalization Group at Large N for Disordered Elastic Systems, and Relation to Replica Symmetry Breaking

We study the replica field theory which describes the pinning of elastic manifolds of arbitrary internal dimension d in a random potential, with the aim of bridging the gap between mean field and renormalization theory. The full effective action is computed exactly in the limit of large embedding space dimension N. The second cumulant of the renormalized disorder obeys a closed self-consistent equation. It is used to derive a Functional Renormalization Group (FRG) equation valid in any dimension d, which correctly matches the Balents-Fisher result to first order in epsilon=4-d. We analyze in detail the solutions of the large-N FRG for both long-range and short-range disorder, at zero and finite temperature. We find consistent agreement with the results of Mezard Parisi (MP) from the Gaussian variational method (GVM) in the case where full replica symmetry breaking (RSB) holds there. We prove that the cusplike non-analyticity in the large N FRG appears at a finite scale, corresponding to the instability of the replica symmetric solution of MP. We show that the FRG exactly reproduces, for any disorder correlator and with no need to invoke Parisi's spontaneous RSB, the non-trivial result of the GVM for small overlap. A formula is found yielding the complete RSB solution for all overlaps. Since our saddle-point equations for the effective action contain both the MP equations and the FRG, it can be used to describe the crossover from FRG to RSB. A qualitative analysis of this crossover is given, as well as a comparison with previous attempts to relate FRG to GVM. Finally, we discuss applications to other problems and new perspectives.

cond-mat↗

arXiv2003

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.

cond-mat↗

arXiv2003

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 $η$ 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 $λ_R \neq λ_C$, 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.

cond-mat↗

arXiv2003

Variable range hopping and quantum creep in one dimension

We study the quantum non linear response to an applied electric field $E$ of a one dimensional pinned charge density wave or Luttinger liquid in presence of disorder. From an explicit construction of low lying metastable states and of bounce instanton solutions between them, we demonstrate quantum creep $v = e^{- c/E^{1/2}}$ as well as a sharp crossover at $E=E^*$ towards a linear response form consistent with variable range hopping arguments, but dependent only on electronic degrees of freedom.

cond-mat.dis-nn↗

arXiv2003

Universal interface width distributions at the depinning threshold

We compute the probability distribution of the interface width at the depinning threshold, using recent powerful algorithms. It confirms the universality classes found previously. In all cases, the distribution is surprisingly well approximated by a generalized Gaussian theory of independant modes which decay with a characteristic propagator G(q)=1/q^(d+2 zeta); zeta, the roughness exponent, is computed independently. A functional renormalization analysis explains this result and allows to compute the small deviations, i.e. a universal kurtosis ratio, in agreement with numerics. We stress the importance of the Gaussian theory to interpret numerical data and experiments.

cond-mat↗

arXiv2003

Higher correlations, universal distributions and finite size scaling in the field theory of depinning

Recently we constructed a renormalizable field theory up to two loops for the quasi-static depinning of elastic manifolds in a disordered environment. Here we explore further properties of the theory. We show how higher correlation functions of the displacement field can be computed. Drastic simplifications occur, unveiling much simpler diagrammatic rules than anticipated. This is applied to the universal scaled width-distribution. The expansion in d=4-epsilon predicts that the scaled distribution coincides to the lowest orders with the one for a Gaussian theory with propagator G(q)=1/q^(d+2 ζ), zeta being the roughness exponent. The deviations from this Gaussian result are small and involve higher correlation functions, which are computed here for different boundary conditions. Other universal quantities are defined and evaluated: We perform a general analysis of the stability of the fixed point. We find that the correction-to-scaling exponent is omega=-epsilon and not -epsilon/3 as used in the analysis of some simulations. A more detailed study of the upper critical dimension is given, where the roughness of interfaces grows as a power of a logarithm instead of a pure power.

cond-mat↗

arXiv2002

Functional renormalization group for anisotropic depinning and relation to branching processes

Using the functional renormalization group, we study the depinning of elastic objects in presence of anisotropy. We explicitly demonstrate how the KPZ-term is always generated, even in the limit of vanishing velocity, except where excluded by symmetry. We compute the beta-function to one loop taking properly into account the non-analyticity. This gives rise to additional terms, missed in earlier studies. A crucial question is whether the non-renormalization of the KPZ-coupling found at 1-loop order extends beyond the leading one. Using a Cole-Hopf-transformed theory we argue that it is indeed uncorrected to all orders. The resulting flow-equations describe a variety of physical situations. A careful analysis of the flow yields several non-trivial fixed points. All these fixed points are transient since they possess one unstable direction towards a runaway flow, which leaves open the question of the upper critical dimension. The runaway flow is dominated by a Landau-ghost-mode. For SR elasticity, using the Cole-Hopf transformed theory we identify a non-trivial 3-dimensional subspace which is invariant to all orders and contains all above fixed points as well as the Landau-mode. It belongs to a class of theories which describe branching and reaction-diffusion processes, of which some have been mapped onto directed percolation.

cond-mat↗

arXiv2002

Energy dynamics in the Sinai model

We study the time dependent potential energy $W(t)=U(x(0)) - U(x(t))$ of a particle diffusing in a one dimensional random force field (the Sinai model). Using the real space renormalization group method (RSRG), we obtain the exact large time limit of the probability distribution of the scaling variable $w=W(t)/(T \ln t)$. This distribution exhibits a {\it nonanalytic} behaviour at $w=1$. These results are extended to a small non-zero applied field. Using the constrained path integral method, we moreover compute the joint distribution of energy $W(t)$ and position $x(t)$ at time $t$. In presence of a reflecting boundary at the starting point, with possibly some drift in the + direction, the RSRG very simply yields the one time and aging two-time behavior of this joint probability. It exhibits differences in behaviour compared to the unbounded motion, such as analyticity. Relations with some magnetization distributions in the 1D spin glass are discussed.

cond-mat.dis-nn↗

arXiv2002

Field theory of statics and dynamics of glasses: rare events and barrier distributions

We study thermally activated dynamics using functional renormalization within the field theory of randomly pinned elastic systems, a prototype for glasses. It appears through an essentially non-perturbative boundary layer in the running effective action, for which we find a consistent scaling ansatz to all orders. We find that in the statics the boundary layer describes the physics of rare low energy metastable states as suggested by a phenomenological droplet picture. The extension to equilibrium dynamics mandates a non-trivial broad distribution of relaxation times for the large scale modes, hitherto neglected in analytic calculations.

cond-mat.dis-nn↗

arXiv2002

2-loop Functional Renormalization Group Theory of the Depinning Transition

We construct the field theory which describes the universal properties of the quasi-static isotropic depinning transition for interfaces and elastic periodic systems at zero temperature, taking properly into account the non-analytic form of the dynamical action. This cures the inability of the 1-loop flow-equations to distinguish between statics and quasi-static depinning, and thus to account for the irreversibility of the latter. We prove two-loop renormalizability, obtain the 2-loop beta-function and show the generation of "irreversible" anomalous terms, originating from the non-analytic nature of the theory, which cause the statics and driven dynamics to differ at 2-loop order. We obtain the roughness exponent zeta and dynamical exponent z to order epsilon^2. This allows to test several previous conjectures made on the basis of the 1-loop result. First it demonstrates that random-field disorder does indeed attract all disorder of shorter range. It also shows that the conjecture zeta=epsilon/3 is incorrect, and allows to compute the violations, as zeta=epsilon/3 (1 + 0.14331 epsilon), epsilon=4-d. This solves a longstanding discrepancy with simulations. For long-range elasticity it yields zeta=epsilon/3 (1 + 0.39735 epsilon), epsilon=2-d (vs. the standard prediction zeta=1/3 for d=1), in reasonable agreement with the most recent simulations. The high value of zeta approximately 0.5 found in experiments both on the contact line depinning of liquid Helium and on slow crack fronts is discussed.

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