arXiv subjects

Pierre Le Doussal

Publications and source records attributed to Pierre Le Doussal.

At least 163 records · Page 9Linked to original sources

arXiv2014

From the sine-Gordon field theory to the Kardar-Parisi-Zhang growth equation

We unveil a remarkable connection between the sine-Gordon quantum field theory and the Kardar-Parisi-Zhang (KPZ) growth equation. We find that the non-relativistic limit of the two point correlation function of the sine-Gordon theory is related to the generating function of the height distribution of the KPZ field with droplet initial conditions, i.e. the directed polymer free energy with two endpoints fixed. As shown recently, the latter can be expressed as a Fredholm determinant which in the large time separation limit converges to the GUE Tracy-Widom cumulative distribution. Possible applications and extensions are discussed.

cond-mat.stat-mech↗

arXiv2014

Interaction quench in a Lieb-Liniger model and the KPZ equation with flat initial conditions

Recent exact solutions of the 1D Kardar-Parisi-Zhang equation make use of the 1D integrable Lieb-Liniger model of interacting bosons. For flat initial conditions, it requires the knowledge of the overlap between the uniform state and arbitrary exact Bethe eigenstates. The same quantity is also central in the study of the quantum quench from a 1D non-interacting Bose-Einstein condensate upon turning interactions. We compare recent advances in both domains, i.e. our previous exact solution, and a new conjecture by De Nardis et al.. This leads to new exact results and conjectures for both the quantum quench and the KPZ problem.

cond-mat.dis-nn↗

arXiv2014

Crossover from droplet to flat initial conditions in the KPZ equation from the replica Bethe ansatz

We show how our previous result based on the replica Bethe ansatz for the Kardar Parisi Zhang (KPZ) equation with the "half-flat" initial condition leads to the Airy$_2$ to Airy$_1$ (i.e. GUE to GOE) universal crossover one-point height distribution in the limit of large time. Equivalently, we obtain the distribution of the free energy of a long directed polymer (DP) in a random potential with one fixed endpoint and the other one on a half-line. We then generalize to a DP when each endpoint is free on its own half-line. It amounts, in the limit of large time, to obtain the distribution of the maximum of the transition process Airy$_{2\to 1}$ (minus a half-parabola) on a half line.

cond-mat.dis-nn↗

arXiv2013

Statistics of shocks in a toy model with heavy tails

We study the energy minimization for a particle in a quadratic well in presence of short-ranged heavy-tailed disorder, as a toy model for an elastic manifold. The discrete model is shown to be described in the scaling limit by a continuum Poisson process model which captures the three universality classes. This model is solved in general, and we give, in the present case (Frechet class), detailed results for the distribution of the minimum energy and position, and the distribution of the sizes of the shocks (i.e. switches in the ground state) which arise as the position of the well is varied. All these distributions are found to exhibit heavy tails with modified exponents. These results lead to an "exotic regime" in Burgers turbulence decaying from a heavy-tailed initial condition.

cond-mat.dis-nn↗

arXiv2013

Collective excitations in a large-d model for graphene

We consider a model of Dirac fermions coupled to flexural phonons to describe a graphene sheet fluctuating in dimension $2+d$. We derive the self-consistent screening equations for the quantum problem, exact in the limit of large $d$. We first treat the membrane alone, and work out the quantum to classical, and harmonic to anharmonic crossover. For the coupled electron-membrane problem we calculate the dressed two-particle propagators of the elastic and electron interactions and find that it exhibits a collective mode which becomes unstable at some wave-vector $q_{\rm c}$ for large enough coupling $g$. The saddle point analysis, exact at large $d$, indicates that this instability corresponds to spontaneous and simultaneous appearance of gaussian curvature and electron puddles. The relevance to ripples in graphene is discussed.

cond-mat.mes-hall↗

arXiv2013

Non-Gaussian effects and multifractality in the Bragg glass

We study, beyond the Gaussian approximation, the decay of the translational order correlation function for a d-dimensional scalar periodic elastic system in a disordered environment. We develop a method based on functional determinants, equivalent to summing an infinite set of diagrams. We obtain, in dimension d=4-epsilon, the even n-th cumulant of relative displacements as <[u(r)-u(0)]^n>^c = A_n ln r, with A_n = -(ε/3)^n Γ(n-1/2) ζ(2n-3)/π^(1/2), as well as the multifractal dimension x_q of the exponential field e^{q u(r)}. As a corollary, we obtain an analytic expression for a class of n-loop integrals in d=4, which appear in the perturbative determination of Konishi amplitudes, also accessible via AdS/CFT using integrability.

cond-mat.dis-nn↗

arXiv2013

Topology trivialization and large deviations for the minimum in the simplest random optimization

Finding the global minimum of a cost function given by the sum of a quadratic and a linear form in N real variables over (N-1)- dimensional sphere is one of the simplest, yet paradigmatic problems in Optimization Theory known as the "trust region subproblem" or "constraint least square problem". When both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field. We first identify and study two distinct large-N scaling regimes in which the linear term (magnetic field) leads to a gradual topology trivialization, i.e. reduction in the total number N_{tot} of critical (stationary) points in the cost function landscape. In the first regime N_{tot} remains of the order $N$ and the cost function (energy) has generically two almost degenerate minima with the Tracy-Widom (TW) statistics. In the second regime the number of critical points is of the order of unity with a finite probability for a single minimum. In that case the mean total number of extrema (minima and maxima) of the cost function is given by the Laplace transform of the TW density, and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the TW law. Though the full form of that distribution is not yet known to us, one of its far tails can be inferred from the large deviation theory for the global minimum. In the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large-deviation approximation by finding both the rate function and the leading pre-exponential factor.

cond-mat.dis-nn↗

arXiv2013

Exact form of the exponential correlation function in the glassy super-rough phase

We consider the random-phase sine-Gordon model in two dimensions. It describes two-dimensional elastic systems with random periodic disorder, such as pinned flux-line arrays, random field XY models, and surfaces of disordered crystals. The model exhibits a super-rough glass phase at low temperature $T<T_{c}$ with relative displacements growing with distance $r$ as $\bar{\langle [θ(r)-θ(0)]^2\rangle} \simeq A(τ) \ln^2 (r/a)$, where $A(τ) = 2 τ^2- 2 τ^3 +\mathcal{O}(τ^4)$ near the transition and $τ=1-T/T_{c}$. We calculate all higher cumulants and show that they grow as $\bar{\langle[θ(r)-θ(0)]^{2n}\rangle}_c \simeq [2 (-1)^{n+1} (2n)! ζ(2n-1) τ^2 + \mathcal{O}(τ^3) ] \ln(r/a)$, $n \geq 2$, where $ζ$ is the Riemann zeta function. By summation, we obtain the decay of the exponential correlation function as $\bar{\langle e^{iq\left[θ(r)-θ(0)\right]}\rangle} \simeq (a/r)^{η(q)} \exp\boldsymbol(-\frac{1}{2}\mathcal{A}(q)\ln^2(r/a)\boldsymbol)$ where $η(q)$ and ${\cal A}(q)$ are obtained for arbitrary $q \leq 1$ to leading order in $τ$. The anomalous exponent is $η(q) = c q^2 - τ^2 q^2 [2γ_E+ψ(q)+ψ(-q)]$ in terms of the digamma function $ψ$, where $c$ is non-universal and $γ_E$ is the Euler constant. The correlation function shows a faster decay at $q=1$, corresponding to fermion operators in the dual picture, which should be visible in Bragg scattering experiments.

cond-mat.stat-mech↗

arXiv2013

Statistics of Avalanches with Relaxation, and Barkhausen Noise: A Solvable Model

We study a generalization of the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model of a particle in a Brownian force landscape, including retardation effects. We show that under monotonous driving the particle moves forward at all times, as it does in absence of retardation (Middleton's theorem). This remarkable property allows us to develop an analytical treatment. The model with an exponentially decaying memory kernel is realized in Barkhausen experiments with eddy-current relaxation, and has previously been shown numerically to account for the experimentally observed asymmetry of Barkhausen-pulse shapes. We elucidate another qualitatively new feature: the breakup of each avalanche of the standard ABBM model into a cluster of sub-avalanches, sharply delimited for slow relaxation under quasi-static driving. These conditions are typical for earthquake dynamics. With relaxation and aftershock clustering, the present model includes important ingredients for an effective description of earthquakes. We analyze quantitatively the limits of slow and fast relaxation for stationary driving with velocity v>0. The v-dependent power-law exponent for small velocities, and the critical driving velocity at which the particle velocity never vanishes, are modified. We also analyze non-stationary avalanches following a step in the driving magnetic field. Analytically, we obtain the mean avalanche shape at fixed size, the duration distribution of the first sub-avalanche, and the time dependence of the mean velocity. We propose to study these observables in experiments, allowing to directly measure the shape of the memory kernel, and to trace eddy current relaxation in Barkhausen noise.

cond-mat.dis-nn↗

arXiv2013

Avalanche dynamics of elastic interfaces

Slowly driven elastic interfaces, such as domain walls in dirty magnets, contact lines, or cracks proceed via intermittent motion, called avalanches. We develop a field-theoretic treatment to calculate, from first principles, the space-time statistics of instantaneous velocities within an avalanche. For elastic interfaces at (or above) their (internal) upper critical dimension d >= d_uc (d_uc = 2, 4 respectively for long-ranged and short-ranged elasticity) we show that the field theory for the center of mass reduces to the motion of a point particle in a random-force landscape, which is itself a random walk (ABBM model). Furthermore, the full spatial dependence of the velocity correlations is described by the Brownian-force model (BFM) where each point of the interface sees an independent Brownian-force landscape. Both ABBM and BFM can be solved exactly in any dimension d (for monotonous driving) by summing tree graphs, equivalent to solving a (non-linear) instanton equation. This tree approximation is the mean-field theory (MFT) for realistic interfaces in short-ranged disorder. Both for the center of mass, and for a given Fourier mode q, we obtain probability distribution functions (PDF's) of the velocity, as well as the avalanche shape and its fluctuations (second shape). Within MFT we find that velocity correlations at non-zero q are asymmetric under time reversal. Next we calculate, beyond MFT, i.e. including loop corrections, the 1-time PDF of the center-of-mass velocity du/dt for dimension d< d_uc. The singularity at small velocity P(du/dt) ~ 1/(du/dt)^a is substantially reduced from a=1 (MFT) to a = 1 - 2/9 (4-d) + ... (short-ranged elasticity) and a = 1 - 4/9 (2-d) + ... (long-ranged elasticity). We show how the dynamical theory recovers the avalanche-size distribution, and how the instanton relates to the response to an infinitesimal step in the force.

cond-mat.dis-nn↗

arXiv2012

Functional renormalization-group approach to decaying turbulence

We reconsider the functional renormalization-group (FRG) approach to decaying Burgers turbulence, and extend it to decaying Navier-Stokes and Surface-Quasi-Geostrophic turbulence. The method is based on a renormalized small-time expansion, equivalent to a loop expansion, and naturally produces a dissipative anomaly and a cascade after a finite time. We explicitly calculate and analyze the one-loop FRG equations in the zero-viscosity limit as a function of the dimension. For Burgers they reproduce the FRG equation obtained in the context of random manifolds, extending previous results of one of us. Breakdown of energy conservation due to shocks and the appearance of a direct energy cascade corresponds to failure of dimensional reduction in the context of disordered systems. For Navier-Stokes in three dimensions, the velocity-velocity correlation function acquires a linear dependence on the distance, zeta_2=1, in the inertial range, instead of Kolmogorov's zeta_2=2/3; however the possibility remains for corrections at two- or higher-loop order. In two dimensions, we obtain a numerical solution which conserves energy and exhibits an inverse cascade, with explicit analytical results both for large and small distances, in agreement with the scaling proposed by Batchelor. In large dimensions, the one-loop FRG equation for Navier-Stokes converges to that of Burgers.

nlin.CD↗

arXiv2012

Counting function fluctuations and extreme value threshold in multifractal patterns: the case study of an ideal $1/f$ noise

To understand the sample-to-sample fluctuations in disorder-generated multifractal patterns we investigate analytically as well as numerically the statistics of high values of the simplest model - the ideal periodic $1/f$ Gaussian noise. By employing the thermodynamic formalism we predict the characteristic scale and the precise scaling form of the distribution of number of points above a given level. We demonstrate that the powerlaw forward tail of the probability density, with exponent controlled by the level, results in an important difference between the mean and the typical values of the counting function. This can be further used to determine the typical threshold $x_m$ of extreme values in the pattern which turns out to be given by $x_m^{(typ)}=2-c\ln{\ln{M}}/\ln{M}$ with $c=3/2$. Such observation provides a rather compelling explanation of the mechanism behind universality of $c$. Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disorder-generated multifractal fields. In particular, we predict that the typical value of the maximum $p_{max}$ of intensity is to be given by $-\ln{p_{max}} = α_{-}\ln{M} + \frac{3}{2f'(α_{-})}\ln{\ln{M}} + O(1)$, where $f(α)$ is the corresponding singularity spectrum vanishing at $α=α_{-}>0$. For the $1/f$ noise we also derive exact as well as well-controlled approximate formulas for the mean and the variance of the counting function without recourse to the thermodynamic formalism.

cond-mat.stat-mech↗

arXiv2012

Rings and Coulomb boxes in dissipative environments

We study a particle on a ring in presence of a dissipative Caldeira-Leggett environment and derive its response to a DC field. We show how this non-equilibrium response is related to a flux averaged equilibrium response. We find, through a 2-loop renormalization group analysis, that a large dissipation parameter ηflows to a fixed point η^R=\hbar/2π. We also reexamine the mapping of this problem to that of the Coulomb box and show that the relaxation resistance, of recent interest, is quantized for large η. For finite η>η^R we find that a certain average of the relaxation resistance is quantized. We propose a Coulomb box experiment to measure a quantized noise.

cond-mat.mes-hall↗

arXiv2012

Short time growth of a KPZ interface with flat initial conditions

The short time behavior of the 1+1 dimensional KPZ growth equation with a flat initial condition is obtained from the exact expressions of the moments of the partition function of a directed polymer with one endpoint free and the other fixed. From these expressions, the short time expansions of the lowest cumulants of the KPZ height field are exactly derived. The results for these two classes of cumulants are checked in high precision lattice numerical simulations. The short time limit considered here is relevant for the study of the interface growth in the large diffusivity/weak noise limit, and describes the universal crossover between the Edwards-Wilkinson and KPZ universality classes for an initially flat interface.

cond-mat.stat-mech↗

arXiv2012

Super-Rough Glassy Phase of the Random Field XY Model in Two Dimensions

We study both analytically, using the renormalization group (RG) to two loop order, and numerically, using an exact polynomial algorithm, the disorder-induced glass phase of the two-dimensional XY model with quenched random symmetry-breaking fields and without vortices. In the super-rough glassy phase, i.e. below the critical temperature $T_c$, the disorder and thermally averaged correlation function $B(r)$ of the phase field $θ(x)$, $B(r) = \bar{<[θ(x) - θ(x+ r) ]^2>}$ behaves, for $r \gg a$, as $B(r) \simeq A(τ) \ln^2 (r/a)$ where $r = |r|$ and $a$ is a microscopic length scale. We derive the RG equations up to cubic order in $τ= (T_c-T)/T_c$ and predict the universal amplitude ${A}(τ) = 2τ^2-2τ^3 + {\cal O}(τ^4)$. The universality of $A(τ)$ results from nontrivial cancellations between nonuniversal constants of RG equations. Using an exact polynomial algorithm on an equivalent dimer version of the model we compute ${A}(τ)$ numerically and obtain a remarkable agreement with our analytical prediction, up to $τ\approx 0.5$.

cond-mat.dis-nn↗

arXiv2012

Directed polymer near a hard wall and KPZ equation in the half-space

We study the directed polymer with fixed endpoints near an absorbing wall, in the continuum and in presence of disorder, equivalent to the KPZ equation on the half space with droplet initial conditions. From a Bethe Ansatz solution of the equivalent attractive boson model we obtain the exact expression for the free energy distribution at all times. It converges at large time to the Tracy Widom distribution $F_4$ of the Gaussian Symplectic Ensemble (GSE). We compare our results with numerical simulations of the lattice directed polymer, both at zero and high temperature.

cond-mat.dis-nn↗

arXiv2012

Super-rough phase of the random-phase sine-Gordon model: Two-loop results

We consider the two-dimensional random-phase sine-Gordon and study the vicinity of its glass transition temperature $T_c$, in an expansion in small $τ=(T_c-T)/T_c$, where $T$ denotes the temperature. We derive renormalization group equations in cubic order in the anharmonicity, and show that they contain two universal invariants. Using them we obtain that the correlation function in the super-rough phase for temperature $T } = \mathcal{A}\ln^2(|x|/a) + \mathcal{O}[\ln(|x|/a)]$, where the amplitude $\mathcal{A}$ is a universal function of temperature $\mathcal{A}=2τ^2-2τ^3+\mathcal{O}(τ^4)$. This result differs at two-loop order, i.e., $\mathcal{O}(τ^3)$, from the prediction based on results from the "nearly conformal" field theory of a related fermion model. We also obtain the correction-to-scaling exponent.

cond-mat.dis-nn↗

arXiv2012

Phase transition of interacting disordered bosons in one dimension

Interacting bosons generically form a superfluid state. In the presence of disorder it can get converted into a compressible Bose glass state. Here we study such transition in one dimension at moderate interaction using bosonization and renormalization group techniques. We derive the two-loop scaling equations and discuss the phase diagram. We find that the correlation functions at the transition are characterized by universal exponents in a finite region around the fixed point.

cond-mat.dis-nn↗