arXiv subjects

Thomas Garel

Publications and source records attributed to Thomas Garel.

At least 37 records · Page 2Linked to original sources

arXiv2010

Random walk in a two-dimensional self-affine random potential : properties of the anomalous diffusion phase at small external force

We consider the random walk of a particle in a two-dimensional self-affine random potential of Hurst exponent $H=1/2$ in the presence of an external force $F$. We present numerical results on the statistics of first-passage times that satisfy closed backward master equations. We find that there exists a zero-velocity phase in a finite region of the external force $0<F<F_c$, where the dynamics follows the anomalous diffusion law $ x(t) \sim ξ(F) \ t^{μ(F)} $. The anomalous exponent $0<μ(F)<1$ and the correlation length $ξ(F)$ vary continuously with $F$. In the limit of vanishing force $F \to 0$, we measure the following power-laws : the anomalous exponent vanishes as $μ(F) \propto F^a$ with $a \simeq 0.6$ (instead of $a=1$ in dimension $d=1$), and the correlation length diverges as $ξ(F) \propto F^{-ν}$ with $ν\simeq 1.29$ (instead of $ν=2$ in dimension $d=1$). Our main conclusion is thus that the dynamics renormalizes onto an effective directed trap model, where the traps are characterized by a typical length $ξ(F)$ along the direction of the force, and by a typical barrier $1/μ(F)$. The fact that these traps are 'smaller' in linear size and in depth than in dimension $d=1$, means that the particle uses the transverse direction to find lower barriers.

cond-mat.dis-nn↗

arXiv2010

Anderson localization of phonons in dimension $d=1,2,3$ : finite-size properties of the Inverse Participation Ratios of eigenstates

We study by exact diagonalization the localization properties of phonons in mass-disordered harmonic crystals of dimension $d=1,2,3$. We focus on the behavior of the typical Inverse Participation Ratio $Y_2(ω,L)$ as a function of the frequency $ω$ and of the linear length $L$ of the disordered samples. In dimensions $d=1$ and $d=2$, we find that the low-frequency part $ω\to 0$ of the spectrum satisfies the following finite-size scaling $L Y_2(ω,L)=F_{d=1}(L^{1/2} ω)$ in dimension $d=1$ and $L^2 Y_2(ω,L)=F_{d=2}((\ln L)^{1/2} ω)$ in dimension $d=2$, with the following conclusions (i) an eigenstate of any fixed frequency $ω$ becomes localized in the limit $L \to +\infty$ (ii) a given disordered sample of size $L^d$ contains a number $N_{deloc}(L)$ of delocalized states growing as $N_{deloc}(L)\sim L^{1/2}$ in $d=1$ and as $N_{deloc}(L)\sim L^2/(\ln L)$ in $d=2$. In dimension $d=3$, we find a localization-delocalization transition at some finite critical frequency $ω_c(W)>0$ (that depends on the disorder strength $W$). Our data are compatible with the finite-size scaling $L^{D(2)} Y_2(ω,L)=F_{d=3}(L^{1/ν} (ω-ω_c))$ with the values $D(2) \simeq 1.3$ and $ν\simeq 1.57$ corresponding to the universality class of the localization transition for the Anderson tight-binding electronic model in dimension $d=3$.

cond-mat.dis-nn↗

arXiv2010

Random cascade models of multifractality : real-space renormalization and travelling-waves

Random multifractals occur in particular at critical points of disordered systems. For Anderson localization transitions, Mirlin and Evers [PRB 62,7920 (2000)] have proposed the following scenario (a) the Inverse Participation Ratios (I.P.R.) $Y_q(L)$ display the following fluctuations between the disordered samples of linear size $L$ : with respect to the typical value $Y^{typ}_q(L) = e^{\bar{\ln Y_q(L)}} \sim L^{- τ_{typ}(q)} $ that involve the typical multifractal spectrum $τ_{typ}(q)$, the rescaled variable $y=Y_q(L)/Y^{typ}_q(L) $ is distributed with a scale-invariant distribution presenting the power-law tail $1/y^{1+β_q}$, so that the disorder-averaged I.P.R. $\bar{Y_q(L)} \sim L^{- τ_{av}(q)} $ have multifractal exponents $τ_{av}(q) $ that differ from the typical ones $τ_{typ}(q)$ whenever $β_q<1$; (b) the tail exponents $β_q$ and the multifractal exponents are related by the relation $β_q τ_{typ}(q)=τ_{av}(q β_q)$. Here we show that this scenario can be understood by considering the real-space renormalization equations satisfied by the I.P.R. For the simplest multifractals described by random cascades, these renormalization equations are formally similar to the recursion relations for disordered models defined on Cayley trees and they admit travelling-wave solutions for the variable $(\ln Y_q)$ in the effective time $t_{eff}=\ln L$ : the exponent $τ_{typ}(q)$ represents the velocity, whereas the tail exponent $β_q$ represents the usual exponential decay of the travelling-wave tail. In addition, we obtain that the relation (b) above can be obtained as a self-consistency condition from the self-similarity of the multifractal spectrum at all scales. Our conclusion is thus that the Mirlin-Evers scenario should apply to other types of random critical points, and even to random multifractals occurring in other fields.

cond-mat.dis-nn↗

arXiv2010

Many-body localization transition in a lattice model of interacting fermions: statistics of renormalized hoppings in configuration space

We consider the one-dimensional lattice model of interacting fermions with disorder studied previously by Oganesyan and Huse [Phys. Rev. B 75, 155111 (2007)]. To characterize a possible many-body localization transition as a function of the disorder strength $W$, we use an exact renormalization procedure in configuration space that generalizes the Aoki real-space RG procedure for Anderson localization one-particle models [H. Aoki, J. Phys. C13, 3369 (1980)]. We focus on the statistical properties of the renormalized hopping $V_L$ between two configurations separated by a distance $L$ in configuration space (distance being defined as the minimal number of elementary moves to go from one configuration to the other). Our numerical results point towards the existence of a many-body localization transition at a finite disorder strength $W_c$. In the localized phase $W>W_c$, the typical renormalized hopping $V_L^{typ} \equiv e^{\bar{\ln V_L}}$ decays exponentially in $L$ as $ (\ln V_L^{typ}) \simeq - \frac{L}{ξ_{loc}}$ and the localization length diverges as $ξ_{loc}(W) \sim (W-W_c)^{-ν_{loc}}$ with a critical exponent of order $ν_{loc} \simeq 0.5$. In the delocalized phase $W<W_c$, the renormalized hopping remains a finite random variable as $L \to \infty$, and the typical asymptotic value $V_{\infty}^{typ} \equiv e^{\bar{\ln V_{\infty}}}$ presents an essential singularity $(\ln V_{\infty}^{typ}) \sim - (W_c-W)^{-κ}$ with an exponent of order $κ\sim 1.4$. Finally, we show that this analysis in configuration space is compatible with the localization properties of the simplest two-point correlation function in real space.

cond-mat.dis-nn↗

arXiv2010

Matching between typical fluctuations and large deviations in disordered systems : application to the statistics of the ground state energy in the SK spin-glass model

For the statistics of global observables in disordered systems, we discuss the matching between typical fluctuations and large deviations. We focus on the statistics of the ground state energy $E_0$ in two types of disordered models : (i) for the directed polymer of length $N$ in a two-dimensional medium, where many exact results exist (ii) for the Sherrington-Kirkpatrick spin-glass model of $N$ spins, where various possibilities have been proposed. Here we stress that, besides the behavior of the disorder-average $E_0^{av}(N)$ and of the standard deviation $ ΔE_0(N) \sim N^{ω_f}$ that defines the fluctuation exponent $ω_f$, it is very instructive to study the full probability distribution $Π(u)$ of the rescaled variable $u= \frac{E_0(N)-E_0^{av}(N)}{ΔE_0(N)}$ : (a) numerically, the convergence towards $Π(u)$ is usually very rapid, so that data on rather small sizes but with high statistics allow to measure the two tails exponents $η_{\pm}$ defined as $\ln Π(u \to \pm \infty) \sim - | u |^{η_{\pm}}$. In the generic case $1< η_{\pm} < +\infty$, this leads to explicit non-trivial terms in the asymptotic behaviors of the moments $\bar{Z_N^n}$ of the partition function when the combination $[| n | N^{ω_f}]$ becomes large (b) simple rare events arguments can usually be found to obtain explicit relations between $η_{\pm}$ and $ω_f$. These rare events usually correspond to 'anomalous' large deviation properties of the generalized form $R(w_{\pm} = \frac{E_0(N)-E_0^{av}(N)}{N^{κ_{\pm}}}) \sim e^{- N^{ρ_{\pm}} {\cal R}_{\pm}(w_{\pm})}$ (the 'usual' large deviations formalism corresponds to $κ_{\pm}=1=ρ_{\pm}$).

cond-mat.dis-nn↗

arXiv2010

Random walk in two-dimensional self-affine random potentials : strong disorder renormalization approach

We consider the continuous-time random walk of a particle in a two-dimensional self-affine quenched random potential of Hurst exponent $H>0$. The corresponding master equation is studied via the strong disorder renormalization procedure introduced in Ref. [C. Monthus and T. Garel, J. Phys. A: Math. Theor. 41 (2008) 255002]. We present numerical results on the statistics of the equilibrium time $t_{eq}$ over the disordered samples of a given size $L \times L$ for $10 \leq L \leq 80$. We find an 'Infinite disorder fixed point', where the equilibrium barrier $Γ_{eq} \equiv \ln t_{eq}$ scales as $Γ_{eq}=L^H u $ where $u$ is a random variable of order O(1). This corresponds to a logarithmically-slow diffusion $ | \vec r(t) - \vec r(0) | \sim (\ln t)^{1/H}$ for the position $\vec r(t)$ of the particle.

cond-mat.dis-nn↗

arXiv2009

Eigenvalue method to compute the largest relaxation time of disordered systems

We consider the dynamics of finite-size disordered systems as defined by a master equation satisfying detailed balance. The master equation can be mapped onto a Schrödinger equation in configuration space, where the quantum Hamiltonian $H$ has the generic form of an Anderson localization tight-binding model. The largest relaxation time $t_{eq}$ governing the convergence towards Boltzmann equilibrium is determined by the lowest non-vanishing eigenvalue $E_1=1/t_{eq}$ of $H$ (the lowest eigenvalue being $E_0=0$). So the relaxation time $t_{eq}$ can be computed {\it without simulating the dynamics} by any eigenvalue method able to compute the first excited energy $E_1$. Here we use the 'conjugate gradient' method to determine $E_1$ in each disordered sample and present numerical results on the statistics of the relaxation time $t_{eq}$ over the disordered samples of a given size for two models : (i) for the random walk in a self-affine potential of Hurst exponent $H$ on a two-dimensional square of size $L \times L$, we find the activated scaling $\ln t_{eq}(L) \sim L^ψ$ with $ψ=H$ as expected; (ii) for the dynamics of the Sherrington-Kirkpatrick spin-glass model of $N$ spins, we find the growth $\ln t_{eq}(N) \sim N^ψ$ with $ψ=1/3$ in agreement with most previous Monte-Carlo measures. In addition, we find that the rescaled distribution of $(\ln t_{eq})$ decays as $e^{- u^η}$ for large $u$ with a tail exponent of order $η\simeq 1.36$. We give a rare-event interpretation of this value, that points towards a sample-to-sample fluctuation exponent of order $ψ_{width} \simeq 0.26$ for the barrier.

cond-mat.dis-nn↗

arXiv2009

Statistics of first-passage times in disordered systems using backward master equations and their exact renormalization rules

We consider the non-equilibrium dynamics of disordered systems as defined by a master equation involving transition rates between configurations (detailed balance is not assumed). To compute the important dynamical time scales in finite-size systems without simulating the actual time evolution which can be extremely slow, we propose to focus on first-passage times that satisfy 'backward master equations'. Upon the iterative elimination of configurations, we obtain the exact renormalization rules that can be followed numerically. To test this approach, we study the statistics of some first-passage times for two disordered models : (i) for the random walk in a two-dimensional self-affine random potential of Hurst exponent $H$, we focus on the first exit time from a square of size $L \times L$ if one starts at the square center. (ii) for the dynamics of the ferromagnetic Sherrington-Kirkpatrick model of $N$ spins, we consider the first passage time $t_f$ to zero-magnetization when starting from a fully magnetized configuration. Besides the expected linear growth of the averaged barrier $\bar{\ln t_{f}} \sim N$, we find that the rescaled distribution of the barrier $(\ln t_{f})$ decays as $e^{- u^η}$ for large $u$ with a tail exponent of order $η\simeq 1.72$. This value can be simply interpreted in terms of rare events if the sample-to-sample fluctuation exponent for the barrier is $ψ_{width}=1/3$.

cond-mat.dis-nn↗

arXiv2009

Statistical properties of two-particle transmission at Anderson transition

The ensemble of $L \times L$ power-law random banded matrices, where the random hopping $H_{i,j}$ decays as a power-law $(b/| i-j |)^a$, is known to present an Anderson localization transition at $a=1$, where one-particle eigenfunctions are multifractal. Here we study numerically, at this critical point, the statistical properties of the transmission $T_2$ for two distinguishable particles, two bosons or two fermions. We find that the statistics of $T_2$ is multifractal, i.e. the probability to have $T_2(L) \sim 1/L^κ$ behaves as $L^{Φ_2(κ)}$, where the multifractal spectrum $Φ_2(κ)$ for fermions is different from the common multifractal spectrum concerning distinguishable particles and bosons. However in the three cases, the typical transmission $T_2^{typ}(L)$ is governed by the same exponent $κ_2^{typ}$, which is much smaller than the naive expectation $2κ_1^{typ}$, where $κ_1^{typ}$ is the typical exponent of the one-particle transmission $T_1(L)$.

cond-mat.dis-nn↗

arXiv2009

Statistics of renormalized on-site energies and renormalized hoppings for Anderson localization models in dimensions d=2 and d=3

For Anderson localization models, there exists an exact real-space renormalization procedure at fixed energy which preserves the Green functions of the remaining sites [H. Aoki, J. Phys. C13, 3369 (1980)]. Using this procedure for the Anderson tight-binding model in dimensions $d=2,3$, we study numerically the statistical properties of the renormalized on-site energies $ε$ and of the renormalized hoppings $V$ as a function of the linear size $L$. We find that the renormalized on-site energies $ε$ remain finite in the localized phase in $d=2,3$ and at criticality ($d=3$), with a finite density at $ε=0$ and a power-law decay $1/ε^2$ at large $| ε|$. For the renormalized hoppings in the localized phase, we find: ${\rm ln} V_L \simeq -\frac{L}{ξ_{loc}}+L^ωu$, where $ξ_{loc}$ is the localization length and $u$ a random variable of order one. The exponent $ω$ is the droplet exponent characterizing the strong disorder phase of the directed polymer in a random medium of dimension $1+(d-1)$, with $ω(d=2)=1/3$ and $ω(d=3) \simeq 0.24$. At criticality $(d=3)$, the statistics of renormalized hoppings $V$ is multifractal, in direct correspondence with the multifractality of individual eigenstates and of two-point transmissions. In particular, we measure $ρ_{typ}\simeq 1$ for the exponent governing the typical decay $\overline{{\rm ln} V_L} \simeq -ρ_{typ} {\rm ln}L$, in agreement with previous numerical measures of $α_{typ} =d+ρ_{typ} \simeq 4$ for the singularity spectrum $f(α)$ of individual eigenfunctions. We also present numerical results concerning critical surface properties.

cond-mat.dis-nn↗

arXiv2009

Statistics of the two-point transmission at Anderson localization transitions

At Anderson critical points, the statistics of the two-point transmission $T_L$ for disordered samples of linear size $L$ is expected to be multifractal with the following properties [Janssen {\it et al} PRB 59, 15836 (1999)] : (i) the probability to have $T_L \sim 1/L^κ$ behaves as $L^{Φ(κ)}$, where the multifractal spectrum $Φ(κ)$ terminates at $κ=0$ as a consequence of the physical bound $T_L \leq 1$; (ii) the exponents $X(q)$ that govern the moments $\overline{T_L^q} \sim 1/L^{X(q)}$ become frozen above some threshold: $X(q \geq q_{sat}) = - Φ(κ=0)$, i.e. all moments of order $q \geq q_{sat}$ are governed by the measure of the rare samples having a finite transmission ($κ=0$). In the present paper, we test numerically these predictions for the ensemble of $L \times L$ power-law random banded matrices, where the random hopping $H_{i,j}$ decays as a power-law $(b/| i-j |)^a$. This model is known to present an Anderson transition at $a=1$ between localized ($a>1$) and extended ($a<1$) states, with critical properties that depend continuously on the parameter $b$. Our numerical results for the multifractal spectra $Φ_b(κ)$ for various $b$ are in agreement with the relation $Φ(κ\geq 0) = 2 [ f(α= d+ \fracκ{2}) -d ]$ in terms of the singularity spectrum $f(α)$ of individual critical eigenfunctions, in particular the typical exponents are related via the relation $κ_{typ}(b)= 2 (α_{typ}(b)-d)$. We also discuss the statistics of the two-point transmission in the delocalized phase and in the localized phase.

cond-mat.dis-nn↗

arXiv2009

Anderson transitions : multifractal or non-multifractal statistics of the transmission as a function of the scattering geometry

The scaling theory of Anderson localization is based on a global conductance $g_L$ that remains a random variable of order O(1) at criticality. One realization of such a conductance is the Landauer transmission for many transverse channels. On the other hand, the statistics of the one-channel Landauer transmission between two local probes is described by a multifractal spectrum that can be related to the singularity spectrum of individual eigenstates. To better understand the relations between these two types of results, we consider various scattering geometries that interpolate between these two cases and analyse the statistics of the corresponding transmissions. We present detailed numerical results for the power-law random banded matrices (PRBM model). Our conclusions are : (i) in the presence of one isolated incoming wire and many outgoing wires, the transmission has the same multifractal statistics as the local density of states of the site where the incoming wire arrives; (ii) in the presence of backward scattering channels with respect to the case (i), the statistics of the transmission is not multifractal anymore, but becomes monofractal. Finally, we also describe how these scattering geometries influence the statistics of the transmission off criticality.

cond-mat.dis-nn↗

arXiv2009

Random wetting transition on the Cayley tree : a disordered first-order transition with two correlation length exponents

We consider the random wetting transition on the Cayley tree, i.e. the problem of a directed polymer on the Cayley tree in the presence of random energies along the left-most bonds. In the pure case, there exists a first-order transition between a localized phase and a delocalized phase, with a correlation length exponent $ν_{pure}=1$. In the disordered case, we find that the transition remains first-order, but that there exists two diverging length scales in the critical region : the typical correlation length diverges with the exponent $ν_{typ}=1$, whereas the averaged correlation length diverges with the bigger exponent $ν_{av}=2$ and governs the finite-size scaling properties. We describe the relations with previously studied models that are governed by the same "Infinite Disorder Fixed Point". For the present model, where the order parameter is the contact density $θ_L=l_a/L$ (defined as the ratio of the number $l_a$ of contacts over the total length $L$), the notion of "infinite disorder fixed point" means that the thermal fluctuations of $θ_L$ within a given sample, become negligeable at large scale with respect to sample-to-sample fluctuations. We characterize the statistics over the samples of the free-energy and of the contact density. In particular, exactly at criticality, we obtain that the contact density is not self-averaging but remains distributed over the samples in the thermodynamic limit, with the distribution ${\cal P}_{T_c}(θ) = 1/(π\sqrt{θ(1-θ)})$.

cond-mat.dis-nn↗

arXiv2008

Anderson transition on the Cayley tree as a traveling wave critical point for various probability distributions

For Anderson localization on the Cayley tree, we study the statistics of various observables as a function of the disorder strength $W$ and the number $N$ of generations. We first consider the Landauer transmission $T_N$. In the localized phase, its logarithm follows the traveling wave form $\ln T_N \simeq \bar{\ln T_N} + \ln t^*$ where (i) the disorder-averaged value moves linearly $\bar{\ln (T_N)} \simeq - \frac{N}{ξ_{loc}}$ and the localization length diverges as $ξ_{loc} \sim (W-W_c)^{-ν_{loc}}$ with $ν_{loc}=1$ (ii) the variable $t^*$ is a fixed random variable with a power-law tail $P^*(t^*) \sim 1/(t^*)^{1+β(W)}$ for large $t^*$ with $0<β(W) \leq 1/2$, so that all integer moments of $T_N$ are governed by rare events. In the delocalized phase, the transmission $T_N$ remains a finite random variable as $N \to \infty$, and we measure near criticality the essential singularity $\bar{\ln (T)} \sim - | W_c-W |^{-κ_T}$ with $κ_T \sim 0.25$. We then consider the statistical properties of normalized eigenstates, in particular the entropy and the Inverse Participation Ratios (I.P.R.). In the localized phase, the typical entropy diverges as $(W-W_c)^{- ν_S}$ with $ν_S \sim 1.5$, whereas it grows linearly in $N$ in the delocalized phase. Finally for the I.P.R., we explain how closely related variables propagate as traveling waves in the delocalized phase. In conclusion, both the localized phase and the delocalized phase are characterized by the traveling wave propagation of some probability distributions, and the Anderson localization/delocalization transition then corresponds to a traveling/non-traveling critical point. Moreover, our results point towards the existence of several exponents $ν$ at criticality.

cond-mat.dis-nn↗

arXiv2008

Non-equilibrium dynamics of finite-dimensional disordered systems : RG flow towards an "infinite disorder" fixed point at large times

To describe the non-equilibrium dynamics of random systems, we have recently introduced (C. Monthus and T. Garel, arxiv:0802.2502) a 'strong disorder renormalization' (RG) procedure in configuration space that can be defined for any master equation. In the present paper, we analyze in details the properties of the large time dynamics whenever the RG flow is towards some "infinite disorder" fixed point, where the width of the renormalized barriers distribution grows indefinitely upon iteration. In particular, we show how the strong disorder RG rules can be then simplified while keeping their asymptotic exactness, because the preferred exit channel out of a given renormalized valley typically dominates asymptotically over the other exit channels. We explain why the present approach is an explicit construction in favor of the droplet scaling picture where the dynamics is governed by the logarithmic growth of the coherence length $l(t) \sim (\ln t)^{1/ψ}$, and where the statistics of barriers corresponds to a very strong hierarchy of valleys within valleys. As an example of application, we have followed numerically the RG flow for the case of a directed polymer in a two-dimensional random medium. The full RG rules are used to check that the RG flow is towards some infinite disorder fixed point, whereas the simplified RG rules allow to study bigger sizes and to estimate the barrier exponent $ψ$ of the fixed point.

cond-mat.dis-nn↗

arXiv2008

Non equilibrium dynamics of disordered systems : understanding the broad continuum of relevant time scales via a strong-disorder RG in configuration space

We show that an appropriate description of the non-equilibrium dynamics of disordered systems is obtained through a strong disorder renormalization procedure in {\it configuration space}, that we define for any master equation with transitions rates $W ({\cal C} \to {\cal C}')$ between configurations. The idea is to eliminate iteratively the configuration with the highest exit rate $W_{out} ({\cal C})= \sum_{{\cal C}'} W ({\cal C} \to {\cal C}')$ to obtain renormalized transition rates between the remaining configurations. The multiplicative structure of the new generated transition rates suggests that, for a very broad class of disordered systems, the distribution of renormalized exit barriers defined as $B_{out} ({\cal C}) \equiv - \ln W_{out}({\cal C})$ will become broader and broader upon iteration, so that the strong disorder renormalization procedure should become asymptotically exact at large time scales. We have checked numerically this scenario for the non-equilibrium dynamics of a directed polymer in a two dimensional random medium.

cond-mat.dis-nn↗

arXiv2008

Driven interfaces in random media at finite temperature : is there an anomalous zero-velocity phase at small external force ?

The motion of driven interfaces in random media at finite temperature $T$ and small external force $F$ is usually described by a linear displacement $h_G(t) \sim V(F,T) t$ at large times, where the velocity vanishes according to the creep formula as $V(F,T) \sim e^{-K(T)/F^μ}$ for $F \to 0$. In this paper, we question this picture on the specific example of the directed polymer in a two dimensional random medium. We have recently shown (C. Monthus and T. Garel, arxiv:0802.2502) that its dynamics for F=0 can be analyzed in terms of a strong disorder renormalization procedure, where the distribution of renormalized barriers flows towards some "infinite disorder fixed point". In the present paper, we obtain that for small $F$, this "infinite disorder fixed point" becomes a "strong disorder fixed point" with an exponential distribution of renormalized barriers. The corresponding distribution of trapping times then only decays as a power-law $P(τ) \sim 1/τ^{1+α}$, where the exponent $α(F,T)$ vanishes as $α(F,T) \propto F^μ$ as $F \to 0$. Our conclusion is that in the small force region $α(F,T)<1$, the divergence of the averaged trapping time $\barτ=+\infty$ induces strong non-self-averaging effects that invalidate the usual creep formula obtained by replacing all trapping times by the typical value. We find instead that the motion is only sub-linearly in time $h_G(t) \sim t^{α(F,T)}$, i.e. the asymptotic velocity vanishes V=0. This analysis is confirmed by numerical simulations of a directed polymer with a metric constraint driven in a traps landscape. We moreover obtain that the roughness exponent, which is governed by the equilibrium value $ζ_{eq}=2/3$ up to some large scale, becomes equal to $ζ=1$ at the largest scales.

cond-mat.dis-nn↗

arXiv2008

Equilibrium of disordered systems : constructing the appropriate valleys in each sample via strong disorder renormalization in configuration space

To describe the equilibrium properties of disordered systems and the possible emergence of various 'phases' at low temperature, we adopt here the 'broken ergodicity' point of view advocated in particular by Palmer [Adv. Phys. 31, 669 (1982)] : the aim is then to construct the valleys of configurations that become separated by diverging barriers and to study their relative weights, as well as their internal properties. To characterize the slow non-equilibrium dynamics of disordered systems, we have recently introduced in [C. Monthus and T. Garel, J. Phys. A 41, 255002 (2008) and arxiv:0804.1847] a strong disorder renormalization procedure in configuration space, based on the iterative elimination of the smallest barrier remaining in the system. In the present paper, we show how this renormalization procedure allows to construct the longest-lived valleys in each disordered sample, and to obtain their free-energies, energies and entropies. This explicit RG formulation is very general since it can be defined for any master equation, and it gives new insights into the main ingredients of the droplet scaling picture. As an application, we have followed numerically the RG flow for the case of a directed polymer in a two-dimensional random medium to obtain histograms of the free-energy, entropy and energy differences between the two longest-lived valleys in each sample.

cond-mat.dis-nn↗