arXiv subjects

Cecile Monthus

Publications and source records attributed to Cecile Monthus.

At least 145 records · Page 8Linked to original sources

arXiv2005

Delocalization transition of the selective interface model: distribution of pseudo-critical temperatures

According to recent progress in the finite size scaling theory of critical disordered systems, the nature of the phase transition is reflected in the distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples $(i)$ of size $L$. In this paper, we apply this analysis to the delocalization transition of an heteropolymeric chain at a selective fluid-fluid interface. The width $ΔT_c(L)$ and the shift $[T_c(\infty)-T_c^{av}(L)]$ are found to decay with the same exponent $L^{-1/ν_{R}}$, where $1/ν_{R} \sim 0.26$. The distribution of pseudo-critical temperatures $T_c(i,L)$ is clearly asymmetric, and is well fitted by a generalized Gumbel distribution of parameter $m \sim 3$. We also consider the free energy distribution, which can also be fitted by a generalized Gumbel distribution with a temperature dependent parameter, of order $m \sim 0.7$ in the critical region. Finally, the disorder averaged number of contacts with the interface scales at $T_c$ like $L^ρ$ with $ρ\sim 0.26 \sim 1/ν_R $.

cond-mat.dis-nn↗

arXiv2005

Distribution of pseudo-critical temperatures and lack of self-averaging in disordered Poland-Scheraga models with different loop exponents

According to recent progresses in the finite size scaling theory of disordered systems, thermodynamic observables are not self-averaging at critical points when the disorder is relevant in the Harris criterion sense. This lack of self-averageness at criticality is directly related to the distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples $(i)$ of size $L$. In this paper, we apply this analysis to disordered Poland-Scheraga models with different loop exponents $c$,corresponding to marginal and relevant disorder. In all cases, we numerically obtain a Gaussian histogram of pseudo-critical temperatures $T_c(i,L)$ with mean $T_c^{av}(L)$ and width $ΔT_c(L)$. For the marginal case $c=1.5$ corresponding to two-dimensional wetting, both the width $ΔT_c(L)$ and the shift $[T_c(\infty)-T_c^{av}(L)]$ decay as $L^{-1/2}$, so the exponent is unchanged ($ν_{random}=2=ν_{pure}$) but disorder is relevant and leads to non self-averaging at criticality. For relevant disorder $c=1.75$, the width $ΔT_c(L)$ and the shift $[T_c(\infty)-T_c^{av}(L)]$ decay with the same new exponent $L^{-1/ν_{random}}$ (where $ν_{random} \sim 2.7 > 2 > ν_{pure}$) and there is again no self-averaging at criticality. Finally for the value $c=2.15$, of interest in the context of DNA denaturation, the transition is first-order in the pure case. In the presence of disorder, the width $ΔT_c(L) \sim L^{-1/2}$ dominates over the shift $[T_c(\infty)-T_c^{av}(L)] \sim L^{-1}$, i.e. there are two correlation length exponents $ν=2$ and $\tilde ν=1$ that govern respectively the averaged/typical loop distribution.

cond-mat.dis-nn↗

arXiv2005

Numerical study of the disordered Poland-Scheraga model of DNA denaturation

We numerically study the binary disordered Poland-Scheraga model of DNA denaturation, in the regime where the pure model displays a first order transition (loop exponent $c=2.15>2$). We use a Fixman-Freire scheme for the entropy of loops and consider chain length up to $N=4 \cdot 10^5$, with averages over $10^4$ samples. We present in parallel the results of various observables for two boundary conditions, namely bound-bound (bb) and bound-unbound (bu), because they present very different finite-size behaviors, both in the pure case and in the disordered case. Our main conclusion is that the transition remains first order in the disordered case: in the (bu) case, the disorder averaged energy and contact densities present crossings for different values of $N$ without rescaling. In addition, we obtain that these disorder averaged observables do not satisfy finite size scaling, as a consequence of strong sample to sample fluctuations of the pseudo-critical temperature. For a given sample, we propose a procedure to identify its pseudo-critical temperature, and show that this sample then obeys first order transition finite size scaling behavior. Finally, we obtain that the disorder averaged critical loop distribution is still governed by $P(l) \sim 1/l^c$ in the regime $l \ll N$, as in the pure case.

cond-mat.dis-nn↗

arXiv2005

Strong disorder RG approach of random systems

There is a large variety of quantum and classical systems in which the quenched disorder plays a dominant rôle over quantum, thermal, or stochastic fluctuations : these systems display strong spatial heterogeneities, and many averaged observables are actually governed by rare regions. A unifying approach to treat the dynamical and/or static singularities of these systems has emerged recently, following the pioneering RG idea by Ma and Dasgupta and the detailed analysis by Fisher who showed that the Ma-Dasgupta RG rules yield asymptotic exact results if the broadness of the disorder grows indefinitely at large scales. Here we report these new developments by starting with an introduction of the main ingredients of the strong disorder RG method. We describe the basic properties of infinite disorder fixed points, which are realized at critical points, and of strong disorder fixed points, which control the singular behaviors in the Griffiths-phases. We then review in detail applications of the RG method to various disordered models, either (i) quantum models, such as random spin chains, ladders and higher dimensional spin systems, or (ii) classical models, such as diffusion in a random potential, equilibrium at low temperature and coarsening dynamics of classical random spin chains, trap models, delocalization transition of a random polymer from an interface, driven lattice gases and reaction diffusion models in the presence of quenched disorder. For several one-dimensional systems, the Ma-Dasgupta RG rules yields very detailed analytical results, whereas for other, mainly higher dimensional problems, the RG rules have to be implemented numerically. If available, the strong disorder RG results are compared with another, exact or numerical calculations.

cond-mat.dis-nn↗

arXiv2005

Two-dimensional wetting with binary disorder: a numerical study of the loop statistics

We numerically study the wetting (adsorption) transition of a polymer chain on a disordered substrate in 1+1 dimension.Following the Poland-Scheraga model of DNA denaturation, we use a Fixman-Freire scheme for the entropy of loops. This allows us to consider chain lengths of order $N \sim 10^5 $ to $10^6$, with $10^4$ disorder realizations. Our study is based on the statistics of loops between two contacts with the substrate, from which we define Binder-like parameters: their crossings for various sizes $N$ allow a precise determination of the critical temperature, and their finite size properties yields a crossover exponent $ϕ=1/(2-α) \simeq 0.5$.We then analyse at criticality the distribution of loop length $l$ in both regimes $l \sim O(N)$ and $1 \ll l \ll N$, as well as the finite-size properties of the contact density and energy. Our conclusion is that the critical exponents for the thermodynamics are the same as those of the pure case, except for strong logarithmic corrections to scaling. The presence of these logarithmic corrections in the thermodynamics is related to a disorder-dependent logarithmic singularity that appears in the critical loop distribution in the rescaled variable $λ=l/N$ as $λ\to 1$.

cond-mat.dis-nn↗

arXiv2004

Spin-glass chain in a magnetic field : influence of the disorder distribution on ground state properties and low-energy excitations

For the spin-glass chain in an external field $h$, a non-zero weight at the origin of the bond distribution $ρ(J)$ is known to induce a non-analytical magnetization at zero temperature : for $ρ(J) \sim A | J |^{μ-1}$ near $J \to 0$, the magnetization follows the Chen-Ma scaling $M \sim h^{μ/(2+μ)}$. In this paper, we numerically revisit this model to obtain detailed statistical information on the ground state configuration and on the low-energy two-level excitations that govern the low temperature properties. The ground state consists of long unfrustrated intervals separated by weak frustrated bonds: We accordingly compute the strength distribution of these frustrated bonds, as well as the length- and magnetization- distributions of the unfrustrated intervals. We find that the low-energy excitations are of two types (i) one frustrated bond of the ground state may have two positions that are nearly degenerate in energy (ii) two neighboring frustrated bonds of the ground state may be annihilated or created with nearly zero energy cost. For each excitation type, we compute its probability density as a function of its length. We moreover show that the contributions of these excitations to various observables (specific heat, Edwards-Anderson order parameter, susceptibility) are in full agreement with direct transfer matrix evaluations at low temperature. Finally, following the special bimodal case $\pm J$, where a Ma-Dasgupta RG procedure has been previously used to compute explicitly the above observables, we discuss the possibility of an extended RG procedure : we find that the ground state can be seen as the result of a hierarchical `fragmentation' procedure that we describe.

cond-mat.dis-nn↗

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

Ma-Dasgupta renormalization studies of various disordered systems

The Ma-Dasgupta real-space renormalization methods allow to study disordered systems which are governed by strong disorder fixed points. After a general introduction to the qualitative ideas and to the quantitative renormalization rules, we describe the explicit exact results that can be obtained in various one-dimensional models, either classical or quantum, either for dynamics or statics. The main part of this dissertation is devoted to statistical physics models, with special attention to (i) the off-equilibrium dynamics of a particle diffusing in a Brownian potential or in a trap landscape, (ii) the coarsening dynamics and the equilibrium of classical disordered spin chains, (iii) the delocalization transition of a random polymer at an interface. The last part of the dissertation deals with two disordered quantum spin chains which exhibit a zero-temperature phase transition as the disorder varies, namely (a) the random antiferromagnetic S=1 spin chain, (b) the random transverse field Ising chain.

cond-mat.dis-nn↗

arXiv2003

Directed polymers and interfaces in random media : free-energy optimization via confinement in a wandering tube

We analyze, via Imry-Ma scaling arguments, the strong disorder phases that exist in low dimensions at all temperatures for directed polymers and interfaces in random media. For the uncorrelated Gaussian disorder, we obtain that the optimal strategy for the polymer in dimension $1+d$ with $0 1/2$ for the wandering of the best favorable tube available. The corresponding free-energy then scales as $F \sim L^ω$ with $ω=2 ν-1$ and the left tail of the probability distribution involves a stretched exponential of exponent $η= (4-d)/2$. These results generalize the well known exact exponents $ν=2/3$, $ω=1/3$ and $η=3/2$ in $d=1$, where the subleading transverse length $R_S \sim L^{1/3}$ is known as the typical distance between two replicas in the Bethe Ansatz wave function. We then extend our approach to correlated disorder in transverse directions with exponent $α$ and/or to manifolds in dimension $D+d=d_{t}$ with $0 0$). In particular, for an interface of dimension $(d_t-1)$ in a space of total dimension $5/3<d_t<3$ with random-bond disorder, our approach yields the confinement exponent $ν_S = (d_t-1)(3-d_t)/(5d_t-7)$. Finally, we study the exponents in the presence of an algebraic tail $1/V^{1+μ}$ in the disorder distribution, and obtain various regimes in the $(μ,d)$ plane.

cond-mat.dis-nn↗

arXiv2003

Finite-size scaling properties of random transverse-field Ising chains : Comparison between canonical and microcanonical ensembles for the disorder

The Random Transverse Field Ising Chain is the simplest disordered model presenting a quantum phase transition at T=0. We compare analytically its finite-size scaling properties in two different ensembles for the disorder (i) the canonical ensemble, where the disorder variables are independent (ii) the microcanonical ensemble, where there exists a global constraint on the disorder variables. The observables under study are the surface magnetization, the correlation of the two surface magnetizations, the gap and the end-to-end spin-spin correlation $C(L)$ for a chain of length $L$. At criticality, each observable decays typically as $e^{- w \sqrt{L}}$ in both ensembles, but the probability distributions of the rescaled variable $w$ are different in the two ensembles, in particular in their asymptotic behaviors. As a consequence, the dependence in $L$ of averaged observables differ in the two ensembles. For instance, the correlation $C(L)$ decays algebraically as 1/L in the canonical ensemble, but sub-exponentially as $e^{-c L^{1/3}}$ in the microcanonical ensemble. Off criticality, probability distributions of rescaled variables are governed by the critical exponent $ν=2$ in both ensembles, but the following observables are governed by the exponent $\tilde ν=1$ in the microcanonical ensemble, instead of the exponent $ν=2$ in the canonical ensemble (a) in the disordered phase : the averaged surface magnetization, the averaged correlation of the two surface magnetizations and the averaged end-to-end spin-spin correlation (b) in the ordered phase : the averaged gap. In conclusion, the measure of the rare events that dominate various averaged observables can be very sensitive to the microcanonical constraint.

cond-mat↗

arXiv2003

Non-linear Response of the trap model in the aging regime : Exact results in the strong disorder limit

We study the dynamics of the one dimensional disordered trap model presenting a broad distribution of trapping times $p(τ) \sim 1/τ^{1+μ}$, when an external force is applied from the very beginning at $t=0$, or only after a waiting time $t_w$, in the linear as well as in the non-linear response regime. Using a real-space renormalization procedure that becomes exact in the limit of strong disorder $μ\to 0$, we obtain explicit results for many observables, such as the diffusion front, the mean position, the thermal width, the localization parameters and the two-particle correlation function. In particular, the scaling functions for these observables give access to the complete interpolation between the unbiased case and the directed case. Finally, we discuss in details the various regimes that exist for the averaged position in terms of the two times and the external field.

cond-mat↗

arXiv2003

Anomalous diffusion, Localization, Aging and Sub-aging effects in trap models at very low temperature

We study in details the dynamics of the one dimensional symmetric trap model, via a real-space renormalization procedure which becomes exact in the limit of zero temperature. In this limit, the diffusion front in each sample consists in two delta peaks, which are completely out of equilibrium with each other. The statistics of the positions and weights of these delta peaks over the samples allows to obtain explicit results for all observables in the limit $T \to 0$. We first compute disorder averages of one-time observables, such as the diffusion front, the thermal width, the localization parameters, the two-particle correlation function, and the generating function of thermal cumulants of the position. We then study aging and sub-aging effects : our approach reproduces very simply the two different aging exponents and yields explicit forms for scaling functions of the various two-time correlations. We also extend the RSRG method to include systematic corrections to the previous zero temperature procedure via a series expansion in $T$. We then consider the generalized trap model with parameter $α\in [0,1]$ and obtain that the large scale effective model at low temperature does not depend on $α$ in any dimension, so that the only observables sensitive to $α$ are those that measure the `local persistence', such as the probability to remain exactly in the same trap during a time interval. Finally, we extend our approach at a scaling level for the trap model in $d=2$ and obtain the two relevant time scales for aging properties.

cond-mat↗

arXiv2003

On a non-linear Fluctuation Theorem for the aging dynamics of disordered trap models

We consider the dynamics of the disordered trap model, which is known to be completely out-of-equilibrium and to present strong localization effects in its aging phase. We are interested into the influence of an external force, when it is applied from the very beginning at $t=0$, or only after a waiting time $t_w$. We obtain a "non-linear Fluctuation Theorem" for the corresponding one-time and two-time diffusion fronts in any given sample, that implies the following consequences : (i) for fixed times, there exists a linear response regime, where the Fluctuation-Dissipation Relation or Einstein relation is valid even in the aging time sector, in contrast with other aging disordered systems; (ii) for a fixed waiting time and fixed external field, the validity of the linear response regime is limited in time by a characteristic time depending on the external force; (iii) in the non-linear response regime, there exists a very simple relation for the asymmetry in the position for the one-time and the two-times disorder averaged diffusion fronts in the presence of an external force, in contrast to other models of random walks in random media. The present non-linear Fluctuation Theorem is a consequence of a special dynamical symmetry of the trap model.

cond-mat↗

arXiv2002

Localization properties of the anomalous diffusion phase $x ~ t^μ$ in the directed trap model and in the Sinai diffusion with bias

We study the anomalous diffusion phase $x ~ t^μ$ with $0<μ<1$ which exists both in the Sinai diffusion at small bias, and in the related directed trap model presenting a large distribution of trapping time $p(τ) \sim 1/τ^{1+μ}$. Our starting point is the Real Space Renormalization method in which the whole thermal packet is considered to be in the same renormalized valley at large time : this assumption is exact only in the limit $μ\to 0$ and corresponds to the Golosov localization. For finite $μ$, we thus generalize the usual RSRG method to allow for the spreading of the thermal packet over many renormalized valleys. Our construction allows to compute exact series expansions in $μ$ of all observables : at order $μ^n$, it is sufficient to consider a spreading of the thermal packet onto at most $(1+n)$ traps in each sample, and to average with the appropriate measure over the samples. For the directed trap model, we show explicitly up to order $μ^2$ how to recover the diffusion front, the thermal width, and the localization parameter $Y_2$. We moreover compute the localization parameters $Y_k$ for arbitrary $k$, the correlation function of two particles, and the generating function of thermal cumulants. We then explain how these results apply to the Sinai diffusion with bias, by deriving the quantitative mapping between the large-scale renormalized descriptions of the two models.

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

Localization of thermal packets and metastable states in Sinai model

We consider the Sinai model describing a particle diffusing in a 1D random force field. As shown by Golosov, this model exhibits a strong localization phenomenon for the thermal packet: the disorder average of the thermal distribution of the relative distance y=x-m(t), with respect to the (disorder-dependent) most probable position m(t), converges in the limit of infinite time towards a distribution P(y). In this paper, we revisit this question of the localization of the thermal packet. We first generalize the result of Golosov by computing explicitly the joint asymptotic distribution of relative position y=x(t)-m(t) and relative energy u=U(x(t))-U(m(t)) for the thermal packet. Next, we compute in the infinite-time limit the localization parameters Y_k, representing the disorder-averaged probabilities that k particles of the thermal packet are at the same place, and the correlation function C(l) representing the disorder-averaged probability that two particles of the thermal packet are at a distance l from each other. We moreover prove that our results for Y_k and C(l) exactly coincide with the thermodynamic limit of the analog quantities computed for independent particles at equilibrium in a finite sample of length L. Finally, we discuss the properties of the finite-time metastable states that are responsible for the localization phenomenon and compare with the general theory of metastable states in glassy systems, in particular as a test of the Edwards conjecture.

cond-mat.dis-nn↗

arXiv2000

Nonequilibrium dynamics of random field Ising spin chains: exact results via real space RG

Non-equilibrium dynamics of classical random Ising spin chains are studied using asymptotically exact real space renormalization group. Specifically the random field Ising model with and without an applied field (and the Ising spin glass (SG) in a field), in the universal regime of a large Imry Ma length so that coarsening of domains after a quench occurs over large scales. Two types of domain walls diffuse in opposite Sinai random potentials and mutually annihilate. The domain walls converge rapidly to a set of system-specific time-dependent positions {\it independent of the initial conditions}. We obtain the time dependent energy, magnetization and domain size distribution (statistically independent). The equilibrium limits agree with known exact results. We obtain exact scaling forms for two-point equal time correlation and two-time autocorrelations. We also compute the persistence properties of a single spin, of local magnetization, and of domains. The analogous quantities for the spin glass are obtained. We compute the two-point two-time correlation which can be measured by experiments on spin-glass like systems. Thermal fluctuations are found to be dominated by rare events; all moments of truncated correlations are computed. The response to a small field applied after waiting time $t_w$, as measured in aging experiments, and the fluctuation-dissipation ratio $X(t,t_w)$ are computed. For $(t-t_w) \sim t_w^{\hatα}$, $\hatα <1$, it equals its equilibrium value X=1, though time translational invariance fails. It exhibits for $t-t_w \sim t_w$ aging regime with non-trivial $X=X(t/t_w) \neq 1$, different from mean field.

cond-mat↗

arXiv1999

On the localization of random heteropolymers at the interface between two selective solvents

To study the localization of random heteropolymers at an interface separating two selective solvents within the model of Garel, Huse, Leibler and Orland, Europhys. Lett. {\bf 8} 9 (1989), we propose an approach based on a disorder-dependent real space renormalization procedure. This approach allows to recover that a chain with a symmetric distribution in hydrophobic/hydrophilic components is localized at any temperature in the thermodynamic limit, whereas a dissymmetric distribution in hydrophobic/hydrophilic components leads to a delocalization phase transition. It yields in addition explicit expressions for the thermodynamic quantities as well as a very detailed description of the statistical properties of the behaviors of the heteropolymers in the high temperature limit. For the case of a small dissymmetry in hydrophobic/hydrophilic components, the renormalization approach yields explicit predictions for the delocalization transition temperature and for the critical behaviors of various quantities : in particular, the free energy presents an essential singularity at the transition, the typical length of blobs in the preferred solvent diverges with an essential singularity, whereas the typical length of blobs in the other solvent diverges algebraically. Finite-size properties are also characterized in details for both cases. In particular, we give the probability distribution of the delocalization temperature for the ensemble of chains of finite (large) length $L$. Finally, we discuss the non-equilibrium dynamics at temperature $T$ starting from a zero-temperature initial condition.

cond-mat↗