arXiv subjects

Christophe Texier

Publications and source records attributed to Christophe Texier.

At least 37 records · Page 2Linked to original sources

arXiv2015

Four-terminal resistances in mesoscopic networks of metallic wires: Weak localisation and correlations

We consider the electronic transport in multi-terminal mesoscopic networks of weakly disordered metallic wires. After a brief description of the classical transport, we analyze the weak localisation (WL) correction to the four-terminal resistances, which involves an integration of the Cooperon over the wires with proper weights. We provide an interpretation of these weights in terms of classical transport properties. We illustrate the formalism on examples and show that weak localisation to four-terminal conductances may become large in some situations. In a second part, we study the correlations of four-terminal resistances and show that integration of Diffuson and Cooperon inside the network involves the same weights as the WL. The formulae are applied to multiconnected wire geometries.

cond-mat.dis-nn↗

arXiv2015

Capacitance and charge relaxation resistance of chaotic cavities - Joint distribution of two linear statistics in the Laguerre ensemble of random matrices

We consider the AC transport in a quantum RC circuit made of a coherent chaotic cavity with a top gate. Within a random matrix approach, we study the joint distribution for the mesoscopic capacitance $C_μ=(1/C+1/C_q)^{-1}$ and the charge relaxation resistance $R_q$, where $C$ is the geometric capacitance and $C_q$ the quantum capacitance. We study the limit of a large number of conducting channels $N$ with a Coulomb gas method. We obtain $\langle R_q\rangle\simeq h/(Ne^2)=R_\mathrm{dc}$ and show that the relative fluctuations are of order $1/N$ both for $C_q$ and $R_q$, with strong correlations $\langle δC_qδR_q\rangle/\sqrt{\langle δC_q^2\rangle\,\langle δR_q^2\rangle}\simeq+0.707$. The detailed analysis of large deviations involves a second order phase transition in the Coulomb gas. The two dimensional phase diagram is obtained.

cond-mat.mes-hall↗

arXiv2014

Fluctuations of random matrix products and 1D Dirac equation with random mass

We study the fluctuations of certain random matrix products $Π_N=M_N\cdots M_2M_1$ of $\mathrm{SL}(2,\mathbb{R})$, describing localisation properties of the one-dimensional Dirac equation with random mass. In the continuum limit, i.e. when matrices $M_n$'s are close to the identity matrix, we obtain convenient integral representations for the variance $Γ_2=\lim_{N\to\infty}\mathrm{Var}(\ln||Π_N||)/N$. The case studied exhibits a saturation of the variance at low energy $\varepsilon$ along with a vanishing Lyapunov exponent $Γ_1=\lim_{N\to\infty}\ln||Π_N||/N$, leading to the behaviour $Γ_2/Γ_1\sim\ln(1/|\varepsilon|)\to\infty$ as $\varepsilon\to0$. Our continuum description sheds new light on the Kappus-Wegner (band center) anomaly.

cond-mat.dis-nn↗

arXiv2014

Wigner time-delay distribution in chaotic cavities and freezing transition

Using the joint distribution for proper time-delays of a chaotic cavity derived by Brouwer, Frahm & Beenakker [Phys. Rev. Lett. {\bf 78}, 4737 (1997)], we obtain, in the limit of large number of channels $N$, the large deviation function for the distribution of the Wigner time-delay (the sum of proper times) by a Coulomb gas method. We show that the existence of a power law tail originates from narrow resonance contributions, related to a (second order) freezing transition in the Coulomb gas.

cond-mat.mes-hall↗

arXiv2013

One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers

We study the one-dimensional Schrödinger equation with a disordered potential of the form $V (x) = ϕ(x)^2+ϕ'(x) + κ(x) $ where $ϕ(x)$ is a Gaussian white noise with mean $μg$ and variance $g$, and $κ(x)$ is a random superposition of delta functions distributed uniformly on the real line with mean density $ρ$ and mean strength $v$. Our study is motivated by the close connection between this problem and classical diffusion in a random environment (the Sinai problem) in the presence of random absorbers~: $ϕ(x)$ models the force field acting on the diffusing particle and $κ(x)$ models the absorption properties of the medium in which the diffusion takes place. The focus is on the calculation of the complex Lyapunov exponent $ Ω(E) = γ(E) - \mathrm{i} πN(E) $, where $N$ is the integrated density of states per unit length and $γ$ the reciprocal of the localisation length. By using the continuous version of the Dyson-Schmidt method, we find an exact formula, in terms of a Hankel function, in the particular case where the strength of the delta functions is exponentially-distributed with mean $v=2g$. Building on this result, we then solve the general case -- in the low-energy limit -- in terms of an infinite sum of Hankel functions. Our main result, valid without restrictions on the parameters of the model, is that the integrated density of states exhibits the power law behaviour $$ N(E) \underset{E\to0+}{\sim} E^ν\hspace{0.5cm} \mbox{where } ν=\sqrt{μ^2+2ρ/g}\:. $$ This confirms and extends several results obtained previously by approximate methods.

cond-mat.dis-nn↗

arXiv2013

Dynamics of a tagged monomer: Effects of elastic pinning and harmonic absorption

We study the dynamics of a tagged monomer of a Rouse polymer for different initial configurations. In the case of free evolution, the monomer displays subdiffusive behavior with strong memory of the initial state. In presence of either elastic pinning or harmonic absorption, we show that the steady state is independent of the initial condition which however strongly affects the transient regime, resulting in non-monotonous behavior and power-law relaxation with varying exponents.

cond-mat.stat-mech↗

arXiv2013

Ergodic vs diffusive decoherence in mesoscopic devices

We report on the measurement of phase coherence length in a high mobility two-dimensional electron gas patterned in two different geometries, a wire and a ring. The phase coherence length is extracted both from the weak localization correction in long wires and from the amplitude of the Aharonov-Bohm oscillations in a single ring, in a low temperature regime when decoherence is dominated by electronic interactions. We show that these two measurements lead to different phase coherence lengths, namely $L_Φ^\mathrm{wire}\propto T^{-1/3}$ and $L_Φ^\mathrm{ring}\propto T^{-1/2}$. This difference reflects the fact that the electrons winding around the ring necessarily explore the whole sample (ergodic trajectories), while in a long wire the electrons lose their phase coherence before reaching the edges of the sample (diffusive regime).

cond-mat.mes-hall↗

arXiv2012

Ordered spectral statistics in 1D disordered supersymmetric quantum mechanics and Sinai diffusion with dilute absorbers

Some results on the ordered statistics of eigenvalues for one-dimensional random Schrödinger Hamiltonians are reviewed. In the case of supersymmetric quantum mechanics with disorder, the existence of low energy delocalized states induces eigenvalue correlations and makes the ordered statistics problem nontrivial. The resulting distributions are used to analyze the problem of classical diffusion in a random force field (Sinai problem) in the presence of weakly concentrated absorbers. It is shown that the slowly decaying averaged return probability of the Sinai problem, $\mean{P(x,t|x,0)}\sim \ln^{-2}t$, is converted into a power law decay, $\mean{P(x,t|x,0)}\sim t^{-\sqrt{2ρ/g}}$, where $g$ is the strength of the random force field and $ρ$ the density of absorbers.

cond-mat.dis-nn↗

arXiv2012

Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices

The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the localisation length. The Lyapunov exponent also features prominently in the theory of products of random matrices pioneered by Furstenberg. After a brief historical survey, we describe some recent work that exploits the close connections between these topics. We review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case. Finally, we point out some limitations of the Lyapunov exponent as a means of studying localisation properties.

cond-mat.dis-nn↗

arXiv2011

Supersymmetric quantum mechanics with Levy disorder in one dimension

We consider the Schroedinger equation with a supersymmetric random potential, where the superpotential is a Levy noise. We focus on the problem of computing the so-called complex Lyapunov exponent, whose real and imaginary parts are, respectively, the Lyapunov exponent and the integrated density of states of the system. In the case where the Levy process is non-decreasing, we show that the calculation of the complex Lyapunov exponent reduces to a Stieltjes moment problem, we ascertain the low-energy behaviour of the density of states in some generality, and relate it to the distributional properties of the Levy process. We review the known solvable cases, where the complex Lyapunov exponent can be expressed in terms of special functions, and discover a new one.

math-ph↗

arXiv2010

Products of random matrices and generalised quantum point scatterers

To every product of $2\times2$ matrices, there corresponds a one-dimensional Schrödinger equation whose potential consists of generalised point scatterers. Products of {\em random} matrices are obtained by making these interactions and their positions random. We exhibit a simple one-dimensional quantum model corresponding to the most general product of matrices in $\text{SL}(2, {\mathbb R})$. We use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms.

cond-mat.dis-nn↗

arXiv2010

$ζ$-regularised spectral determinants on metric graphs

Several general results for the spectral determinant of the Schrödinger operator on metric graphs are reviewed. Then, a simple derivation for the $ζ$-regularised spectral determinant is proposed, based on the Roth trace formula. Two types of boundary conditions are studied: functions continuous at the vertices and functions whose derivative is continuous at the vertices. The $ζ$-regularised spectral determinant of the Schrödinger operator acting on functions with the most general boundary conditions is conjectured in conclusion. The relation to the Ihara, Bass and Bartholdi formulae obtained for combinatorial graphs is also discussed.

math-ph↗

arXiv2009

Effect of boundaries on the spectrum of a one-dimensional random mass Dirac Hamiltonian

The average density of states (DoS) of the one-dimensional Dirac Hamiltonian with a random mass on a finite interval [0,L] is derived. Our method relies on the eigenvalues distributions (extreme value statistics problem) which are explicitly obtained. The well-known Dyson singularity \sim-L/|epsilon|ln^3|ε| is recovered above the crossover energy epsilon_c\sim exp-sqrt{L}. Below epsilon_c we find a log-normal suppression of the average DoS \sim 1/(|epsilon|sqrt(L))exp(-(ln^2|epsilon|)/L).

cond-mat.dis-nn↗

arXiv2009

Quantum oscillations and decoherence due to electron-electron interaction in metallic networks and hollow cylinders

We have studied the quantum oscillations of the conductance for arrays of connected mesoscopic metallic rings, in the presence of an external magnetic field. Several geometries have been considered: a linear array of rings connected with short or long wires compared to the phase coherence length, square networks and hollow cylinders. Compared to the well-known case of the isolated ring, we show that for connected rings, the winding of the Brownian trajectories around the rings is modified, leading to a different harmonics content of the quantum oscillations. We relate this harmonics content to the distribution of winding numbers. We consider the limits where coherence length $L_φ$ is small or large compared to the perimeter $L$ of each ring constituting the network. In the latter case, the coherent diffusive trajectories explore a region larger than $L$, whence a network dependent harmonics content. Our analysis is based on the calculation of the spectral determinant of the diffusion equation for which we have a simple expression on any network. It is also based on the hypothesis that the time dependence of the dephasing between diffusive trajectories can be described by an exponential decay with a single characteristic time $τ_φ$ (model A) . At low temperature, decoherence is limited by electron-electron interaction, and can be modelled in a one-electron picture by the fluctuating electric field created by other electrons (model B). It is described by a functional of the trajectories and thus the dependence on geometry is crucial. Expressions for the magnetoconductance oscillations are derived within this model and compared to the results of model A. It is shown that they involve several temperature-dependent length scales.

cond-mat.dis-nn↗

arXiv2009

One-dimensional classical diffusion in a random force field with weakly concentrated absorbers

A one-dimensional model of classical diffusion in a random force field with a weak concentration $ρ$ of absorbers is studied. The force field is taken as a Gaussian white noise with $\mean{ϕ(x)}=0$ and $\mean{ϕ(x)ϕ(x')}=g δ(x-x')$. Our analysis relies on the relation between the Fokker-Planck operator and a quantum Hamiltonian in which absorption leads to breaking of supersymmetry. Using a Lifshits argument, it is shown that the average return probability is a power law $\smean{P(x,t|x,0)}\sim{}t^{-\sqrt{2ρ/g}}$ (to be compared with the usual Lifshits exponential decay $\exp{-(ρ^2t)^{1/3}}$ in the absence of the random force field). The localisation properties of the underlying quantum Hamiltonian are discussed as well.

cond-mat.dis-nn↗

arXiv2008

Localization for one-dimensional random potentials with large local fluctuations

We study the localization of wave functions for one-dimensional Schrödinger Hamiltonians with random potentials $V(x)$ with short range correlations and large local fluctuations such that $\int\D{x} \smean{V(x)V(0)}=\infty$. A random supersymmetric Hamiltonian is also considered. Depending on how large the fluctuations of $V(x)$ are, we find either new energy dependences of the localization length, $\ell_\mathrm{loc}\propto{}E/\ln{E}$, $\ell_\mathrm{loc}\propto{}E^{μ/2}$ with $0<μ<2$ or $\ell_\mathrm{loc}\propto\ln^{μ-1}E$ for $μ>1$, or superlocalization (decay of the wave functions faster than a simple exponential).

cond-mat.dis-nn↗

arXiv2008

Breaking supersymmetry in a one-dimensional random Hamiltonian

The one-dimensional supersymmetric random Hamiltonian $H_{susy}=-\frac{d^2}{dx^2}+ϕ^2+ϕ'$, where $ϕ(x)$ is a Gaussian white noise of zero mean and variance $g$, presents particular spectral and localization properties at low energy: a Dyson singularity in the integrated density of states (IDoS) $N(E)\sim1/\ln^2E$ and a delocalization transition related to the behaviour of the Lyapunov exponent (inverse localization length) vanishing like $γ(E)\sim1/|\ln{}E|$ as $E\to0$. We study how this picture is affected by breaking supersymmetry with a scalar random potential: $H=H_{susy}+V(x)$ where $V(x)$ is a Gaussian white noise of variance $σ$. In the limit $σ\ll{g}^3$, a fraction of states $N(0)\sim{g}/\ln^2(g^3/σ)$ migrate to the negative spectrum and the Lyapunov exponent reaches a finite value $γ(0)\sim{g}/\ln(g^3/σ)$ at E=0. Exponential (Lifshits) tail of the IDoS for $E\to-\infty$ is studied in detail and is shown to involve a competition between the two noises $ϕ$ and $V$ whatever the larger is. This analysis relies on analytic results for $N(E)$ and $γ(E)$ obtained by two different methods: a stochastic method and the replica method. The problem of extreme value statistics of eigenvalues is also considered (distribution of the n-th excited state energy). The results are analyzed in the context of classical diffusion in a random force field in the presence of random annihilation/creation local rates.

cond-mat.dis-nn↗

arXiv2008

On the spectrum of the Laplace operator of metric graphs attached at a vertex -- Spectral determinant approach

We consider a metric graph $\mathcal{G}$ made of two graphs $\mathcal{G}_1$ and $\mathcal{G}_2$ attached at one point. We derive a formula relating the spectral determinant of the Laplace operator $S_\mathcal{G}(γ)=\det(γ-Δ)$ in terms of the spectral determinants of the two subgraphs. The result is generalized to describe the attachment of $n$ graphs. The formulae are also valid for the spectral determinant of the Schrödinger operator $\det(γ-Δ+V(x))$.

math-ph↗