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 $S_\mathcal{G}(\gamma)=\det(\gamma-\Delta)$ 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 Schr\"odinger operator $\det(\gamma-\Delta+V(x))$.Comment: LaTeX, 8 pages, 7 eps figures, v2: new appendix, v3: discussions and ref adde

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 Schr\"odinger 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{2\rho/g}}, where $g$ is the strength of the random force field and $\rho$ the density of absorbers.Comment: 10 pages ; LaTeX ; 4 pdf figures ; Proceedings of the meeting "Fundations and Applications of non-equilibrium statistical mechanics", Nordita, Stockholm, october 2011 ; v2: appendix added ; v3: figure 2.left adde

Conditional stability of unstable viscous shock waves in compressible gas dynamics and MHD

Extending our previous work in the strictly parabolic case, we show that a linearly unstable Lax-type viscous shock solution of a general quasilinear hyperbolic--parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small $L^1\cap H^3$ perturbations, converging time-asymptotically to a translate of the unperturbed wave. That is, for a shock with $p$ unstable eigenvalues, we establish conditional stability on a codimension-$p$ manifold of initial data, with sharp rates of decay in all $L^p$. For $p=0$, we recover the result of unconditional stability obtained by Mascia and Zumbrun. The main new difficulty in the hyperbolic--parabolic case is to construct an invariant manifold in the absence of parabolic smoothing.Comment: 32p

NMR study of electronic correlations in Mn-doped Ba(Fe 1 − x Co x ) 2 As 2 and BaFe 2 (As 1 − x P x ) 2

International audienceWe probe the real space electronic response to a local magnetic impurity in isovalent and het-erovalent doped BaFe2As2 (122) using Nuclear Magnetic Resonance (NMR). The local moments carried by Mn impurities doped into Ba(Fe1−xCox)2As2 (Co-122) and BaFe2(As1−xPx)2 (P-122) at optimal doping induce a spin polarization in the vicinity of the impurity. The amplitude, shape and extension of this polarisation is given by the real part of the susceptibility χ (r) of FeAs layers, and is consequently related to the nature and strength of the electronic correlations present in the system. We study this polarisation using 75 As NMR in Co-122 and both 75 As and 31 P NMR in P-122. The NMR spectra of Mn-doped materials is made of two essential features. First is a satellite line associated with nuclei located as nearest neighbor of Mn impurities. The analysis of the temperature dependence of the shift of this satellite line shows that Mn local moments behave as isolated Curie moments. The second feature is a temperature dependent broadening of the central line. We show that the broadening of the central line follows the susceptibility of Mn local moments, as expected from typical RKKY-like interactions. This demonstrates that the susceptibility χ (r) of FeAs layers does not make significant contribution to the temperature dependent broadening of the central line. χ (r) is consequently only weakly temperature dependent in optimally doped Co-122 and P-122. This behaviour is in contrast with that of strongly correlated materials such as under-doped cuprate high-Tc superconductors where the central line broadens faster than the impurity susceptibility grows, because of the development of strong magnetic correlations when T is lowered. Moreover, the FeAs layer susceptibility is found quantitatively similar in both heterovalent doped and isolvalent doped BaFe2As2

Mn local moments prevent superconductivity in iron-pnictides Ba(Fe 1-x Mn x)2As2

75As nuclear magnetic resonance (NMR) experiments were performed on Ba(Fe1-xMnx)2As2 (xMn = 2.5%, 5% and 12%) single crystals. The Fe layer magnetic susceptibility far from Mn atoms is probed by the75As NMR line shift and is found similar to that of BaFe2As2, implying that Mn does not induce charge doping. A satellite line associated with the Mn nearest neighbours (n.n.) of 75As displays a Curie-Weiss shift which demonstrates that Mn carries a local magnetic moment. This is confirmed by the main line broadening typical of a RKKY-like Mn-induced staggered spin polarization. The Mn moment is due to the localization of the additional Mn hole. These findings explain why Mn does not induce superconductivity in the pnictides contrary to other dopants such as Co, Ni, Ru or K.Comment: 6 pages, 7 figure

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.Comment: LaTeX, 23 pages, 3 pdf figures ; review for a special issue on "Lyapunov analysis" ; v2 : typo corrected in eq.(3) & minor change

Sinai model in presence of dilute absorbers

We study the Sinai model for the diffusion of a particle in a one dimension random potential in presence of a small concentration $\rho$ of perfect absorbers using the asymptotically exact real space renormalization method. We compute the survival probability, the averaged diffusion front and return probability, the two particle meeting probability, the distribution of total distance traveled before absorption and the averaged Green's function of the associated Schrodinger operator. Our work confirms some recent results of Texier and Hagendorf obtained by Dyson-Schmidt methods, and extends them to other observables and in presence of a drift. In particular the power law density of states is found to hold in all cases. Irrespective of the drift, the asymptotic rescaled diffusion front of surviving particles is found to be a symmetric step distribution, uniform for $|x| < {1/2} \xi(t)$, where $\xi(t)$ is a new, survival length scale ($\xi(t)=T \ln t/\sqrt{\rho}$ in the absence of drift). Survival outside this sharp region is found to decay with a larger exponent, continuously varying with the rescaled distance $x/\xi(t)$. A simple physical picture based on a saddle point is given, and universality is discussed.Comment: 21 pages, 2 figure

Functionals of the Brownian motion, localization and metric graphs

We review several results related to the problem of a quantum particle in a random environment. In an introductory part, we recall how several functionals of the Brownian motion arise in the study of electronic transport in weakly disordered metals (weak localization). Two aspects of the physics of the one-dimensional strong localization are reviewed : some properties of the scattering by a random potential (time delay distribution) and a study of the spectrum of a random potential on a bounded domain (the extreme value statistics of the eigenvalues). Then we mention several results concerning the diffusion on graphs, and more generally the spectral properties of the Schr\"odinger operator on graphs. The interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated. Finally, we consider a two-dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices. We recall the connection between spectral properties of this model and winding functionals of the planar Brownian motion.Comment: Review article. 50 pages, 21 eps figures. Version 2: section 5.5 and conclusion added. Several references adde

Induced vs Spontaneous Breakdown of S-matrix Unitarity: Probability of No Return in Quantum Chaotic and Disordered Systems

We investigate systematically sample-to sample fluctuations of the probability $\tau$ of no return into a given entrance channel for wave scattering from disordered systems. For zero-dimensional ("quantum chaotic") and quasi one-dimensional systems with broken time-reversal invariance we derive explicit formulas for the distribution of $\tau$, and investigate particular cases. Finally, relating $\tau$ to violation of S-matrix unitarity induced by internal dissipation, we use the same quantity to identify the Anderson delocalisation transition as the phenomenon of spontaneous breakdown of S-matrix unitarity.Comment: This is the published version, with a few modifications added to the last par