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Christian Sadel

Publications and source records attributed to Christian Sadel.

At least 19 recordsLinked to original sources
arXiv2024

Scaling of the Lyapunov exponent at a balanced hyperbolic critical point

In both the random hopping model and at topological phase transitions in one-dimensional chiral systems, the Lyapunov exponent vanishes at zero energy, but is here shown to have an inverse logarithmic increase with a coefficient that is computed explicitly. This is the counterpart of the Dyson spike in the density of states. The argument also transposes to the free energy density of the random field Ising model, and more generally to many so-called balanced hyperbolic critical points. It is based on the fact that the Furstenberg measure in rescaled logarithmic Dyson-Schmidt variables can be well-approximated by an absolutely continuous measure with trapezoidal density.

math-ph↗
arXiv2024

On absolutely continuous spectrum for one-channel unitary operators

In this paper, we develop the radial transfer matrix formalism for unitary one-channel operators. This generalizes previous formalisms for CMV matrices and scattering zippers. We establish an analog of Carmona's formula and deduce criteria for absolutely continuous spectrum which we apply to random Hilbert Schmidt perturbations of periodic scattering zippers.

math-ph↗
arXiv2023

Footprint of a topological phase transition on the density of states

For a generalized Su-Schrieffer-Heeger model the energy zero is always critical and hyperbolic in the sense that all reduced transfer matrices commute and have their spectrum off the unit circle. Disorder driven topological phase transitions in this model are characterized by a vanishing Lyapunov exponent at the critical energy. It is shown that the integrated density of states away from a transition has a pseudogap with an explicitly computable Hölder exponent, while it has a characteristic divergence (Dyson spike) at the transition points. The proof is based on renewal theory for the Prüfer phase dynamics and the optional stopping theorem for martingales of suitably constructed comparison processes.

math-ph↗
arXiv2020

Transfer matrices for discrete Hermitian operators and absolutely continuous spectrum

We introduce a transfer matrix method for the spectral analysis of discrete Hermitian operators with locally finite hopping. Such operators can be associated with a locally finite graph structure and the method works in principle on any such graph. The key result is a spectral averaging formula well known for Jacobi or 1-channel operators giving the spectral measure at a root vector by a weak limit of products of transfer matrices. Here, we assume an increase in the rank for the connections between spherical shells which is a typical situation and true on finite dimensional lattices $\mathbb{Z}^d$. The product of transfer matrices are considered as a transformation of the relations of 'boundary resolvent data' along the shells. The trade off is that at each level or shell with more forward then backward connections (rank-increase) we have a set of transfer matrices at a fixed spectral parameter. Still, considering these products we can relate the minimal norm growth over the set of all products with the spectral measure at the root and obtain several criteria for absolutely continuous spectrum. Finally, we give some example of operators on stair-like graphs (increasing width) which has absolutely continuous spectrum with a sufficiently fast decaying random shell-matrix-potential.

math.SP↗
arXiv2018

GOE statistics for Anderson models on antitrees and thin boxes in $\mathbb{Z}^3$ with deformed Laplacian

Sequences of certain finite graphs, antitrees, are constructed along which the Anderson model shows GOE statistics, i.e. a re-scaled eigenvalue process converges to the ${\rm Sine}_1$ process. The Anderson model on the graph is a random matrix being the sum of the adjacency matrix and a random diagonal matrix with independent identically distributed entries along the diagonal. The strength of the randomness stays fixed, there is no re-scaling with matrix size. These considered random matrices giving GOE statistics can also be viewed as random Schrödinger operators $\mathcal{P}Δ+\mathcal{V}$ on thin finite boxes in $\mathbb{Z}^3$ where the Laplacian $Δ$ is deformed by a projection $\mathcal{P}$ commuting with $Δ$.

math-ph↗
arXiv2018

Spectral Theory of one-channel operators and application to absolutely continuous spectrum for Anderson type models

A one-channel operator is a self-adjoint operator on $\ell^2(\mathbb{G})$ for some countable set $\mathbb{G}$ with a rank 1 transition structure along the sets of a quasi-spherical partition of $\mathbb{G}$. Jacobi operators are a very special case. In essence, there is only one channel through which waves can travel across the shells to infinity. This channel can be described with transfer matrices which include scattering terms within the shells and connections to neighboring shells. Not all of the transfer matrices are defined for some countable set of energies. Still, many theorems from the world of Jacobi operators are translated to this setup. The results are then used to show absolutely continuous spectrum for the Anderson model on certain finite dimensional graphs with a one-channel structure. This result generalizes some previously obtained results on antitrees.

math-ph↗
arXiv2017

Topological boundary invariants for Floquet systems and quantum walks

A Floquet systems is a periodically driven quantum system. It can be described by a Floquet operator. If this unitary operator has a gap in the spectrum, then one can define associated topological bulk invariants which can either only depend on the bands of the Floquet operator or also on the time as a variable. It is shown how a K-theoretic result combined with the bulk-boundary correspondence leads to edge invariants for the half-space Floquet operators. These results also apply to topological quantum walks.

math-ph↗
arXiv2017

Singular analytic linear cocycles with negative infinite Lyapunov exponents

We show that linear analytic cocycles where all Lyapunov exponents are negative infinite are nilpotent. For such one-frequency cocycles we show that they can be analytically conjugated to an upper triangular cocycle or a Jordan normal form. As a consequence, an arbitrarily small analytic perturbation leads to distinct Lyapunov exponents. Moreover, in the one-frequency case where the $k$-th Lyapunov exponent is finite and the $k+1$st negative infinite, we obtain a simple criterion for domination in which case there is a splitting into a nilpotent part and an invertible part.

math.DS↗
arXiv2017

Complex one-frequency cocycles

We show that on a dense open set of analytic one-frequency complex valued cocycles in arbitrary dimension Oseledets filtration is either dominated or trivial. The underlying mechanism is different from that of the Bochi-Viana Theorem for continuous cocycles, which links non-domination with discontinuity of the Lyapunov exponent. Indeed, in our setting the Lyapunov exponents are shown to depend continuously on the cocycle, even if the initial irrational frequency is allowed to vary. On the other hand, this last property provides a good control of the periodic approximations of a cocycle, allowing us to show that domination can be characterized, in the presence of a gap in the Lyapunov spectrum, by additional regularity of the dependence of sums of Lyapunov exponents.

math.DS↗
arXiv2015

A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes

We consider products of random matrices that are small, independent identically distributed perturbations of a fixed matrix $T_0$. Focusing on the eigenvalues of $T_0$ of a particular size we obtain a limit to a SDE in a critical scaling. Previous results required $T_0$ to be a (conjugated) unitary matrix so it could not have eigenvalues of different modulus. From the result we can also obtain a limit SDE for the Markov process given by the action of the random products on the flag manifold. Applying the result to random Schrödinger operators we can improve some result by Valko and Virag showing GOE statistics for the rescaled eigenvalue process of a sequence of Anderson models on long boxes. In particular we solve a problem posed in their work.

math-ph↗
arXiv2015

Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel

We show that the Anderson model has a transition from localization to delocalization at exactly 2 dimensional growth rate on antitrees with normalized edge weights which are certain discrete graphs. The kinetic part has a one-dimensional structure allowing a description through transfer matrices which involve some Schur complement. For such operators we introduce the notion of having one propagating channel and extend theorems from the theory of one-dimensional Jacobi operators that relate the behavior of transfer matrices with the spectrum. These theorems are then applied to the considered model. In essence, in a certain energy region the kinetic part averages the random potentials along shells and the transfer matrices behave similar as for a one-dimensional operator with random potential of decaying variance. At $d$ dimensional growth for $d>2$ this effective decay is strong enough to obtain absolutely continuous spectrum, whereas for some uniform $d$ dimensional growth with $d<2$ one has pure point spectrum in this energy region. At exactly uniform $2$ dimensional growth also some singular continuous spectrum appears, at least at small disorder. As a corollary we also obtain a change from singular spectrum ($d\leq 2$) to absolutely continuous spectrum ($d\geq 3)$ for random operators of the type $\mathcal{P}_r Δ_d \mathcal{P}_r+λ\mathcal{V}$ on $\mathbb{Z}^d$, where $\mathcal{P}_r$ is an orthogonal radial projection, $Δ_d$ the discrete adjacency operator (Laplacian) on $\mathbb{Z}^d$ and $λ\mathcal{V}$ a random potential.

math-ph↗
arXiv2014

Localization for transversally periodic random potentials on binary trees

We consider a random Schrödinger operator on the binary tree with a random potential which is the sum of a random radially symmetric potential, $Q_r$, and a random transversally periodic potential, $κQ_t$, with coupling constant $κ$. Using a new one-dimensional dynamical systems approach combined with Jensen's inequality in hyperbolic space (our key estimate) we obtain a fractional moment estimate proving localization for small and large $κ$. Together with a previous result we therefore obtain a model with two Anderson transitions, from localization to delocalization and back to localization, when increasing $κ$. As a by-product we also have a partially new proof of one-dimensional Anderson localization at any disorder.

math-ph↗
arXiv2013

A Herman-Avila-Bochi formula for higher dimensional pseudo-unitary and hermitian-symplectic cocycles

A Herman-Avila-Bochi type formula is obtained for the average sum of the top d Lyapunov exponents over a one-parameter family of G-cocycles, where G is the group that leaves a certain, non-degenerate hermitian form of signature (c,d) invariant. The generic example of such a group is the pseudo-unitary group U(c,d) or in the case c=d, the hermitian-symplectic group HSp(2d) which naturally appears for cocycles related to Schrödinger operators. In the case d=1, the formula for HSp(2d) cocycles reduces to the Herman-Avila-Bochi formula for SL(2,R) cocycles.

math.DS↗
arXiv2012

Absolutely Continuous Spectrum for Random Schroedinger Operators on the Bethe Strip

The Bethe Strip of width $m$ is the cartesian product $\B\times\{1,...,m\}$, where $\B$ is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have "extended states" for small disorder. More precisely, we consider Anderson-like Hamiltonians $\;H_λ=\frac12 Δ\otimes 1 + 1 \otimes A + λ\Vv$ on a Bethe strip with connectivity $K \geq 2$, where $A$ is an $m\times m$ symmetric matrix, $\Vv$ is a random matrix potential, and $λ$ is the disorder parameter. Given any closed interval $I\subset (-\sqrt{K}+a_{\mathrm{max}},\sqrt{K}+a_{\mathrm{min}})$, where $a_{\mathrm{min}}$ and $a_{\mathrm{max}}$ are the smallest and largest eigenvalues of the matrix $A$, we prove that for $λ$ small the random Schrödinger operator $\;H_λ$ has purely absolutely continuous spectrum in $I$ with probability one and its integrated density of states is continuously differentiable on the interval $I$.

math-ph↗
arXiv2012

Relations between Transfer and Scattering Matrices in the presence of Hyperbolic Channels

We consider a cable described by a discrete, space-homogeneous, quasi one-dimensional Schrödinger operator $H_0$. We study the scattering by a finite disordered piece (the scatterer) inserted inside this cable. For energies $E$ where $H_0$ has only elliptic channels we use the Lippmann-Schwinger equations to show that the scattering matrix and the transfer matrix, written in an appropriate basis, are related by a certain polar decomposition. For energies $E$ where $H_0$ has hyperbolic channels we show that the scattering matrix is related to a reduced transfer matrix and both are of smaller dimension than the transfer matrix. Moreover, in this case the scattering matrix is determined from a limit of larger dimensional scattering matrices, as follows: We take a piece of the cable of length $m$, followed by the scatterer and another piece of the cable of length $m$, consider the scattering matrix of these three joined pieces inserted inside an ideal lead at energy $E$ (ideal means only elliptic channels), and take the limit $m\to\infty$.

math-ph↗