A Kinetic Theory for Quantum Transport in Aperiodic Media - Schulz-Baldes, Hermann; Bellissard, Jean (Journal of Statistical Physics, 1998)
Transport theory is reinvestigated in a one-particle model including dissipation mechanisms through a phenomenological time-dependent part. Using the description of an aperiodic crystal in terms of noncommutative geometry, it is then possible to rigorously derive Kubo formulas for the electric conductivity. Within the framework of the relaxation time approximation, a fractal analysis applied to the spectral measures entering in the Kubo formula allows us to examine anomalous transport owing to quantum interferences. This leads to anomalies in Drude's formula.
Keywords: Kubo formula in aperiodic; media anomalous transport

Analysis of Sparse Recovery Algorithms via the Replica Method - Bereyhi, Ali; Müller, Ralf; Schulz-Baldes, Hermann (Compressed Sensing in Information Processing, 2022)
This chapter goes through the fundamental connections between statistical mechanics and estimation theory by focusing on the particular problem of compressive sensing. We first show that the asymptotic analysis of a sparse recovery algorithm is mathematically equivalent to the problem of calculating the free energy of a spin glass in the thermodynamic limit. We then use the replica method from statistical mechanics to evaluate the performance in the asymptotic regime. The asymptotic results have several applications in communications and signal processing. We briefly go through two instances of these applications: Characterization of joint sparse recovery algorithms used in distributed compressive sensing and tuning of receivers employed for detection of spatially modulated signals.
CRIS-Team Scopus Importer:2022-11-04

Analyticity properties of the scattering matrix for matrix Schrödinger operators on the discrete line - Ballesteros, Miguel; Franco, Gerardo; Schulz-Baldes, Hermann (Journal of Mathematical Analysis and Applications, 2021)
Explicit formulas for the analytic extensions of the scattering matrix and the time delay of a quasi-one-dimensional discrete Schrödinger operator with a potential of finite support are derived. This includes a careful analysis of the band edge singularities and allows to prove a Levinson-type theorem. The main algebraic tool are the plane wave transfer matrices.
CRIS-Team Scopus Importer:2021-01-29
Keywords: Discrete Schrödinger operator; Scattering theory; Spectral theory

Anomalous Drude Model - Schulz-Baldes, Hermann (Physical Review Letters, 1997)
A generalization of the Drude model is studied. On the one hand, the free motion of the particles is allowed to be sub- or superdiffusive; on the other hand, the distribution of the time delay between collisions is allowed to have a long tail and even a nonvanishing first moment. The collision averaged motion is either regular diffusive or Lévy-flight-like. The anomalous diffusion coefficients show complex scaling laws. The conductivity can be calculated in the diffusive regime. The model is of interest for the phenomenological study of electronic transport in quasicrystals.
UnivIS-Import:2015-03-05:Pub.1997.nat.dma.pma22.anomal

Anomalous transport: a mathematical framework - Schulz-Baldes, Hermann; Bellissard, Jean (Reviews in Mathematical Physics, 1998)
We develop a mathematical framework allowing to study anomalous transport in homogeneous solids. The main tools characterizing the anomalous transport properties are spectral and diffusion exponents associated to the covariant Hamiltonians describing these media. The diffusion exponents characterize the spectral measures entering in Kubo's formula for the conductivity and hence lead to anomalies in Drude's formula. We give several formulas allowing to calculate these exponents and treat, as an example, Wegner's n-orbital model as well as the Anderson model in coherent potential approximation.
UnivIS-Import:2015-03-05:Pub.1998.nat.dma.pma22.anomal

Anomalous transport in aperiodic media - Schulz-Baldes, Hermann (Proceedings of the ICMP 2000, 2001)
UnivIS-Import:2015-04-16:Pub.2001.nat.dma.pma22.anomal

Anomalous transport in presence of singular continuous spectral measures - Barbaroux, Jean-Marie; Schulz-Baldes, Hermann (Annales de l'Institut Henri Poincaré - Physique Théorique, 1999)
UnivIS-Import:2015-03-05:Pub.1999.nat.dma.pma22.anomal

Anomalous transport in quasicrystals - Bellissard, Jean; Schulz-Baldes, Hermann (Proceedings of the 5th International Conference on Quasicrystals, 1995)
UnivIS-Import:2015-04-16:Pub.1995.nat.dma.pma22.anomal

Application of Semifinite Index Theory to Weak Topological Phases - Bourne, Christopher; Schulz-Baldes, Hermann (2016 MATRIX Annals, 2018)
Recent work by Prodan and the second author showed that weak invariants of topological insulators can be described using Kasparov’s KK-theory. In this note, a complementary description using semifinite index theory is given. This provides an alternative proof of the index formulae for weak complex topological phases using the semifinite local index formula. Real invariants and the bulk-boundary correspondence are also briefly considere

Applications to Solid State Systems - Schulz-Baldes, Hermann; Stoiber, Tom (Springer, 2022)
This chapter is about the applications of the results in the earlier chapters to solid state systems. There are numerous new results, and furthermore new or at least improved arguments of statements that can be already be found in the literature. The overview in the Preface essentially only mentions two new results (the bulk-boundary correspondence for irrational edges and the existence of flat bands of edge states for chiral Hamiltonians with a pseudogap and non-vanishing weak invariants). As this chapter is rather long and contains much more than that, let us give a brief more detailed outline of its contents.
CRIS-Team Scopus Importer:2024-12-16

Approximate symmetries and conservation laws in topological insulators and associated Z-invariants - Doll, Nora; Schulz-Baldes, Hermann (Annals of Physics, 2020)
Solid state systems with time reversal symmetry and/or particle–hole symmetry often only have Z2-valued strong invariants for which no general local formula is known. For physically relevant values of the parameters, there may exist approximate symmetries or almost conserved observables, such as the spin in a quantum spin Hall system with small Rashba coupling. It is shown in a general setting how this allows to define robust integer-valued strong invariants stemming from the complex theory, such as the spin Chern numbers, which modulo 2 are equal to the Z2-invariants. Moreover, these integer invariants can be computed using twisted versions of the spectral localizer.
CRIS-Team Scopus Importer:2020-07-10
Keywords: Approximate laws; Spectral localizer; Strong invariants

Band Edge Limit of the Scattering Matrix for Quasi-One-Dimensional Discrete Schrodinger Operators - Ballesteros, Miguel; Franco, Gerardo; Garro, Guillermo; Schulz-Baldes, Hermann (Complex Analysis and Operator Theory, 2022)
This paper is about the scattering theory for one-dimensional matrix Schrodinger operators with a matrix potential having a finite first moment. The transmission coefficients are analytically continued and extended to the band edges. An explicit expression is given for these extensions. The limits of the reflection coefficients at the band edges are also calculated.
CRIS-Team WoS Importer:2022-03-11

Bayesian Inference with Nonlinear Generative Models: Comments on Secure Learning - Bereyhi, Ali; Loureiro, Bruno; Krzakala, Florent; Müller, Ralf; Schulz-Baldes, Hermann (IEEE Transactions on Information Theory, 2023)
Unlike the classical linear model, nonlinear generative models have been addressed sparsely in the literature of statistical learning. This work aims to shed light on these models and their secrecy potential. To this end, we invoke the replica method to derive the asymptotic normalized cross entropy in an inverse probability problem whose generative model is described by a Gaussian random field with a generic covariance function. Our derivations further demonstrate the asymptotic statistical decoupling of the Bayesian estimator and specify the decoupled setting for a given nonlinear model. The replica solution depicts that strictly nonlinear models establish an all-or-nothing phase transition: there exists a critical load at which the optimal Bayesian inference changes from perfect to an uncorrelated learning. Based on this finding, we design a new secure coding scheme which achieves the secrecy capacity of the wiretap channel. This interesting result implies that strictly nonlinear generative models are perfectly secured without any secure coding. We justify this latter statement through the analysis of an illustrative model for perfectly secure and reliable inference.
CRIS-Team Scopus Importer:2023-11-03
Keywords: Bayes methods; Bayesian inference; decoupling principle; Encoding; Gaussian random fields; Glass; Information theory; information-theoretically secure learning; Load modeling; Nonlinear generative models; Nonlinear optics; replica method; Statistical learning

Besov Spaces for Isometric G-Actions - Schulz-Baldes, Hermann; Stoiber, Tom (Springer, 2022)
Classical Sobolev, Besov and Triebel-Lizorkin spaces measure smoothness and integrability properties of functions on Rn simultaneously. From several points of view, the Besov spaces are particularly well-behaved [1] which is why they appear naturally in numerous contexts. Relevant for the present context is Peller’s characterization of traceclass properties of Hankel operators by Besov regularity of their symbols [2]. Furthermore, it is remarkable that in trace theorems there is no loss of Besov regularity (other than for Sobolev spaces) [1].
CRIS-Team Scopus Importer:2024-12-16

Boundary maps for C**-crossed products with R with an application to the quantum Hall effect - Kellendonk, Johannes; Schulz-Baldes, Hermann (Communications in Mathematical Physics, 2004)
The boundary map in K-theory arising from the Wiener-Hopf extension of a crossed product algebra with <img alt="https://static-content.springer.com/image/art%3A10.1007%2Fs00220-004-1122-7/MediaObjects/s00220-004-1122-7flb2.gif" src="https://static-content.springer.com/image/art%3A10.1007%2Fs00220-004-1122-7/MediaObjects/s00220-004-1122-7flb2.gif" /> is the Connes-Thom isomorphism. In this article the Wiener Hopf extension is combined with the Heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces (and cyclic cohomology). It then follows directly from a non-commutative Stokes theorem that this map is dual w.r.t. Connes’ pairing of cyclic cohomology with K-theory. As an application, we prove equality of quantized bulk and edge conductivities for the integer quantum Hall effect described by continuous magnetic Schrödinger operators.
UnivIS-Import:2015-03-09:Pub.2004.nat.dma.pma22.bounda

Bulk and Boundary Invariants for Complex Topological Insulators - Prodan, Emil; Schulz-Baldes, Hermann (Springer International Publishing, 2016)
This monograph offers an overview of rigorous results on fermionic topological insulators from the complex classes, namely, those without symmetries or with just a chiral symmetry. Particular focus is on the stability of the topological invariants in the presence of strong disorder, on the interplay between the bulk and boundary invariants and on their dependence on magnetic fields. The first part presents motivating examples and the conjectures put forward by the physics community, together with a brief review of the experimental achievements. The second part develops an operator algebraic approach for the study of disordered topological insulators. This leads naturally to the use of analytical tools from K-theory and non-commutative geometry, such as cyclic cohomology, quantized calculus with Fredholm modules and index pairings. New results include a generalized Streda formula and a proof of the delocalized nature of surface states in topological insulators with non-trivial invariants. The concluding chapter connects the invariants to measurable quantities and thus presents a refined physical characterization of the complex topological insulators.  This book is intended for advanced students in mathematical physics and researchers alike.

Callias-type operators associated to spectral triples - Schulz-Baldes, Hermann; Stoiber, Tom (Journal of Noncommutative Geometry, 2023)
Callias-type (or Dirac-Schrödinger) operators associated to abstract semifinite spectral triples are introduced and their indices are computed in terms of an associated index pairing derived from the spectral triple. The result is then interpreted as an index theorem for a non-commutative analogue of spectral flow. Both even and odd spectral triples are considered, and both commutative and non-commutative examples are given.
Keywords: index theorem; Callias operator; spectral flow

Chern numbers as half-signature of the spectral localizer - Lozano Viesca, Edgar; Schober, Jonas; Schulz-Baldes, Hermann (Journal of Mathematical Physics, 2019)
Two recent papers proved that complex index pairings can be calculated as the half-signature of a finite dimensional matrix, called the spectral localizer. This paper contains a new proof of this connection for even index pairings based on a spectral flow argument. It also provides a numerical study of the spectral gap and the half-signature of the spectral localizer for a typical two-dimensional disordered topological insulator in the regime of a mobility gap at the Fermi energy. This regime is not covered by the above mathematical results (which suppose a bulk gap), but, nevertheless, the half-signature of the spectral localizer is a clear indicator of a topological phase.
CRIS-Team Scopus Importer:2019-08-06

Delocalization in random polymer models - Jitomirskaya, Svetlana; Schulz-Baldes, Hermann; Stolz, Gunter (Communications in Mathematical Physics, 2003)
A random polymer model is a one-dimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer model. It is proven that the Lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there. Large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport, showing that it is almost surely overdiffusive even though the models are known to have pure-point spectrum with exponentially localized eigenstates for almost every configuration of the polymers. Furthermore, the level spacing is shown to be regular at the critical energy.
UnivIS-Import:2015-03-09:Pub.2003.nat.dma.pma22.deloca

Dimensional Reduction and Scattering Formulation for Even Topological Invariants - Schulz-Baldes, Hermann; Toniolo, Daniele (Communications in Mathematical Physics, 2021)
Strong invariants of even-dimensional topological insulators of independent Fermions are expressed in terms of an invertible operator on the Hilbert space over the boundary. It is given by the Cayley transform of the boundary restriction of the half-space resolvent. This dimensional reduction is routed in new representation for the K-theoretic exponential map. It allows to express the invariants via the reflection matrix at the Fermi energy, for the scattering set-up of a wire coupled to the half-space insulator.
CRIS-Team Scopus Importer:2020-11-20

Duality for Toeplitz Extensions - Schulz-Baldes, Hermann; Stoiber, Tom (Springer, 2022)
Let (A, G, ξ) be a C∗ -dynamical system, where A is a separable C∗ -algebra and G= R or G= T. This section will relate A and A⋊ ξG in an exact sequence of C∗ -algebras and then show a duality for pairings of cyclic cocycles with Kj(A) respectively Kj + 1(A⋊ ξG).
CRIS-Team Scopus Importer:2024-12-16

Edge channels and Chern numbers in the integer quantum Hall effect - Kellendonk, Johannes; Richter, Thomas; Schulz-Baldes, Hermann (Reviews in Mathematical Physics, 2002)
A quantization theorem for the edge currents is proven for discrete magnetic half-plane operators. Hence the edge channel number is a valid concept also in presence of a disordered potential. Under a gap condition on the corresponding planar model, this quantum number is shown to be equal to the quantized Hall conductivity as given by the Kubo-Chern formula. For the proof of this equality, we consider an exact sequence of C*-algebras (the Toeplitz extension) linking the half-plane and the planar problem, and use a duality theorem for the pairings of K-groups with cyclic cohomology.
UnivIS-Import:2015-03-09:Pub.2002.nat.dma.pma22.edgech

Edge versus Bulk Currents in the Integer Quantum Hall Effect - Schulz-Baldes, Hermann; Kellendonk, Johannes; Richter, Thomas (Journal of Physics A: Mathematical and Theoretical, 2000)
UnivIS-Import:2015-03-09:Pub.2000.nat.dma.pma22.edgeve

Erratum: Oscillation theory for the density of states of high dimensional random operators (International Mathematics Research Notices DOI: 10.1093/imrn/rnx246) - Großmann, Julian; Schulz-Baldes, Hermann; Villegas-Blas, Carlos (International Mathematics Research Notices, 2019)
In "Oscillation Theory for the Density of States of High Dimensional Random Operators" by Julian Großmann, Hermann Schulz-Baldes, and Carlos Villegas-Blas, the acknowledgments statement in Section 1 was omitted. This correction has been implemented in print and online.
CRIS-Team Scopus Importer:2021-04-20

Finite volume calculation of K-theory invariants - Schulz-Baldes, Hermann; Loring, Terry (New York Journal of Mathematics, 2017)
Odd index pairings of K1-group elements with Fredholm modules are of relevance in index theory, differential geometry and applications such as to topological insulators. For the concrete setting of operators on a Hilbert space over a lattice, it is shown how to calculate the resulting index as the signature of a suitably constructed finite-dimensional matrix, more precisely the finite volume restriction of what we call the spectral localizer. In presence of real symmetries, secondary ℤ2-invariants can be obtained as the sign of the Pfaffian of the spectral localizer. These results reconcile two complementary approaches to invariants of topological insulators.

Footprint of a topological phase transition on the density of states - De Moor, Joris; Sadel, Christian; Schulz-Baldes, Hermann (Letters in Mathematical Physics, 2023)
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 away from such a transition the density of states vanishes at zero energy 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.
Keywords: Dyson spike; Topological phase transition; Hyperbolic critical energy

Gaussian fluctuations for random matrices with correlated entries - Schulz-Baldes, Hermann; Schenker, Jeffrey (International Mathematics Research Notices, 2007)
For randommatrix ensembles with non-gaussian matrix elements that may exhibit some correlations, it is shown that centered traces of polynomials in the matrix converge in distribution to a Gaussian process whose covariance matrix is diagonal in the basis of Chebyshev polynomials. The proof is combinatorial and adapts Wigner's argument showing the convergence of the density of states to the semicircle law. © The Author 2007.
UnivIS-Import:2015-03-09:Pub.2007.nat.dma.pma22.gaussi

Gaussian fluctuations of products of random matrices distributed close to the identity - Drabkin, Maxim; Schulz-Baldes, Hermann (Journal of Difference Equations and Applications, 2015)
Products of random 2 × 2 matrices exhibit Gaussian fluctuations around almost surely convergent Lyapunov exponents. In this paper, the distribution of the random matrices is supported by a small neighbourhood of order of the identity matrix. The Lyapunov exponent and the variance of the Gaussian fluctuations are calculated perturbatively in and this requires a detailed analysis of the associated random dynamical system on the unit circle and its invariant measure. The result applies to anomalies and band edges of one-dimensional random Schrödinger operators.
Keywords: products of random matrices; Lyapunov exoponent; variances; perturbation theory

Generalized Connes-Chern characters in KK-theory with an application to weak invariants of topological insulators - Prodan, Emil; Schulz-Baldes, Hermann (Reviews in Mathematical Physics, 2016)
We use constructive bounded Kasparov K" role="presentation">K-theory to investigate the numerical invariants stemming from the internal Kasparov products Ki(풜)×KKi(풜,ℬ)→K0(ℬ)→ℝ" role="presentation">Ki(𝒜)×KKi(𝒜,ℬ)→K0(ℬ)→ℝ, i=0,1" role="presentation">i=0,1, where the last morphism is provided by a tracial state. For the class of properly defined finitely-summable Kasparov (풜,ℬ)" role="presentation">(𝒜,ℬ)-cycles, the invariants are given by the pairing of K" role="presentation">K-theory of ℬ" role="presentation">ℬ with an element of the periodic cyclic cohomology of ℬ" role="presentation">ℬ, which we call the generalized Connes–Chern character. When 풜" role="presentation">𝒜 is a twisted crossed product of ℬ" role="presentation">ℬ by ℤk" role="presentation">ℤk, 풜=ℬ⋊ξθℤk" role="presentation">𝒜=ℬ⋊θξℤk, we derive a local formula for the character corresponding to the fundamental class of a properly defined Dirac cycle. Furthermore, when ℬ=C(Ω)⋊ξ′ϕℤj" role="presentation">ℬ=C(Ω)⋊ϕξ′ℤj, with C(Ω)" role="presentation">C(Ω) the algebra of continuous functions over a disorder configuration space, we show that the numerical invariants are connected to the weak topological invariants of the complex classes of topological insulators, defined in the physics literature. The end products are generalized index theorems for these weak invariants, which enable us to predict the range of the invariants and to identify regimes of strong disorder in which the invariants remain stable. The latter will be reported in a subsequent publicatio
Keywords: Operator Algebras; Disordered Systems and Neural Networks; Mathematical Physics; K-Theory and Homology

Geometry of Weyl theory for Jacobi matrices with matrix entries - Schulz-Baldes, Hermann (Journal D Analyse Mathematique, 2010)
A Jacobi matrix with matrix entries is a self-adjoint block tridiagonal matrix with invertible blocks on the off-diagonals. The Weyl surface describing the dependence of Green's matrix on the boundary conditions is interpreted as the set of maximally isotropic subspaces of a quadratic form given by the Wronskian. Analysis of the possibly degenerate limit quadratic form leads to the limit point/limit surface theory of maximal symmetric extensions for semi-infinite Jacobi matrices with matrix entries with arbitrary deficiency indices. The resolvent of the extensions is calculated explicitly. © 2010 Hebrew University Magnes Press.
UnivIS-Import:2015-03-09:Pub.2010.nat.dma.pma22.geomet

Harmonic analysis in operator algebras and its applications to index theory and topological solid state systems - Schulz-Baldes, Hermann; Stoiber, Tom (Springer, 2022)
This book contains a self-consistent treatment of Besov spaces for W*-dynamical systems, based on the Arveson spectrum and Fourier multipliers. Generalizing classical results by Peller, spaces of Besov operators are then characterized by trace class properties of the associated Hankel operators lying in the W*-crossed product algebra. These criteria allow to extend index theorems to such operator classes. This in turn is of great relevance for applications in solid-state physics, in particular, Anderson localized topological insulators as well as topological semimetals. The book also contains a self-contained chapter on duality theory for R-actions. It allows to prove a bulk-boundary correspondence for boundaries with irrational angles which implies the existence of flat bands of edge states in graphene-like systems. This book is intended for advanced students in mathematical physics and researchers alike.

Homotopy arguments for quantized Hall conductivity - Richter, Thomas; Schulz-Baldes, Hermann (Journal of Mathematical Physics, 2001)
Using the strong localization bounds obtained by the Aizenman-Molcanov method for a particle in a magnetic field and a disordered potential, we show that the zero-temperature Hall conductivity of a gas of such particles is quantized and constant as long as both Fermi energy and disorder coupling parameter vary in a region of strong localization of the corresponding two-dimensional phase diagram. © 2001 American Institute of Physics.
UnivIS-Import:2015-03-09:Pub.2001.nat.dma.pma22.homoto

Index Pairings in Presence of Symmetries with Applications to Topological Insulators - Großmann, Julian; Schulz-Baldes, Hermann (Communications in Mathematical Physics, 2016)
In a basic framework of a complex Hilbert space equipped with a complex conjugation and an involution, linear operators can be real, quaternionic, symmetric or anti-symmetric, and orthogonal projections can furthermore be Lagrangian. This paper investigates index pairings of projections and unitaries submitted to such symmetries. Various scenarios emerge: Noether indices can take either arbitrary integer values or only even integer values or they can vanish and then possibly have secondary Z2-invariants. These general results are applied to prove index theorems for the strong invariants of disordered topological insulators. The symmetries come from the Fermi projection (K-theoretic part of the pairing) and the Dirac operator (K-homological part of the pairing depending on the dimension of physical space).

Intermittent lower bound on quantum diffusion - Guarneri, Italo; Schulz-Baldes, Hermann (Letters in Mathematical Physics, 1999)
An intermittent lower bound on quantum diffusion is proven in presence of a multifractal spectral measure.
UnivIS-Import:2015-03-05:Pub.1999.nat.dma.pma22.interm
Keywords: Multifractal measures; Quantum diffusion

Invariants of disordered semimetals via the spectral localizer - Schulz-Baldes, Hermann; Stoiber, Tom (EPL - Europhysics Letters, 2022)
The spectral localizer consists in placing the Hamiltonian in a Dirac trap. For topological insulators its spectral asymmetry is equal to the topological invariants, providing a highly efficient tool for numerical computation. Here this technique is extended to disordered semimetals and allows to access the number of Dirac or Weyl points as well as weak invariants. These latter invariants imply the existence of surface state

Krein signatures of transfer operators for half-space topological insulators - Schulz-Baldes, Hermann; Villegas-Blas, Carlos (Journal of Physics A: Mathematical and Theoretical, 2016)
We propose a complementary point of view on the topological invariants of two-dimensional tight-binding models restricted to half-spaces. The transfer operators for such systems are J-unitary on a infinite dimensional Krein space (,J) and, for energies in the bulk gap, only have discrete spectrum on the unit circle. These eigenvalues have Krein inertia which can be used to define topological invariants determining the nature of the surface modes and allowing to distinguish different topological phases. This is illustrated by numerical results.

Levinson theorem for discrete Schrödinger operators on the line with matrix potentials having a first moment - Ballesteros, Miguel; Franco, Gerardo; Naumkin, Ivan; Schulz-Baldes, Hermann (Communications in Contemporary Mathematics, 2023)
This paper proves new results on spectral and scattering theory for matrix-valued Schrödinger operators on the discrete line with non-compactly supported perturbations whose first moments are assumed to exist. In particular, a Levinson theorem is proved, in which a relation between scattering data and spectral properties (bound and half bound states) of the corresponding Hamiltonians is derived. The proof is based on stationary scattering theory with prominent use of Jost solutions at complex energies that are controlled by Volterra-type integral equations.

Lifshitz tails for the 1D Bernoulli-Anderson model - Schulz-Baldes, Hermann (Markov Processes and Related Fields, 2004)
UnivIS-Import:2015-03-09:Pub.2004.nat.dma.pma22.lifshi
Keywords: Lifshitz tails; Anderson model

Localization and Chern numbers for weakly disordered BdG operators - de Nittis, Giuseppe; Drabkin, Maxim; Schulz-Baldes, Hermann (Markov Processes and Related Fields, 2015)
After a short discussion of various random Bogoliubov-de Gennes (BdG) model operators and the associated physics, the Aizenman - Molchanov method is applied to prove Anderson localization in the weak disorder regime for the spectrum in the central gap. This allows to construct random BdG operators which have localized states in an interval centered at zero energy. Furthermore, techniques for the calculation of Chern numbers are reviewed and applied to two non-trivial BdG operators, the p + ip wave and d + id wave superconductors.
UnivIS-Import:2015-04-15:Pub.2013.nat.dma.pma22.locali
Keywords: Anderson localization; Random Bogoliubov-de Gennes operators; Weak disorder

Local Markers for Crystalline Topology - Cerjan, Alexander; Loring, Terry A.; Schulz-Baldes, Hermann (Physical Review Letters, 2024)
Over the last few years, crystalline topology has been used in photonic crystals to realize edge- and corner-localized states that enhance light-matter interactions for potential device applications. However, the band-theoretic approaches currently used to classify bulk topological crystalline phases cannot predict the existence, localization, or spectral isolation of any resulting boundary-localized modes. While interfaces between materials in different crystalline phases must have topological states at some energy, these states need not appear within the band gap, and thus may not be useful for applications. Here, we derive a class of local markers for identifying material topology due to crystalline symmetries, as well as a corresponding measure of topological protection. As our real-space-based approach is inherently local, it immediately reveals the existence and robustness of topological boundary-localized states, yielding a predictive framework for designing topological crystalline heterostructures. Beyond enabling the optimization of device geometries, we anticipate that our framework will also provide a route forward to deriving local markers for other classes of topology that are reliant upon spatial symmetries.
CRIS-Team Scopus Importer:2024-03-15

Low density expansion for Lyapunov exponents - Schulz-Baldes, Hermann (Mathematical Physics of Quantum Mechanics, 2006)
A perturbative formula for the Lyapunov exponent of a one-dimensional random medium for weakly coupled disorder was first given by Thouless [12] and then proven rigorously by Pastur and Figotin [9]. Anomalies in the perturbation theory at the band center were discovered by Kappus and Wegner [7] and further discussed by various other authors [2,3,11]. The Lyapunov exponent is then identified with the inverse localization length of the system. This short note concerns the behavior of the Lyapunov exponent for a low density of impurities, each of which may, however, be large. The presented method is as [6,10,11] a further application of diagonalizing the transfer matrices without perturbation (here the low density of impurities) and then rigorously controlling the error terms by means of oscillatory sums of rotating modified Prüer phases. Some of the oscillatory sums remain large if the rotation phases (here the quasi-momenta) are rational. This leads to supplementary contributions of the Kappus-Wegner type. © Springer 2006.
UnivIS-Import:2015-04-20:Pub.2006.nat.dma.pma22.lowden

Lower bounds on wave packet propagation by packing dimensions of spectral measures - Guarneri, Italo; Schulz-Baldes, Hermann (Mathematical Physics Electronic Journal, 1999)
We prove that, for any quantum evolution in l2(ZD), there exist arbitrarily long time scales on which the qth moment of the position operator increases at least as fast as a power of time given by q/D times the packing dimension of the spectral measure. Packing dimensions of measures and their connections to scaling exponents and box-counting dimensions are also discussed.
UnivIS-Import:2015-03-05:Pub.1999.nat.dma.pma22.lowerb

Lyapunov Exponents at Anomalies of SL(2, ℝ)-actions - Schulz-Baldes, Hermann (Operator Theory, Analysis and Mathematical Physics, 2007)
Anomalies are known to appear in the perturbation theory for the one-dimensional Anderson model. A systematic approach to anomalies at critical points of products of random matrices is developed, classifying and analysing their possible types. The associated invariant measure is calculated formally. For an anomaly of so-called second degree, it is given by the ground-state of a certain Fokker-Planck equation on the unit circle. The Lyapunov exponent is calculated to lowest order in perturbation theory with rigorous control of the error terms.
UnivIS-Import:2015-03-09:Pub.2007.nat.dma.pma22.lyapun

Lyapunov spectra for all symmetry classes of quasi-one-dimensional disordered systems of non-interacting Fermions - Ludwig, Andreas W. W.; Schulz-Baldes, Hermann; Stolz, Michael (Journal of Statistical Physics, 2013)
A random phase property is proposed for products of random matrices drawn from any one of the classical groups associated with the ten Cartan symmetry classes of non-interacting disordered Fermion systems. It allows to calculate the Lyapunov spectrum explicitly in a perturbative regime. These results apply to quasi-one-dimensional random Dirac operators which can be constructed as representatives for each of the ten symmetry classes. For those symmetry classes that correspond to two-dimensional topological insulators or superconductors, the random Dirac operators describing the one-dimensional boundaries have vanishing Lyapunov exponents and almost surely an absolutely continuous spectrum, reflecting the gapless and conducting nature of the boundary degrees of freedom. © 2013 Springer Science+Business Media New York.
UnivIS-Import:2015-03-09:Pub.2013.nat.dma.pma22.lyapun
Keywords: Delocalization; Lyapunov spectrum; Symmetry classes

Mott law as lower bound for a random walk in a random environment - Faggionato, Alessandra; Schulz-Baldes, Hermann; Spehner, D. (Communications in Mathematical Physics, 2006)
We consider a random walk on the support of an ergodic stationary simple point process on ℝ d, d < 2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.
UnivIS-Import:2015-03-09:Pub.2006.nat.dma.pma22.mottla

Non-commutative odd Chern numbers and topological phases of disordered chiral systems - Prodan, Emil; Schulz-Baldes, Hermann (Journal of Functional Analysis, 2016)
An index theorem for higher Chern characters of odd Fredholm modules over crossed product algebras is proved, together with a local formula for the associated cyclic cocycle. The result generalizes the classic Noether–Gohberg–Krein index theorem, which in its simplest form states that the winding number of a complex-valued function over the circle is equal to the index of the associated Toeplitz operator. When applied to the non-commutative Brillouin zone, this generalization allows to define topological invariants for all condensed matter phases from the chiral unitary (or AIII-symmetry) class in the presence of strong disorder and magnetic fields, whenever the Fermi level lies in a region of Anderson localized spectrum.
UnivIS-Import:2015-04-02:Pub.2014.nat.dma.pma22.noncom
Keywords: Index theorem; Local formula for Chern character; Topological insulators; Chiral unitary class

On Z(2)-indices for ground states of fermionic chains - Bourne, Chris; Schulz-Baldes, Hermann (Reviews in Mathematical Physics, 2020)
For parity-conserving fermionic chains, we review how to associate Z(2)-indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the Z(2)-valued spectral flow provides a topological obstruction for two systems to have the same Z(2)-index. A rudimentary definition of a Z(2)-phase label for a class of parity-invariant and pure ground states of the one-dimensional infinite CAR algebra is also provided. Ground states with differing phase labels cannot be connected without a closing of the spectral gap of the infinite GNS Hamiltonian.
CRIS-Team WoS Importer:2020-11-06

Orbital polarization and magnetization for independent particles in disordered media - Schulz-Baldes, Hermann; Teufel, Stefan (Communications in Mathematical Physics, 2013)
Formulas for the contribution of the conduction electrons to the polarization and magnetization are derived for disordered systems and within a one-particle framework. These results generalize known formulas for Bloch electrons and the presented proofs considerably simplify and strengthen prior justifications. The new formulas show that orbital polarization and magnetization are of geometric nature. This leads to quantization for a periodically driven Piezo effect as well as the derivative of the magnetization w. r. t. the chemical potential. It is also shown how the latter is connected to boundary currents in Chern insulators. The main technical tools in the proofs are an adaption of Nenciu's super-adiabatic theory to C*-dynamical systems and Bellissard's Ito derivatives w. r. t. the magnetic field. © 2012 Springer-Verlag Berlin Heidelberg.
UnivIS-Import:2015-03-09:Pub.2013.nat.dma.pma22.orbita

Oscillation theory for the density of states of high dimensional random operators - Großmann, Julian; Schulz-Baldes, Hermann; Villegas-Blas, Carlos (International Mathematics Research Notices, 2017)
Sturm-Liouville oscillation theory is studied for Jacobi operators with block entries given by covariant operators on an infinite dimensional Hilbert space. It is shown that the integrated density of states of the Jacobi operator is approximated by the winding of the Pruefer phase w.r.t. the trace per unit volume. This rotation number can be interpreted as a spectral flow in a von Neumann algebra with finite trace.

Parity as Z2-valued spectral flow - Doll, Nora; Schulz-Baldes, Hermann; Waterstraat, Nils (Bulletin of the London Mathematical Society, 2019)
This note is about the topology of the path space of linear Fredholm operators on a real Hilbert space. Fitzpatrick and Pejsachowicz introduced the parity of such a path, based on the Leray-Schauder degree of a path of parametrices. Here an alternative analytic approach is presented which reduces the parity to the Z2-valued spectral flow of an associated path of chiral skew-adjoints. Furthermore, the related notion of Z2-index of a Fredholm pair of chiral complex structures is introduced and connected to the parity of a suitable path. Several non-trivial examples are provided. One of them concerns topological insulators, another an application to the bifurcation of a non-linear partial differential equation.
CRIS-Team WoS Importer:2019-09-17

Partially hyperbolic random dynamics on Grassmannians - De Moor, Joris; Dorsch, Florian; Schulz-Baldes, Hermann (Journal of Mathematical Physics, 2023)
A sequence of invertible matrices given by a small random perturbation around a fixed diagonal partially hyperbolic matrix induces a random dynamics on the Grassmann manifolds. Under suitable weak conditions, it is known to have a unique invariant (Furstenberg) measure. The main result gives concentration bounds on this measure, showing that with high probability, the random dynamics stays in the vicinity of stable fixed points of the unperturbed matrix, in a regime where the strength of the random perturbation dominates the local hyperbolicity of the diagonal matrix. As an application, bounds on sums of Lyapunov exponents are obtained.
CRIS-Team Scopus Importer:2023-09-01

Persistence of spin edge currents in disordered quantum spin Hall systems - Schulz-Baldes, Hermann (Communications in Mathematical Physics, 2013)
For a disordered two-dimensional model of a topological insulator (such as a Kane-Mele model with disordered potential) with small coupling of spin invariance and time-reversal symmetry breaking terms (such as a Rashba spin-orbit coupling and a Zeeman term), it is proved that the spin edge currents persist provided there is a spectral gap and the spin Chern numbers are well-defined and non-trivial. These are sufficient conditions for being in the quantum spin Hall phase. The result materializes the general philosophy that topological insulators are topologically non-trivial bulk systems with persistent edge or surface currents. © 2013 Springer-Verlag Berlin Heidelberg.
UnivIS-Import:2015-03-09:Pub.2013.nat.dma.pma22.persis

Perturbation theory for Lyapunov exponents of an Anderson model on a strip - Schulz-Baldes, Hermann (Geometric and Functional Analysis, 2004)
It is proven that the inverse localization length of an Anderson model on a strip of width L is bounded above by L/λ2 for small values of the coupling constant λ of the disordered potential. For this purpose, a formalism is developed in order to calculate the bottom Lyapunov exponent associated with random products of large symplectic matrices perturbatively in the coupling constant of the randomness.
UnivIS-Import:2015-03-09:Pub.2004.nat.dma.pma22.pertur_0

Perturbative test of single parameter scaling for 1D random media - Schrader, Robert; Schulz-Baldes, Hermann; Sedrakyan, Ara (Annales Henri Poincaré, 2004)
Products of random matrices associated to one-dimensional random media satisfy a central limit theorem assuring convergence to a gaussian centered at the Lyapunov exponent. The hypothesis of single parameter scaling states that its variance is equal to the Lyapunov exponent. We settle discussions about its validity for a wide class of models by proving that, away from anomalies, single parameter scaling holds to lowest order perturbation theory in the disorder strength. However, it is generically violated at higher order. This is explicitly exhibited for the Anderson model.
UnivIS-Import:2015-03-09:Pub.2004.nat.dma.pma22.pertur_7

Phase-averaged transport for quasiperiodic Hamiltonians - Bellissard, Jean; Guarneri, Italo; Schulz-Baldes, Hermann (Communications in Mathematical Physics, 2002)
For a class of discrete quasi-periodic Schroedinger operators defined by covariant re- presentations of the rotation algebra, a lower bound on phase-averaged transport in terms of the multifractal dimensions of the density of states is proven. This result is established under a Diophantine condition on the incommensuration parameter. The relevant class of operators is distinguished by invariance with respect to symmetry automorphisms of the rotation algebra. It includes the critical Harper (almost-Mathieu) operator. As a by-product, a new solution of the frame problem associated with Weyl-Heisenberg-Gabor lattices of coherent states is given.
UnivIS-Import:2015-03-09:Pub.2002.nat.dma.pma22.phasea

Pointwise dimensions on real Julia sets - Schulz-Baldes, Hermann (Journal of Physics A: Mathematical and General, 1999)
From a formula for the pointwise dimensions of Bernoulli measures on real quadratic Julia sets, we deduce several aspects of their multifractal analysis. In particular, we prove a lower bound for the scaling function and give explicit formulae for its borders.
UnivIS-Import:2015-03-05:Pub.1999.nat.dma.pma22.pointw

Positive Lyapunov exponents and localization bounds for strongly mixing potentials - Sadel, Christian Hermann; Schulz-Baldes, Hermann (Advances in Theoretical and Mathematical Physics, 2008)
For a one-dimensional discrete Schrödinger operator with a weakly coupled potential given by a strongly mixing dynamical system with power law decay of correlations, we derive for all energies including the band edges and the band center a perturbative formula for the Lyapunov exponent. Under adequate hypothesis, this shows that the Lyapunov exponent is positive on the whole spectrum. This in turn implies that the Hausdorff dimension of the spectral measure is zero and that the associated quantum dynamics grows at most logarithmically in time. © 2008 International Press.
UnivIS-Import:2015-03-09:Pub.2008.nat.dma.pma22.positi

Preface and Overview - Schulz-Baldes, Hermann; Stoiber, Tom (Springer, 2022)
CRIS-Team Scopus Importer:2024-12-16

Preliminaries on Crossed Products - Schulz-Baldes, Hermann; Stoiber, Tom (Springer, 2022)
This chapter reviews several facts about crossed products. Many aspects are covered in the standard reference [1], but some are taken from other works like [2–7]. Many results hold for general locally compact abelian group actions, but for sake of concreteness we will only restrict to the case of abelian n-parameter groups which are relevant for the applications that we have in mind. Hence throughout G=Tn0⊕Rn1 where n= n0+ n1 and we make the identification T= R Z= [ 0, 1 ] / ∼. Unless there is an explicitly different choice of normalization (which only applies in Chap. 5), we fix the Haar measures on Td and Rd to be the usual Lebesgue measure and on Zd the counting measure. Combined with the conventions for the Fourier transform as given below, one can then identify the dual groups T^ = Z, Z^ = T and R^ = R in such a way that the Plancherel theorem holds without proportionality constants.
CRIS-Team Scopus Importer:2024-12-16

Pseudo-gaps for random hopping models - Dorsch, Florian; Schulz-Baldes, Hermann (Journal of Physics A: Mathematical and Theoretical, 2020)
For one-dimensional random Schrodinger operators, the integrated density of states is known to be given in terms of the (averaged) rotation number of the Prufer phase dynamics. This paper develops a controlled perturbation theory for the rotation number around an energy at which all the transfer matrices commute and are hyperbolic. Such a hyperbolic critical energy appears in random hopping models. The main result is a Holder continuity of the rotation number at the critical energy that implies the existence of a pseudo-gap. The proof uses renewal theory. The result is illustrated by numerics.
CRIS-Team WoS Importer:2020-05-05

Quantization of edge currents of continuous magnetic operators - Kellendonk, Johannes; Schulz-Baldes, Hermann (Journal of Functional Analysis, 2004)
For a magnetic Hamiltonian on a half-plane given by the sum of the Landau operator with Dirichlet boundary conditions and a random potential, a quantization theorem for the edge currents is proven. This shows that the concept of edge channels also makes sense in presence of disorder. Moreover, Gaussian bounds on the heat kernel and its covariant derivatives are obtained. © 2003 Elsevier Inc. All rights reserved.
UnivIS-Import:2015-03-09:Pub.2004.nat.dma.pma22.quanti

Quantization of interface currents - Kotani, Motoko; Schulz-Baldes, Hermann; Villegas-Blas, Carlos (Journal of Mathematical Physics, 2014)
At the interface of two two-dimensional quantum systems, there may exist interface currents similar to edge currents in quantum Hall systems. It is proved that these interface currents are macroscopically quantized by an integer that is given by the difference of the Chern numbers of the two systems. It is also argued that at the interface between two time-reversal invariant systems with half-integer spin, one of which is trivial and the other non-trivial, there are dissipationless spin-polarized interface current
UnivIS-Import:2015-04-02:Pub.2015.nat.dma.pma22.quanti

Quantum Differentiation and Index Theorems - Schulz-Baldes, Hermann; Stoiber, Tom (Springer, 2022)
As already stressed in the overview in Preface, this chapter is the mathematical core of this book. In the next Sect. 3.1 the Hankel and Toeplitz operators associated to a W∗ -dynamical system are introduced and then the traceclass Peller criterion is proved. Combining an L2 -criterion with interpolation theory, Sect. 3.2 then proves Peller criteria for higher Schatten classes.
CRIS-Team Scopus Importer:2024-12-16

Random Dirac operators with time reversal symmetry - Sadel, Christian Hermann; Schulz-Baldes, Hermann (Communications in Mathematical Physics, 2010)
Quasi-one-dimensional stochastic Dirac operators with an odd number of channels, time reversal symmetry but otherwise efficiently coupled randomness, are shown to have one conducting channel and absolutely continuous spectrum of multiplicity two. This follows by adapting the criteria of Guivarch-Raugi and Goldsheid-Margulis to the analysis of random products of matrices in the group SO*(2L), and then a version of Kotani theory for these operators. Absence of singular spectrum can be shown by adapting an argument of Jaksic-Last if the potential contains random Dirac peaks with absolutely continuous distribution. © The Author(s) 2009.
UnivIS-Import:2015-03-09:Pub.2010.nat.dma.pma22.random_1

Random Lie Group actions on compact manifolds: a perturbative analysis - Sadel, Christian Hermann; Schulz-Baldes, Hermann (Annals of Probability, 2010)
A random Lie group action on a compact manifold generates a discrete time Markov process. The main object of this paper is the evaluation of associated Birkhoff sums in a regime of weak, but sufficiently effective coupling of the randomness. This effectiveness is expressed in terms of random Lie algebra elements and replaces the transience or Furstenberg's irreducibility hypothesis in related problems. The Birkhoff sum of any given smooth function then turns out to be equal to its integral w.r.t. a unique smooth measure on the manifold up to errors of the order of the coupling constant. Applications to the theory of products of random matrices and a model of a disordered quantum wire are presented. © Institute of Mathematical Statistics, 2010.
UnivIS-Import:2015-03-09:Pub.2010.nat.dma.pma22.random_9
Keywords: Birkhoff sum; Group action; Invariant measure

RANDOM MOBIUS DYNAMICS ON THE UNIT DISC AND PERTURBATION THEORY FOR LYAPUNOV EXPONENTS - Dorsch, Florian; Schulz-Baldes, Hermann (Discrete and Continuous Dynamical Systems-Series B, 2022)
Randomly drawn 2 x 2 matrices induce a random dynamics on the Riemann sphere via the Mobius transformation. Considering a situation where this dynamics is restricted to the unit disc and given by a random rotation perturbed by further random terms depending on two competing small parameters, the invariant (Furstenberg) measure of the random dynamical system is determined. The results have applications to the perturbation theory of Lyapunov exponents which are of relevance for one-dimensional discrete random Schrodinger operators.
CRIS-Team WoS Importer:2021-11-26

Random perturbations of hyperbolic dynamics - Dorsch, Florian; Schulz-Baldes, Hermann (Electronic Journal of Probability, 2019)
A sequence of large invertible matrices given by a small random perturbation around a fixed diagonal and positive matrix induces a random dynamics on a high-dimensional sphere. For a certain class of rotationally invariant random perturbations it is shown that the dynamics approaches the stable fixed points of the unperturbed matrix up to errors even if the strength of the perturbation is large compared to the relative increase of nearby diagonal entries of the unperturbed matrix specifying the local hyperbolicit

Random phase property and the Lyapunov spectrum for disordered multi-channel systems - Römer, Rudolf A.; Schulz-Baldes, Hermann (Journal of Statistical Physics, 2010)
A random phase property establishing in the weak coupling limit a link between quasi-one-dimensional random Schrödinger operators and full random matrix theory is advocated. Briefly summarized it states that the random transfer matrices placed into a normal system of coordinates act on the isotropic frames and lead to a Markov process with a unique invariant measure which is of geometric nature. On the elliptic part of the transfer matrices, this measure is invariant under the unitaries in the hermitian symplectic group of the universality class under study. While the random phase property can up to now only be proved in special models or in a restricted sense, we provide strong numerical evidence that it holds in the Anderson model of localization. A main outcome of the random phase property is a perturbative calculation of the Lyapunov exponents which shows that the Lyapunov spectrum is equidistant and that the localization lengths for large systems in the unitary, orthogonal and symplectic ensemble differ by a factor 2 each. In an Anderson-Ando model on a tubular geometry with magnetic field and spin-orbit coupling, the normal system of coordinates is calculated and this is used to derive explicit energy dependent formulas for the Lyapunov spectrum. © 2010 Springer Science+Business Media, LLC.
UnivIS-Import:2015-03-09:Pub.2010.nat.dma.pma22.random
Keywords: Anderson-Ando model; Localization length; Random phase property

Reduced transfer operators for singular difference equations - Schulz-Baldes, Hermann (Journal of Difference Equations and Applications, 2022)
For tridiagonal block Jacobi operators, the standard transfer operator techniques only work if the off-diagonal entries are invertible. Under suitable assumptions on the range and kernel of these off-diagonal operators which assure a homogeneous minimal coupling between the blocks, it is shown how to construct reduced transfer operators that have the usual Krein space unitarity property and also a crucial monotonicity in the energy variable. This allows to extend the results of oscillation theory to such systems.
CRIS-Team WoS Importer:2022-12-16

Replica Symmetry Breaking in Compressive Sensing - Bereyhi, Ali; Schulz-Baldes, Hermann; Müller, Ralf (Information Theory and Applications Workshop (ITA), 2017)
For noisy compressive sensing systems, the asymptotic distortion with respect to an arbitrary distortion function is determined when a general class of least-square based reconstruction schemes is employed. The sampling matrix is considered to belong to a large ensemble of random matrices including i.i.d. and projector matrices, and the source vector is assumed to be i.i.d. with a desired distribution. We take a statistical mechanical approach by representing the asymptotic distortion as a macroscopic parameter of a spin glass and employing the replica method for the large-system analysis. In contrast to earlier studies, we evaluate the general replica ansatz which includes the RS ansatz as well as RSB. The generality of the solution enables us to study the impact of symmetry breaking. Our numerical investigations depict that for the reconstruction scheme with the "zero-norm" penalty function, the RS fails to predict the asymptotic distortion for relatively large compression rates; however, the one-step RSB ansatz gives a valid prediction of the performance within a larger regime of compression rates.
Keywords: STATISTICAL-MECHANICS; SPIN-GLASSES; CODES

Rigorous spectral analysis of the metal-insulator transition in a limit-periodic potential - Schulz-Baldes, Hermann; Zarrouati, Marc (Journal of Statistical Physics, 1998)
We consider the limit-periodic Jacobi matrices associated with the real Julia sets of fλ(z) = z2-λ. for which λε[2, ∝) can be seen as the strength of the limit-periodic coefficients. The typical local spectral exponent of their spectral measures is shown to be a harmonic function in λ decreasing logarithmically from 1 to 0.
UnivIS-Import:2015-03-05:Pub.1998.nat.dma.pma22.rigoro
Keywords: Julia matrices; Local spectral exponents

Rotation numbers for Jacobi matrices with matrix entries - Schulz-Baldes, Hermann (Mathematical Physics Electronic Journal, 2007)
A Jacobi matrix with matrix entries is a selfadjoint block tridiagonal matrix with positive definite blocks on the off-diagonals. A rotation number calculation for its eigenvalues is presented. This is a matricial generalization of the oscillation theorem for the discrete analogues of Sturm-Liouville operators. The three universality classes of time reversal invariance are dealt with by implementing the corresponding symmetries. For Jacobi matrices with random matrix entries, this leads to a formula for the integrated density of states which can be calculated perturbatively in the coupling constant of the randomness with an optimal control on the error terms.
UnivIS-Import:2015-03-09:Pub.2007.nat.dma.pma22.rotati

RSB decoupling property of MAP estimators - Bereyhi, Ali; Müller, Ralf; Schulz-Baldes, Hermann (2016 IEEE Information Theory Workshop (ITW), 2016)
The large-system decoupling property of a MAP estimator is studied when it estimates the i.i.d. vector x from the observation y=Ax+z with A being chosen from a wide range of matrix ensembles, and the noise vector z being i.i.d. and Gaussian. Using the replica method, we show that the marginal joint distribution of any two corresponding input and output symbols converges to a deterministic distribution which describes the input-output distribution of a single user system followed by a MAP estimator. Under the bRSB assumption, the single user system is a scalar channel with additive noise where the noise term is given by the sum of an independent Gaussian random variable and b correlated interference terms. As the bRSB assumption reduces to RS, the interference terms vanish which results in the formerly studied RS decoupling principle.

Scaling diagram for the localization length at a band edge - Sadel, Christian Hermann; Schulz-Baldes, Hermann (Annales Henri Poincaré, 2007)
A weak-coupling scaling diagram for the Lyapunov exponent and the integrated density of states near a band edge of a random Jacobi matrix is obtained. The analysis is based on the use of a Fokker-Planck operator describing the drift-diffusion of the Prüfer phases. © 2007 Birkhaueser.
UnivIS-Import:2015-03-09:Pub.2007.nat.dma.pma22.scalin

Scattering theory for lattice operators in dimension d ≥ 3 - Bellissard, Jean; Schulz-Baldes, Hermann (Reviews in Mathematical Physics, 2012)
This paper analyzes the scattering theory for periodic tight-binding Hamiltonians perturbed by a finite range impurity. The classical energy gradient flow is used to construct a conjugate (or dilation) operator to the unperturbed Hamiltonian. For dimension d ≥ 3, the wave operator is given by an explicit formula in terms of this dilation operator, the free resolvent and the perturbation. From this formula, the scattering and time delay operators can be read off. Using the index theorem approach, a Levinson theorem is proved which also holds in the presence of embedded eigenvalues and threshold singularities.
UnivIS-Import:2015-03-09:Pub.2012.nat.dma.pma22.scatte

Scattering zippers and their spectral theory - Laurent, Marin; Schulz-Baldes, Hermann (Journal of Spectral Theory, 2013)
A scattering zipper is a system obtained by concatenation of scattering events with equal even number of incoming and out going channels. The associated scattering zipper operator is the unitary equivalent of Jacobi matrices with matrix entries and generalizes Blatter-Browne and Chalker-Coddington models and CMV matrices. Weyl discs are analyzed and used to prove a bijection between the set of semi-infinite scattering zipper operators and matrix valued probability measures on the unit circle. Sturm-Liouville oscillation theory is developed as a tool to calculate the spectra of finite and periodic scattering zipper operators.
UnivIS-Import:2015-03-09:Pub.2013.nat.dma.pma22.scatte

Secure Coding via Gaussian Random Fields - Bereyhi, Ali; Loureiro, Bruno; Krzakala, Florent; Müller, Ralf; Schulz-Baldes, Hermann (IEEE International Symposium on Information Theory - Proceedings, 2022)
Inverse probability problems whose generative models are given by strictly nonlinear Gaussian random fields show the all-or-nothing behavior: There exists a critical rate at which Bayesian inference exhibits a phase transition. Below this rate, the optimal Bayesian estimator recovers the data perfectly, and above it the recovered data becomes uncorrelated. This study uses the replica method from the theory of spin glasses to show that this critical rate is the channel capacity. This interesting finding has a particular application to the problem of secure transmission: A strictly nonlinear Gaussian random field along with random binning can be used to securely encode a confidential message in a wiretap channel. Our large-system characterization demonstrates that this secure coding scheme asymptotically achieves the secrecy capacity of the Gaussian wiretap channel.
CRIS-Team Scopus Importer:2022-09-02
Keywords: decoupling principle; information-theoretic secrecy; Nonlinear Gaussian random fields; replica method

Semicircle law and freeness for random matrices with symmetries or correlations - Schulz-Baldes, Hermann; Schenker, Jeffrey (Mathematical Research Letters, 2005)
For a class of random matrix ensembles with correlated matrix elements, it is shown that the density of states is given by the Wigner semi-circle law. This is applied to effective Hamiltonians related to the Anderson model in dimensions greater than or equal to two.
UnivIS-Import:2015-03-09:Pub.2005.nat.dma.pma22.semici

Signature and spectral flow for J-unitary S^1 Fredholm operators - Schulz-Baldes, Hermann (Integral Equations and Operator Theory, 2014)
UnivIS-Import:2015-03-09:Pub.2014.nat.dma.pma22.signat
Keywords: J-unitary operator; Fredholm property; Intersection index

Signatures for J-hermitians and J-unitaries on Krein spaces with Real structures - Schulz-Baldes, Hermann; Villegas-Blas, Carlos (Mathematische Nachrichten, 2017)
For J-hermitian operators on a Krein space (,J) satisfying an adequate Fredholm property, a global Krein signature is shown to be a homotopy invariant. It is argued that this global signature is a generalization of the Noether index. When the Krein space has a supplementary Real structure, the sets of J-hermitian Fredholm operators with Real symmetry can be retracted to certain of the classifying spaces of Atiyah and Singer. Secondary ℤ2-invariants are introduced to label their connected components. Related invariants are also analyzed for J-unitary operators.

Simultaneous quantization of edge and bulk Hall conductivity - Schulz-Baldes, Hermann; Kellendonk, Johannes; Richter, Thomas (Journal of Physics A: Mathematical and General, 2000)
The edge Hall conductivity is shown to be an integer multiple of e 2 /h which is almost surely independent of the choice of the disordered configuration. Its equality to the bulk Hall conductivity given by the Kubo-Chern formula follows from K -theoretic arguments. This leads to quantization of the Hall conductance for any redistribution of the current in the sample. It is argued that in experiments at most a few per cent of the total current can be carried by edge states.

Skew localizer and Z2-flows for real index pairings - Doll, Nora; Schulz-Baldes, Hermann (Advances in Mathematics, 2021)
Real index pairings of projections and unitaries on a separable Hilbert space with a real structure are defined when the projections and unitaries fulfill symmetry relations invoking the real structure, namely projections can be real, quaternionic, even or odd Lagrangian and unitaries can be real, quaternionic, symmetric or anti-symmetric. There are 64 such real index pairings of real K-theory with real K-homology. For 16 of them, the index of the Fredholm operator representing the pairing vanishes, but there is a secondary Z2-valued invariant. The first set of results provides index formulas expressing each of these 16 Z2-valued pairings as either an orientation flow or a half-spectral flow. The second and main set of results constructs the skew localizer for a pairing stemming from an unbounded Fredholm module and shows that the Z2-invariant can be computed as the sign of the Pfaffian of the skew localizer and in 8 of the cases as the sign of the determinant of the off-diagonal entry of the skew localize. This is of relevance for the numerical computation of invariants of topological insulator
CRIS-Team Scopus Importer:2021-10-22
Keywords: Index theory; Spectral flow; Z2-invariant

Space versus energy oscillations of prÜfer phases for matrix sturm-liouville and jacobi operators - Schulz-Baldes, Hermann; Urban, Liam (Electronic Journal of Differential Equations, 2020)
This note considers Sturm oscillation theory for regular matrix Sturm-Liouville operators on finite intervals and for matrix Jacobi operators. The number of space oscillations of the eigenvalues of the matrix Prüfer phases at a given energy, defined by a suitable lift in the Jacobi case, is shown to be equal to the number of eigenvalues below that energy. This results from a positivity property of the Prüfer phases, namely they cannot cross −1 in the negative direction, and is also shown to be closely linked to the positivity of the matrix Prüfer phase in the energy variable. The theory is illustrated by numerical calculations for an explicit example.
CRIS-Team Scopus Importer:2020-07-31
Keywords: Jacobi operators; Matrix Prüfer phases; Oscillation theory; Sturm-Liouville operators

Spectral averaging techniques for Jacobi matrices - del Rio, Rafael; Martinez, Carmen; Schulz-Baldes, Hermann (Journal of Mathematical Physics, 2008)
Spectral averaging techniques for one-dimensional discrete Schrödinger operators are revisited and extended. In particular, simultaneous averaging over several parameters is discussed. Special focus is put on proving lower bounds on the density of the averaged spectral measures. These Wegner-type estimates are used to analyze stability properties for the spectral types of Jacobi matrices under local perturbations. © 2008 American Institute of Physics.
UnivIS-Import:2015-03-09:Pub.2008.nat.dma.pma22.spectr

Spectral averaging techniques for Jacobi matrices with matrix entries - Sadel, Christian Hermann; Schulz-Baldes, Hermann (Journal of Physics A: Mathematical and Theoretical, 2009)
A Jacobi matrix with matrix entries is a self-adjoint block tridiagonal matrix with invertible blocks on the off-diagonals. Averaging over boundary condi- tions leads to explicit formulae for the averaged spectral measure which can potentially be useful for spectral analysis. Furthermore, another variant of spectral averaging over coupling constants for these operators is presented. © 2009 IOP Publishing Ltd.
UnivIS-Import:2015-03-09:Pub.2009.nat.dma.pma22.spectr

Spectral flow: A functional analytic and index-Theoretic approach - Doll, Nora; Schulz-Baldes, Hermann; Waterstraat, Nils (De Gruyter, 2023)
This is the first treatment entirely dedicated to an analytic study of spectral flow for paths of selfadjoint Fredholm operators, possibly unbounded or understood in a semifinite sense. The importance of spectral flow for homotopy and index theory is discussed in detail. Applications concern eta-invariants, the Bott-Maslov and Conley-Zehnder indices, Sturm-Liouville oscillation theory, the spectral localizer and bifurcation theory.
CRIS-Team Scopus Importer:2023-07-07
Keywords: Indextheorie von Fredholm-Paaren; Oszillationstheorie für Jacobi-Operatoren und Streutheorie; Selbstadjungierte Fredholm-Operatoren; Variationsbifurkationstheorie

Spectral flow argument localizing an odd index pairing - Loring, Terry; Schulz-Baldes, Hermann (Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques, 2019)
An odd Fredholm module for a given invertible operator on a Hilbert space is specified by an unbounded so-called Dirac operator with compact resolvent and bounded commutator with the given invertible. Associated to this is an index pairing in terms of a Fredholm operator with Noether index. Here it is shown by a spectral flow argument how this index can be calculated as the signature of a finite dimensional matrix called the spectral localizer.

Spectral flow for skew-adjoint Fredholm operators - Carey, Alan L.; Phillips, John; Schulz-Baldes, Hermann (Journal of Spectral Theory, 2019)
An analytic definition of a Z 2 -valued spectral flow for paths of real skew-adjoint Fredholm operators is given. It counts the parity of the number of changes in the orientation of the eigenfunctions at eigenvalue crossings through 0 along the path. The Z 2 -valued spectral flow is shown to satisfy a concatenation property and homotopy invariance, and it provides an isomorphism on the fundamental group of the real skew-adjoint Fredholm operators. Moreover, it is connected to a Z 2 -index pairing for suitable paths. Applications concern the zero energy bound states at defects in a Majorana chain and a spectral flow interpretation for the Z 2 -polarization in these models.
CRIS-Team Scopus Importer:2019-05-28
Keywords: Skew-symmetric operators; Spectral flow; Topological insulators

Spectral Flow of Monopole Insertion in Topological Insulators - Carey, Alan L.; Schulz-Baldes, Hermann (Communications in Mathematical Physics, 2019)
Inserting a magnetic flux into a two-dimensional one-particle Hamiltonian leads to a spectral flow through a given gap which is equal to the Chern number of the associated Fermi projection. This paper establishes a generalization to higher even dimension by inserting non-abelian monopoles of the Wu-Yang type. The associated spectral flow is then equal to a higher Chern number. For the study of odd spacial dimensions, a new so-called ‘chirality flow’ is introduced which, for the insertion of a monopole, is then linked to higher winding numbers. This latter fact follows from a new index theorem for the spectral flow between two unitaries which are conjugates of each other by a self-adjoint unitary.
CRIS-Team Scopus Importer:2019-09-03

Spectral flows associated to flux tubes - de Nittis, Giuseppe; Schulz-Baldes, Hermann (Annales Henri Poincaré, 2016)
When a flux quantum is pushed through a gapped two-dimensional tight-binding operator, there is an associated spectral flow through the gap which is shown to be equal to the index of a Fredholm operator encoding the topology of the Fermi projection. This is a natural mathematical formulation of Laughlin's Gedankenexperiment. It is used to provide yet another proof of the bulk-edge correspondence. Furthermore, when applied to systems with time reversal symmetry, the spectral flow has a characteristic Z_2 signature, while for particle-hole symmetric systems it leads to a criterion for the existence of zero energy modes attached to half-flux tubes. Combined with other results, this allows to explain all strong invariants of two-dimensional topological insulators in terms of a single Fredholm operator.
UnivIS-Import:2015-04-15:Pub.2015.nat.dma.pma22.spectr

Spectral flows of dilations of Fredholm operators - de Nittis, Giuseppe; Schulz-Baldes, Hermann (Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques, 2015)
UnivIS-Import:2015-04-15:Pub.2015.nat.dma.pma22.spectr_6
Keywords: spectral flow; Fredholm operators; Z2 indices

Spectral localization for semimetals and Callias operators - Schulz-Baldes, Hermann; Stoiber, Tom (Journal of Mathematical Physics, 2023)
A semiclassical argument is used to show that the low-lying spectrum of a self-adjoint operator, the so-called spectral localizer, determines the number of Dirac or Weyl points of an ideal semimetal. Apart from the ion-mobility spectrometer localization procedure, an explicit computation for the local toy models given by a Dirac or Weyl point is the key element of proof. The argument has numerous similarities to Witten’s reasoning leading to the strong Morse inequalities. The same techniques allow to prove a spectral localization for Callias operators associated with potentials with isolated gap-closing points.
CRIS-Team Scopus Importer:2023-09-29

Spectral localizer for line-gapped non-Hermitian systems - Cerjan, Alexander; Koekenbier, Lars; Schulz-Baldes, Hermann (Journal of Mathematical Physics, 2023)
Short-ranged and line-gapped non-Hermitian Hamiltonians have strong topological invariants given by an index of an associated Fredholm operator. It is shown how these invariants can be accessed via the signature of a suitable spectral localizer. This numerical technique is implemented in an example with relevance to the design of topological photonic systems, such as topological lasers.
CRIS-Team Scopus Importer:2023-09-29

Statistical Mechanics of MAP Estimation: General Replica Ansatz - Bereyhi, Ali; Müller, Ralf; Schulz-Baldes, Hermann (IEEE Transactions on Information Theory, 2019)
The large-system performance of maximum-a-poste-rior estimation is studied considering a general distortion function when the observation vector is received through a linear system with additive white Gaussian noise. The analysis considers the system matrix to be chosen from the large class of rotationally invariant random matrices. We take a statistical mechanical approach by introducing a spin glass corresponding to the estimator, and employing the replica method for the large-system analysis. In contrast to earlier replica based studies, our analysis evaluates the general replica ansatz of the corresponding spin glass and determines the asymptotic distortion of the estimator for any structure of the replica correlation matrix. Consequently, the replica symmetric as well as the replica symmetry breaking ansatz with b steps of breaking is deduced from the given general replica ansatz. The generality of our distortion function lets us derive a more general form of the maximum-a-posterior decoupling principle. Based on the general replica ansatz, we show that for any structure of the replica correlation matrix, the vector-valued system decouples into a bank of equivalent decoupled scalar systems followed by maximum-a-posterior estimators. The structure of the decoupled system is further studied under both the replica symmetry and the replica symmetry breaking assumptions. For b steps of symmetry breaking, the decoupled system is found to be an additive system with a non-Gaussian noise term given as the sum of an independent Gaussian random variable with b non-Gaussian impairment terms which depend on the input symbol. The general decoupling property of the maximum-a-posterior estimator leads to the idea of a replica simulator which represents the replica ansatz through the state evolution of a transition system described by its corresponding decoupled system. As an application of our study, we investigate large compressive sensing systems by considering the ℓp norm minimization recovery schemes. Our numerical investigations show that the replica symmetric ansatz for ℓ0 norm recovery fails to give an accurate approximation of the mean square error as the compression rate grows, and therefore, the replica symmetry breaking ansätze are needed in order to assess the performance precisely.
CRIS-Team Scopus Importer:2020-02-21
Keywords: compressive sensing; decoupling principle; equivalent single-user system; linear vector channel; Maximum-a-posterior estimation; replica method; replica simulator; replica symmetry breaking; statistical physics; zero norm

Sturm intersection theory for periodic Jacobi matrices and linear Hamiltonian systems - Schulz-Baldes, Hermann (Linear Algebra and Its Applications, 2012)
Sturm-Liouville oscillation theory for periodic Jacobi operators with matrix entries is discussed and illustrated. The proof simplifies and clarifies the use of intersection theory of Bott, Maslov and Conley-Zehnder. It is shown that the eigenvalue problem for linear Hamiltonian systems can be dealt with by the same approach. © 2011 Elsevier Inc. All rights reserved.
UnivIS-Import:2015-03-09:Pub.2012.nat.dma.pma22.sturmi
Keywords: Bott-Maslov index; Sturm-Liouville oscillation theory

Subdiffusive quantum transport for 3D-Hamiltonians with absolutely continuous spectra - Bellissard, Jean; Schulz-Baldes, Hermann (Journal of Statistical Physics, 2000)
We exhibit a class of Hamiltonians in dimension D ≥ 3, describing a quantum particle in an aperiodic medium with absolutely continuous spectrum and sub-diffusive behavior. The diffusion exponent, which characterizes the time growth of the mean square displacement, can be chosen slightly bigger than Guarneri's lower bound. These models are built out of 1D Hamiltonians with well-understood spectral and transport properties.
UnivIS-Import:2015-03-09:Pub.2000.nat.dma.pma22.subdif
Keywords: Anomalous quantum transport; Fractal measures

The density of surface states as the total time delay - Schulz-Baldes, Hermann (Letters in Mathematical Physics, 2016)
For a scattering problem of tight-binding Bloch electrons by a weak random surface potential, a generalized Levinson theorem is put forward showing the equality of the total density of surface states and the density of the total time delay. The proof uses explicit formulas for the wave operators in the new rescaled energy and interaction (REI) representation, as well as an index theorem for adequate associated operator algebra
UnivIS-Import:2015-04-02:Pub.2013.nat.dma.pma22.theden
Keywords: surface scattering; Levinson theorem

The generators of the K-groups of the sphere - Schulz-Baldes, Hermann; Stoiber, Tom (Expositiones Mathematicae, 2023)
This note presents an elementary iterative construction of the generators for the complex K-groups Ki(C(Sd)) of the d-dimensional spheres. These generators are explicitly given as the restrictions of Dirac or Weyl Hamiltonians to the unit sphere. Connections to solid state physics are briefly elaborated on.
CRIS-Team Scopus Importer:2023-11-24
Keywords: Connecting maps; Dirac Hamiltonian; Generators; K-groups; Weyl Hamiltonian

The non-commutative geometry of the quantum Hall effect - Bellissard, Jean; van Elst, A.; Schulz-Baldes, Hermann (Journal of Mathematical Physics, 1994)
An overview of the integer quantum Hall effect is given. A mathematical framework using nonommutative geometry as defined by Connes is prepared. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localized states.
UnivIS-Import:2015-03-05:Pub.1994.nat.dma.pma22.thenon

The non-commutative topology of two-dimensional dirty superconductors - de Nittis, Giuseppe; Schulz-Baldes, Hermann (Journal of Geometry and Physics, 2018)
Non-commutative analysis tools have successfully been applied to the integer quantum Hall effect, in particular for a proof of the stability of the Hall conductance in an Anderson localization regime and of the bulk-boundary correspondence. In this work, these techniques are implemented to study two-dimensional dirty superconductors described by Bogoliubov-de Gennes Hamiltonians. After a thorough presentation of the basic framework and the topological invariants, Kubo formulas for the thermal, thermoelectric and spin Hall conductance are analyzed together with the corresponding edge currents. (C) 2017 Elsevier B.V. All rights reserved.
Keywords: Topological insulators; Thermal quantum Hall effect; Bulk and boundary correspondence

The spectral localizer for even index pairings - Loring, Terry A.; Schulz-Baldes, Hermann (Journal of Noncommutative Geometry, 2020)
Even index pairings are integer-valued homotopy invariants combining an even Fredholm module with a Ko-class specified by a projection. Numerous classical examples are known from differential and non-commutative geometry and physics. Here it is shown how to construct a finite-dimensional self-adjoint and invertible matrix, called the spectral localizer, such that half its signature is equal to the even index pairing. This makes the invariant numerically accessible. The index-theoretic proof heavily uses fuzzy spheres.
CRIS-Team WoS Importer:2020-07-24

The spectral localizer for semifinite spectral triples - Schulz-Baldes, Hermann; Stoiber, Tom (Proceedings of the American Mathematical Society, 2021)
The notion of a spectral localizer is extended to pairings with semifinite spectral triples. By a spectral flow argument, any semifinite index pairing is shown to be equal to the signature of the spectral localizer. As an application, a formula for the weak invariants of topological insulators is derived. This provides a new approach to their numerical evaluation.
CRIS-Team Scopus Importer:2020-11-27

Topological boundary invariants for Floquet systems and quantum walks - Sadel, Christian Hermann; Schulz-Baldes, Hermann (Mathematical Physics Analysis and Geometry, 2017)
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.
Keywords: Floquet topological insulators Quantum walks Connecting maps in K-theory

Topological edge states for disordered bosonic systems - Peano Cavasola, Vittorio; Schulz-Baldes, Hermann (Journal of Mathematical Physics, 2018)
Quadratic bosonic Hamiltonians over a one-particle Hilbert space can be described by a Bogoliubov-de Gennes (BdG) Hamiltonian on a particle-hole Hilbert space. In general, the BdG Hamiltonian is not self-adjoint, but only J-self-adjoint on the particle-hole space viewed as a Krein space. Nevertheless, its energy bands can have non-trivial topological invariants like Chern numbers or winding numbers. By a thorough analysis for tight-binding models, it is proved that these invariants lead to bosonic edge modes which are robust to a large class of possibly disordered perturbations. Furthermore, general scenarios are presented for these edge states to be dynamically unstable even though the bulk modes are stable. Published by AIP Publishing.

Topological Indices in Condensed Matter - Schulz-Baldes, Hermann (Elsevier, 2024)
This contribution describes the mathematical theory of topological indices in solid state systems composed of non-interacting Fermions. In particular, this covers the spectral localizer and the bulk-boundary correspondence.
CRIS-Team Scopus Importer:2025-04-01
Keywords: Bulk-boundary correspondence; Index theory; Spectral localizer; Topological insulators

Topological insulators from the perspective of non-commutative geometry and index theory - Schulz-Baldes, Hermann (Jahresbericht der deutschen Mathematiker-Vereinigung, 2016)
Topological insulators are solid state systems of independent electrons for which the Fermi level lies in a mobility gap, but the Fermi projection is nevertheless topologically non-trivial, namely it cannot be deformed into that of a normal insulator. This non-trivial topology is encoded in adequately defined invariants and implies the existence of surface states that are not susceptible to Anderson localization. This non-technical review reports on recent progress in the understanding of the underlying mathematical structures, with a particular focus on index theor

Topological invariants of edge states for periodic two-dimensional models - Avila, Julio Cesar; Schulz-Baldes, Hermann; Villegas-Blas, Carlos (Mathematical Physics Analysis and Geometry, 2013)
Transfer matrix methods and intersection theory are used to calculate the bands of edge states for a wide class of periodic two-dimensional tight-binding models including a sublattice and spin degree of freedom. This allows to define topological invariants by considering the associated Bott-Maslov indices which can be easily calculated numerically. For time-reversal symmetric systems in the symplectic universality class this leads to a ℤ2-invariant for the edge states. It is shown that the edge state invariants are related to Chern numbers of the bulk systems and also to (spin) edge currents, in the spirit of the theory of topological insulators. © 2012 Springer Science+Business Media Dordrecht.
UnivIS-Import:2015-03-09:Pub.2013.nat.dma.pma22.topolo
Keywords: Bloch theory; Edge states; Topological invariants

Transfer matrix analysis of non-Hermitian Hamiltonians: asymptotic spectra and topological eigenvalues - Koekenbier, Lars; Schulz-Baldes, Hermann (Journal of Spectral Theory, 2024)
Transfer matrix techniques are used to provide a new proof of Widom’s results on the asymptotic spectral theory of finite block Toeplitz matrices. Furthermore, a rigorous treatment of the skin effect, spectral outliers, the generalized Brillouin zone and the bulk-boundary correspondence in such systems is given. This covers chiral Hamiltonians with topological eigenvalues close to zero, but no line-gap.
CRIS-Team Scopus Importer:2024-11-15
Keywords: banded Toeplitz matrices; topological bound states; transfer matrices

Transport in the random Kronig-Penney model - Drabkin, Maxim; Kirsch, Werner; Schulz-Baldes, Hermann (Journal of Mathematical Physics, 2012)
The Kronig-Penney model with random Dirac potentials on the lattice has critical energies at which the Lyapunov exponent vanishes the density of states has a van Hove singularity. This leads to a non-trivial quantum diffusion even though the spectrum is known to be pure-point. © 2012 American Institute of Physics.
UnivIS-Import:2015-03-09:Pub.2012.nat.dma.pma22.transp_3

Upper bounds for quantum dynamics governed by Jacobi matrices with self-similar measures - Guarneri, Italo; Schulz-Baldes, Hermann (Reviews in Mathematical Physics, 1999)
We study a class of one-sided Hamiltonian operators with spectral measures given by invariant and ergodic measures of dynamical systems of the interval. We analyse dimensional properties of the spectral measures and prove upper bounds for the asymptotic spread in time of wavepackets. These bounds involve the Hausdorff dimension of the spectral measure, multiplied by a correction calculated from the dynamical entropy, the density of states, and the capacity of the support. For Julia matrices, the correction disappears and the growth is ruled by the fractal dimension.
UnivIS-Import:2015-03-05:Pub.1999.nat.dma.pma22.upperb

Upper bounds on wavepacket spreading for random Jacobi matrices - Jitomirskaya, Svetlana; Schulz-Baldes, Hermann (Communications in Mathematical Physics, 2007)
A method is presented for proving upper bounds on the moments of the position operator when the dynamics of quantum wavepackets is governed by a random (possibly correlated) Jacobi matrix. As an application, one obtains sharp upper bounds on the diffusion exponents for random polymer models, coinciding with the lower bounds obtained in a prior work. The second application is an elementary argument (not using multiscale analysis or the Aizenman-Molchanov method) showing that under the condition of uniformly positive Lyapunov exponents, the moments of the position operator grow at most logarithmically in time. © Springer-Verlag 2007.
UnivIS-Import:2015-03-09:Pub.2007.nat.dma.pma22.upperb

Weak disorder expansion for localization lengths of quasi-1D systems - Römer, Rudolf A.; Schulz-Baldes, Hermann (EPL - Europhysics Letters, 2004)
A perturbative formula for the lowest Lyapunov exponent of an Anderson model on a strip is presented. It is expressed in terms of an energy-dependent doubly stochastic matrix, the size of which is proportional to the strip width. This matrix and the resulting perturbative expression for the Lyapunov exponent are evaluated numerically. Dependence on energy, strip width and disorder strength are thoroughly compared with the results obtained by the standard transfer matrix method. Good agreement is found for all energies in the band of the free operator and this even for quite large values of the disorder strength.
UnivIS-Import:2015-03-09:Pub.2004.nat.dma.pma22.weakdi

Z_2 indices and factorization properties for odd symmetric Fredholm operators - Schulz-Baldes, Hermann (Documenta Mathematica, 2015)
UnivIS-Import:2015-04-02:Pub.2013.nat.dma.pma22.z_2ind