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.}, author = {Prodan, Emil and Schulz-Baldes, Hermann}, doi = {10.1007/978-3-319-29351-6}, faupublication = {yes}, isbn = {978-3-319-29350-9}, peerreviewed = {unknown}, publisher = {Springer International Publishing}, series = {Mathematical Physics Studies}, title = {{Bulk} and {Boundary} {Invariants} for {Complex} {Topological} {Insulators}}, year = {2016} } @article{faucris.312091207, abstract = {

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.

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.

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.

This
paper proves new results on spectral and scattering theory for
matrix-valued Schr\"odinger 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.

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.

}, author = {Bellissard, Jean and Schulz-Baldes, Hermann}, doi = {10.1142/S0129055X12500201}, faupublication = {yes}, journal = {Reviews in Mathematical Physics}, note = {UnivIS-Import:2015-03-09:Pub.2012.nat.dma.pma22.scatte}, pages = {1250020-1250071}, peerreviewed = {Yes}, title = {{Scattering} theory for lattice operators in dimension d ≥ 3}, url = {http://de.arxiv.org/abs/1109.5459}, volume = {24}, year = {2012} } @article{faucris.113269244, abstract = {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.}, author = {Laurent, Marin and Schulz-Baldes, Hermann}, faupublication = {yes}, journal = {Journal of Spectral Theory}, note = {UnivIS-Import:2015-03-09:Pub.2013.nat.dma.pma22.scatte}, pages = {47-82}, peerreviewed = {unknown}, title = {{Scattering} zippers and their spectral theory}, url = {http://de.arxiv.org/abs/1112.4959}, volume = {3}, year = {2013} } @inproceedings{faucris.281179195, abstract = {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.}, author = {Bereyhi, Ali and Loureiro, Bruno and Krzakala, Florent and Müller, Ralf and Schulz-Baldes, Hermann}, booktitle = {IEEE International Symposium on Information Theory - Proceedings}, date = {2022-06-26/2022-07-01}, doi = {10.1109/ISIT50566.2022.9834788}, faupublication = {yes}, isbn = {9781665421591}, keywords = {decoupling principle; information-theoretic secrecy; Nonlinear Gaussian random fields; replica method}, note = {CRIS-Team Scopus Importer:2022-09-02}, pages = {1241-1246}, peerreviewed = {unknown}, publisher = {Institute of Electrical and Electronics Engineers Inc.}, title = {{Secure} {Coding} via {Gaussian} {Random} {Fields}}, venue = {Espoo}, volume = {2022-June}, year = {2022} } @article{faucris.111530144, abstract = {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.}, author = {Schulz-Baldes, Hermann and Schenker, Jeffrey}, faupublication = {yes}, journal = {Mathematical Research Letters}, note = {UnivIS-Import:2015-03-09:Pub.2005.nat.dma.pma22.semici}, pages = {531-542}, peerreviewed = {Yes}, title = {{Semicircle} law and freeness for random matrices with symmetries or correlations}, url = {http://de.arxiv.org/abs/math-ph/0505003}, volume = {12}, year = {2005} } @article{faucris.117661324, author = {Schulz-Baldes, Hermann}, doi = {10.1007/s00020-013-2094-9}, faupublication = {yes}, journal = {Integral Equations and Operator Theory}, keywords = {J-unitary operator; Fredholm property; Intersection index}, note = {UnivIS-Import:2015-03-09:Pub.2014.nat.dma.pma22.signat}, pages = {323-374}, peerreviewed = {Yes}, title = {{Signature} and spectral flow for {J}-unitary {S}^1 {Fredholm} operators}, url = {http://de.arxiv.org/abs/1210.0184}, volume = {78}, year = {2014} } @article{faucris.115607184, abstract = {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.}, author = {Schulz-Baldes, Hermann and Villegas-Blas, Carlos}, faupublication = {yes}, journal = {Mathematische Nachrichten}, pages = {1840-1858}, peerreviewed = {Yes}, title = {{Signatures} for {J}-hermitians and {J}-unitaries on {Krein} spaces with {Real} structures}, volume = {290}, year = {2017} } @article{faucris.119138404, abstract = {The edge Hall conductivity is shown to be an integer multiple of

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.

}, author = {Guarneri, Italo and Schulz-Baldes, Hermann}, doi = {10.1142/S0129055X99000398}, faupublication = {no}, journal = {Reviews in Mathematical Physics}, note = {UnivIS-Import:2015-03-05:Pub.1999.nat.dma.pma22.upperb}, pages = {1249-1268}, peerreviewed = {Yes}, title = {{Upper} bounds for quantum dynamics governed by {Jacobi} matrices with self-similar measures}, url = {http://www.ma.utexas.edu/mp{\_}arc-bin/mpa?yn=98-382}, volume = {11}, year = {1999} } @article{faucris.111791504, abstract = {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.}, author = {Jitomirskaya, Svetlana and Schulz-Baldes, Hermann}, doi = {10.1007/s00220-007-0252-0}, faupublication = {yes}, journal = {Communications in Mathematical Physics}, note = {UnivIS-Import:2015-03-09:Pub.2007.nat.dma.pma22.upperb}, pages = {601-618}, peerreviewed = {Yes}, title = {{Upper} bounds on wavepacket spreading for random {Jacobi} matrices}, url = {http://de.arxiv.org/abs/math-ph/0607029}, volume = {273}, year = {2007} } @article{faucris.111269884, abstract = {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.}, author = {Römer, Rudolf A. and Schulz-Baldes, Hermann}, doi = {10.1209/epl/i2004-10190-9}, faupublication = {yes}, journal = {EPL - Europhysics Letters}, note = {UnivIS-Import:2015-03-09:Pub.2004.nat.dma.pma22.weakdi}, pages = {247-253}, peerreviewed = {Yes}, title = {{Weak} disorder expansion for localization lengths of quasi-{1D} systems}, url = {http://de.arxiv.org/abs/cond-mat/0405125}, volume = {68}, year = {2004} } @article{faucris.120730324, author = {Schulz-Baldes, Hermann}, faupublication = {yes}, journal = {Documenta Mathematica}, note = {UnivIS-Import:2015-04-02:Pub.2013.nat.dma.pma22.z{\_}2ind}, pages = {1481-1500}, peerreviewed = {Yes}, title = {{Z{\_}2} indices and factorization properties for odd symmetric {Fredholm} operators}, url = {https://www.math.uni-bielefeld.de/documenta/vol-20/vol-20.html}, volume = {20}, year = {2015} }