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Modern technological advances allow for the study of systems with additional synthetic dimensions. Higher-order topological insulators in topological states of matters have been pursued in lower physical dimensions by exploiting synthetic dimensions with phase transitions. While synthetic dimensions can be rendered in the photonics and cold atomic gases, little to no work has been succeeded in acoustics because acoustic wave-guides cannot be weakly coupled in a continuous fashion. Here, we formulate the theoretical principles and manufacture acoustic crystals composed of arrays of acoustic cavities strongly coupled through modulated channels to evidence one-dimensional (1D) and two-dimensional (2D) dynamic topological pumpings. In particular, the higher-order topological edge-bulk-edge and corner-bulk-corner transport are physically illustrated in finite-sized acoustic structures. We delineate the generated 2D and four-dimensional (4D) quantum Hall effects by calculating first and second Chern numbers and physically demonstrate robustness against the geometrical imperfections. Synthetic dimensions could provide a powerful way for acoustic topological wave steering and open up a platform to explore any continuous orbit in higher-order topological matter in dimensions four and higher., The authors create synthetic dimensions in acoustic crystals composed of cavity arrays, strongly coupled through modulated channels. They provide evidence for 1D and 2D dynamic topological pumping, and show that the higher-order topological sound transport is robust against the geometrical imperfections.

Consider a finite triangulation of a surface $M$ of genus $g$ and assume that spin-less fermions populate the edges of the triangulation. The quantum dynamics of such particles takes place inside the algebra of canonical anti-commutation relations (CAR). Following Kitaev's work on toric models, we identify a sub-algebra of CAR generated by elements associated to the triangles and vertices of the triangulation. We show that any Hamiltonian drawn from this sub-algebra displays topological spectral degeneracy. More precisely, if $\mathcal P$ is any of its spectral projections, the Booleanization of the fundmental group $\pi_1(M)$ can be embedded inside the group of invertible elements of the corner algebra $\mathcal P \, {\rm CAR} \, \mathcal P$. As a consequence, $\mathcal P$ decomposes in $4^g$ lower projections. Furthermore, a projective representation of $\mathbb Z_2^{4g}$ is also explicitly constructed inside this corner algebra. Key to all these is a presentation of CAR as a crossed product with the Boolean group $(2^X,\Delta)$, where $X$ is the set of fermion sites and $\Delta$ is the symmetric difference., Comment: Ann. Henri Poincar\'e (2021)

Topological phases supported by quasi-periodic spin-chain models and their bulk-boundary principles are investigated by numerical and K-theoretic methods. We show that, for both the un-correlated and correlated phases, the operator algebras that generate the Hamiltonians are non-commutative tori, hence the quasi-periodic chains display physics akin to the quantum Hall effect in two and higher dimensions. The robust topological edge modes are found to be strongly shaped by the interaction and, generically, they have hybrid edge-localized and chain-delocalized structures. Our findings lay the foundations for topological spin pumping using the phason of a quasi-periodic pattern as an adiabatic parameter, where selectively chosen quantized bits of magnetization can be transferred from one edge of the chain to the other., Comment: 23 figures

We demonstrate the existence of a conceptually distinct topological pumping phenomenon in one-dimensional chains undergoing topological adiabatic cycles. Specifically, for a stack of two semi-infinite chains cycled in opposite directions and coupled at one edge by a gapping potential, we derive a higher-order bulk-boundary correspondence that relates the bulk Chern number associated with the adiabatic cycle of a single infinite chain and the number of electrons transferred between the semi-infinite chains. The relation is formulated using the relative index of two projections and proven using K-theoretic calculations. The phenomenon is exemplified using the Rice-Mele model and possible experimental implementations with classical and quantum degrees of freedom are discussed., Comment: 6 pages and Supplemental Material. Latest version for publication

Quantum Hall physics has been theoretically predicted in four dimensions and higher. In hypothetical 2n dimensions, the topological characters of both the bulk and the boundary are manifested as quantized nonlinear transport coefficients that respectively connect to the $n\mathrm{th}$ Chern number of the bulk gap projection and to the $n\mathrm{th}$ winding number of the Weyl spectral singularities on the ($2n\ensuremath{-}1$)-dimensional boundaries. Here, we introduce the concept of phason engineering in metamaterials and use it as a vehicle to access and apply the quantum Hall physics in arbitrary dimensions. Using these specialized design principles, we fabricate a reconfigurable two-dimensional aperiodic acoustic crystal with a phason living on a 2-torus, giving us access to the four-dimensional quantum Hall physics. Also, we supply a direct experimental confirmation that the topological boundary spectrum assembles in a Weyl singularity when mapped as a function of the quasimomenta. We also demonstrate topological wave steering enabled by the Weyl physics of the three-dimensional boundaries.

We consider the algebra $\dot\Sigma(\mathcal L)$ generated by the inner-limit derivations over the ${\rm GICAR}$ algebra of a fermion gas populating an aperiodic Delone set $\mathcal L$. Under standard physical assumptions such as finite interaction range, Galilean invariance and continuity with respect to the aperiodic lattice, we demonstrate that the image of $\dot \Sigma(\mathcal L)$ through the Fock representation can be completed to a groupoid-solvable pro-$C^\ast$-algebra. Our result is the first step towards unlocking the $K$-theoretic tools available for separable $C^\ast$-algebra for applications in the context of interacting fermions.

The bulk-boundary and a new bulk-defect correspondence principles are formulated using groupoid algebras. The new strategy relies on the observation that the groupoids of lattices with boundaries or defects display spaces of units with invariant accumulation manifolds, hence they can be naturally split into disjoint unions of open and closed invariant sub-sets. This leads to standard exact sequences of groupoid $C^\ast$-algebras that can be used to associate a Kasparov element to a lattice defect and to formulate an extremely general bulk-defect correspondence principle. As an application, we establish a correspondence between topological defects of a 2-dimensional square lattice and Kasparov's group $KK^1 (C^\ast(\mathbb Z^3),\mathbb C)$. Numerical examples of non-trivial bulk-defect correspondences are supplied.

A Thouless pump can be regarded as a dynamical version of the integer quantum Hall effect. In a finite-size configuration, such topological pump displays edge modes that emerge dynamically from one bulk-band and dive into the opposite bulk-band, an effect that can be reproduced with both quantum and classical systems. Here, we report the first un-assisted dynamic energy transfer across a metamaterial, via pumping of such topological edge modes. The system is a topological aperiodic acoustic crystal, with a phason that can be fast and periodically driven in adiabatic cycles. When one edge of the metamaterial is excited in a topological forbidden range of frequencies, a microphone placed at the other edge starts to pick up a signal as soon as the pumping process is set in motion. In contrast, the microphone picks no signal when the forbidden range of frequencies is non-topological., Comment: Embedded, are the audio files of the first acoustic signal topologically pumped from one edge to the other

This work represents the first chapter of a project on the foundations of first-principle calculations of the electron transport in crystals at finite temperatures. We are interested in the range of temperatures, where most electronic components operate, that is, room temperature and above. The aim is a predictive first-principle formalism that combines ab-initio molecular dynamics and a finite-temperature Kubo-formula for homogeneous thermodynamic phases. The input for this formula is the ergodic dynamical system $(\Omega,\mathbb G,{\rm d}\mathbb P)$ defining the crystalline phase, where $\Omega$ is the configuration space for the atomic degrees of freedom, $\mathbb G$ is the space group acting on $\Omega$ and ${\rm d}\mathbb P$ is the ergodic Gibbs measure relative to the $\mathbb G$-action. The present work develops an algorithmic method for quantifying $(\Omega,\mathbb G,{\rm d}\mathbb P)$ from first principles. Using the silicon crystal as a working example, we find the Gibbs measure to be extremely well characterized by a multivariate normal distribution, which can be quantified using a small number of parameters. The latter are computed at various temperatures and communicated in the form of a table. Using this table, one can generate large and accurate thermally-disordered atomic configurations to serve, for example, as input for subsequent simulations of the electronic degrees of freedom., Comment: 33 pages, 16 figures

The hyperbolic space affords an infinite number of regular tessellations, as opposed to the Euclidean space. Thus, the hyperbolic space significantly extends the design space lattices, potentially providing access to unexplored wave phenomena. Here we investigate the dynamic behavior of hyperbolic tessellations governed by interactions whose strengths depend upon the distances between neighboring nodes. We find eigen-modes that are primarily localized either at the center or towards the boundary of the Poincare disk, where hyperbolic lattices are represented. Hyperbolic translations of the seeding polygon produce distorted lattices, leading to a redistribution of the eigen-modes akin to edge-to-edge transitions. The spectral flow associated with these deformed lattices reveals a rich behavior that is characterized by modes that are spatially asymmetric and localized. The strength of the localization can be predicted from the slopes of the corresponding spectral branches, suggesting a potential topological origin for the observed phenomena. The rich yet predictable spectral flow and the high modal density of these lattices, along with the propensity of their modes to be strongly localized, suggest potential applications of hyperbolic lattices as vibration sensors, which operate over a large range of frequencies and exploit the sensitivity of localized modes to perturbations. In addition, hyperbolic lattices can inform the design of architected structural components with strong vibration attenuation and isolation capabilities.

Mechanical systems can display topological characteristics similar to that of topological insulators. Here we report a large class of topological mechanical systems related to the BDI symmetry class. These are self-assembled chains of rigid bodies with an inversion centre and no reflection planes. The particle-hole symmetry characteristic to the BDI symmetry class stems from the distinct behaviour of the translational and rotational degrees of freedom under inversion. This and other generic properties led us to the remarkable conclusion that, by adjusting the gyration radius of the bodies, one can always simultaneously open a gap in the phonon spectrum, lock-in all the characteristic symmetries and generate a non-trivial topological invariant. The particle-hole symmetry occurs around a finite frequency, and hence we can witness a dynamical topological Majorana edge mode. Contrasting a floppy mode occurring at zero frequency, a dynamical edge mode can absorb and store mechanical energy, potentially opening new applications of topological mechanics., Properties of topological insulators can be realized in mechanical systems, opening potential applications of topological mechanics. Here, Prodan et al. report a dynamical topological Majorana edge mode in self-assembled chains of rigid bodies with particle-hole symmetry.

We derive explicit closed-form matrix representations of Hamiltonians drawn from tensored algebras, such as quantum spin Hamiltonians. These formulas enable us to soft-code generic Hamiltonian systems and to systematize the input data for uniformly structured as well as for un-structured Hamiltonians. The result is an optimal computer code that can be used as a black box that takes in certain input files and returns spectral information about the Hamiltonian. The code is tested on Kitaev's toric code deployed on triangulated surfaces of genus 0 and 1. The input file corresponding to the minimal triangulation of genus 2 is also supplied., optimized code can be executed on a laptop

The emergence of a fractal energy spectrum is the quintessence of the interplay between two periodic parameters with incommensurate length scales. crystals can emulate such interplay and also exhibit a topological bulk-boundary correspondence, enabled by their nontrivial topology in virtual dimensions. Here we propose, fabricate and experimentally test a reconfigurable one-dimensional (1D) acoustic array, in which the resonant frequencies of each element can be independently fine-tuned by a piston. We map experimentally the full Hofstadter butterfly spectrum by measuring the acoustic density of states distributed over frequency while varying the long-range order of the array. Furthermore, by adiabatically changing the phason of the array, we map topologically protected fractal boundary states, which are shown to be pumped from one edge to the other. This reconfigurable crystal serves as a model for future extensions to electronics, photonics and mechanics, as well as to quasi-crystalline systems in higher dimensions. Hofstadter’s butterfly is a fractal pattern which pictorially represents the behavior of electrons under an applied magnetic field in a 2D lattice as a pair of butterfly wings. Here, the authors recreate this pattern by measuring the acoustic density of states in a fine-tuned one-dimensional acoustic array.

We use magnetic flux-tubes to stabilize zero-energy modes in a lattice realization of a 2-dimensional superconductor from class D of classification table of topological condensed matter systems. The zero modes are exchanged by slowly displacing the flux-tubes and an application of the adiabatic theorem demonstrates the geometric nature of the resulting unitary time-evolution operators. Furthermore, an explicit numerical evaluation reveals that the evolutions are in fact topological, hence supplying a representation of the braid group, which turns out to be non-abelian. This physical representation is further formalized using single-strand planar diagrams. Lastly, we discuss how these predictions can be implemented with and observed in classical meta-materials and how the standard Majorana representation of the braid group can be generated by measuring derived physical observables.

Topological boundary and interface modes are generated in an acoustic waveguide by simple quasi-periodic patternings of the walls. The procedure opens many topological gaps in the resonant spectrum and qualitative as well as quantitative assessments of their topological character are supplied. In particular, computations of the bulk invariant for the continuum wave equation are performed. The experimental measurements reproduce the theoretical predictions with high fidelity. In particular, acoustic modes with high Q-factors localized in the middle of a breathable waveguide are engineered by a simple patterning of the walls.

Nontrivial braid-group representations appear as non-Abelian quantum statistics of emergent Majorana zero modes in one- and two-dimensional topological superconductors. Here, we generate such representations with topologically protected domain-wall modes in a classical analog of the Kitaev superconducting chain, with a particle-holelike symmetry and a ${\mathbb{Z}}_{2}$ topological invariant. The midgap modes are found to exhibit distinct fusion channels and rich non-Abelian braiding properties, which are investigated using a $T$-junction setup. We employ the adiabatic theorem to explicitly calculate the braiding matrices for one and two pairs of these midgap topological defects.

This is the second part of a project on the foundations of first-principle calculations of the electron transport in crystals at finite temperatures, aiming at a predictive first-principles platform that combines ab-initio molecular dynamics (AIMD) and a finite-temperature Kubo-formula with dissipation for thermally disordered crystalline phases. The latter are encoded in an ergodic dynamical system $(\Omega,\mathbb G,{\rm d}\mathbb P)$, where $\Omega$ is the configuration space of the atomic degrees of freedom, $\mathbb G$ is the space group acting on $\Omega$ and ${\rm d}\mathbb P$ is the ergodic Gibbs measure relative to the $\mathbb G$-action. We first demonstrate how to pass from the continuum Kohn-Sham theory to a discrete atomic-orbitals based formalism without breaking the covariance of the physical observables w.r.t. $(\Omega,\mathbb G,{\rm d}\mathbb P)$. Then we show how to implement the Kubo-formula, investigate its self-averaging property and derive an optimal finite-volume approximation for it. We also describe a numerical innovation that made possible AIMD simulations with longer orbits and elaborate on the details of our simulations. Lastly, we present numerical results on the transport coefficients of crystal silicon at different temperatures.

The quantum valley-Hall effect (QVHE) manifests in both classical and quantum materials as the emergence of pairs of quasichiral bands along certain interfaces. This bulk-boundary principle is well understood in the limit where the valleys result from a slight splitting of Dirac singularities. However, using a versatile experimental platform based on magnetically coupled spinners, we demonstrate that this regime is not suitable for metamaterial applications due to the delocalization of the interface modes. We also find that a strong splitting of the Dirac singularities washes away the QVHE. We then propose that the enlargement of the bulk gap to be accompanied by a Berry curvature engineering that keeps it localized near the valleys. This is a new regime, entirely outside the umbrella of Dirac physics, which we call the valley-Chern effect (VCE). By establishing an exact relation between VCE and quantum spin-Hall effect, we demonstrate a robust bulk-boundary principle, which could be the foundation of a new wave of applications of topological metamaterials.

Topological condensed matter systems from class A and class AII of the classification table have received classical electromagnetic and mechanical analogs and protected wave-guiding with such systems has been demonstrated experimentally. Here we introduce a map which generates classical analogs for all entries of the classification table, using only passive elements. Physical mechanical models are provided for all strong topological phases in dimension 2, as well as for three classes in dimension 3. This includes topological super-conducting phases, which have never been attempted with classical systems.

The research in topological materials and meta-materials reached maturity and is now gradually entering the phase of practical applications and devices. However, scaling down the experimental demonstrations definitely presents a challenge. In this work, we study coupled identical resonators whose collective dynamics is fully determined by the pattern in which the resonators are arranged. We call a pattern topological if boundary resonant modes fully fill all existing spectral gaps whenever the pattern is halved. This is a characteristic of the pattern and is entirely independent of the structure of the resonators and the details of the couplings. Existence of such patterns is proven using $K$-theory and exemplified using a novel experimental platform based on magnetically coupled spinners. Topological meta-materials built on these principles can be easily engineered at any scale, providing a practical platform for applications and devices., video recordings of the topological modes

Large classes of electronic, photonic, and acoustic crystals and quasi-crystals have been predicted to support topological wave-modes. Some of these modes are stabilized by certain symmetries but others occur as pure wave phenomena, hence they can be observed in many other media that support wave propagation. Surface water-waves are mechanical in nature but very different from the elastic waves, hence they can provide a new platform for studying topological wave-modes. Motivated by this perspective, we report theoretical and experimental characterizations of water-wave crystals obtained by periodic patterning of the water surface. In particular, we demonstrate the band structure of the spectra and existence of spectral gaps.

We study bulk-boundary correspondences and related surface phenomena stabilized by the second Chern number in three-dimensional insulators driven in adiabatic cycles. Magnetic fields and disorder effects are incorporated in our analysis using operator algebraic methods. We use the connecting maps between the $K$-theories of bulk and boundary algebras as engines for the bulk-boundary correspondences. We discovered that both the exponential and the index connecting maps are relevant for the context considered here as they lead to distinct experimentally observable surface phenomena, such as pumping and transfer of quantum surface Hall states or proximity induced Hall effect. The surface Hall physics of time-reversal symmetric topological insulators is also investigated using the new tools, which can model irrational magnetic fluxes and arbitrary large surface disorder., Comment: 29 pages, 6 figures; final version submitted for journal publication

The valley Chern-effect is theoretically demonstrated with a novel alternating current circuitry, where closed-loop LC-resonators sitting at the nodes of a honeycomb lattice are inductively coupled along the bonds. This enables us to generate a dynamical matrix which copies identically the Hamiltonian driving the electrons in graphene. The valley-Chern effect is generated by splitting the inversion symmetry of the lattice. After a detailed study of the Berry curvature landscape and of the localization of the interface modes, we derive an optimal configuration of the circuit. Furthermore, we show that Q-factors as high as $10^4$ can be achieved with reasonable materials and configurations.

Starting from a dynamical system $(\Omega,G)$, with $G$ a generic topological group, we devise algorithms that generate families of patterns in the Euclidean space, which densely embed $G$ and on which $G$ acts continuously by rigid shifts. We refer to such patterns as being dynamically generated. For $G=\mathbb Z^d$, we adopt Bellissard's $C^\ast$-algebraic formalism to analyze the dynamics of coupled resonators arranged in dynamically generated point patterns. We then use the standard connecting maps of $K$-theory to derive precise conditions that assure the existence of topological boundary modes when a sample is halved. We supply four examples for which the calculations can be carried explicitly. The predictions are supported by many numerical experiments., Comment: 52 pages, 27 Figures

A model for two-dimensional electronic, photonic, and mechanical metamaterial systems is presented, which has flat one-dimensional zero-mode energy bands and stable localized states of a topological origin confined within twin boundaries, antiphase boundaries, and at open edges. Topological origins of these flat bands are analyzed for an electronic system as a specific example, using a two-dimensional extension of the Su-Schrieffer-Heeger Hamiltonian with alternating shift of the chains. It is demonstrated that the slow group velocities of the localized flat band states are sensitively controlled by the distance between the boundaries and the propagation can be guided through designed paths of these boundaries. We also discuss how to realize this model in metamaterials., Comment: 5 page main text, 2 page supplemental material

Doped semiconductor nanocrystals represent a new type of quantum plasmonic material with optical resonances in the infrared. These nanocrystals are fundamentally different from the metal nanoparticles because the electron density in a semiconductor can be tuned over a wide interval. Using the DFT-based time-dependent formalism, we computed the absorption spectra of doped quantum dots as a function of the number of carriers in the dot. The dynamic properties of doped quantum dots undergo an interesting transition from the size-quantization regime to the classical regime of plasmon oscillations. We demonstrated this quantum-to-classical transition for both self-doped copper chalcogenide dots and impurity-doped II–VI nanocrystals. We also showed that the plasmon frequency and electron density in the dot depend on the type of doping, which can be bulk-like or surface-like. The results obtained in this study can be used to predict and describe optical properties of a variety of semiconductor nanocrystals with ...

This Chapter introduces and characterizes the approximating algebras used later to define the canonical finite-volume approximations and to analyze the numerical errors. In particular, the approximating finite algebra generates the ordinary finite super-cell models with periodic boundary conditions, which are commonly used in the numerical investigations. Formalizing them in an algebraic setting will enable us to bridge more naturally with the thermodynamic limit, to put forward a canonical finite-volume approximation and to ultimately resolve the error estimates with minimal effort. Several key ideas of the Chapter already appeared in [3] but other are reported here for the first time.

The search for strong topological phases in generic aperiodic materials and meta-materials is now vigorously pursued by the condensed matter physics community. In this work, we first introduce the concept of patterned resonators as a unifying theoretical framework for topological electronic, photonic, phononic etc. (aperiodic) systems. We then discuss, in physical terms, the philosophy behind an operator theoretic analysis used to systematize such systems. A model calculation of the Hall conductance of a 2-dimensional amorphous lattice is given, where we present numerical evidence of its quantization in the mobility gap regime. Motivated by such facts, we then present the main result of our work, which is the extension of the Chern number formulas to Hamiltonians associated to lattices without a canonical labeling of the sites, together with index theorems that assure the quantization and stability of these Chern numbers in the mobility gap regime. Our results cover a broad range of applications, in particular, those involving quasi-crystalline, amorphous as well as synthetic (i.e. algorithmically generated) lattices., Comment: 44 pages, 4 figures. v2: typos corrected and references updated. v3: Minor changes, to appear in J. Phys. A (Mathematical and Theoretical)

In this Chapter we analyze correlation functions of the type: $$\begin{aligned} {\mathcal T}( \partial ^{\alpha _1} G_1(h) \, \partial ^{\alpha _2} G_2(h) \ldots ) \; , \quad h \in {\mathcal A}_d \; , \end{aligned}$$ (8.1) where \(G_i\)’s are only piecewise continuous and the discontinuities occur inside the essential spectrum of h. We establish that, if the discontinuities occur inside the mobility gaps of h, then the canonical finite-volume algorithm presented in Chap. 5 continues to display a rapid convergence to the thermodynamic limit. Key to the error estimates is the Aizenman–Molchanov bound (Aizenman and Molchanov, Commun Math Phys 157:245–278, 1993, [2], Aizenman and Warzel, Random operators: disorder effects on quantum spectra and dynamics, 2015, [3]), which is discussed and adapted to the present context.

Our main goal for this Chapter is to introduce the periodic table of topological insulators and superconductors (see Table 1.1). Since the physical principles behind this table involve the robustness of certain physical properties against disorder, we will take a short detour and introduce first the class of homogeneous materials and then the sub-class of homogeneous disordered crystals. Our brief exposition introduces the main physical characteristics of these materials and puts forward a possible physical definition of the class of homogeneous materials. In parallel, it introduces general concepts from the mathematical program pioneered by Jean Bellissard and his collaborators (Bellissard, Lect. Notes Phys. 257, 99–156, 1986, [4], Bellissard, Geometric and Topological Methods for Quantum Field Theory, 2003, [7], Bellissard et al., Directions in Mathematical Quasicrystals, 2000, [5]), which will become central to our exposition.

This Chapter introduces and characterizes the so called lattice models, which integrate naturally in the general framework for homogeneous materials introduced in Chap. 1. It also fixes the notations and the conventions used throughout. A model of disorder is introduced and the various regimes of disorder are discussed. The Chapter concludes with the introduction and characterization of the topological invariants corresponding to the phases classified by \({\mathbb Z}\) and \(2{\mathbb Z}\) in Table 1.1.

As we have seen in Sect. 3.8, the response functions and the various thermodynamic coefficients are expressed as correlation functions of the type: $$\begin{aligned} {\mathcal T}\big ( \partial ^{\alpha _1} G_1(h) \, \partial ^{\alpha _2} G_2(h) \ldots \big ) \; , \quad h \in {\mathcal A}_d \; , \end{aligned}$$ (5.1) and variations of it, such as the Kubo formula treated in Sect. 7.1. In this Chapter, we use the previously introduced auxiliary algebras and propose a canonical finite-volume algorithm for computing the above correlation functions. Our main goals for the Chapter are to integrate the algorithm in a broader context and then to summarize for the reader the concrete steps of the algorithm, as applied to disordered homogeneous solids. The error estimates are the subjects of the following Chapters.

We consider one dimensional tight binding models on $\ell^2(\mathbb Z)$ whose spatial structure is encoded by a Sturmian sequence $(\xi_n)_n\in \{a,b\}^\mathbb Z$. An example is the Kohmoto Hamiltonian, which is given by the discrete Laplacian plus an onsite potential $v_n$ taking value $0$ or $1$ according to whether $\xi_n$ is $a$ or $b$. The only non-trivial topological invariants of such a model are its gap-labels. The bulk-boundary correspondence we establish here states that there is a correspondence between the gap label and a winding number associated to the edge states, which arises if the system is augmented and compressed onto half space $\ell^2(\mathbb N)$. This has been experimentally observed with polaritonic waveguides. A correct theoretical explanation requires, however, first a smoothing out of the atomic motion via phason flips. With such an interpretation at hand, the winding number corresponds to the mechanical work through a cycle which the atomic motion exhibits on the edge states., Comment: 37 pages, 7 figures

This Chapter reports computer assisted calculations of the topological invariants for strongly disordered models of topological insulators from class A and AIII in the classification Table 1.1. The computations are based on the canonical finite-volume algorithm presented in Chap. 5. The expressions of the topological invariants involve functional calculus with discontinuous step-wise functions, hence their numerical simulations are covered by the analysis in Chap. 8, provided the discontinuities occur inside the Anderson localized spectrum. The performance and the accuracy of the algorithms will be quantified and analyzed in various ways.

In this Chapter we derive upper bounds on the error introduced by restricting to a finite-volume, and show that they decay to zero faster than any inverse power of the volume. The error bounds derived here are relevant for the finite-temperature correlation functions, which in general involve only the smooth functional calculus with the Hamiltonian. In these cases, the fast convergence to the thermodynamic limit of the numerical algorithms is not conditioned by the localized or delocalized character of the energy spectrum. This is important, for example, when studying the metal-insulator transition, where inherently the character of the spectrum changes.

In this Chapter, we review the fundamental theoretical tools, starting with the space of disordered configurations and its associated dynamical systems, the \(C^*\)-algebra \({\mathcal A}_d\) of the physical observables, together with its Fourier and differential calculus. The latter is provided by a set of commuting derivations \(\partial \) and a trace \({\mathcal T}\). The triple \(({\mathcal A}_d,\partial ,{\mathcal T})\) defines a non-commutative manifold known as the non-commutative Brillouin torus. We reformulate the topological invariants and other response functions in this new framework. We also introduce the magnetic derivations and investigate the behavior of the correlation functions w.r.t. the magnetic fields. This Chapter also fixes the notation and defines the precise settings for the rest of our calculations.

In the first part of the Chapter, we show that the non-commutative Kubo-formula for the transport coefficients is covered by the theory developed so far. In the second part, we present computer assisted calculations of the finite-temperature transport coefficients, for model Hamiltonians relevant for the integer quantum Hall effect and Chern insulators. The focus will be on the physical signature of the Anderson transition occurring at the boundary between the topological phases.

Using exact diagonalization and quantum Monte Carlo calculations we investigate the effects of disorder on the phase diagram of both non-interacting and interacting models of two-dimensional topological insulators. In the fermion sign problem-free interacting models we study, electron-electron interactions are described by an on-site repulsive Hubbard interaction and disorder is included via the one-body hopping operators. In both the non-interacting and interacting models we make use of recent advances in highly accurate real-space numerical evaluation of topological invariants to compute phase boundaries, and in the non-interacting models determine critical exponents of the transitions. We find different models exhibit distinct stability conditions of the topological phase with respect to interactions and disorder. We provide a general analytical theory that accurately predicts these trends., 13 pages, 5 figures

The plasmon resonances in metallic nanorods are investigated using fully quantum mechanical time-dependent density functional theory. The computed optical absorption curves display well-defined longitudinal and transverse plasmon resonances whose energies depend on the aspect ratio of the rods, in excellent agreement with classical electromagnetic modeling. The field enhancements obtained from the quantum mechanical calculations, however, differ significantly from classical predictions for distances shorter than 0.5 nm from the nanoparticle surfaces. These deviations can be understood as arising from the nonlocal screening properties of the conduction electrons at the nanoparticle surface.