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Institute
Graphene is a strictly two-dimensional honeycomb lattice of carbon atoms whose low-energy charge-carrier dynamics obey the massless pseudospin-1/2 Dirac-Weyl equation (or chiral Weyl equation) where the chiral centers (or valleys) are the corners K and K‘ of the Brillouin zone. The linear spectrum near the Dirac nodal points lends graphene its exotic and ultra-relativistic properties.
However, condensed matter systems can possess fermionic excitations with linear dispersions that have no analog in high-energy physics since the crystal space group - instead of the Poincare group - constrains the energy dispersions. Perhaps the first example in this regard is the T_3 lattice (Dice Gitter), a honeycomb-like lattice with an extra atom placed at the center of each hexagon and coupled to only one of the sublattices. The spectrum features a strictly flat band that crosses the two conical intersections of the Dirac cones at K and K' inherited from graphene. The enlarged pseudospin-1 Dirac-Weyl equation describes the low-energy dynamics. By rescaling the transfer amplitude of the additional atoms in the T_3 lattice with a parameter 0<α<1, the resulting α-T_3 lattice continously interpolates between graphene and the T_3 lattice.
In this work, we explore the behavior of generalized Dirac-Weyl quasiparticles in external magnetic and valley-dependent pseudoelectromagnetic fields induced by out-of-plane strain. First, we studied Dirac-Weyl quasiparticles in external fields confined to circular quantum dots by generalizing the infinite-mass boundary condition to the α-T_3 lattices. We verified the analytically derived valley-anisotropic eigenstates of the quantum dot by numerically solving the tight-binding lattice-model in closed (isolated) and open (contacted) systems.
Second, we considered strain fields in the α-T_3 lattices to modify the low-energy transport properties by an effective pseudo-gauge field with opposite signs at the K and K‘ valley. In particular, we showed that the inhomogeneous pseudomagnetic field generated by Gaussian out-of-plane strain at the center of a four-terminal Hall bar setup acts as a valley filter. Most interestingly, the valley polarization is most dominant when incoming electrons are excited to pseudo-Landau level subbands. These bands are linked to different iso-field orbits encircling the lobes of the pseudomagnetic field. Addittionaly, any intermediate α breaks the inversion symmetry of the α-T_3 lattice and thus splits the pseudo-Landau levels into sublattice-polarized bands.
Third, we equipped the out-of-plane strain with a time-periodic drive to induce a valley-dependent pseudoelectric field perpendicular to the pseudomagnetic field. We assessed the steady-state transport properties and found – besides the static regime for small energies – two α-dependent valley-filtering regimes due to the periodic drive. Firstly, we found an additional valley-polarization plateau at the Floquet-zone boundary between the central and first Floquet copy that also displayed a “flower”-like pattern in the local density of states. Secondly, we detected a series of transmission gaps at the center of every Floquet sideband 2mΩ related to the Floquet coupling of the flat band with the central Floquet copy. Under certain strain parameters, a novel valley-filtering regime appears near the transmission gaps where the incoming K electrons are focused through the bump by the pseudoelectric field, instead of encircling the lobes of the pseudomagnetic field. A stability analysis demonstrated that the polarization regimes are tunable by the driving frequency.
Lastly, we demonstrated that the flat band in the Haldane-dice lattice modified by a uniaxial strain along the zigzag orientation remains singular at all band crossings where the model undergoes a topological phase transition between C=+-2 and C=0. To show this, we computed the compact localized eigenstates and the quantum distance of the Bloch wave function around the band-touching points. We derived the resulting non-contractible loop states and an extended state whose components are tunabe by the system parameters.
This thesis contains studies on a special class of topological insulators, so called anomalous Floquet topological insulators, which exclusively occur in periodically driven systems. At the boundary of an anomalous Floquet topological insulator, topologically protected transport occurs even though all of the Floquet bands are topologically trivial. This is in stark contrast to ordinary topological insulators of both static and Floquet type, where the topological invariants of the bulk bands completely determine the chiral boundary states via the bulk-boundary correspondence. In anomalous Floquet topological insulators, the boundary states are instead characterized by bulk invariants that account for the full dynamical evolution of the Floquet system.
Here, we explore the interplay between topology, symmetry, and non-Hermiticity in two-dimensional anomalous Floquet topological insulators. The central results of this exploration are (i) new expressions for the topological invariants of symmetry-protected anomalous Floquet topological phases which can be efficiently computed numerically, (ii) the construction of a universal driving protocol for symmetry-protected anomalous Floquet topological phases and its experimental implementation in photonic waveguide lattices, (iii) the discovery of non-Hermitian boundary state engineering which provides unprecedented possibilities to control and manipulate the topological transport of anomalous Floquet topological insulators.
Matrix-product-state based methods, in particular the density-matrix renormalization group, are used to numerically investigate several one-dimensional systems, focusing on models with symmetry-protected topological phases that generalize the spin-1 Haldane chain. In the first part, ground state properties such as topological order parameters and the criticality at quantum phase transitions are studied.
The second part deals with dynamic properties of spin chains. Using time-dependent matrix-product-state calculations, the dynamic structure factor, and the transport properties of contacted spin chains are analyzed.
Optomechanical (om) systems are characterized by their nonlinear light-matter interaction. This is responsible for unique dynamic properties and allows the detection of a variety of classical and quantum mechanical phenomena on a microscopic as well as on a macroscopic scale. In this work we have studied the dynamic behavior of two laser-driven om systems, the single om cell ("cavity optomechanics / membrane-in-the-middle setup") and a two-dimensional hexagonal array of these cells ("om graphene"). The first case was motivated by the possibility to detect the transition from quantum mechanics to classical mechanics directly on the basis of the dynamic behavior. For this we focus on multistability effects of the optical and mechanical degrees of freedom, that are modeled by harmonic oscillators. Our description is based on the quantum optical master equation, which takes into account the environmental interaction assuming a vanishing temperature. As a consequence of decoherence, the dynamics occur near the semiclassical limit, i.e. it is characterized by quantum fluctuations. The quantum-to-classical transition is realized formally by rescaling the equations of motion. In the classical limit, quantum fluctuations disappear and the mean field equations were evaluated by analytical and numerical methods. We found that classical multistability is characterized by stationary signatures on the route to chaos, as well as by the coexistence of single-periodic orbits for the mechanical degree of freedom. The latter point was extensively evaluated by means of a self-consistent approach. For the dynamics in the quantum regime quantum fluctuations cannot be neglected. For this purpose, the master equation was solved by means of a numerical implementation of the Quantum State Diffusion (QSD) method. Based on Wigner and autocorrelation functions, we were able to show that quantum multistability is a dynamic effect: chaotic dynamics is suppressed and there is a time-dependent distribution of the phase space volume on classical simple-periodic orbits. The results can be interpreted within a semiclassical picture, which makes use of the single QSD quantum trajectory. Accordingly, the quantum-classical transition is explained as a time-scale effect, which is determined by tunneling probabilities in an effective mean-field potential. The subject of the second part of the work is the transport of low-energy Dirac quasiparticles in om graphene, propagating as light and sound waves. For this purpose, we investigated the scattering of a plane light wave by laser-induced photon-phonon coupling planar and circular barriers. The starting point is the om Dirac equation, which results from the continuum approximation of the Hamiltonian description of the two-dimensional array near the semiclassical limit. This work was motivated by the rich and interesting relativistic transport and tunneling phenomena found for electrons in graphene, which now appear in a new way. The reason is the presence of the new spin degree of freedom, which distinguishes the optical and mechanical excitations. In this spin space, the om interaction can be understood as a potential, which in our analysis consists of a time-independent and a time-dependent sinusoidal part. For the first case of a static barrier, the transport is elastic and is characterized by stationary scattering signatures. After solving the scattering problem via continuity conditions we were able to identify different scattering regimes depending on scattering parameters. In addition to relativistic phenomena such as Klein tunneling, simple parameter variation allows to use the barrier as a resonant light-sound interconverter and angle-dependent emitter. For the oscillating barrier, the transport is inelastic and is characterized by dynamic scattering signatures. To solve the time-periodic scattering problem, we have applied the Floquet theory for an effective two-level system. As a result of the barrier oscillation, photons and phonons can get and give away energy portions in the form of integer multiples of the oscillation frequency. The interference of short (classical) and long-wave (quantum) components leads to mixing of the scattering regimes. This allows to use the barrier as a time-periodic light-sound interconverter with interesting radiation characteristics. In addition, we have argued that the oscillating barrier provides the necessary energetic conditions for detecting zitterbewegung.