Topological Edge States with Zero Hall Conductivity in a Dimerized Hofstadter Model

The Hofstadter model is a simple yet powerful Hamiltonian to study quantum Hall physics in a lattice system, manifesting its essential topological states. Lattice dimerization in the Hofstadter model opens an energy gap at half filling. Here we show that even if the ensuing insulator has a Chern number equal to zero, concomitantly a doublet of edge states appear that are pinned at specific momenta. We demonstrate that these states are topologically protected by inversion symmetry in specific one-dimensional cuts in momentum space, define and calculate the corresponding invariants and identify a platform for the experimental detection of these novel topological states.Comment: 5 pages, 3 figures + Supplemental Material (6 pages

Symmetry-enriched topological states of matter in insulators and semimetals

Topological states of matter are a novel family of phases that elude the conventional Landau paradigm of phase transitions. Topological phases are characterized by global topological invariants which are typically reflected in the quantization of physical observables. Moreover, their characteristic bulk-boundary correspondence often gives rise to robust surface modes with exceptional features, such as dissipationless charge transport or non-Abelian statistics. In this way, the study of topological states of matter not only broadens our knowledge of matter but could potentially lead to a whole new range of technologies and applications. In this light, it is of great interest to find novel topological phases and to study their unique properties. In this work, novel manifestations of topological states of matter are studied as they arise when materials are subject to additional symmetries. It is demonstrated how symmetries can profoundly enrich the topology of a system. More specifically, it is shown how symmetries lead to additional nontrivial states in systems which are already topological, drive trivial systems into a topological phase, lead to the quantization of formerly non-quantized observables, and give rise to novel manifestations of topological surface states. In doing so, this work concentrates on weakly interacting systems that can theoretically be described in a single-particle picture. In particular, insulating and semi-metallic topological phases in one, two, and three dimensions are investigated theoretically using single-particle techniques

Problem-Based Learning for the Teaching of Skills: Results of a Practical Use

Digitalisation and COVID-19 have been changing the way we teach. Last year distance learning was often the only possibility to transfer knowledge. Based on an intensive literature review, the paper presents best practices to enable knowledge transfer and analyses the possibilities and shortcomings of two approaches by applying them in several case studies: 1) Blended learning: provides content and knowledge mainly with the help of learning videos and feedback/reflection sessions. 2) Problem-based learning: secures that the students can apply the learned content and methodologies. Beside the results of the case studies the paper provides the feedback of the students

Electron interferometry with nano-gratings

We present an electron interferometer based on near-field diffraction from two nanostructure gratings. Lau fringes are observed with an imaging detector, and revivals in the fringe visibility occur as the separation between gratings is increased from 0 to 3 mm. This verifies that electron beams diffracted by nanostructures remain coherent after propagating farther than the Talbot length $z_T = 2d^2/\lambda$ = 1.2 mm, and hence is a proof of principle for the function of a Talbot-Lau interferometer for electrons. Distorted fringes due to a phase object demonstrates an application for this new type of electron interferometer.Comment: 4 pgs, 6 figure

Effects of counterion fluctuations in a polyelectrolyte brush

We investigate the effect of counterion fluctuations in a single polyelectrolyte brush in the absence of added salt by systematically expanding the counterion free energy about Poisson-Boltzmann mean field theory. We find that for strongly charged brushes, there is a collapse regime in which the brush height decreases with increasing charge on the polyelectrolyte chains. The transition to this collapsed regime is similar to the liquid-gas transition, which has a first-order line terminating at a critical point. We find that for monovalent counterions the transition is discontinuous in theta solvent, while for multivalent counterions the transition is generally continuous. For collapsed brushes, the brush height is not independent of grafting density as it is for osmotic brushes, but scales linear with it.Comment: 9 pages, 9 figure