# Research

## Higher-order topological insulators

We study topological states of matter that are insulating in the bulk and on the surfaces, but have protected conducting channels on the hinges of a crystal. We showed that elementary bismuth has this property (read more here and the original article here). We also work on classical realizations of such states in the lab, in particular in electric circuits (read more here and the original article here).

## Topological order

**Lattice models of itinerant electrons** The fractional quantum Hall effect is a paradigmatic state of matter with topological order and anyon excitations. We showed that lattice models with topological band structures, so-called Chern insulators, can exhibit the same physics without an external magnetic field [article]. The topologically ordered states emerge most favorably if the topological band is relatively flat, but if the electronic interactions are strong enough, this flatness condition is not necessary [article]. The concept of a fractional Chern insulator can also be extended to fractional topological insulators with time-reversal symmetry [article]. See also our review for the 2014 Nobel Symposium on Topological States of Matter [article].

**Condensation transitions in topological quantum field theory** Bosonic anyons in topologically ordered states of matter can in principle undergo a Bose-Einstein condensation, triggering a transition to a different type of topological order. In [article] we show how to determine the universal properties of the topological order after a boson condensed. Further, we showed that certain topological bosons cannot condense — a no-go theorem that has consequences for the classification of partition functions in conformal field theories.

## Superconductivity and Majorana fermions

**SrPtAs — a potential Weyl superconductor** Muon spin-rotation experiments found that SrPtAs, a superconductor with hexagonal crystal structure, breaks time-reversal symmetry in the superconducting state [article]. An analysis of the possible superconducting states, together with a numerical functional renormalization group calculation indicates that this material realizes a sought-after chiral d-wave superconductor with chiral Majorana surface states. At the same time, it has Majorana-Weyl excitations at isolated points in the bulk where the gap function is nodal [article].

**Majorana fermions in Shiba systems** Superconductors supporting Majorana fermion excitations can be engineered by decorating the surface of a conventional superconductor with a chain of magnetic adatoms. We showed how Majorana the Majorana bound states can be created and manipulated in such a setup using an external magnetic field (Link Nature com). Furthermore, we illustrated that this concept also applies to two-dimensional lattices of adatoms to create a chiral p-wave superconductor [article] and to chains of non-magnetic adatoms to create time-reversal symmetric topological superconductors [article].

## Topological metals

**Weyl semimetals** We explored several aspects of the physics of Weyl semimetal materials. Weyl semimetals have a linear dispersion near a band degeneracy at generic points in momentum space. They show Fermi arc surface states [article] which are spin polarized and do not form closed lines as regular two-dimensional fermi surfaces do. In transport experiments, a negative longitudinal magnetoresistance related to the chiral anomaly is one of their key signatures [article].

**Nodal-line semimetals **Aside from degeneracy points, electronic bands can also feature symmetry-protected degenerate along lines in momentum space. We identified PbTaSe as such a topological line node semimetal with drumhead surface states [article]. In addition, this material is an interesting superconductor with nontrivial topological band structure. It may host Majorana fermions in the vortex core localized near the surface.

**2D nonsymmorphic topological semimetals** Motivated by the electronic structure of MoTe2 and WTe2, we introduce the concept of a topological band-inverted semimetal with non-symmorphic symmetries in two spatial dimensions * *[arXiv:1604.01398]. In these materials, the resulting Dirac electrons are strongly tilted to form electron and hole pockets with a topologically protected touching point. We characterize the band structure with a non-Abelian Wilson-loop invariant. Our findings identify possible origins of the exotic electronic properties of MoTe2 and WTe2, such as their titanic magnetoresistance.