# Research

## 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.