# Topology in Quantum Matter

## Novel Topological Phases

**Project Leader: **Titus Neupert, Luka Trifunovic

**Main** **Contributors**: Mark H Fischer, Glenn Wagner, Martina Soldini

Our group studies novel forms of topological phases that go beyond the ten-fold way of topological insulators. Important examples pioneered in our group include higher-order topological phases and the concept of delicate topology.

The bulk-boundary correspondence is central to topological phases of matter: a nontrivial bulk implies protected boundary modes. Examples of such boundary modes are the chiral edge states of a quantum Hall system or the Dirac surface electrons of a three-dimensional topological insulators. Higher-order topology refers to a more intricate bulk-boundary correspondence: instead of surface and edge states, higher-order topological matter features topological edge and corner modes.

In a collaboration with experimental groups, we have shown that elementary bismuth features higher-order topology and we conjectured the same for SnTe. Besides seeking to identify more bulk materials with higher-order topology, we study the appearance of corner modes in two-dimensional systems and the interplay of higher-order topology and surface topological order.

We introduced a family of topological insulators that would be considered trivial in the standard paradigm of topological insulators. In particular, these novel insulators have a symmetric, exponentially-localized Wannier representation. However, each Wannier function cannot be localized to a single primitive unit cell in the bulk. We showed that such *multicellular topology *is neither stable, nor fragile, but *delicate*, in other words the topology is unstable towards addition of trivial bands as either valence or conduction band.

## Topological States in Artificial Lattices

**PI:** Titus Neupert

**Main Contributors:** Glenn Wagner, Martina Soldini

Artificial Lattices, not naturally occurring but created in the lab, can exhibit novel and intriguing physics. A particularly prominent example of recent years that our group studies is twisted bilayer graphene, which, for the right ‘magic’ angles, shows both correlated insulating behavior and superconductivity.

Novel physics can emerge when placing atoms on a material in a periodic fashion. Decorating a single layer of graphene with non-magnetic transition metal ions, for examples, results in a band structure with many features similar to the ones found in magic-angle twisted bilayer graphene, namely flat bands, responsible for the importance of correlations, and non-trivial topology. Depositing magnetic atoms on the surface of a (conventional) superconductor, on the other hand, can lead to the emergence of topological bands formed by the resulting Shiba bound states.

## Topological Quantum Materials

**Project Leader: Titus Neupert
Main Contributors: **Mark H. Fischer,Songbo Zhang, Nikita Astrakhantsev, Xiaoxiong Liu

It is our goal to relate the concepts of topological band theory and the phenomena enabled by strong electron correlations as well as magnetic order to the realm of crystalline materials. To achieve this, we develop advanced numerical tools to analyze and predict the properties of materials and we work closely with experimental groups to corroborate and analyze their data.

Based on first-principles calculations, we are developing a software-package for the high-accuracy and efficient computation of response functions that arise from Berry-phase properties of Bloch states, such as the anomalous Hall conductivity, the Nernst effect, and various nonlinear responses. We also employ a variety of tools to analyze topological magnetic phases, starting from density functional theory all the way to slave-boson calculations for Kondo lattice systems.

The range of materials we target include topological insulators, Weyl and other topological semimetals, Moire heterostructures of Van-der Waals materials, and topological magnets.

## Unconventional and Topological Superconductivity

**Project Leaders: **Mark H. Fischer, Titus Neupert

**Main Contributors: ** Nikita Astrakhantsev, Martina Soldini

Superconductivity takes a particular role among correlated phases of quantum matter in that quantum physics is dramatically manifested a macroscopic ‘classical’ observable, the (vanishing) resistivity. The Department of Physics at the University of Zurich has a strong tradition in the study of superconductors: the Nobel laureate Karl Alex Müller, who discovered high-temperature superconductivity in the Cuprates, is an emeritus and Andreas Schilling holds the world record for the highest transition temperature under ambient pressure.

In our theoretical research, we are primarily interested in the interplay of superconducting order with topological features of materials, crystalline symmetries and strong spin-orbit coupling. For example, we are seeking conditions under which so-called Majorana bound states appear - exotic quasiparticles that may be useful for quantum computing and quantum