# Topology in Quantum Matter

## Higher Order Topological Phases

**Project Leader: **Titus Neupert

**Main** **Contributors**: Apoorv Tiwari, Marta Brzezinska , Mark H Fischer, Nastya Skurativska

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.

## Topological Order

**PI:** Titus Neupert

**Main Contributors: **Apoorv Tiwari, Bartholomew Andrews

The fractional quantum Hall effect is a paradigmatic state of matter exhibiting topological order. The defining feature of this exotic quantum order is a set of degenerate ground states that cannot be distinguished by any local measurement. Because of this property, the degenerate ground-state manifold could be used to store quantum information in a way resilient against perturbations. In addition, fractional quantum Hall states feature universal anyonic excitations.

Previously, we have shown that lattice models with topological band structures, so-called Chern insulators, can exhibit the same physics as the quantum Hall effect, but without the need for an external magnetic field. Here, the topologically ordered states emerge most favorably if the band is relatively flat. If the electronic interactions are strong enough, this flatness condition is, however, not necessary. The concept of a fractional Chern insulator can also be extended to fractional topological insulators with time-reversal symmetry.

In our group, we are furthermore interested in more abstract descriptions that capture the universal aspects of topologically ordered states. For two-dimensional systems, a rather complete framework has been developed using the mathematics of category theory as well as through Chern-Simons theories and bosonization. Among other research directions, we employ this methodology to understand phase transitions between topologically ordered states through a generalized notion of Bose-Einstein condensation and to study topological order on the surface of three-dimensional systems.

## Topological Quantum Materials

**Project Leader: Titus Neupert
Main Contributors: **Mark H. Fischer, Stepan Tsirkin, Arkadiy Davydov, Marta Brzezinska, 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: **Anastasiia Skurativska, Nikita Astrakhantsev

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