We work on a range of different topics, currently mainly focusing on mechanisms and properties of unconventional and topological superconductors. We use a wide range of theoretical techniques, from analytically tractable Green’s function methods to large-scale numerical calculations of inhomogeneous superconducting systems and strongly correlated materials, also including complementary ab-initio calculations. Below are shorter descriptions of several of the currently funded projects.

### Topological superconductors for robust quantum computation

Topological superconductors are a newly discovered class of materials with features uniquely advantageous for quantum computing. They have lately generated an immense amount of attention due to the possibility of them having effective Majorana fermions at surfaces, vortices, and other defects. Approximately one can say that a Majorana fermion is half an electron, or more accurately, in a system with Majorana fermions the wave function of an electron has split up into two separate parts, see figure below. This non-local property of two Majorana fermions can be used for exceptionally fault-tolerant quantum computing. A quantum computer uses the quantum nature of matter to represent data and perform calculations and can be exponentially faster than any classical supercomputer. However, quantum systems are generally extremely sensitive to disturbances and we are still far from being able to construct useful quantum computers. Topological superconductors with Majorana fermions avoid this extreme sensitivity by using the non-local nature of the Majorana fermions, which automatically make them very robust.

*Single electronic state at zero energy divided up into two spatially separated Majorana fermions: one at the edge of the sample and one in a superconducting vortex core. States at non-zero energy are not spatially split and form regular electronic levels. Adapted from Björnson and Black-Schaffer, Phys. Rev. B 88, 024501 (2013).*

The goal of this project is to theoretically discover new and experimentally viable TSCs with Majorana fermions and to determine the properties of the Majorana fermions, aiming at future topological quantum computation applications. In this project we focus both on the currently most promising proposals for TSCs found in superconducting hybrid structures of well-known spin-orbit coupled materials and on discovering new experimentally feasible TSCs in graphene and related materials.

### New topological superconductors using flat bands, orbital effects, and quantum critical points

This project aims at theoretically discover and characterize entirely new quantum states of matter by novel combinations of global topology and local superconducting order. More specifically, the project takes a new and systematic approach to finding new topological superconductors, by (A) starting from complex but common normal electronic structures and by (B) significantly expanding the class of known superconductors spontaneously breaking time-reversal symmetry. In part (A) we study materials with normal-state flat energy bands, emerging due to non-trivial topology, and materials with multiple low-energy orbitals, which each harbor large possibilities for topological superconductivity. In part (B) we focus on two very promising new possibilities for time-reversal symmetry breaking superconductivity; as an intermediary phase in the quantum critical region of nodal superconductors, such as the example shown in the figure below, and in flat band surface states of nodal-line superconductors. Specific targets range from achieving high-temperature superconductivity in topological flat band materials, including different graphene-like systems, to finding new time-reversal symmetry breaking chiral states in strongly correlated materials and new exotic phenomena in nematic multiorbital superconductors.

*Phase diagram for spin-singlet superconductivity in the hyperhoneycomb lattice as a function of doping (μ) and coupling constant (J). In-between a nodal (blue) and a fully gapped (red) phase, a time-reversal symmetry breaking phase (green) appears. Adapted from Schmidt, Bouhon, and Black-Schaffer, Phys. Rev. B 94, 104513 (2016).*

### New mechanisms and materials for odd-frequency superconductivity

Odd-frequency superconductivity is a very unique superconducting state that is odd in time or, equivalently, frequency, which is opposite to the ordinary behavior of superconductivity. It has been realized to be the absolute key to understand the surprising physics of superconductor-ferromagnet (SF) structures and has also enabled the whole emerging field of superconducting spintronics. In this project we discover and explore entirely new mechanisms and materials for odd-frequency superconductivity, to both generate a much deeper understanding of superconductivity and open for entirely new functionalities. Importantly, we generalize and apply our initial discoveries of two new odd-frequency mechanisms, present in bulk multiband superconductors and in hybrid structures between topological insulators and conventional superconductors, respectively. In both cases odd-frequency superconductivity is generated without any need for ferromagnets or interfaces, completely different from the situation in SF structures.

The goal is to significantly expand the concept and importance of odd-frequency superconductivity to a very wide class of materials, ranging from multiband, bilayer, and nanoscale superconductors to topological and Weyl superconductors. One example is the chiral and multiband superconductor Sr2RuO4 as illustrated in the figure below. We also aim to establish the connection between topology and odd-frequency pairing, which needs to be addressed in order to understand topological superconductors, as well as incorporate new materials and functionality into traditional SF structures. To achieve these goals we develop a novel methodological framework for large-scale and fully quantum mechanical studies with atomic level resolution, solving self-consistently for the superconducting state and incorporating quantum transport calculations.

*Chiral p+ip superconductivity in Sr2RuO4 divided into intraorbital contributions within the three low-energy orbitals and even- and odd-frequency interorbital pairing. **Adapted from Komendova and Black-Schaffer, Phys. Rev. Lett. 119, 087001 (2017).*

### Functional Dirac materials

Dirac Materials (DM), with low energy properties controlled by excitations in the Dirac nodes, have proved to hold strong potential for basic science and applications. These materials, including unconventional, high Tc (d-wave) superconductors, topological insulators, and graphene, have been at the forefront of modern condensed matter for the last two decades. The d-wave state in cuprate oxide superconductors exhibits strong correlations, non-Fermi liquid behavior, and spatially modulated states to name a few of the unconventional properties. Recent theoretical prediction and discovery of topological insulators is another example of a major development in the field of Dirac materials. Topological insulators constitute a new state of quantum matter that has unique properties that lead to the prediction of new effects and possible applications. A topological insulator is a material that is a bulk insulator/semiconductor but has helical Dirac states at the surface. As a consequence, electrons traveling in opposite directions have opposite spin orientations and there exists therefore a dissipation free spin current circulating on the surface of the material. These topological states can host new particles like Majorana fermions and the display new properties such as axion dynamics. Nodes are universally responsible for the emergent states and control the topological protection of Dirac Materials. Opening of an excitation gap of the Dirac node states allows one to functionalize, manipulate, and drastically alter optical and electric, magnetic, structural, and thermal responses. This collaborative project between several theory and experimental groups aims at achieving new functionalities by proper design of nano-structures, films, and interfaces of DMs. To enable these advances we will build on our proven expertise in the growth, nanostructuring, characterization, and theoretical modeling of these materials.