Our research spans and combines several contemporary areas in modern physics, such as unconventional superconductivity, topological matter, and strongly correlated systems. Much of our current focus is on mechanisms and properties of unconventional, topological, or inhomogeneous superconductivity. We often use detailed low-energy effective models to study the properties of the superconducting state in many different novel materials and superconducting hybrid structures.
We use a wide range of theoretical techniques, from analytically tractable Green’s function methods to large-scale numerical calculations of inhomogeneous systems and strongly correlated materials, also including complementary ab-initio calculations. Below are short descriptions of our present larger activities.
Using Nanoscale Inhomogeneity to Enhance Superconductivity
Superconductivity is a uniquely quantum mechanical phenomenon that despite decades of intense research is still very hard to control. It is heavily dependent on the density of states (DOS) around zero energy, but this DOS tunability has so far remained largely untapped. This activity aims to theoretically create and enhance superconductivity by producing large DOS peaks at zero energy using nanoscale inhomogeneity. A rare, not yet understood, example is twisted bilayer graphene, an all-carbon material not assumed to be superconducting. Here small twist angles produce an inhomogeneous moiré structure hosting large zero-energy DOS peaks that have recently been shown to create superconductivity.
Within this activity we aim to understand, as well as enhance, superconductivity in moiré structures in both graphene and topological insulators. Importantly, we also work on finding other new and distinct mechanisms of nanoscale inhomogeneity and zero-energy DOS peaks enhancing superconductivity, such as disorder or quasicrystal structures. We also study how zero-energy DOS peaks may create a superconducting phase crystal in different superconductors, generalizing findings from high-temperature cuprate superconductor surfaces, see figure. Due to its nanoscale phase modulations and spontaneous supercurrents, the phase crystal dramatically enriches the superconducting properties. Taken together, this project aims to create an entirely new, inhomogeneous, landscape for superconductivity.
Superconducting phase crystal at the [110] cuprate high-temperature superconductor surface visualized by plotting in color scale the phase sin(theta) of the d-wave order parameter. Later dimensions are given in terms of the lattice constant. Adapted from Chakraborty, Löfwander, Fogelström, and Black-Schaffer, npj Quantum Mater. 7, 44 (2022)
Using non-Hermiticity for Enhanced Superconductivity
This activity aims to build a theoretical framework for creating, enhancing, and understanding electronic ordering, especially superconductivity, in open systems due to their non-Hermitian (NH) effects, thereby launching a new open era of ordering. A central theme in material physics is emergent behavior, in particular emergence of electronic ordering such as magnetism and superconductivity. Ordering requires interactions between electrons and generally a large density of states (DOS) at zero energy. A material is usually also not just by itself, but coupled to an outside environment, creating an open system. However, such openness is usually thought of as destroying quantum behavior, including ordering. This activity aims to change this view.
Many effects of openness can be treated by introducing NH terms in the otherwise normal, Hermitian, description. One central NH effect is exceptional points (EPs), where not only energies are degenerate but the wavefunctions also become parallel. EPs at zero energy may thus be able to generate the necessary DOS for ordering and fundamentally new behavior due to their unique wavefunctions. As a proof-of-principle, we have already demonstrated that NH effects in the 1D Kitaev model, the prototype 1D topological superconductor with Majorana fermions (see also below), can dramatically enhance superconductivity, see figure.
Superconducting gap induced in a NH 1D Kitaev model (Hatano-Nelson-Kitaev model) as function of applied superconducting order parameter Δ and NH term Γ in units of the nearest neighbor hopping parameter t. (Left) The induced superconducting gap d is of the order 1 even if the superconducting order Δ is many magnitudes larger, showing an exceptional enhanced. (Right) Local density of states (LDOS) along the chain showing well-localized Majorana states despite vanishingly small superconducting order Δ. Adapted from Arouca, Cayao, and Black-Schaffer, Phys. Rev. B 108, L060506 (2023)
We are pursuing several distinct goals within this project. One is to establish and understand exceptionally enhanced ordering, especially superconductivity, created by EPs, as well as discover new emergent orders. Another is to utilize the quantum geometric tensor to understand superconductivity, topology, and phase transitions in systems with NH-induced ordering. Finally, we aim to use disorder-induced NH effects to expand Fermi surfaces and thus promote ordering. Each part has the potential for substantial impact on realizations and understanding of emergence in nature, while at the same time together accomplishing the overall goal of launching a new open era of ordering.
Quantum Geometry and Flat Bands towards Room-Temperature Superconductivity
The quest for room-temperature superconductivity is considered a holy grail of modern physics. Not only would it contribute to solving many of today’s energy challenges, but also facilitate the integration of quantum computers into society. The most promising material so far are the high-temperature cuprate superconductors. However, their strong electron correlations and lack of tunability have hindered progress. In this activity we are inspired by twisted bilayer graphene, where at certain magic twisting angles a moiré pattern with flat bands is created, which in turn generates both strong electron correlations and superconductivity. The activity aims to explore how similar moiré patterns and the quantum metric they carry can be used to tune and design the band structure and create flat bands in both high-temperature cuprate superconductors and graphene. By combining these two material platforms, we aim to increase our understanding of the principles behind high-temperature superconductivity and promote superconductivity at higher temperatures, ultimately aiming to achieve room-temperature superconductivity.
This activity is a joint experiment-theory project with Floriana Lombardi, Sergey Kubatkin, Ulf Gran (Chalmers), and Johannes Hofmann (Gothenburg).
New Mechanisms and Materials for Odd-Frequency Superconductivity
Odd-frequency superconductivity is a 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. Within this activity we aim to 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 in this activity 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.
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).
Dynamic Quantum Matter
Quantum dynamics as a material design principle is an emerging paradigm in condensed matter physics. The answers to basic questions about the nature of the orders, dynamics, coherence, and dissipation of dynamic quantum matter are still unknown. Within this activity, we aim at addressing some of the most basic questions about dynamic quantum matter: the nature of the orders in the time domain of quantum mechanics, the emergence of entangled orders, and the role of dissipation on quantum orders.
Our goal is to predict, engineer, and probe non-equilibrium phases that have no equilibrium analogue. To accomplish these goals, we investigate external drivers that stabilize existing states by enhancing their robustness against noise and ultimately also investigate the non-equilibrium states that do not have analogues in equilibrium, as well as study energy dissipation and localization, including through non-Hermitian effects, that play fundamental roles in limiting and enabling non-equilibrium states. Through a collaborative effort we aim at developing new theoretical models and methods able to understand and accurately predict the creation and control of novel dynamical quantum states of matter. Our theoretical efforts are supplemented by key experimental measurements using ultrafast, coherent photon probes with energies from the meV (terahertz) to the keV (x-rays) range.
This activity is a joint theory-experiment project together with the research groups of Alexander Balatsky (Nordita), Jens Bardarson (KTH), Emil Bergholtz and Stefano Bonetti (Stockholm).
New Frontiers for Topological States of Matter
This activity aims to theoretically discover and characterize entirely new quantum states of matter by novel combinations of global topology and local superconducting order. The discovery of a vast amount of different topological states of matter has completely revolutionized physics the last ten years. The defining property of topological matter is the global non-trivial topology of the electronic structure. This is fundamentally different from the traditional way of classifying matter, where local order parameters instead are the key concept, such as the magnetization in a magnet. In topological superconductors these two disparate views of matter naturally merge, as they have both a global non-trivial topology and a local (superconducting) order.
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).
We are pursuing several systematic approaches to find new topological superconductors, using different but interlinked concepts. These range from focusing on using complex but still common normal-state electronic structures, such as flat energy dispersions and multiorbital configurations, to exploring how time-reversal symmetry can be spontaneously broken in known superconductors and thus generate non-trivial topology. We are also using nanostructuring to achieve both novel phases and sculpture artificial boundaries in topological matter.