# Condensed Matter Theory

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“More is different” — the title of Philip Anderson’s seminal 1972 paper captures the essence of the principle of emergence that underpins all natural sciences: Phenomena that appear in systems at different levels of complexity can lead to entirely new properties, entities, and behaviors, depending on which scales the system is studied. A complex system is more than the sum of its parts.

Condensed matter systems, among which we mainly focus on the study of crystalline solids, host particularly well-understood examples for emergent phenomena, mostly at low temperatures, where quantum mechanics becomes important. In many cases they result from the interaction of mobile electrons or localized magnetic moments in the crystal. Among the types of emergent phenomena that we study are

## Topological order

Quantum statistical mechanics classifies all free particles in nature in two groups: fermions and bosons. Topologically ordered states of matter have particle-like excitations that are neither fermions nor bosons and behave entirely different. They are called anyons and are a prime example of emergence as they cannot exist without the “background” of the quantum system. Topologically ordered systems can also host string excitations, a concept even more strikingly different than all ingredients of the standard model of particle physics, which only knows particles as free objects. We study general field theories and concrete lattice models of topological order to expand our yet incomplete understanding of these phases of matter and to find material examples that realize them.

## Superconductivity

The resistivity drops to exactly zero and magnetic fields are pushed out of the system: These are the macroscopic characteristics of superconductivity — a coherent quantum-mechanical state that interacting electrons can from at low temperatures. The list of superconducting materials is ever growing, and many of them show particularly interesting forms of superconductivity, by breaking additional symmetries of the system in the superconducting state such as time-reversal symmetry or translational symmetry. Such exotic superconductors and nano-scale devices that involve superconductivity can also be used to engineer localized anyon excitations, which normally appear in topologically ordered systems. These anyons, called Majorana fermions, could be used as topological quantum memories. We study superconducting materials in order to identify examples of the exotic forms of superconductivity and develop concepts for superconducting nano-devices that support anyon excitations.

## Topological insulators and metals

In high school we learn that the kinetic energy of a particle is proportional to the velocity (or momentum) squared. This is also true for free electrons. However, in a crystal the dependence of the energy on the momentum can be severely modified. For example, in graphene, a monolayer of carbon atoms, the electron energy grows linearly with the momentum. This is reminiscent of free massless particles, like photons. Topological metals can feature a variety of energy-momentum dependences that are protected by the symmetries of the crystal and the topological properties of the electronic quantum state, yielding for instance massless Dirac fermions or Weyl fermions. Topological insulators, in contrast, possess such exotic states on their boundary, while their bulk is insulating. The conducing boundary states are robust against large classes of perturbations. We identify new materials with these exotic properties and analyze the new physical phenomena associated with their topological electronic states.

Besides their intrinsic scientific value, materials with novel emergent phenomena can help to build new computer chips, memories, solar cells, and sensors. Furthermore, they may be used as building blocks for future quantum computers and quantum memories, which work fundamentally different from the classical computers that we use today. Our research makes contact with several other fields of physics, such as quantum field theory, in particular topological and conformal field theory, optical quantum systems, as well as quantum information.