One of the central paradigms in modern science is to focus on collective behaviour and organising principles. In many scientific disciplines ranging from natural to social sciences it is a key issue to understand the complex behaviour of many entities compared to the properties of a single one, e.g., besides many others, in studying the climate of the earth, our brain, in investigating swarm intelligence, in predicting changes in our society or trends in financial markets. In short, this can be expressed by "more is different".
The collective behaviour of matter is therefore one of the most important topics of modern science. Understanding it is crucial in basic research, as it holds the key to a variety of correlated manybody states  realised for example in spin liquids or superconductors. At the same time, it is clear that this understanding forms the basis for many technological applications which define the modern era. In particular, it is decisive to gain a systematic understanding of collective phenomena in order to identify fundamentally new behaviour.
My general research interests are emergent collective phenomena of quantum manybody systems which are relevant for condensed matter physics, atomic physics, and quantum information. Interactions between various degrees of freedom lead to novel states of matter with fascinating properties and interesting collective properties. Furthermore, it is an important aspect of my research to develop modern theoretical tools to treat microscopic models of correlated systems in order to predict and to pinpoint emergent quantum phenomena in real systems.
Topologicallyordered quantum phases represent a class of twodimensional ground states where elementary properties such as the groundstate degeneracy depend on the topology of the system. Elementary excitations above such a ground state carry fractional quantum numbers and obey an exotic statistics which is neither fermionic nor bosonic. Such particles are called anyons where one distinguishes Abelian and nonAbelian anyons. The latter are the central objects for the field of topological quantum computation.
The concept of topological order is theoretically and experimentally relevant for the fractional quantum Hall effect, for the class of topological insulators, for graphene, and for quantum magnetism. In recent years we have for the first time developped a true quasiparticle picture for Abelian as well as nonAbelian anyons. This allows the characterization of topological quantum phase transitions. Currently we are working on the robustness of threedimensional topological order including fracton phases.
We study the robustness of 3D intrinsic topogical order under external perturbations by investigating the paradigmatic microscopic model, the 3D toric code in an external magnetic field. Exact dualities as well as variational calculations reveal a groundstate phase diagram with first and secondorder quantum phase transitions. The variational approach can be applied without further approximations only for certain field directions. In the general field case, an approximative scheme based on an expansion of the variational energy in orders of the variational parameters is developed. For the breakdown of the 3D intrinsic topological order, it is found that the (im)mobility of the quasiparticle excitations is crucial in contrast to their fractional statistics.
References
We examine the zerotemperature phase diagram of the twodimensional LevinWen stringnet model with Fibonacci (see left figure) and Ising anyons in the presence of competing interactions. Combining highorder series expansions and exact diagonalizations, we find that the nonAbelian topological phases can be separated from nontopological phases by secondorder quantum critical points, the positions of which are computed accurately. The evaluation of critical exponents suggests unusual universality classes.
References
We investigated the stability of the topological phase of the toric code model in the presence of a uniform magnetic field. The corresponding phase diagram is shown on the left [3]. We find that when this perturbation is strong enough, the system undergoes a topological phase transition whose first or secondorder nature depends on the field orientation. When this transition is of second order, it is in the Ising universality class except for the special red line on which the criticality is different, since charges and fluxes condense simultaneously.
ReferencesThere are many important quantum systems where longrange interactions are relevant. Important examples are dipolar interactions between spins in spinice materials giving rise to emergent magnetic monopoles, effective longrange magnetic interactions between
zigzag edges in graphene, and especially trapped coldion systems in quantum optics for which the nature of interactions can be varied flexibly. The paradigmatic model in this context is the transversefield Ising model with algebraically decaying Ising interactions.
We study the critical breakdown of twodimensional quantum magnets in the presence of algebraically decaying longrange interactions by investigating the transversefield Ising model on the square and triangular lattice. This is achieved technically by combining perturbative continuous unitary trans formations with classical Monte Carlo simulations to extract highorder series for the oneparticle excitations in the highfield quantum paramagnet. We find that the unfrustrated systems change from mean field to nearestneighbor universality with continuously varying critical exponents. In the frustrated case on the square lattice the system remains in the universality class of the nearestneighbor model independent of the longrange nature of the interaction, while we argue that the quantum criticality for the triangular lattice is terminated by a firstorder phase transition line.
The Hubbard model represents one of the most studied microscopic models in condensed matter physics. On a very simple level it describes the interplay between the kinetics and the Coulomb interaction of electrons in solids. Despite its simplicity, it displays a very rich phase diagram and it is very important for the microscopic modelling of many condensed matter materials.
One particular intriguing aspect is the interplay between strongly correlated fermionic degrees of freedom and geometrical frustration of the underlying lattice topology. In the past years there are more and more evidences for the existence of insulating but nonmagnetic intermediate phases with exotic properties at half filling. Experimentally, this is for example relevant for a class of organic superconductors which can be understood in terms of a Hubbard model on an anisotropic triangular lattice. We developped efficient tools to derive effective spin models which describe the Mott phase including the fascinating nonmagnetic phases in a quantitative fashion. Recently, we derived the relevant lowenergy model for the Hubbard model on the isotropic triangular lattice and we showed that it comprises a nonmagnetic intermediate phase. These are the first quantitative calculations which suggest the existence of a spinBose metal in a twodimensional model of interacting electrons.
For many problems we have applied and develop the method of perturbative continuous unitary transformations (pCUTs) as well as other linked cluster expansions. In essence, such techniques provides analytical effective lowenergy models in the thermodynamic limit up to high orders in a small parameter. The knowledge of a highorder series expansion allows in many cases to determine nonperturbative properties like critical points and critical exponents.
Nonperturbative extensions of the pCUT method would be very helpful. In practice, such extensions are often problematic since uncontrolled truncations of the flow equations lead to divergencies of the flow. Recently, we developped a robust nonperturbative scheme for gapped quantum phases. The essential idea is to combine continuous unitary transformations and graph theory.
References
We identify a fundamental challenge for nonperturbative linked cluster expansions (NLCEs) resulting from the reduced symmetry on graphs, most importantly the breaking of translational symmetry, when targeting the properties of excited states. A generalized notion of cluster additivity is introduced, which is used to formulate an optimized scheme of graphbased continuous unitary transformations (gCUTs) allowing to solve and to physically understand this fundamental challenge. Most importantly, it demands to go beyond the paradigm of using the exact eigenvectors on graphs.
Reference: European Physics Letters 110, 20006 (2015)













