Picture of Kai Schmidt

research interests

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 many-body 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 many-body 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.

topologically ordered quantum systems

Topologically-ordered quantum phases represent a class of two-dimensional ground states where elementary properties such as the ground-state 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 non-Abelian 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 quasi-particle picture for Abelian as well as non-Abelian anyons. This allows the characterization of topological quantum phase transitions. Currently we are working on the robustness of three-dimensional topological order including fracton phases.

Phase diagram

quantum robustness and phase transitions of the 3D Toric Code in a field

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 ground-state phase diagram with first and second-order 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.

[1] SciPost Phys. 6, 078 (2019)

Phase diagram

topological phase transitions in non-Abelian string-net models

We examine the zero-temperature phase diagram of the two-dimensional Levin-Wen string-net model with Fibonacci (see left figure) and Ising anyons in the presence of competing interactions. Combining high-order series expansions and exact diagonalizations, we find that the non-Abelian topological phases can be separated from nontopological phases by second-order quantum critical points, the positions of which are computed accurately. The evaluation of critical exponents suggests unusual universality classes.

[1] Physical Review Letters 110, 147203 (2013)
[2] Physical Review B 89, 201103(R) (2014)

Phase diagram

toric code in a magnetic field

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 second-order 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.

[1] Physical Review B 79, 033109 (2009)
[2] Physical Review B 80, 081104 (2009)
[3] Physical Review Letters 106 107203 (2011)

quantum magnets with long-range interactions

There are many important quantum systems where long-range interactions are relevant. Important examples are dipolar interactions between spins in spin-ice materials giving rise to emergent magnetic monopoles, effective long-range magnetic interactions between zigzag edges in graphene, and especially trapped cold-ion systems in quantum optics for which the nature of interactions can be varied flexibly. The paradigmatic model in this context is the transverse-field Ising model with algebraically decaying Ising interactions.


quantum criticality of 2d quantum magnets with long-range interactions

We study the critical breakdown of two-dimensional quantum magnets in the presence of algebraically decaying long-range interactions by investigating the transverse-field 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 high-order series for the one-particle excitations in the high-field quantum paramagnet. We find that the unfrustrated systems change from mean- field to nearest-neighbor universality with continuously varying critical exponents. In the frustrated case on the square lattice the system remains in the universality class of the nearest-neighbor model independent of the long-range nature of the interaction, while we argue that the quantum criticality for the triangular lattice is terminated by a first-order phase transition line.

[1] Physical Review Letters 122, 017203 (2019)
[2] Physical Review B 94, 075156 (2016)

interacting electronic systems

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.

Phase diagram

spin liquid on the triangular lattice

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 non-magnetic 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 non-magnetic phases in a quantitative fashion. Recently, we derived the relevant low-energy model for the Hubbard model on the isotropic triangular lattice and we showed that it comprises a non-magnetic intermediate phase. These are the first quantitative calculations which suggest the existence of a spin-Bose metal in a two-dimensional model of interacting electrons.

[1] Physical Review Letters 105, 267204 (2010)
[2] New Journal of Physics 14, 115027 (2012)
[3] Chiral spin liquids of the SU(3) Hubbard model on the triangular lattice: arXiv:1802.03179

effective models - development of theoretical tools

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 low-energy models in the thermodynamic limit up to high orders in a small parameter. The knowledge of a high-order series expansion allows in many cases to determine non-perturbative properties like critical points and critical exponents.

Phase diagram

graph-based continuous unitary transformations (gCUTs)

Non-perturbative 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 non-perturbative scheme for gapped quantum phases. The essential idea is to combine continuous unitary transformations and graph theory.

[1] European Physics Letters 94, 17004 (2011)
[2] New Journal of Physics 14, 115027 (2012)

Phase diagram

A generalized perspective on non-perturbative linked cluster expansions

We identify a fundamental challenge for non-perturbative 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 graph-based 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)

topics and methods

  • Topological order - Topological phase transitions - Anyonic quasi-particles - Topolgical quantum computation
  • Frustrated magnetic and electronic systems
  • Low-dimensional quantum magnetism
  • Dynamical correlations - Spectroscopy
  • Bosonic Phases - Supersolids
  • Cold atoms in optical lattices
  • Mott physics of Hubbard models - Spin liquids
  • Organic superonductors
  • High-temperature superonductors
  • Cluster phases - Measurement based quantum computations
  • Derivation of effective models
  • Continuous unitary transformations
  • Series expansion techniques
  • Quantum Monte Carlo simulations