Institute for Theoretical Physics • University of Cologne

The prospect of intrinsically protected quantum computing is driving the research on topological quantum systems. Materials with a topologically nontrivial band structure are however difficult to manufacture experimentally. We recently showed that topological materials can be engineered by quite simple means, when considering a different space: superconducting junctions have the phase difference and the number of transported charges across the junction as conjugate variables. In the main part of the talk, I will present the example of a four-terminal junction whose energy spectrum exhibits Weyl singularities in the space of three independent phases, which thus play the role of quasimomenta. The nonzero Chern number can be directly probed through a transport measurement, namely, through a quantised transconductance. In the final part of the talk, I will provide an outlook on future directions. For one, electric circuits may be used to study topological phase transitions specific to open systems out of equilibrium, and secondly, they may exhibit a so far unexplored connection between topology and quantum thermodynamics.

In this talk I will discuss recent experimental and theoretical advances combining one dimensional topological nanowires into small mesoscopic islands, pushing Majorana physics into the Coulomb blockade regime, and creating new quantum phases governed by correlations. I will elucidate that transport through such islands gives access to a rather unstudied type of quantum impurity physics, where the electrons in the leads combine with the Majorana fermions in the island to yield effective bosonic charge carriers. This gives rise to novel non-equilibrium transport behavior. Pushing these islands using a gate volatge into charge degeneracy points presents very interesting regimes allowing to probe strongly correlated Majorana fermions at short time scales and at high energy scales. In addition, creating arrays of such islands may form a highly convenient platform for topological quantum computing schemes based on the surface code. Here, electron transport can be used for quantum measurements and manipulation of the stored information in the Majorana qubits.

The grand canonical ensemble lies at the core of quantum and classical  statistical mechanics. A small system thermalizes to this ensemble while exchanging heat and particles with a bath. A quantum system may exchange  quantities represented by operators that fail to commute. Whether such  a ​ ​ system thermalizes and what form the thermal state has are questions  about ​ ​ truly quantum thermodynamics. Here we investigate this thermal  state from ​ ​ three perspectives. First, we introduce an approximate microcanonical ensemble. If this ensemble characterizes the system- and-bath composite, tracing out the bath yields the system's thermal  state. This state is expected to be the equilibrium point, we argue,  of typical dynamics. Finally, we define a resource-theory model for thermodynamic exchanges of noncommuting observables. Complete passivity ​-- ​ the inability to extract work from equilibrium states ​ ​ -- ​, ​ implies  the thermal state's form, too. Our work opens new avenues into  equilibrium in the presence of quantum noncommutation. ​ [Based on arXiv:1512.01189]​

Topological states of quantum matter such as topological insulators and superconductors have been an active field of research in physics for many years. The recent experimental progress on their realization with ultracold atomic gases raises natural questions about the notion of topological quantum matter far from thermal equilibrium. In this talk, we present our theoretical findings in this context. We first discuss the non-equilibrium Hall response of a system initialized in a topologically trivial state before its Hamiltonian is ramped into a Chern insulator phase, comparing the coherent dynamics with effects of dephasing. In the second part of the talk, we discuss non-equilibrium properties of driven systems. In particular, we show how a perfect spin momentum locking, so far only known from edge states of two-dimensional topological insulators can emerge in the stroboscopic dynamics of one-dimensional Floquet lattice systems. Finally, we discuss dynamically defined topological quantum numbers that have no non-equilibrium counterpart, and report on their very recent experimental observation.

An interesting topic in the theory of quantum phase transitions is that of the transition between a classically ordered state, such as a Néel state, and a quantum ordered state, such as a valence bond crystal. On the classically ordered side of the transition, we have a well developed understanding based on the spontaneous breaking of a continuous symmetry and the occurrence of associated Goldstone modes (spin waves). On the quantum ordered side, things are rather less clear. Even the description of the quantum ordered state itself is not straightforward, since it is not a saddle-point of the usual spin-coherent-state path integral. In this talk, I shall present an approach designed to address this problem by constructing the path integral over matrix product states rather than spin coherent states. This allows both classically ordered states (which are matrix product states of bond dimension 1) and quantum ordered states (which are matrix product states of higher bond dimension) to be captured on an equal footing. I shall use this approach to show that, for at least one example of such a transition, a Landau-Ginzburg-Wilson-type description can be given, where the ‘order parameter’ is a field representing the nearest-neighbour entanglement of the spins. The talk is designed to be entirely self-contained: in particular, no prior knowledge of matrix product states or quantum ordered phases is assumed. The work about which I shall speak was undertaken in collaboration with Andrew Green (University College London), Jonathan Keeling (St Andrews), and Steve Simon (Oxford), under the auspices of the TOPNES programme. Most of the details can be found at https://arxiv.org/abs/1607.01778.

Spin-polarized scanning tunneling microscopy at low temperature (8 K) and in high magnetic fields (6 T) is a powerful technique to investigate magnetic properties of individual nanoscale objects ranging in size form single atoms to several thousand atoms. I focus on the magnetization reversal and the spin-dependent electronic properties of bilayer Co, Fe-decorated Co and Fe islands on Cu(111). We find a novel noncollinear, helical magnetic order in the Fe islands, which is identified by a magnetic stripe contrast with a period of 1.28 nm. The high spatial resolution of the scanning tunneling spectroscopy reveals the significance of structural and electronic relaxation for the magnetic anisotropy and the spin-dependent transport properties of single islands. In an outlook I present first results from our new STM, which operates at 0.3 K in a vector magnetic field, on superconductivity of Pb monolayers and on the proximity effect.