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The hexagonal manganites play host to a range of properties from the technologically relevant – ferroelectricity, frustrated magnetism, magnetoelectric coupling, multiferroism, functional domains and domain walls – to being a model system for testing high- and low energy theories. Recent experiments using piezoresponse force microscopy (PFM), high angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) and second harmonic generation (SHG) revealed an intriguing cloverleaf pattern caused by topological defects.
The formation of topological defects is central to understanding both the functional and exotic properties in these materials. The Kibble-Zurek mechanism, which remains an open question in cosmology, predicts a scaling law for the number of defects formed during a phase transition. Herein we pursue a complementary line of questioning by combining symmetry analysis, first-principles calculations, and phenomenological models. We show that hexagonal manganites form one-dimensional topologically-protected vortices. We then apply the Kibble-Zurek theory of topological defect formation to the hexagonal manganites to quantitatively corroborate our predictions arising from first-principles electronic structure theory with recent literature data. Finally we explore the crossover out of the Kibble-Zurek regime [1].
We next apply the developed topological description of hexagonal manganites to explain the formation of dual domains and domain walls in InMnO3. Again using a combination of theory and calculations, we give a universal description of topological defects in both ferroelectric and non-polar domains and predict the resulting domain wall structures.
Finally, we propose a new class of materials with the hexagonal manganite structure to test the Hubbard Hamiltonian. We take a top-down approach to design a material ab initio with a half-filled non-degenerate band. We then characterize the electronic properties of the candidate materials, demonstrating Mott-insulating behavior and potential exotic superconductivity [2].

_{[1] S. M. Griffin et al., Phys. Rev. X 2 (4), 041022 (2012)
[2] S. M. Griffin et al., Phys. Rev. B 93, 075115 (2016)}

We theoretically investigate different aspects of dynamic mesoscopic conductors with the ultimate goal of contributing to the development of quantum technologies. Our contributions can be grouped into three domains:

I) Controlled emission and entanglement of individual electrons:
We propose experiments to generate entanglement using single-electron sources. Investigating the noise properties of coherent single-electron excitations, we show that a single electron partitioned at a beam-splitter is entangled, making such systems potentially useful for quantum computation.

II) Statistics of current fluctuations and electron waiting times for phase-coherent conductors:
We develop a novel theory for joint electron waiting times and use this theory to describe single- electron sources. Furthermore, we connect the negative values that arise in full counting statistics (FCS) to an interference effect, showing that the FCS can be used to detect non-classical behavior.

III) Thermal machines at the nano-scale:
We propose two heat engines, one relying on the wave-nature of electrons, the other on the particle- nature of photons. The latter shows an intriguing separation of heat and work. We further propose a refrigerator that exhibits coherence-enhanced cooling, outperforming any classical analogue.

Topological quantum state represents a new class of materials with unique ground-state protected by topological invariant, which is not only fundamentally important in condensed-matter physics but also offers a promising opportunity for realizing the energy-saving electronics and quantum computer. Recently, topological classification of quantum phases has been extended from non-interacting insulators to strongly correlated insulators, and further to semimetals. Using state-of-the-art angle-resolved photoemission spectroscopy at Swiss Light Source, we have proved direct spectroscopy evidences of new topological quantum states, including:

Direct observations of the metallic surface state and its helical spin texture in SmB6 as evidences of the topological Kondo insulator [1].
Direct observations of bulk Weyl cones, surface Fermi arcs and their correspondence in transition-metal monophosphides as experimental evidences of the Weyl semimetal [2].

_{[1] B. Q. Lv*, N. Xu*, H. M. Weng* et. al., Observation of Weyl nodes in TaAs. Nature Physics 11, 724-727 (2015). (*contributed equally); N. Xu et. al., Observation of Weyl nodes and Fermi arcs in TaP. Nature Communications 7, 11006 (2016); N. Xu et. al., Distinct evolutions of Weyl fermion quasiparticles and Fermi arcs with bulk band topology in Weyl semimetals. Physical Review Letters 118, 106406 (2017).
[2] N. Xu et. al., Surface and bulk electronic structure of the strongly correlated system SmB6 and implications for a topological Kondo insulator. Phys. Rev. B 88, 121102(R) (2013); N. Xu et. al., Direct observation of the spin texture in SmB6 as evidence of the topological Kondo insulator. Nature Communications 5, 4566 (2014).}

Catalysts, which are often metal nanoparticles, are of vital importance in large-scale production of fuel and chemicals. Fabrication of catalytic model systems in a controlled manner with well-defined shape, size and position of catalytic particles as well as their study at the single particle level is necessary to gain deeper insight into chemical mechanisms in catalysis. We develop novel model surfaces using state-of-the-art top-down nanofabrication techniques such as extreme ultraviolet (EUV) lithography [1] and electron beam lithography (EBL) [2,3] to achieve nanometer precision over particle size and its positioning. Step-and-repeat exposures using EUV-achromatic Talbot lithography, which is robust to individual defects on the transmission mask, has enabled very high throughput fabrication of nanoparticles down to 15 nm feature size and 100 nm pitch, spread over an area of many cm2 in less than a few minutes [1].

We developed a new strategy to combine top-down nanofabrication together with X-ray photoemission electron microscope (XPEEM) at the Swiss Light Source to study catalytic nanoparticles [2,3]. EBL is used to achieve well-defined metal nanoparticles down to six nanometers and in-situ visualization of chemical action is done at the single nanoparticle level. This development is used to investigate the mechanism of hydrogen spillover, a critical phenomenon in heterogeneous catalysis [3]. Hydrogen spillover is the surface migration of hydrogen atoms from the catalyst onto and away from the catalyst support. Discovered in 1960s, it has since been widely controversial subject and evidence of its occurrence is disputed. Direct experimental proof of its existence does not exist due the lack of well-defined model systems and the inability to observe the effect directly. We employ EBL to place pairs of nanoparticles close to each other with an unprecedented accuracy of one nanometer and single-particle in-situ X-ray absorption spectromicroscopy was done to visualize hydrogen spillover. For the first time, distance dependence of hydrogen spillover has been experimentally visualized [3], and the hydrogen diffusion and migration mechanisms are elucidated by DFT calculations.

_{[1] W. Karim, S. A. Tschupp, M. Oezaslan, T. J. Schmidt, J. Gobrecht, J. A. van Bokhoven, and Y. Ekinci, Nanoscale, 7, 7386 (2015).
[2] W. Karim, A. Kleibert, U. Hartfelder, A. Balan, J. Gobrecht, J. A. van Bokhoven, and Y. Ekinci, Scientific Reports, 6, 18818 (2016).
[3] W. Karim, S. Clelia, A. Kleibert, J. Gobrecht, J. VandeVondele, Y. Ekinci, and J. A. van Bokhoven, Nature, 541, 68-71 (2017).}

The giant leaps in miniaturization have enabled us to witness how mere thought experiments of single atom devices are suddenly close to becoming a physical reality. In magnetic storage media, by way of example, the smallest individually accessible magnetic bits contain few atom large clusters with magnetic lifetimes in the seconds range at cryogenic temperatures, but a recent report of magnetic remanence in ensembles of holmium atoms on magnesium oxide (MgO) promised a path toward data storage at the atomic limit [1]. It had been unclear, however, how the magnetic state of individual single atom magnets could be accessed. In the present work [2], we demonstrate the reading and writing of individual Ho atoms on MgO. We read the state of the Ho single atom magnet by tunnel magnetoresistance and write the state by a current pulse, using a scanning tunneling microscope. We are able to unambiguously prove the magnetic origin of the two bistable Ho states through STM enabled electron spin resonance on nearby sensor atoms. Via STM-ESR, we determine a large Ho out-of-plane magnetic moment of (10.1 ± 0.1) µB. We furthermore built a prototypical 2-Ho bit array to which we write all four states and which we read out directly via TMR and remotely on a sensor atom using ESR. We observe that the Ho single atom magnets independently retain their magnetic state over many hours. Our work marks the ultimate goal of miniaturization, and the realization of a programmable single atom magnet means that the thought experiment of single atom bits has now become a physical reality.

_{[1] F. Donati, S. Rusponi, S. Stepanow, C. Wäckerlin, A. Singha, L. Persichetti, R. Baltic, K. Diller, F. Patthey, E. Fernandes, J. Dreiser, Ž. Šljivan?anin, K. Kummer, C. Nistor, P. Gambardella, and H. Brune, Science 352, 318 (2016).
[2] F. D. Natterer, K. Yang, W. Paul, P. Willke, T. Choi, T. Greber, A. J. Heinrich, and C. P. Lutz, Nature 543, 226 (2017).}

Understanding the various phases of matter and the transitions between them is the central idea in condensed matter physics. Together with the concept of a phase of matter comes the concept of an order parameter that identifies it. Over the recent years, order parameters have become increasingly more complex (i.e. non-local), since the phases they identify have become more exotic. Identifying an order parameter can be a notoriously hard problem. Since it is nowadays less difficult to obtain large volumes of data -- either by experiment or by simulations -- it has become feasible to use machine learning techniques to identify order parameters. A well-known machine learning technique is that of supervised learning with a neural network. The neural network can be taught to classify inputs into distinct classes, by repeatedly showing it examples of pre-labeled pairs. As an example, one may train a machine to recognize low- and high-temperature snapshots of the 2-dimensional Ising model as being 'below' and 'above' the critical temperature. After it has been trained, one may ask it to judge where the transition point is by trying to classify snapshots close to the transition [1]. For more complicated models however, the phases may be unknown and hence pre-labelling the data is impossible. In such cases supervised learning must be replaced by unsupervised learning. We demonstrate a method for identifying the transition points using an unsupervised method [2]. We guess a labeling of the input data, and attempt to teach this to a network. If the classification is highly inconsistent (i.e. giving similar input data very different classification labels), the network is confused and has difficulties learning to classify inputs. If the guess of labels is consistent with the input, the network performs well in classifying inputs. The maximum of the performance curve as a function of guessed classification then identifies the most consistent labeling and hence the transition point. We demonstrate this approach by identifying the classical transition point in the Ising model, the topological transition in the Kitaev chain and the non-trivial eigenstate transition in a many-body localized system.

_{[1] Juan Carrasquilla and Roger G. Melko, Nature Physics, 13, 431-434 (2017)
[2] Evert P. L. van Nieuwenburg, Ye-Hua Liu and Sebastian D. Huber, Nature Physics, 13, 435-439 (2017)}