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Department of Physics
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PhD Defense of Gary Schmiedinghoff

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Applications and Extensions of Flow Equations to Closed and Open Quantum Systems

Flow equations, also known as continuous unitary transformations, provide a powerful renormalization tool to transform a Hamiltonian and observables to an effective basis, where they take a more amenable form. However, unitary transformations often fail for non-Hermitian Hamiltonians, which appear, for instance, in dissipative systems. Furthermore, flow equation approaches often struggle in the vicinity of critical points.

This thesis aims to cover three separate problems regarding flow equations:
Spin ladders are crucial models for the description of strongly correlated quantum systems. An advanced method of probing such systems in various excitation channels is resonant inelastic X-ray scattering, but the theoretical prediction of the corresponding spectral densities is intricate. In this thesis, we compute the spectral densities of a spin-1/2 Heisenberg ladder with the flow equation method and predict novel three-triplon bound states. We demonstrate that these bound states only arise in the presence of irreducible three-triplon interactions by exploiting the strengths of our method.

Flow equations often fail in the vicinity of a critical point due to the divergent correlation length. The method could be improved by performing it in momentum space, where strongly delocalized physics can be described more easily. To this end, we investigate the transverse-field Ising model and show that the flows of various coefficients have a common convergence behavior, which offers a prospect for considerable improvements in future works. Additionally, we propose and test truncation schemes in momentum space, which could prove useful to describe low-energy physics.

Another current problem is the description of open quantum systems, i.e. quantum systems which are affected by dissipation because they couple to an external bath. Dissipative flow equations provide a framework to treat the non-Hermitian Hamiltonians and Lindbladians appearing in such systems. We propose a novel generator scheme based on the particle-conserving generator and benchmark the convergence speed and accuracy in spite of truncation compared to previously considered generators. We demonstrate that our proposed generator scheme provides high convergence speed and excellent accuracy. Furthermore, we encapsulate all currently known dissipative generator schemes in a universal framework, which can be used to propose various novel generator schemes favoring either convergence speed or accuracy.
 

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Location & approach

The campus of TU Dort­mund University is located close to interstate junction Dort­mund West, where the Sauerlandlinie A 45 (Frankfurt-Dort­mund) crosses the Ruhrschnellweg B 1 / A 40. The best interstate exit to take from A 45 is “Dort­mund-Eichlinghofen” (closer to South Campus), and from B 1 / A 40 “Dort­mund-Dorstfeld” (closer to North Campus). Signs for the uni­ver­si­ty are located at both exits. Also, there is a new exit before you pass over the B 1-bridge leading into Dort­mund.

To get from North Campus to South Campus by car, there is the connection via Vogelpothsweg/Baroper Straße. We recommend you leave your car on one of the parking lots at North Campus and use the H-Bahn (suspended monorail system), which conveniently connects the two campuses.

TU Dort­mund University has its own train station (“Dort­mund Uni­ver­si­tät”). From there, suburban trains (S-Bahn) leave for Dort­mund main station (“Dort­mund Hauptbahnhof”) and Düsseldorf main station via the “Düsseldorf Airport Train Station” (take S-Bahn number 1, which leaves every 20 or 30 minutes). The uni­ver­si­ty is easily reached from Bochum, Essen, Mülheim an der Ruhr and Duisburg.

You can also take the bus or subway train from Dort­mund city to the uni­ver­si­ty: From Dort­mund main station, you can take any train bound for the Station “Stadtgarten”, usually lines U41, U45, U 47 and U49. At “Stadtgarten” you switch trains and get on line U42 towards “Hombruch”. Look out for the Station “An der Palmweide”. From the bus stop just across the road, busses bound for TU Dort­mund University leave every ten minutes (445, 447 and 462). Another option is to take the subway routes U41, U45, U47 and U49 from Dort­mund main station to the stop “Dort­mund Kampstraße”. From there, take U43 or U44 to the stop “Dort­mund Wittener Straße”. Switch to bus line 447 and get off at “Dort­mund Uni­ver­si­tät S”.

The AirportExpress is a fast and convenient means of transport from Dortmund Airport (DTM) to Dortmund Central Station, taking you there in little more than 20 minutes. From Dortmund Central Station, you can continue to the university campus by interurban railway (S-Bahn). A larger range of international flight connections is offered at Düsseldorf Airport (DUS), which is about 60 kilometres away and can be directly reached by S-Bahn from the university station.

The H-Bahn is one of the hallmarks of TU Dort­mund University. There are two stations on North Campus. One (“Dort­mund Uni­ver­si­tät S”) is directly located at the suburban train stop, which connects the uni­ver­si­ty directly with the city of Dort­mund and the rest of the Ruhr Area. Also from this station, there are connections to the “Technologiepark” and (via South Campus) Eichlinghofen. The other station is located at the dining hall at North Campus and offers a direct connection to South Campus every five minutes.

The facilities of TU Dortmund University are spread over two campuses, the larger Campus North and the smaller Campus South. Additionally, some areas of the university are located in the adjacent “Technologiepark”.

Site Map of TU Dortmund University (Second Page in English).