@phdthesis{Tutschku2021,
  author    = {Tutschku, Christian Klaus},
  title     = {Anomaly Induced Transport And Hall Viscous Effects In 2+1 Space-Time Dimensions},
  doi       = {10.25972/OPUS-23913},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-239131},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2021},
  abstract  = {The main goal of this thesis is to elucidate the sense in which recent experimental progress in condensed matter physics, namely the verification of two-dimensional Dirac-like materials and their control in ballistic- as well as hydrodynamic transport experiments enables the observation of a well-known 'high-energy' phenomenon: The parity anomaly of planar quantum electrodynamics (QED\(_{2+1}\)). In a nutshell, the low-energy physics of two-dimensional Quantum Anomalous Hall (QAH) insulators like (Hg,Mn)Te quantum wells or magnetically doped (Bi,Sb)Te thin films can be described by the combined response of two 2+1 space-time dimensional Chern insulators with a linear dispersion in momentum. Due to their Dirac-like spectra, each of those Chern insulators is directly related to the parity anomaly of planar quantum electrodynamics. However, in contrast to a pure QED\(_{2+1}\) system, the Lagrangian of each Chern insulator is described by two different mass terms: A conventional momentum-independent Dirac mass \(m\), as well as a momentum-dependent so-called Newtonian mass term \(B \vert \mathbf{k} \vert^2\). According to the parity anomaly it is not possible to well-define a parity- and U(1) gauge invariant quantum system in 2+1 space-time dimensions. More precisely, starting with a parity symmetric theory at the classical level, insisting on gauge-invariance at the quantum level necessarily induces parity-odd terms in the calculation of the quantum effective action. The role of the Dirac mass term in the calculation of the effective QED\(_{2+1}\) action has been initially studied in Phys. Rev. Lett. 51, 2077 (1983). Even in the presence of a Dirac mass, the associated fermion determinant diverges and lacks gauge invariance. This requires a proper regularization/renormalizaiton scheme and, as such, transfers the peculiarities of the parity anomaly to the massive case. In the scope of this thesis, we connect the momentum-dependent Newtonian mass term of a Chern insulator to the parity anomaly. In particular, we reveal, that in the calculation of the effective action, before renormalization, the Newtonian mass term acts similarly to a parity-breaking element of a high-energy regularization scheme. This calculation allows us to derive the finite frequency correction to the DC Hall conductivity of a QAH insulator. We derive that the leading order AC correction contains a term proportional to the Chern number. This term originates from the Newtonian mass and can be measured via electrical or via magneto-optical experiments. The Newtonian mass, in particular, significantly changes the resonance structure of the AC Hall conductivity in comparison to pure Dirac systems like graphene. In addition, we study the effective action of the aforementioned Chern insulators in external out-of-plane magnetic fields. We show that as a consequence of the parity anomaly the QAH phase in (Hg,Mn)Te quantum wells or in magnetically doped (Bi,Sb)Te thin films survives in out-of-plane magnetic fields, violates the Onsager relation, and can therefore be distinguished from a conventional quantum Hall (QH) response. As a smoking-gun of the QAH phase in increasing magnetic fields, we predict a transition from a quantized Hall plateau with \(\sigma_\mathrm{xy}= -\mathrm{e}^2/\mathrm{h}\) to a not perfectly quantized plateau which is caused by scattering processes between counter-propagating QH and QAH edge states. This transition is expected to be of significant relevance in paramagnetic QAH insulators like (Hg,Mn)Te/CdTe quantum wells, in which the exchange interaction competes against the out-of-plane magnetic field. All of the aforementioned results do not incorporate finite temperature effects. In order to shed light on such phenomena, we further analyze the finite temperature Hall response of 2+1 dimensional Chern insulators under the combined influence of a chemical potential and an out-of-plane magnetic field. As we have mentioned above, this non-dissipative transport coefficient is directly related to the parity anomaly of planar quantum electrodynamics. Within the scope of our analysis we show that the parity anomaly itself is not renormalized by finite temperature effects. However, the parity anomaly induces two terms of different physical origin in the effective Chern-Simons action of a QAH insulator, which are directly proportional to its Hall conductivity. The first term is temperature and chemical potential independent and solely encodes the intrinsic topological response. The second term specifies the non-topological thermal response of conduction- and valence band modes, respectively. We show that the relativistic mass \(m\) of a Chern insulator counteracts finite temperature effects, whereas its non-relativistic Newtonian mass \(B \vert \mathbf{k} \vert^2 \) enhances these corrections. In addition, we are extending our associated analysis to finite out-of-plane magnetic fields, and relate the thermal response of a Chern insulator therein to the spectral asymmetry, which is a measure of the parity anomaly in out-of-plane magnetic fields. In the second part of this thesis, we study the hydrodynamic properties of two-dimensional electron systems with a broken time-reversal and parity symmetry. Within this analysis we are mainly focusing on the non-dissipative transport features originating from a peculiar hydrodynamic transport coefficient: The Hall viscosity \(\eta_\mathrm{H}\). In out-of-plane magnetic fields, the Hall viscous force directly competes with the Lorentz force, as both mechanisms contribute to the overall Hall voltage. In our theoretical considerations, we present a way of uniquely distinguishing these two contributions in a two-dimensional channel geometry by calculating their functional dependencies on all external parameters. We are in particular deriving that the ratio of the Hall viscous contribution to the Lorentz force contribution is negative and that its absolute value decreases with an increasing width, slip-length and carrier density. Instead, it increases with the electron-electron mean free path in the channel geometry considered. We show that in typical materials such as GaAs the Hall viscous contribution can dominate the Lorentz signal up to a few tens of millitesla until the total Hall voltage vanishes and eventually is exceeded by the Lorentz contribution. Last but not least, we derive that the total Hall electric field has a parabolic form originating from Lorentz effects. Most remarkably, the offset of this parabola is directly characterized by the Hall viscosity. Therefore, in summary, our results pave the way to measure and to identify the Hall viscosity via both global and local measurements of the entire Hall voltage.},
  subject      = {Anomalie},
  language  = {en}
}
@phdthesis{Poerner2018,
  author    = {P{\"o}rner, Frank},
  title     = {Regularization Methods for Ill-Posed Optimal Control Problems},
  edition   = {1. Auflage},
  publisher = {W{\"u}rzburg University Press},
  address   = {W{\"u}rzburg},
  isbn      = {978-3-95826-086-3 (Print)},
  doi       = {10.25972/WUP-978-3-95826-087-0},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-163153},
  school      = {W{\"u}rzburg University Press},
  pages     = {xiii, 166},
  year      = {2018},
  abstract  = {This thesis deals with the construction and analysis of solution methods for a class of ill-posed optimal control problems involving elliptic partial differential equations as well as inequality constraints for the control and state variables. The objective functional is of tracking type, without any additional \(L^2\)-regularization terms. This makes the problem ill-posed and numerically challenging. We split this thesis in two parts. The first part deals with linear elliptic partial differential equations. In this case, the resulting solution operator of the partial differential equation is linear, making the objective functional linear-quadratic. To cope with additional control constraints we introduce and analyse an iterative regularization method based on Bregman distances. This method reduces to the proximal point method for a specific choice of the regularization functional. It turns out that this is an efficient method for the solution of ill-posed optimal control problems. We derive regularization error estimates under a regularity assumption which is a combination of a source condition and a structural assumption on the active sets. If additional state constraints are present we combine an augmented Lagrange approach with a Tikhonov regularization scheme to solve this problem. The second part deals with non-linear elliptic partial differential equations. This significantly increases the complexity of the optimal control as the associated solution operator of the partial differential equation is now non-linear. In order to regularize and solve this problem we apply a Tikhonov regularization method and analyse this problem with the help of a suitable second order condition. Regularization error estimates are again derived under a regularity assumption. These results are then extended to a sparsity promoting objective functional.},
  subject      = {Optimale Steuerung},
  language  = {en}
}
@phdthesis{Axmacher2014,
  author    = {Axmacher, Franz},
  title     = {Die SVM-gest{\"u}tzte Pr{\"a}diktabilit{\"a}t der Bindungsspezifit{\"a}t ‎von SH3-Dom{\"a}nen anhand ihrer Aminos{\"a}uresequenz},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-113349},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2014},
  abstract  = {Die Identifikation der Bindungsspezifit{\"a}ten von Proteininteraktionsdom{\"a}nen und damit letztlich auch ‎die F{\"a}higkeit potentielle Bindungspartner dieser in vivo vorherzusagen bildet ein grundlegendes ‎Element f{\"u}r das Verst{\"a}ndnis der biologischen Funktionen dieser Dom{\"a}nen. In dieser Arbeit wurde ‎untersucht, inwieweit solche Vorhersagen bez{\"u}glich der SH3-Dom{\"a}ne - als Beispiel f{\"u}r eine ‎Proteininteraktionsdom{\"a}ne - mithilfe von Support-Vector-Machines (SVMs) m{\"o}glich sind, wenn ‎diesen als Informationsquelle ausschließlich die innerhalb der Aminos{\"a}uresequenz der Dom{\"a}ne ‎konservierten Informationen zur Verf{\"u}gung stehen. Um den SVM-basierten Klassifikator zu ‎trainieren und zu validieren, wurde ein Satz aus 51 SH3-Dom{\"a}nen verwendet, die zuvor ‎entsprechend ihrer Ligandenpr{\"a}ferenz in ein System aus acht verschiedenen Klassen eingeteilt ‎worden waren. Da die innerhalb der Aminos{\"a}uresequenzen konservierten Informationen in ‎abstrakte Zahlenwerte konvertiert werden mussten (Voraussetzung f{\"u}r mathematisch basierte ‎Klassifikatoren wie SVMs), wurde jede Aminos{\"a}uresequenz durch ihren jeweiligen Fisher-Score-‎Vektor ausgedr{\"u}ckt. Die Ergebnisse erbrachten einen Klassifikationserror, welcher weit unterhalb des ‎Zufallsniveaus lag, was darauf hindeutet, dass sich die Bindungsspezifit{\"a}t (Klasse) einer SH3-Dom{\"a}ne ‎in der Tat von seiner Aminos{\"a}uresequenz ableiten lassen d{\"u}rfte. Mithilfe klassenspezifisch ‎emittierter, artifizieller Sequenzen, implementiert in den Trainingsprozess des Klassifikators, um ‎etwaigen nachteiligen Auswirkungen von Overfitting zu entgegenzuwirken, sowie durch ‎Ber{\"u}cksichtigung taxonomischer Informationen des Klassensystems w{\"a}hrend Training und ‎Validierung, ließ sich der Klassifikationserror sogar noch weiter senken und lag schließlich bei lediglich ‎‎35,29\% (vergleiche Zufall: 7/8 = 87.50\%). Auch die Nutzung von Feature Selections zur Abmilderung ‎Overfitting-bedingter, negativer Effekte lieferte recht vielversprechende Ergebnisse, wenngleich ihr ‎volles Potential aufgrund von Software-Beschr{\"a}nkungen nicht ausgenutzt werden konnte.‎ Die Analyse der Positionen im Sequence-Alignment, welche f{\"u}r den SVM- basierten Klassifikator am ‎relevantesten waren, zeigte, dass diese h{\"a}ufig mit Positionen korrelierten, von denen angenommen ‎wird auch in vivo eine Schl{\"u}sselrolle bei der Determination der Bindungsspezifit{\"a}t (Klasse) zu spielen. ‎Dies unterstreicht nicht nur die Reliabilit{\"a}t des pr{\"a}sentierten Klassifikators, es gibt auch Grund zur ‎Annahme, dass das Verfahren m{\"o}glicherweise auch als Supplement anderer Ans{\"a}tze genutzt werden ‎k{\"o}nnte, welche zum Ziel haben die Positionen zu identifizieren, die die Ligandenpr{\"a}ferenz in vivo ‎determinieren. Informationen, die nicht nur f{\"u}r ein besseres Verst{\"a}ndnis der SH3-Dom{\"a}ne (und ‎m{\"o}glicherweise auch anderer Proteininteraktionsdom{\"a}nen) von grundlegender Bedeutung sind, ‎sondern auch aus pharmakologischer Sicht von großem Interesse sein d{\"u}rften.‎},
  subject      = {Support-Vektor-Maschine},
  language  = {de}
}