# Theoretische Physik I

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Item Capsule rheology and machine learning(2024) Kratz, Felix Sebastian; Kierfeld, Jan; Risselada, Herre JelgerShow more apsules and their properties have provoked an increasing interest in several fields of the sciences and industry. In the sciences, several relevant biological system are modeled as a liquid core encapsulated by a skin of some sort, e.g. red blood cells. In industry, capsules are usually used the other way around -- not to model nature, but rather to design for functionality, e.g. in medical application or the food industry. Given their ubiquitous application, we discuss and investigate the solution of shape equations for freely pendant droplets, capsules and derive a method to incorporate viscous dissipation for time dependent deformation sequences. These theoretical investigations are supplemented with a novel numerical framework which allows us to solve the shape equations, fit them to experimental images, and therefore infer information from experiments. We apply the theoretical and numerical insights gained during the course of this work to investigate the properties of complex interfaces, such as multi-layer systems. While an individual capsule has interesting applications, the reality often is that a capsule can not be isolated from other capsules or some constraining boundaries. We therefore investigate -- for the first time in literature -- the contact problem of a pressurized, bending-stiff, adhesive, elastic capsule under an external force both with a solid wall and with another capsule of this kind. The resulting shape equations give us access to the shape-parameter diagram and allow us to understand the contact problem without performing any experiment. We rather integrate the shape equations numerically and find the solutions nature realizes, together with all relevant derived quantities, such as the contact force. Additionally, we design a meta-material (theoretically) from an elastic capsule unit-cell by extending the contact theory to a columnar structure. Several problems encountered in physics, especially in inverse problems, can be considered ill-conditioned. An ill-conditioned problem reacts sensitive to perturbations of the input data and usually needs to be regularized or otherwise constrained to produce stable predictions or results. In this thesis we explore the potential of machine learning approaches for exactly this task. With liquid droplet and elastic capsule shape fitting, as well as traction force microscopy, as example problems, we convincingly show that machine learning approaches for these ill-conditioned problems are suitable and outperform conventional methods by orders of magnitude in speed, allowing for a entirely new applications.Show more Item Synchronization of Higgs oscillations in long-range interacting superconductors(2024-03-06) Fauseweh, BenediktShow more Item Dynamics and length distributions of microtubules with a multistep catastrophe mechanism(2023-01-23) Schwietert, Felix; Heydenreich, Lina; Kierfeld, JanShow more Regarding the experimental observation that microtubule (MT) catastrophe can be described as a multistep process, we extend the Dogterom–Leibler model for dynamic instability in order to discuss the effect that such a multistep catastrophe mechanism has on the distribution of MT lengths in the two regimes of bounded and unbounded growth. We show that in the former case, the steady state length distribution is non-exponential and has a lighter tail if multiple steps are required to undergo a catastrophe. If rescue events are possible, we detect a maximum in the distribution, i.e. the MT has a most probable length greater than zero. In the regime of unbounded growth, the length distribution converges to a Gaussian distribution whose variance decreases with the number of catastrophe steps. We extend our work by applying the multistep catastrophe model to MTs that grow against an opposing force and to MTs that are confined between two rigid walls. We determine critical forces below which the MT is in the bounded regime, and show that the multistep characteristics of the length distribution are largely lost if the growth of an MT in the unbounded regime is restricted by a rigid wall. All results are verified by stochastic simulations.Show more Item Understanding the dynamics of randomly positioned dipolar spin ensembles(2023-12-01) Gräßer, Timo; Rezai, Kristine; Sushkov, Alexander O.; Uhrig, Götz S.Show more Dipolar spin ensembles with random spin positions are attracting much attention because they help us to understand decoherence as it occurs in solid-state quantum bits in contact with spin baths. Also, these ensembles are systems which may show many-body localization, at least in the sense of very slow spin dynamics. We present measurements of the autocorrelations of spins on diamond surfaces at infinite temperature in a doubly rotating frame which eliminates local disorder. Strikingly, the timescales in the longitudinal and the transversal channel differ by more than one order of magnitude, which is a factor much greater than one would have expected from simulations of spins on lattices. A previously developed dynamic mean-field theory for spins (spinDMFT) fails to explain this phenomenon. Thus, we improve it by extending it to clusters (CspinDMFT). This theory does capture the striking mismatch up to two orders of magnitude for random ensembles. Without positional disorder, however, the mismatch is only moderate with a factor below 4. The pivotal role of positional disorder suggests that the strong mismatch is linked to precursors of many-body localization.Show more Item Switching the magnetization in quantum antiferromagnets(2023-09-05) Uhrig, Götz S.; Bolsmann, Katrin; Khudoyberdiev, AsliddinShow more The orientation of the order parameter of quantum magnets can be used to store information in a dense and efficient way. Switching this order parameter corresponds to writing data. To understand how this can be done, we study a precessional reorientation of the sublattice magnetization in an (an)isotropic quantum antiferromagnet induced by an applied magnetic field. For this intriguing nonequilibrium issue, we introduce a description including the leading quantum and thermal fluctuations, namely time-dependent Schwinger boson mean-field theory, because this theory allows us to describe both ordered phases and the phases in between them, as is crucial for switching. An activation energy has to be overcome, requiring a minimum applied field ht that is given essentially by the spin gap. It can be reduced significantly for temperatures approaching the Néel temperature, facilitating switching. The time required for switching diverges when the field approaches ht, which is the signature of an inertia in the magnetization dynamics. The temporal evolution of the magnetization and of the energy reveals signs of dephasing. The switched state has lost a part of its coherence because the magnetic modes do not evolve in phase.Show more Item Applications and extensions of flow equations to closed and open quantum systems(2022) Schmiedinghoff, Gary; Uhrig, Götz S.; Anders, FrithjofShow more 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.Show more Item Dynamics and forces in the mitotic spindle(2022) Schwietert, Felix; Kierfeld, Jan; Risselada, Herre JelgerShow more Mikrotubuli sind zylinderförmige Filamente und Teil des Zytoskeletts. Ihre Polymerisationsdynamik zeichnet sich durch eine dynamische Instabilität von Wachstums- und Schrumpfphasen aus. Die zufälligen Wechsel vom schrumpfenden in den wachsenden Zustand und umgekehrt werden als Rettungen bzw. Katastrophen bezeichnet. Letztere können experimentellen Beobachtungen zufolge als Mehrschrittprozesse beschrieben werden. Im ersten Teil dieser Arbeit wird das empirische Dogterom-Leibler-Modell der dynamischen Instabilität erweitert, um auszuarbeiten, welche Auswirkungen eine Mehrschrittkatastrophe auf die Längenverteilung eines Mikrotubulus in den Regimen gebundenen und ungebundenen Wachstums hat. Es zeigt sich, dass die Mikrotubuluslängen im gebundenen Regime nicht mehr exponentiell und weniger endlastig verteilt sind, wenn eine Katastrophe aus mehreren Schritten besteht. Wenn Rettungen möglich sind, hat die Verteilung ein Maximum und der Mikrotubulus somit eine wahrscheinlichste Länge, die größer ist als 0. Im Regime ungebundenen Wachstums nähert sich die Längenverteilung einer Normalverteilung an, die mit steigender Anzahl der Katastrophenschritte schmaler wird. In der Mitosespindel sind Mikrotubuli durch Kinetochore mit den Chromosomen verbunden und üben so Kräfte aus, die in der Metaphase zu stochastischen Oszillationen der Chromosomen führen. Im zweiten Teil dieser Arbeit untersuchen wir in Modellen der Mitosespindel die kollektive Dynamik von Mikrotubuli, die durch elastische Federn an Kinetochore gebunden sind. Die Modelle beinhalten die dynamische Instabilität der Mikrotubuli und die Kräfte, die durch die elastischen Verbindungen wirken. Für ein einseitiges Modell mit nur einem Kinetochor, das einer externen Kraft ausgesetzt ist, können mithilfe einer Molekularfeldnäherung Fokker-Planck-Gleichungen aufgestellt und gelöst werden. Aus der Lösung folgt eine bistabile Abhängigkeit der Kinetochorgeschwindigkeit von der externen Kraft. Im zweiseitigen Modell mit zwei elastisch gekoppelten Kinetochoren führt die Bistabilität zu Oszillationen, die denen der Chromosomen in der Metaphase gleichen. Das Modell kann erklären, warum in Zellen mit einem schnellen polwärtigen Mikrotubulusfluss keine Oszillationen beobachtet wurden. Polare Auswurfkräfte gewährleisten im Modell eine Anordnung der Kinetochore am Spindeläquator und führen zu geregelteren Oszillationen mit verringerter Amplitude. Wenn das Modell so geändert wird, dass die Mikrotubuli nur Zugkräfte auf das Kinetochor ausüben können, treten Oszillationen nur unter der Voraussetzung auf, dass in der Nähe der Kinetochore Katastrophen induziert werden. Die Modellparameter können so angepasst werden, dass die modellierten Oszillationen auch in quantitativer Hinsicht mit Messungen in PtK1-Zellen übereinstimmen. Ein wichtiger Bestandteil des Kinetochors sind stäbchenförmige Ndc80-Komplexe, die den Mikrotubulus binden und deren elastischen Eigenschaften als wichtig für die Kraftübertragung vom Mikrotubulus auf das Chromosom erachtet werden. Im letzten Teil dieser Arbeit wird eine Methode präsentiert, die es erlaubt, den zeitlichen Verlauf der effektiven Steifigkeit von Ndc80-Komplexen zu ermitteln, die in einer optischen Falle entgegen einer Kraft dem schrumpfenden Ende eines Mikrotubulus folgen. Die Anwendung der Methode auf mehrere Experimente zeigt, dass sowohl der Wildtyp als auch drei weitere Ndc80-Varianten steifer werden, wenn der schrumpfende Mikrotubulus sie unter Spannung setzt. Die gemessene Steifigkeit hat eine annähernd lineare Abhängigkeit von der angelegten Kraft und ist unabhängig vom dynamischen Zustand des Mikrotubulus. Mithilfe eines elastischen Modells kann die Versteifung auf die spezielle Architektur des Ndc80-Komplexes sowie auf das Biegen gekrümmter Protofilamente zurückgeführt werden. Ein Modell mit einer kraftabhängigen Bindungsaffinität reproduziert die lineare Beziehung zwischen Steifigkeit und Kraft.Show more Item Non-equilibrium dynamics of a driven-dissipative dimerized spin-1/2 chain(2022) Yarmohammadi, Mohsen; Uhrig, Götz S.; Haas, Stephan W.Show more Due to the rise in experimental progress in several photonic facilities, theoretical addressing the non-equilibrium behavior in driven-dissipative quantum systems has triggered considerable interest in recent times. This thesis is devoted to the analysis of dynamics of a dimerized spin chain model which is driven out-of-equilibrium by the presence of a classical steady laser field. A particular study is given on the spin-phonon coupling effect treated as weak-to-strong perturbations, that the infrared-active phonon is driven by the laser. All systems in nature are interacting with their surroundings and the effects of the environment have to be approximated. To begin with, we employ the quantum Markovian master equation, which follows the construction of the dissipation path to a phononic bath for both phonon and spin sectors in the driven coupled spin-lattice system. We approach this thesis by exploring how the non-equilibrium steady states are created, con-trolled, and preserved by the internal and external interactions. This includes a detailed study of non-equilibrium dynamics of driven-dissipative quantum magnetic materials. First, we prepare the tools, protocols, and approximations needed to model a dimerized spin-1/2 chain as a chain of non-interacting triplons. The spin-phonon coupling is treated by the theoretical framework of the mean-field formalism. Second, we approximate the phononic bath with constant damping for each sector to easily derive the master equations of motion for the physical observables in the entire system. Third, we discuss the validity of such approximative master equations by considering many physical degrees of freedom. These settings produce a large variety of interesting phenomena and physical insights.Show more Item Magnetic blue shift of Mott gaps enhanced by double exchange(2021-12-30) Hafez-Torbati, Mohsen; Bossini, Davide; Anders, Frithjof B.; Uhrig, Götz S.Show more A substantial energy gap of charge excitations induced by strong correlations is the characteristic feature of Mott insulators. We study how the Mott gap is affected by long-range antiferromagnetic order. Our key finding is that the Mott gap is increased by the magnetic ordering: A magnetic blue shift (MBS) occurs. Thus the effect is proportional to the exchange coupling in the leading order in the Hubbard model. In systems with additional localized spins the double-exchange mechanism induces an additional contribution to the MBS which is proportional to the hopping in the leading order. The coupling between spin and charge degrees of freedom bears the potential to enable spin-to-charge conversion in Mott systems on extreme time scales determined by hopping and exchange only, since a spin-orbit-mediated transfer of angular momentum is not involved in the process. In view of spintronic and magnonic applications, it is highly promising to observe that several entire classes of compounds show exchange and double-exchange effects. Exemplarily, we show that the magnetic contribution to the band-gap blue shift observed in the optical conductivity of α-MnTe is correctly interpreted as the MBS of a Mott gap.Show more Item Dynamic mean-field theory for dense spin systems at infinite temperature(2021-12-10) Gräßer, Timo; Bleicker, Philip; Hering, Dag-Björn; Yarmohammadi, Mohsen; Uhrig, Götz S.Show more A dynamic mean-field theory for spin ensembles (spinDMFT) at infinite temperatures on arbitrary lattices is established. The approach is introduced for an isotropic Heisenberg model with S=12 and external field. For large coordination numbers, it is shown that the effect of the environment of each spin is captured by a classical time-dependent random mean field which is normally distributed. Expectation values are calculated by averaging over these mean fields, i.e., by a path integral over the normal distributions. A self-consistency condition is derived by linking the moments defining the normal distributions to spin autocorrelations. In this framework, we explicitly show how the rotating-wave approximation becomes a valid description for increasing magnetic field. We also demonstrate that the approach can easily be extended. Exemplarily, we employ it to reach a quantitative understanding of a dense ensemble of spins with dipolar interaction which are distributed randomly on a plane including static Gaussian noise as well.Show more Item The Fermi-Hubbard model and its limiting cases as a testbed for techniques and phenomena(2021) Bleicker, Philip; Uhrig, Götz S.; Schmidt, Kai PhillipShow more Trotz mittlerweile etwa sechs Jahrzehnten Forschung am Fermi-Hubbard-Modell (FHM) geht von kaum einem anderen Modell eine vergleichbar hohe Faszination aus. Das Modell ist von konzeptionell einfacher Struktur und enthält dabei doch die wesentliche Grundzutat dessen, was Festkörper ausmacht: Wechselwirkung. Ebendiese Wechselwirkung ist es, die abschließenden Lösungen des FHM im Gleich- sowie Nichtgleichgewicht diametral entgegensteht und hohe Ansprüche an die verwendeten methodischen Zugänge stellt. In dieser Arbeit wenden wir uns dem FHM sowie einigen hieraus ableitbaren Modellen zu, etwa dem t-J-Modell oder dem Heisenberg-Modell, diskutieren Kernfragen der aktuellen Forschung und nutzen bekannte Ansätze wie die CET oder TPQS sowie neue Techniken wie die iEoM, um zur Klärung einiger zentraler Fragestellungen beizutragen. Wir starten mit Analysen von Äquilibration und Thermalisierung in gequenchten von der Umgebung abgeschlossenen Quantensysteme und bestätigen und erweitern bisherige Annahmen. Ferner widerlegen wir die Vermutung eines dynamischen Phasenübergangs in einer Dimension. In einem zweiten Schritt reduzieren wir das FHM auf ein effektives t-J-Modell, das besonders im Kontext von Hochtemperatursupraleitung Beachtung findet, und betrachten hierin die Ladungsträgerdynamik. Darüber hinaus bestätigen wir Annahmen zur quantitativen Vorhersagbarkeit der Autokorrelation in dichten Spin-Systemen. Im letzten Schritt schlagen wir einen neuartigen theoretischen Ansatz mittels iEoM für die systematische Berechnung von Greenfunktionen in reduzierten Operator-Unterräumen vor und motivieren seine Anwendbarkeit im Rahmen einer exemplarischen Rechnung.Show more Item Nonequilibrium spin phenomena in quantum dots induced by periodic optical excitation(2021) Schering, Philipp; Uhrig, Götz S.; Anders, FrithjofShow more The coherent control of a charge carrier spin that is localized in a semiconductor quantum dot and the generation of long-lived states for information storage are of particular interest for quantum information processing. This spin interacts predominantly with the surrounding nuclear spins in the quantum dot, which can be described by the central spin model. The periodic application of circularly polarized laser pulses induces nonequilibrium spin dynamics in the quantum dot, giving rise to various phenomena that can be observed in experiments. In this thesis, models and semiclassical approaches are developed to simulate the driven spin dynamics in this system under experimental conditions. For the case where a transverse magnetic field is applied, it is found that the part of the spin mode locking effect stemming from nuclei-induced frequency focusing depends nonmonotonically on the strength of the magnetic field, with strong similarities to experimental observations. The complex behavior is related to various nuclear magnetic resonances with respect to the repetition rate of the laser pulses, which can be exploited for novel kind of nuclear magnetic resonance spectroscopy of the emerging nonequilibrium steady states. For the case where a longitudinal magnetic field is applied, the influence of the pump pulse power on the spin inertia and on the polarization recovery effect is analyzed. With the help of the developed model, the related experiments can be understood and described quantitatively. In this context, a novel effect termed resonant spin amplification in Faraday geometry is predicted, which enables the direct measurement of the longitudinal g factor of the resident charge carriers. Model calculations are used to find the optimal conditions for its detection and ways to improve its visibility are pointed out. The comparison with recent experiments that demonstrate the realization of the effect shows a remarkable agreement.Show more Item Surfactant-loaded capsules as Marangoni microswimmers at the air–water interface: symmetry breaking and spontaneous propulsion by surfactant diffusion and advection(2021-03-08) Ender, Hendrik; Froin, Ann-Kathrin; Rehage, Heinz; Kierfeld, JanShow more We present a realization of a fast interfacial Marangoni microswimmer by a half-spherical alginate capsule at the air–water interface, which diffusively releases water-soluble spreading molecules (weak surfactants such as polyethylene glycol (PEG)), which act as “fuel” by modulating the air–water interfacial tension. For a number of different fuels, we can observe symmetry breaking and spontaneous propulsion although the alginate particle and emission are isotropic. The propulsion mechanism is similar to soap or camphor boats, which are, however, typically asymmetric in shape or emission to select a swimming direction. We develop a theory of Marangoni boat propulsion starting from low Reynolds numbers by analyzing the coupled problems of surfactant diffusion and advection and fluid flow, which includes surfactant-induced fluid Marangoni flow, and surfactant adsorption at the air–water interface; we also include a possible evaporation of surfactant. The swimming velocity is determined by the balance of drag and Marangoni forces. We show that spontaneous symmetry breaking resulting in propulsion is possible above a critical dimensionless surfactant emission rate (Peclet number). We derive the relation between Peclet number and swimming speed and generalize to higher Reynolds numbers utilizing the concept of the Nusselt number. The theory explains the observed swimming speeds for PEG–alginate capsules, and we unravel the differences to other Marangoni boat systems based on camphor, which are mainly caused by surfactant evaporation from the liquid–air interface. The capsule Marangoni microswimmers also exhibit surfactant-mediated repulsive interactions with walls, which can be qualitatively explained by surfactant accumulation at the wall.Show more Item Chemomechanical simulation of microtubule dynamics(2020) Schmidt, Matthias; Kierfeld, Jan; Löw, UteShow more Microtubules are hollow cylindrical filaments that are part of the cytoskeleton. During their growth, they exhibit “dynamic instability”, i.e., they can suddenly switch from growing to shrinking (catastrophe) and vice versa (rescue). Obtaining a microscopic understanding of this dynamic instability will also provide a better insight into basic cell functions like cell division and can provide a pathway to manipulate such processes. We introduce and parameterize a microtubule model that combines a mechanical, three-dimensional model of the microtubule's structure with the chemical processes happening during its growth. These chemical processes are the polymerization and depolymerization of tubulin dimers, the building blocks of microtubules, the formation and rupture of lateral bonds between neighboring tubulin monomers, and random hydrolysis. By ensuring that the simulation is computationally efficient despite the need for energy minimizations after each event, we are able to simulate microtubule growth for realistic timescales to observe catastrophes and rescues. In addition to analyzing different properties of the simulated microtubules and investigating the effects of dilution, i.e., the sudden decrease of the concentration of free tubulin dimers around the microtubule, we also consider mechanical feedback on the hydrolysis rate. We find this mechanical feedback to result in an effective anti-vectorial hydrolysis mechanism and that there are GTP-tubulin dimers much further away from the microtubule's plus end.Show more Item Quasiparticle decay induced by spin anisotropies in the frustrated spin ladder system BiCu2PO6(2021) Müller, Leanna Blanche; Uhrig, Götz S.; Schmidt, Kai PhillipShow more The inorganic compound BiCu2PO6 contains tubelike structures, which are described magnetically by weakly coupled frustrated spin ladders with a finite energy gap. The elementary excitations are triplons of which the degeneracy is lifted due to Dzyaloshinskii-Moriya interactions. In certain regions of the Brillouin zone the lifetime of the triplon excitation modes becomes finite due to the hybridization of the single-triplon with the two-triplon states. In addition, the dispersions of these triplon modes show a striking down-bending before ceasing to exist. In experiment, BiCu2PO6 shows various types of decay processes, which can be caused by different symmetry breaking interactions. In previous studies, we established a minimal model to include all symmetry-allowed interactions, such as the Dzyaloshinskii-Moriya interaction. Based on this minimal model, we show in this thesis that isotropic and anisotropic effects are responsible for noticeable quasiparticle decay and certain down-shifts of the single-triplon energies. The analyses are based on a deepCUT approach for the isotropic case augmented by a perturbative treatment of the anisotropic couplings inducing quasiparticle decay at zero temperature.Show more Item From diffusive mass transfer in Stokes flow to low Reynolds number Marangoni boats(2021-02-12) Ender, Hendrik; Kierfeld, JanShow more We present a theory for the self-propulsion of symmetric, half-spherical Marangoni boats (soap or camphor boats) at low Reynolds numbers. Propulsion is generated by release (diffusive emission or dissolution) of water-soluble surfactant molecules, which modulate the air–water interfacial tension. Propulsion either requires asymmetric release or spontaneous symmetry breaking by coupling to advection for a perfectly symmetrical swimmer. We study the diffusion–advection problem for a sphere in Stokes flow analytically and numerically both for constant concentration and constant flux boundary conditions. We derive novel results for concentration profiles under constant flux boundary conditions and for the Nusselt number (the dimensionless ratio of total emitted flux and diffusive flux). Based on these results, we analyze the Marangoni boat for small Marangoni propulsion (low Peclet number) and show that two swimming regimes exist, a diffusive regime at low velocities and an advection-dominated regime at high swimmer velocities. We describe both the limit of large Marangoni propulsion (high Peclet number) and the effects from evaporation by approximative analytical theories. The swimming velocity is determined by force balance, and we obtain a general expression for the Marangoni forces, which comprises both direct Marangoni forces from the surface tension gradient along the air–water–swimmer contact line and Marangoni flow forces. We unravel whether the Marangoni flow contribution is exerting a forward or backward force during propulsion. Our main result is the relation between Peclet number and swimming velocity. Spontaneous symmetry breaking and, thus, swimming occur for a perfectly symmetrical swimmer above a critical Peclet number, which becomes small for large system sizes. We find a supercritical swimming bifurcation for a symmetric swimmer and an avoided bifurcation in the presence of an asymmetry.Show more Item Chaining of hard disks in nematic needles: particle-based simulation of colloidal interactions in liquid crystals(2020-07-29) Müller, David; Kampmann, Tobias Alexander; Kierfeld, JanShow more Colloidal particles suspended in liquid crystals can exhibit various effective anisotropic interactions that can be tuned and utilized in self-assembly processes. We simulate a two-dimensional system of hard disks suspended in a solution of dense hard needles as a model system for colloids suspended in a nematic lyotropic liquid crystal. The novel event-chain Monte Carlo technique enables us to directly measure colloidal interactions in a microscopic simulation with explicit liquid crystal particles in the dense nematic phase. We find a directional short-range attraction for disks along the director, which triggers chaining parallel to the director and seemingly contradicts the standard liquid crystal field theory result of a quadrupolar attraction with a preferred 45∘ angle. Our results can be explained by a short-range density-dependent depletion interaction, which has been neglected so far. Directionality and strength of the depletion interaction are caused by the weak planar anchoring of hard rods. The depletion attraction robustly dominates over the quadrupolar elastic attraction if disks come close. Self-assembly of many disks proceeds via intermediate chaining, which demonstrates that in lyotropic liquid crystal colloids depletion interactions play an important role in structure formation processes.Show more Item Bistability and oscillations in cooperative microtubule and kinetochore dynamics in the mitotic spindle(2020-05-01) Schwietert, Felix; Kierfeld, JanShow more In the mitotic spindle microtubules attach to kinetochores via catch bonds during metaphase, and microtubule depolymerization forces give rise to stochastic chromosome oscillations. We investigate the cooperative stochastic microtubule dynamics in spindle models consisting of ensembles of parallel microtubules, which attach to a kinetochore via elastic linkers. We include the dynamic instability of microtubules and forces on microtubules and kinetochores from elastic linkers. A one-sided model, where an external force acts on the kinetochore is solved analytically employing a mean-field approach based on Fokker–Planck equations. The solution establishes a bistable force–velocity relation of the microtubule ensemble in agreement with stochastic simulations. We derive constraints on linker stiffness and microtubule number for bistability. The bistable force–velocity relation of the one-sided spindle model gives rise to oscillations in the two-sided model, which can explain stochastic chromosome oscillations in metaphase (directional instability). We derive constraints on linker stiffness and microtubule number for metaphase chromosome oscillations. Including poleward microtubule flux into the model we can provide an explanation for the experimentally observed suppression of chromosome oscillations in cells with high poleward flux velocities. Chromosome oscillations persist in the presence of polar ejection forces, however, with a reduced amplitude and a phase shift between sister kinetochores. Moreover, polar ejection forces are necessary to align the chromosomes at the spindle equator and stabilize an alternating oscillation pattern of the two kinetochores. Finally, we modify the model such that microtubules can only exert tensile forces on the kinetochore resulting in a tug-of-war between the two microtubule ensembles. Then, induced microtubule catastrophes after reaching the kinetochore are necessary to stimulate oscillations. The model can reproduce experimental results for kinetochore oscillations in PtK1 cells quantitatively.Show more Item Robustness and variation of low-dimensional signal transmission in topological phases(2019) Malki, Maik; Uhrig, Götz S.; Schmidt, Kai PhillipShow more This thesis investigates the variation of signal transmission in topological phases as well as their robustness in one- and two-dimensional systems. For this purpose, multiple approaches in different systems are pursued. First, the possibility of designed modifications at the boundaries is explored in order to change the Fermi velocity at edge states of topological phases. The Fermi velocity as a quantity of the transport behavior describes the speed of signal transmission. The main idea is to hybridize local modes with dispersive edge modes in a controllable way so that the signal speed can be significantly slowed down. In the beginning, the Haldane model is modified. Thereafter, the findings are extended by the spin degree of freedom, yielding to Kane–Mele model with helical edge states. In addition, the robustness of edge states against local disorder is investigated by reconstructing the dispersion of the edge modes in the Haldane model. As a result, certain limits regarding the protection of topological edge states become apparent. Triggered by the results for lattice systems, the central idea is carried over to the integer quantum Hall effect of a free two-dimensional electron gas. The local modes are generated by periodically aligned bays at the boundary of the sample. The Hamiltonian of a free two-dimensional electron gas subjected to a perpendicular magnetic field is approximated by a finely discretized lattice. Hence the dispersion of arbitrary periodic geometries becomes numerically accessible. The application of a gate voltage brings the weakly hybridized edge states into resonance with the Fermi energy. Therefore the Fermi velocity can be varied by up to two orders of magnitude. To extend the research approach, graphene is investigated as another possible implementation due to its special properties and the technical possibility to realize desired geometries. The numerical results indicate that possible applications such as delay lines or interferometers are feasible. The investigation of the topological properties of triplon excitations in BiCu2PO6 reveals new insights into the bulk-boundary correspondence. BiCu2PO6 is described by frustrated quantum spin-1/2 ladders which are weakly coupled to form a two-dimensional lattice. The eigenenergies and eigenmodes of single-triplons are determined by applying deepCUTs and Bogoliubov transformations. The one-triplon dispersions are used to fit the inelastic neutron scattering data by adjusting the coupling constants. Based on that, BiCu2PO6 is shown to be the first disordered quantum antiferromagnet exhibiting a gap and a non-trivial Zak phase. Additionally the topological character of BiCu2PO6 is established by a finite winding number. Despite the bulk-boundary correspondence, no localized edge states can be found due to the absence of an indirect gap. The investigation of the Su–Schrieffer–Heeger model confirms that the disappearance of the indirect gap leads to delocalized in-gap states. In order to further explore the localization of edge states regarding the indirect gap, two-dimensional topological systems such as the Haldane model and the topological checkerboard model are investigated as well. Finally, the investigation of the ferromagnetic Shastry–Sutherland lattices reveals the existence of topological magnon excitations. Using exact spin wave theory, finite Chern numbers of the magnon bands are determined which give rise to chiral edge states. The thermal Hall conductivity as an experimental signature of the topological phase is calculated. Various promising compounds are discussed as possible physical realizations of ferromagnetic Shastry–Sutherland lattices.Show more Item Deformation of ferrofluid-filled elastic capsules and swelling elastic disks as microswimmers(2019) Wischnewski, Christian Jochen; Kierfeld, Jan; Rehage, HeinzShow more This thesis includes two parts presenting the results on the research on the deformation of ferrofluid-filled elastic capsules in external magnetic fields as well as on the possibility of the usage of swelling elastic disks as microswimmers by triggering hysteretic shape transformations. The first part is a primarily theoretical study of deformations and shape transitions of ferrofluid capsules induced by homogeneous and inhomogeneous magnetic fields. The thin elastic shell is modelled by a non-linear shallow shell theory resulting in a system of non-linear shape equations. The properties of the ferrofluid are represented in these equations by the magnetic surface force density that is derived from the magnetic stress tensor by Rosensweig. The elastic part and the magnetic part form a coupled self-consistent problem that is solved iteratively. With that model, two different general cases are investigated. The objective of the first investigation are linearly magnetizable ferrofluid capsules in homogeneous external fields. These capsules are elongated into spheroidal shapes at first and show a transition into shapes with conical tips for stronger magnetic fields and higher susceptibilities. This behavior is shown to be completely analogous to the behavior of simple ferrofluid drops. After that, the model is tested in a comparison with the deformation behavior of real ferrofluid-filled alginate capsules in an experiment with an inhomogeneous external field. All parameters of the capsules and the magnetic field are determined from the experiment. With the surface Poisson ratio as a fit parameter, a good agreement between the numerics and the experiment can be observed. The Poisson ratio of the given alginate capsules is found to be surprisingly close to one. In the second part, the primary objective is to prove the concept that a flat elastic disk can be used as a microswimmer when a periodical swelling process triggers a shape transformation. Therefore, the disk is modelled by a simple spring mesh with an additional bending energy. A swelling pattern changing the springs’ rest length is defined and the total energy of the disk is minimized. This leads to either elliptic dome-like shapes or hyperbolic saddle-like shapes for sufficiently strong swelling. This transition into a curved shape shows numerical pseudohysteretic effects that can be expanded to a real hysteresis by adding an additional potential energy. These hysteretic effects are used to break the time invariance of the deformation and bypass the scallop theorem and allow the creation of a microswimmer. Since the swimmer is assumed to operate a low Reynolds numbers, where time invariant deformations do not cause any effective propulsion, these hysteretic effects are they key component in the swimming mechanism. In order to model the hydrodynamic interaction with a surrounding fluid, a Rotne-Prager model for the interaction between small spheres, that model the disk, is used. We find that the elliptic shape is able to move into the direction of its opening, while the hyperbolic shape cannot swim due to its symmetry. In addition, a simplified five-sphere model is introduced in order to imitate the deformation behavior of the disk and finally, the fluid velocity fields of the disk and the simplified model are analyzed.Show more