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Tensor network methods for low-dimensional quantum systems
Get PDFThis thesis contributes to developing and applying tensor network methods to simulate correlated many-body quantum systems.
Numerical simulations of correlated quantum many-body systems are challenging. To describe a many-body wavefunction, the required number of parameters grows exponentially with respect to the system size. This exponential wall fundamentally limits our progress on correlated quantum systems in low dimensions. Tensor network methods in recent years have proven to be a useful framework to understand, control and possibly reduce this intrinsic complexity.
The basic idea of tensor network methods is to decompose a many-body wave function as a network of small, multi-index tensors. A one-dimensional (1D) tensor network factorizes a wave function into a train of three-index tensors. This 1D tensor network ansatz is called a matrix product state (MPS) or a tensor train. A two-dimensional (2D) tensor network state is called a projected entangled-pair state (PEPS). This peculiar name PEPS comes from a quantum information perspective, where each local tensor is interpreted as a projector and correlates with the rest of the tensor network through (auxiliary) maximally entangled pairs.
In the first part, we consider MPSs to study 1D and quasi-2D quantum systems. The key parameter of an MPS is its bond dimension, which controls the numerical accuracy. How large a bond dimension can be reached highly depends on the algorithms employed. The contemporary algorithms, although widely used, have to limit the bond dimension due to their high numerical costs. We develop a controlled bond expansions (CBE) scheme that allows us to grow the bond dimensions with marginal computational efforts. This CBE scheme stems from a geometric point of view to parametrize the variational space of an MPS and can be applied in various contexts. Here, we focus on applying the CBE scheme to two types of problems. The first are optimization problems, like solving the extremal eigenvalue problem. This is relevant for the ground state search, and we show that CBE can accelerate the convergence of MPS in terms of CPU time. The second is to solve ordinary differential equations, such as the time-dependant Schrödinger equation. With the help of CBE, it becomes feasible to use MPS to simulate long-time dynamics that could not be accurately computed hitherto.
In the second part, we employ PEPS to simulate 2D quantum systems. PEPS is an expensive but powerful tool to simulate 2D lattices directly in the thermodynamic limit. The PEPS on infinite lattices is acronymed iPEPS. For completeness, a pedagogical review of iPEPS based on Benedikt Bruognolo’s PhD work, which I helped polsih for publication in Scipost, is included to cover the algorithmic details. Using iPEPS methods, we study the two-dimensional t-J model on square lattices at the small doping. In this work, we uncover the importance of spin rotational symmetry. Our numerics suggest that by allowing spontaneous spin-symmetry breaking or not, we can supress or permit the emergence of superconducting order in the thermodynamic limit. This finding provides useful insight to cuprate materials. Also, we use iPEPS to investigate the ground state nature of the honeycomb Kitaev-Γ model. Through a joint effort of classical and iPEPS simulations, we identify an exotic magnetic order in the parameter regime relevant to α-RuCl3 materials.
In the third and final part, we study the parton construction of tensor network states. Here, we do not simulate the ground state of a given many-body Hamiltonian. Instead, we take an indirect route that first constructs a parton state in an enlarged Hilbert space, and then applies the Gutzwiller projection to return to the original physical Hilbert space. Such a parton approach has been an important theoretical technique to treat electron-electron correlations nonperturbatively in condensed matter physics. Its marriage with tensor network methods furthers its influence. Various properties of parton wave functions, which are difficult to compute previously, can now be easily accessed. We first use the parton approach to construct MPSs that harbor SU(N) chiral topological orders. The MPS representation of these Gutzwiller projected parton states allows us to compute entanglement spectra, which hold crucial information to characterize different chiral topological orders. We also develop a method to construct parton states using PEPSs. In this project, we use PEPS to approximate parton states of the π-flux models that host U(1)-Dirac spin liquids. Our approach enables us to compute the critical exponent of the spin-spin correlations for the spin-half system, whose value is still currently under debate
Eine systematische Untersuchung der nicht-pharmakologischen Therapiemethoden bei ängstlichen Patienten in der Zahnmedizin
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Der Einfluss des Fernsehens auf Bildungsmotivation und -entscheidung
Get PDFDie Zahlen sprechen eine deutliche Sprache: Von 100 Kindern in Deutschland besuchen 79 Kinder aus Akademikerfamilien die Universität, während nur 27 Kinder aus nichtakademischen Familien ein Studium beginnen. Woran liegt das und wie kann dieser sozialen Ungleichheit in der Bildung abgeholfen werden?
Medien prägen unsere Vorstellung von Realität, sie liefern Orientierung und Rollenvorbilder und beeinflussen unsere Einstellungen. Fernsehcharaktere dienen häufig als Vorbilder für den eigenen Berufswunsch. Die Frage, die diese Arbeit versucht, zu beantworten, lauten deshalb: „Welche Rolle spielt die Fernsehnutzung bei der Bildungsmotivation und -entscheidung junger Erwachsener?“
Mithilfe eines eigens entwickelten Wirkungsmodells wird der Einfluss des Elternhauses und der Einfluss des Fernsehens auf die Bildungsmotivation und -entscheidung junger Erwachsener untersucht und es zeigt sich: Wer mehr akademische Serien schaut, ist motivierter, ein Studium zu beginnen.
Damit zeigt diese Arbeit erstmalig einen direkten Zusammenhang zwischen Fernsehen und Bildungsmotivation, und identifiziert die Medien im Prozess der Bildungsentscheidung als potenziell ausgleichenden Faktor gegen die soziale Ungleichheit in der Bildung.
Jutta Haubenreich ist seit 2009 Bildungsreferentin bei der Fraunhofer Academy, der Weiterbildungseinrichtung der Fraunhofer-Gesellschaft, und promovierte von 2015 bis 2022 berufsbegleitend am Institut für Kommunikationswissenschaft und Medienforschung der Ludwig-Maximilians-Universität München. Ihr Interesse gilt der Frage, was Menschen motiviert, sich lebenslang zu bilden
Towards more reliable machine learning: conceptual insights and practical approaches for unsupervised manifold learning and supervised benchmark studies
Get PDFGrain boundaries in ultrathin organic semiconductors
Get PDFOrganic semiconductors stand out over their inorganic counterparts, since they can be processed in thinner films and at lower energies, onto nearly any surface, and with tunable optical and electrical properties for specific purposes. These unique properties of organic semiconductors allow to save material, energy, space and cost and make them ideal for applications in customer-specific end-products.
Although organic semiconductors have been implemented in semiconductor devices such as organic solar-cells, organic light-emitting diodes, organic field-effect transistors or sensors for years, these devices still show inferior performances compared to inorganic devices, especially in terms of reduced mobility, efficiency, reproducibility and stability. It is widely accepted that grain boundaries in organic semiconductors are one of the main responsibles for these
drawbacks, since they act as trapping, recombination and/or degradation sites. However, why and how grain boundaries emerge (structurally and energetically), and which properties of grain boundaries mainly influence the device performance is still under investigation.
Since addressing these questions will help to control charge transport in organic semiconductors and improve device performance, this work presents a fundamental investigation of grain boundaries in monolayer-thin films of an organic small molecule. These films stand out due to high crystallinity and atomically smoothness across grain boundaries. This, as well as their thinness, allows to characterize single grains and grain boundaries at the location where charge
transport takes place in organic field-effect transistors, namely at the semiconductor-insulator interface.
By Kelvin probe force microscopy (KPFM) grain boundaries are found as a first result to act as energy barriers or valleys, and different thin-film application techniques are presented resulting in films in which either a specific type of grain boundary predominates, or in films where barriers and valleys coexist. While it is particularly advantageous for future experiments to be able to control the existence of different types of grain boundaries in organic materials, the films with both types prove the fundamental difference between energy barriers and valleys.
KPFM measurements not only allow a qualitative differentiation of barriers and valleys, but also a quantitative description of „grain boundary heights“. Valley depths and barrier heights can both be decreased by increasing the charge-carrier density in the organic semiconductor-film. However, they only vanish at charge-carrier densities above the typical operating regime of organic solar-cells and organic light-emitting diodes, which underlines the relevance of
investigating charge transport at grain boundaries.
Consequently, time-resolved KPFM measurements are conducted to investigate the trapping and detrapping mechanisms at grain boundaries and other local impurities, as well as their influence on global device parameters. While valleys trap charge carriers in deep traps, barriers backscatter electrons, but also indicate an increased trap-state density at the organic-semiconductor interface, thereby leading to a stronger reduction of charge transport than valleys. Valleys, on the contrary, are found to mainly define the global device parameters such as the turn-on and threshold voltage or the qualitative behavior of hysteresis. This finding underlines the need to be able to control not only the grain-boundary density in organic semiconductors, but also their type and absolute height.
However, since it is challenging to control the emergence and electric properties of grain boundaries in organic semiconductors by experimental methods, an alternative experiment is presented with the aim to manipulate charge transport across grain boundaries by illumination with far-infrared light.
It is assumed that photons from this light source are absorbed, leading to the excitation of charge carriers out of valleys or across barriers and thus to a measurable photocurrent. This photocurrent can be measured energy-resolved by using a modified Fourier transform infrared spectrometer, which allows to detect and characterize grain boundaries even in bulk-like materials.
Finally, charge transport in a novel metal-organic framework is investigated directionally, globally and locally, to put the role of grain boundaries in organic semiconductors into a context. It is found that in this special material grain boundaries do not play an as important role as the stacking direction of single planes of the metal-organic framework.
To summarize, the findings of this work lead toward controlling the properties of grain boundaries in organic semiconductors and their role in organic semiconductor devices such as field-effect transistors, organic solar-cells or organic light-emitting diodes
German secondary data: opportunities and challenges for assessing routine cancer care from a health institution's perspective
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