Constructing topological models by symmetrization: A projected entangled pair states study

Symmetrization of topologically ordered wave functions is a powerful method for constructing new topological models. Here we study wave functions obtained by symmetrizing quantum double models of a group G in the projected entangled pair states (PEPS) formalism. We show that symmetrization naturally gives rise to a larger symmetry group G˜ which is always non-Abelian. We prove that by symmetrizing on sufficiently large blocks, one can always construct wave functions in the same phase as the double model of G˜. In order to understand the effect of symmetrization on smaller patches, we carry out numerical studies for the toric code model, where we find strong evidence that symmetrizing on individual spins gives rise to a critical model which is at the phase transitions of two inequivalent toric codes, obtained by anyon condensation from the double model of G˜.

Conformal invariants

This master thesis investigates the moduli of families of curves and the capacities of the Gr¨otzsch and Teichm¨uller rings, which are applied in the main parts of this master thesis. The extremal properties of these rings are discussed in connection with the spherical symmetrization. Applications are given to the study of distortion of quasiconformal maps in the euclidean n-dimensional space.

Magnetic Resonance Experiments with Atomic Hydrogen

In this thesis three experiments with atomic hydrogen (H) at low temperatures T<1 K are presented. Experiments were carried out with two- (2D) and three-dimensional (3D) H gas, and with H atoms trapped in solid H2 matrix. The main focus of this work is on interatomic interactions, which have certain specific features in these three systems considered. A common feature is the very high density of atomic hydrogen, the systems are close to quantum degeneracy. Short range interactions in collisions between atoms are important in gaseous H. The system of H in H₂ differ dramatically because atoms remain fixed in the H₂ lattice and properties are governed by long-range interactions with the solid matrix and with H atoms. The main tools in our studies were the methods of magnetic resonance, with electron spin resonance (ESR) at 128 GHz being used as the principal detection method. For the first time in experiments with H in high magnetic fields and at low temperatures we combined ESR and NMR to perform electron-nuclear double resonance (ENDOR) as well as coherent two-photon spectroscopy. This allowed to distinguish between different types of interactions in the magnetic resonance spectra. Experiments with 2D H gas utilized the thermal compression method in homogeneous magnetic field, developed in our laboratory. In this work methods were developed for direct studies of 3D H at high density, and for creating high density samples of H in H₂. We measured magnetic resonance line shifts due to collisions in the 2D and 3D H gases. First we observed that the cold collision shift in 2D H gas composed of atoms in a single hyperfine state is much smaller than predicted by the mean-field theory. This motivated us to carry out similar experiments with 3D H. In 3D H the cold collision shift was found to be an order of magnitude smaller for atoms in a single hyperfine state than that for a mixture of atoms in two different hyperfine states. The collisional shifts were found to be in fair agreement with the theory, which takes into account symmetrization of the wave functions of the colliding atoms. The origin of the small shift in the 2D H composed of single hyperfine state atoms is not yet understood. The measurement of the shift in 3D H provides experimental determination for the difference of the scattering lengths of ground state atoms. The experiment with H atoms captured in H2 matrix at temperatures below 1 K originated from our work with H gas. We found out that samples of H in H2 were formed during recombination of gas phase H, enabling sample preparation at temperatures below 0.5 K. Alternatively, we created the samples by electron impact dissociation of H₂ molecules in situ in the solid. By the latter method we reached highest densities of H atoms reported so far, 3.5(5)x10¹⁹ cm^-3. The H atoms were found to be stable for weeks at temperatures below 0.5 K. The observation of dipolar interaction effects provides a verification for the density measurement. Our results point to two different sites for H atoms in H2 lattice. The steady-state nuclear polarizations of the atoms were found to be non-thermal. The possibility for further increase of the impurity H density is considered. At higher densities and lower temperatures it might be possible to observe phenomena related to quantum degeneracy in solid.

GROUND STATE AND NON-GROUND STATE SOLUTIONS OF SOME STRONGLY COUPLED ELLIPTIC SYSTEMS

We study an elliptic system of the form Lu = vertical bar v vertical bar(p-1) v and Lv = vertical bar u vertical bar(q-1) u in Omega with homogeneous Dirichlet boundary condition, where Lu := -Delta u in the case of a bounded domain and Lu := -Delta u + u in the cases of an exterior domain or the whole space R-N. We analyze the existence, uniqueness, sign and radial symmetry of ground state solutions and also look for sign changing solutions of the system. More general non-linearities are also considered.

Messung der CP-verletzenden Asymmetrie von K + - pi + - pi 0 pi 0 mit dem NA48-Detektor

Bis heute ist die Frage nicht geklärt, warum bei der Entstehung des Universums Materie gegenüber der Antimaterie bevorzugt war und das heutige Materieuniversum entstanden ist. Eine Voraussetzung für die Entstehung dieser Materie-Antimaterie-Asymmetrie ist die Verletzung der Kombination von Ladungs- (C) und Punktsymmetrie (P), die CP-Verletzung. CP-Verletzung kann sich unter anderem in den Zerfällen K+- -> pi+- pi0 pi0 zeigen. Die NA48/2"=Kollaboration zeichnete während den Jahren 2003 und 2004 über 200~TB Daten von Zerfällen geladener Kaonen auf. In dieser Arbeit wurde die CP"=verletzende Asymmetrie der Zerfälle K+- -> pi+- pi0 pi0 mit über 90~Millionen ausgewählten Ereignissen aus diesem Datensatz gemessen. Vorhersagen im Standardmodell der Teilchenphysik sagen hier eine CP"=verletzende Asymmetrie in der Größenordnung zwischen $10^{-6}$ und $10^{-5}$ voraus. In Modellen außerhalb des Standardmodells kann es aber auch größere Asymmetrien geben. Das NA48/2"=Experiment war darauf ausgelegt, mögliche systematische Unsicherheiten zu begrenzen. Um dies zu erreichen, wurden positive und negative Kaonen simultan an einem Target erzeugt und ihr Impuls durch ein Strahlsystem mit zwei Strahlengängen auf ca. $60~GeV/c$ begrenzt. Die Strahlen wurden auf wenige Millimeter genau überlagert in die Zerfallsregion geleitet. Die Strahlengänge von positiven und negativen Kaonen sowie die Polarität des Magneten des Impulsspektrometers wurden regelmäßig gewechselt. Dies erlaubte eine Symmetrisierung von Strahlführung und Detektor für positive und negative Kaonen während der Analyse. Durch ein Vierfachverhältnis der vier Datensätze mit den unterschiedlichen Konfigurationen konnte sichergestellt werden, dass alle durch Strahlführung oder Detektor erzeugten Asymmetrien sich in erster Ordnung aufheben. Um die unterschiedlichen Produktionsspektren von positiven und negativen Kaonen auszugleichen wurde in dieser Arbeit eine Ereignisgewichtung durchgeführt. Die Analyse wurde auf mögliche systematische Unsicherheiten untersucht. Dabei zeigte sich, dass die systematischen Unsicherheiten in der Analyse deutlich kleiner als der statistischer Fehler sind. Das Ergebnis der Messung des die CP-verletzende Asymmetrie beschreibenden Parameters $A_g$ ist: begin{equation} A_g= (1,2 pm 1,7_{mathrm{(stat)}} pm 0,7_{mathrm{(sys)}}) cdot 10^{-4}. end{equation} Diese Messung ist fast zehnmal genauer als bisherige Messungen und stimmt innerhalb ihrer Unsicherheit mit dem Standardmodell überein. Modelle, die eine größere CP-Verletzung in diesem Zerfall vorhersagen, können ausgeschlossen werden.

Raman and infrared spectra of dimethyl ether 13C-isotopologue (CH3O13CH3) from a CCSD(T) potential energy surface

So far, no experimental data of the infrared and Raman spectra of 13C isotopologue of dimethyl ether are available. With the aim of providing some clues of its low-lying vibrational bands and with the hope of contributing in a next spectral analysis, a number of vibrational transition frequencies below 300 cm−1 of the infrared spectrum and around 400 cm−1 of the Raman spectrum have been predicted and their assignments were proposed. Calculations were carried out through an ab initio three dimensional potential energy surface based on a previously reported one for the most abundant dimethyl ether isotopologue (M. Villa et al., J. Phys. Chem. A 115 (2011) 13573). The potential function was vibrationally corrected and computed with a highly correlated CCSD(T) method involving the COC bending angle and the two large amplitude CH3 internal rotation degrees of freedom. Also, the Hamiltonian parameters could represent a support for the spectral characterization of this species. Although the computed vibrational term values are expected to be very accurate, an empirical adjustment of the Hamiltonian has been performed with the purpose of anticipating some workable corrections to any possible divergence of the vibrational frequencies. Also, the symmetry breaking derived from the isotopic substitution of 13C in the dimethyl ether was taken into account when the symmetrization procedure was applied.

Problems for P-Monge-Ampere Equations

2010 Mathematics Subject Classification: 35A23, 35B51, 35J96, 35P30, 47J20, 52A40.