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- Kvorning, Thomas, 1986-
(författare)
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Non-local behaviour from local interactions
- 2017
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- With the discovery of the quantum Hall effect more than thirty years ago, a whole new field emerged—that of topological quantum matter. This field is now a very mature one, and many different aspects are covered in the literature. The main text of this thesis introduces the field and gives a background to topological quantum matter, as well as topological aspects of superconductivity and the Abelian fractional quantum Hall (FQH) states. Together with the main text there are five articles that address five different questions, all connected to topological quantum matter.In the first article, representative wave functions for the Abelian FQH states are calculated using conformal field theory methods. Before this paper was published, similar constructions had been restricted to flat geometries, but in this paper we generalize the analysis to the simplest curved geometry, namely the sphere. On top of being of interest for numerical studies (which usually are performed on a sphere), the response of the FQH liquids to curvature can be used to detect a topological quantity, the shift, which is the average orbital spin of the constituent electrons.In the second article, we construct an effective field theory for the two-dimensional spinless, chiral p-wave superconductor that faithfully describes the topological properties of the bulk state, and also provides a model for the subgap states at vortex cores and edges. In particular, it captures the topologically protected zero-modes and has the correct ground state degeneracy on the torus.In the third paper, tools for a hydrodynamic theory for insulators in three dimensions are derived. Specifically, we use functional bosonization to write insulators as a condensation phase of the U(1) gauge theory obtained in the functional bosonization language.In the fourth paper, we investigate the edge Majorana modes in the two-dimensional chiral p-wave superconductor. We define the model on surfaces with different geometries—the annulus, the cylinder, the Möbius band, and a cone—and with different configurations of magnetic flux threading holes in these surfaces. In particular, we address the following question: Given that, in the absence of magnetic flux, the ground state on the annulus does not support Majorana modes, while the one on the cylinder does, how is it possible that the conical geometry can interpolate smoothly between the two?In the fifth and last article, we demonstrate that two-dimensional chiral superconductors on curved surfaces spontaneously develop magnetic flux. We propose this geo-Meissner effect as an unequivocal signature of chiral superconductivity that could be observed in layered materials under stress. We also employ the effect to explain some puzzling questions related to the location of Majorana modes.
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| 2. |
- Spånslätt, Christian, 1987-
(författare)
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Topological states of matter in low dimensions
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- A central theme in condensed matter physics is the classification and characterization of states of matter. In the recent decades, it has become evident that there exists a large class of quantum mechanical systems that must be classified according to properties deeply rooted in the mathematical field of topology, rather than in terms of which symmetries they break. Together with rapid technological developments, such topological states of matter pose a promising path for a fundamentally new generation of quantum devices and exotically engineered materials, with applications in quantum metrology, quantum sensing, and quantum computations.This doctoral thesis comprises a study of a general theoretical framework describing topological states of matter, followed by applications in systems of low dimension. Together, these two parts form the foundation for the following accompanying papers:PAPERS I-II concerns Majorana zero modes in various Josephson junction setups of one-dimensional topological superconductors. In addition to determining the mathematical conditions for the existence of such exotic states, the papers also provide experimental proposals for their detection. PAPER III deals with the nature of a widely used model of a synthetically engineered one-dimensional topological superconductor. It is shown that in a certain parameter limit, the superconducting order parameter obtains a geometric contribution, which originates from the directional nature of the Rashba spin-orbit coupling. This geometrical dependence is argued to manifest itself in various Josephson junction setups, and in particular as a direct connection between the charge current density and the local curvature.PAPER IV proposes a method of constructing non-local order parameters for two-dimensional Chern insulators, and describes how such operators can be used to distinguish between different topological sectors.
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