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C 2020 by Karmela Padavic-Callaghan. All Rights Reserved c 2020 by Karmela Padavic-Callaghan. All rights reserved. HOLLOW CONDENSATES, TOPOLOGICAL LADDERS AND QUASIPERIODIC CHAINS BY KARMELA PADAVIC-CALLAGHAN DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 2020 Urbana, Illinois Doctoral Committee: Professor Michael Stone, Chair Professor Smitha Vishveshwara, Director of Research Assistant Professor Bryce Gadway Professor Alexey Bezryadin Abstract This thesis presents three distinct topics pertaining to the intesection of condensed matter and atomic, molec- ular and optical (AMO) physics. We theoretically address the physics of hollow Bose-Einstein condensates and the behavior of vortices within them then discuss localization-delocalization physics of one-dimensional quasiperiodic models, and end by focusing on the physics of localized edge modes and topological phases in quasi-one-dimensional ladder models. For all three topics we maintain a focus on experimentally acces- sible, physically realistic systems and explicitly discuss experimental implementations of our work or its implications for future experiments. First, we study shell-shaped Bose-Einstein condensates (BECs). This work is motivated by experiments aboard the International Space Station (ISS) in the Cold Atom Laboratory (CAL) where hollow condensates are being engineered [1]. Additionally, shell-like structures of superfluids form in interiors of neutron stars and with ultracold bosons in three-dimensional optical lattices. Our work serves as a theoretical parallel to CAL studies and a step towards understanding these more complex systems. We model hollow BECs as confined by a trapping potential that allows for transitions between fully-filled and hollow geometries. Our study is the first to consider such a real-space topological transition. We find that collective mode frequencies of spherically symmetric condensates show non-monotonic features at the hollowing-out point. We further determine that for fully hollow spherically symetric BECs effects of Earth's gravity are very destructive and consequently focus on microgravity environments. Finally, we study quantized vortices on hollow condensate shells and their response to system rotation. Vortex behavior is interesting as a building block for studies of more complicated quantum fluid equilibration processes and physics of rotating neutron stars interiors. Condensate shell's closed and hollow geometry constrains possible vortex configurations. We find that those configurations are stable only for high rotation rates. Further, we determine that vortex lines nucleate at lower rotation rates for hollow condensates than those that are fully-filled. Second, we analyze the effects of quasiperiodicity in one-dimensional systems. Distinct from truly disor- dered systems, these models exhibit delocalization in contrast to well-known facts about Anderson localiza- tion [2]. We study the famous Aubry-Andre-Harper (AAH) model, a one-dimensional tight-binding model ii that localizes only for sufficiently strong quasiperiodic on-site modulation and is equivalent to the Hofstadter problem at its critical point. Generalizations of the AAH model have been studied numerically and a gen- eralized self-dual AAH model has been proposed and analytically analyzed by S. Ganeshan, J. Pixley and S. Das Sarma (GPD) in Ref. [3]. For extended and generalized AAH models the appearance of a mobility edge i.e. an energy cut-off dictating which wavefunctions undergo the localization-delocalization transition is expected. For the GPD model this critical energy has been theoretically determined. We employ transfer matrices to study one-dimensional quasiperiodic systems. Transfer matrices characterize localization physics through Lyapunov exponents. The symplectic nature of transfer matrices allows us to represent them as points on a torus. We then obtain information about wavefunctions of the system by studying toroidal curves corresponding to transfer matrix products. Toroidal curves for localized, delocalized and critical wavefunctions are distinct, demonstrating a geometrical characterization of localization physics. Applying the transfer matrix method to AAH-like models, we formulate a geometrical picture that captures the emer- gence of the mobility edge. Additionally, we connect with experimental findings concerning a realization of the GPD model in an interacting ultracold atomic system. Third, we consider a generalization of the Su-Schrieffer-Heeger (SSH) model. The SSH chain is a one- dimensional tight-binding model that can host localized bound states at its ends. It is celebrated as the simplest model having topological properties captured by invariants calculated from its band-structure. We study two coupled SSH chains i.e. the SSH ladder. The SSH ladder has a complex phase diagram determined by inter-chain and intra-chain couplings. We find three distinct phases: a topological phase hosting localized zero energy modes, a topologically trivial phase having no edge modes and a phase akin to a weak topological insulator where edge modes are not robust. The topological phase of the SSH ladder is analogous to the Kitaev chain, which is known to support localized Majorana fermion end modes. Bound states of the SSH ladder having the same spatial wavefunction profiles as these Majorana end modes are Dirac fermions or bosons. The SSH ladder is consequently more suited for experimental observation than the Kitaev chain. For quasiperiodic variations of the inter-chain coupling, the SSH ladder topological phase diagram reproduces Hofstadter's butterfly pattern. This system is thus a candidate for experimental observation of the famous fractal. We discuss one possible experimental setup for realizing the SSH ladder in its Kitaev chain-like phase in a mechanical meta-material system. This approach could also be used to experimentally study the Hofstadter butterfly in the future. Presented together, these three topics illustrate the richness of the intersection of condensed matter and AMO physics and the many exciting prospects of theoretical work in the realm of the former combining with experimental advances within the latter. iii To Mom, Dad and BT. iv Acknowledgments I wrote one of my college admissions essays about the first physics class I had ever taken. I described what I learned in that class { that the real world can be modeled through mathematical equations { as inspiration for every piece of work I had undertaken afterwards. The tie between my junior-in-high-school self, having just moved to a far away foreign country while my family remained in Croatia, and physics was so strong that I wanted high-powered admissions officers to judge me based on those first few moments of amazent with it that I had experienced in the seventh grade. In other words, for more than a decade I have wanted much else other than to be a physicist and do physics. This thesis is a culmination of that desire and there are many who have made it possible for me to grow into the physicist I decided I would become all the way back then. First and foremost, my gratitude goes to Prof. Smitha Vishveshwara who has advised, mentored and guided me since my first day in Urbana-Champaign. The norm within her group has always been to dream big, to chase odd topics whenever we really cared for them, to never say no to an unconventional idea and always work in good faith, with a cheery attitude and lots of optimism. In the early years of my doctoral work, this was very valuable to me. Having worked with Prof. Vishveshawara on many outreach projects involving the intersection of physics and the arts, I will also always appreciate her ability to wear many hats, play many roles (sometimes quite literally) and pull it all off with a smile. I am thankful for her showing me that this is who a physicst can be in addition to all the academic excellence she has corralled my way. Regardless of friendly attitudes and long-dreamed dreams, graduate school has been a long and chal- lenging process, full of self-discovery and sometimes painful growth along all axes of my being a person. I do not think I would have been able to take this all on without the endless emotional and practical support from my family and spouse. As they have all lived far away from me for the vast majority of my time in Urbana-Champaign, even just the endless conversations and infinite text-message chains have meant a world to me. I have traveled a tremendous amount in the past six years and my husband's willingness to tag along or take his own doctoral work on trains, buses and planes has been immensely helpful. Working on our degrees at the same time has so often been both inspiring and comforting. (Bennett, you are the true MVP v of this endeavor.) Over the years I have had many great teachers and collaborators. In this vein, I owe thanks to Profs. B. Gadway, E. Fradkin, R. Leigh, T. Hughes and M. Stone at UIUC and Profs. C. Prodan at NJIT, C. Lannert at Smith College, N. Lundblad at Bates College, D. Sen at IIS, Dr. Kuei Sun at Univeristy of Texas and Dr. Wade DeGottardi. Especially I want to thank the members of my thesis committe for devoting some of their time to my work. I have also been inspired and humbled by many discussions with my group-mates Varsha Subrahmanyan and Suraj S. Hegde. I mentored Varsha in her first year of graduate school and have always been grateful for the coincidence of her subsequently joining the same research group { I have always been amazed by her work as a physicist and her curiosity and kindness as a person. I fondly remember a spring semester in which we attended a comic book convention together, and our many lunches at the Red Herring. Suraj, on the other hand, has been something of an informal mentor to me for years and truly showed me the ropes on so much of the work presented below.
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