Rotational Symmetry and Phase Dynamics in Topological Superconductors

Topological superconductors are phases of matter that cannot be adiabatically connected to a trivial superconductor without closing the bulk gap or breaking some fundamental symmetry. Intimately tied to topological superconductors are Majorana bound states. These are neutral zero-energy quasiparticles, localised to boundaries or defects, that are highly sought after as ingredients for quantum computation. In the first part of this thesis, we consider two-dimensional crystalline superconductors with discrete rotational symmetry. These can host zero-dimensional Majorana bound states at their corners, indicative of so-called second-order topology. We establish a bulk-boundary correspondence linking the presence of such Majorana bound states to bulk topological invariants based on momentum-space rotation representations. We thus establish when a topological crystalline superconductor protected by rotational symmetry displays second-order topological superconductivity. Our approach is based on stacked Dirac Hamiltonians, using which we relate transitions between topological phases to the transformation properties between adjacent gapped boundaries. We find that in addition to the bulk rotational invariants, the presence of Majorana bound states in a given geometry depends on the interplay between weak topological invariants and the location of the rotation centre relative to the lattice. In the second part of the thesis, we couple a quantum particle to a topology-changing fermionic bath. Generically, coupling to a bath suppresses the particle's amplitude to tunnel between potential minima, even at zero temperature. While this effect can be neglected for gapped baths, our bath has minima that correspond to different bath topologies. This enforces the bath to undergo gap closing along the tunnelling path. We develop a field theory for this quantum tunnelling problem, linking the instantons describing tunnelling in a bath of d space dimensions to topological boundary modes of systems in d+1 dimensions. We study in detail a d=1 example, inspired by planar Josephson junctions where the particle coordinate is the superconducting phase whose value sets the electronic topology. We find that the topology change suppresses tunnelling by a factor scaling exponentially with the system size. This translates to a correspondingly enhanced suppression of the energy splitting for the lowest-lying states, despite these being linear combinations of states near potential minima where the bath is gapped. Our results help to estimate the influence of charging energy on topological phases arising due to the Josephson effect and, conversely, to assess the potential utility of such topological systems as superconducting qubits. For moderate-sized baths, the incomplete suppression of tunnelling opens the prospect of quantum-mechanical superpositions of many-body states of different topology, including superpositions of states with and without Majorana fermions.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2022

Author dc:creator

• Roberts, Elis

Advisor dc:contributor.advisor

• Beri, Benjamin

• Condensed matter
• Topological phases of matter
• Majorana bound states
• Topological superconductivity

Licence

• https://www.repository.cam.ac.uk/bitstreams/36c9441a-475e-46e1-bd1d-4a6be9bee3ae/download
• http://purl.org/NET/rdflicense/allrightsreserved

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.117682

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/383153