Topological Effects in Tunneling-Coupled Systems of One-Dimensional Quantum Rings

Abstract

Using a model of idealized, crossed one-dimensional quantum wires we construct a novel model for a single electron on tunneling-coupled systems of one-dimensional quantum rings. We explore and find that topology can affect the energetics of the system, and can introduce frustration in the three ring case. We also study the special cases of an external magnetic field that controls the complex phase of the tunneling matrix element, and introduce the knot theory concept of "writhe" as a new topological quantity for distinguishing one hole systems. We find that writhe not only determines the energetic response of the system to magnetic field strength, but is also responsible for a single particle topological quantum phase transition involving the ground state wave function winding number.

Automatic Learning of Topological Phase Boundaries

“…One such configuration is a single self-connected wire wrapped into a trefoil knot (Figure 6). If one allows δ-function tunnel coupling where the wire crosses itself, interesting effects of frustration and topology appear [32]. These effects are made richer by applying a magnetic field such that the tunnel-coupling matrix elements pick up a complex Aharonov-Bohm phase factor.…”