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

TL;DR

This paper models tunneling-coupled one-dimensional quantum rings, revealing how topology influences system energetics, induces frustration, and leads to a topological quantum phase transition controlled by magnetic field and writhe.

Contribution

It introduces a novel topological model for quantum rings, incorporating writhe as a key quantity affecting energetics and phase transitions.

Findings

Topology affects the energetics of quantum ring systems.

Writhe determines the system's response to magnetic fields.

A topological quantum phase transition involving the ground state wave function is identified.

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.