Numerical Investigation of Stub Length Influence on Dispersion Relations and Parity Effect in Aharonov-Bohm Rings

TL;DR

This study numerically explores how varying stub length ratios in Aharonov-Bohm rings affects their dispersion relations and parity effects, revealing sensitive dependence and breakdown of simple parity predictions.

Contribution

It provides a detailed numerical analysis of how stub length influences dispersion relations and parity effects in mesoscopic rings, challenging previous theoretical predictions.

Findings

Changing stub length ratio shifts dispersion branches.

Parity breakdown occurs at or below v/u=0.205.

Stub length tuning affects persistent current behavior.

Abstract

Aharonov-Bohm (AB) rings with side-attached stubs are model systems for quantum-interference studies in mesoscopic physics. The geometry of such systems, particularly the ratio of stub length ($v$) to ring circumference ($u$), can significantly alter their electronic states. In this work, we solve Deo's transcendental mode-condition equation (Eq. 2.15 from Deo, 2021 [Deo2021]) numerically -- using Python's NumPy and SciPy libraries -- for ring-stub geometries with $v/u = 0.200, 0.205,$ and $0.210$ to generate dispersion relations ($ku$ vs. $\Phi/\Phi_{0}$) and the underlying function $\text{Re}(1/T)$. We find that changing $v/u$ shifts several of the six lowest calculated dispersion branches, with $\Delta(ku)$ up to approximately $0.34$ for the 6th branch at $\Phi=0$ when comparing $v/u=0.200$ and $v/u=0.210$. This also alters gap widths. Notably, for $v/u=0.205$ and $v/u=0.210$, the 5th and 6th consecutive calculated modes both exhibit paramagnetic slopes near zero Aharonov-Bohm flux, indicating the parity breakdown initiates at or below $v/u=0.205$. This directly demonstrates a breakdown of the simple alternating parity effect predicted by Deo (2021) [Deo2021]. These results highlight the sensitivity of mesoscopic ring spectra to fine-tuning of stub length, with potential implications for experimental control of persistent currents, as further illustrated by calculations of the net current.