## Abstract

Microscopic response theory is used to study the linear electromagnetics of a Möbius jellium quantum wire. In the energy range of interest ($\hbar \omega \sim 1 {-} 200 \;{\rm meV}$), the wire can be considered as a Möbius band with complete transverse electron confinement (${\sim}$ one-level quantum well with width $w \sim k_F^{- 1} \ll \lambda = c/\omega$). The electronic energy eigenvalues and eigenstates are calculated numerically via a diagonalization of a scalar determinant in which the relevant matrix elements of the geometrical potential (given in terms of string curvature and torsion) occur. The exact lowest lying states for a 50 nm long heavily doped GaAs wire are compared to those obtained in lowest-order perturbation theory. The sequence (from the bottom) EOEEOEOO… of the even (E) and odd (O) parity states is not the even–odd sequence EOEOEOEO… of a harmonic potential. The interchanges originate in the peak structure of the geometrical potential. Significant energy splitting of the lowest lying eigenvalues of the Möbius wire is observed when compared to the degenerate energy eigenvalues of a corresponding circular wire. The frequency dependence of the conductivity tensor components are calculated, and the peak structure is identified from the $E \to E$, and $O \to O$ and $E \leftrightarrow O$ transitions. It is shown that elastic light scattering might be a versatile probe for determining the Möbius wire’s electronic-state structure.

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