Let us look at a very naive model of an open carbon nanotube↓ (a schematic representation of a “capped” tube is given in Fig. 4.4↑). This is a cylindrical structure made of carbon; a extremely simplified model would be a thin conducting shell, which we shall investigate here. Suppose the radius of the tube is $\rho_{c}$ , and that electrons are confined inside the tube’s wall. Working in cylindrical coordinates, where the $z$ -axis is parallel to the axis of the tube, we find that the electronic states are described by the time-independent Schrödinger equation $$-\frac{\hbar^{2}}{2m}\left[\frac{1}{\rho_{c}^{2}}(\partial_{\phi})^{2}+(\partial_{z})^{2}\right]\psi(\phi,z)=E\psi(\phi,z),$$ with the standard solution by separation of variables, using periodic boundary conditions on $\phi$ , $$\begin{equation} \psi(\phi+2\pi,z)=\psi(\phi,z),\label{eq:CE:pbc}\end{equation}$$ we find $$\begin{align} \psi & =e^{ikz}e^{im\phi},\quad k\in\mathbb{R},m\in\mathbb{Z},\label{eq:CE:sepAHB1}\\ E & =\frac{\hbar^{2}}{2m}\left[k_{z}^{2}+\frac{m^{2}}{\rho_{c}^{2}}\right].\nonumber \end{align}$$ We now thread a homogeneous magnetic field through the inside of the tube, parallel to the $z$ -axis. This can’t penetrate the conducting wall, so can it influence the electrons? It is straightforward to show [example sheet] that inside the skin of the tube we have, using continuity of $\vec{A}$ , $$\vec{A}=\frac{B}{2}\rho_{c}\hat{\phi}.$$ This means that we have to modify the Schrödinger equation to read $$-\frac{\hbar^{2}}{2m}\left[\left(\frac{1}{\rho_{c}}\partial_{\phi}+\frac{ieB\rho_{c}}{2\hbar}\right)^{2}+(\partial_{z})^{2}\right]\psi(\phi,z)=E\psi(\phi,z).$$ Separating variables again, we find the same behaviour for the $z$ variable as (↓), but for the $\phi$ variable we have a different equation $$\left(\partial_{\phi}+\frac{ieB\rho_{c}^{2}}{2\hbar}\right)^{2}\Phi(\phi)=\kappa^{2}\rho_{c}^{2}\Phi(\phi).$$ Thus $$\Phi=\exp\left[i\kappa\phi-\frac{ieB\rho_{c}^{2}}{2\hbar}\phi\right].$$ Imposing the periodic boundary conditions (↓), we find that $$\kappa=m+\frac{eB\rho_{c}^{2}}{2\hbar},\quad m\in\mathbb{Z}.$$ So the spectrum has changed to $$E=\frac{\hbar^{2}}{2m}\left[k_{z}^{2}+\frac{\left(m+\frac{eB\rho_{c}^{2}}{2\hbar}\right)^{2}}{\rho_{c}^{2}}\right].$$ The quantity $$\frac{eB\rho_{c}^{2}}{2\hbar}=\frac{e}{2h}B\pi\rho_{c}^{2}$$ is exactly the number of magnetic flux quanta passing through the cross-sectional area of the tube ($e/2h$ is the “quantum of flux”). The spectrum is unchanged if this equals an integer, but for fractional number of flux quanta we have an observable effect. [V]  [V] For a real experiment on the effect discussed, see A. Bachtold et al, Nature 397 (1999) 673.

We reach the surprising conclusion that even though no magnetic field penetrates where the electrons are, the coupling to $\vec{A}$ rather than the physical field means that the wave function is sensitive to enclosed flux, since the space is “topologically nontrivial”. This is a bound state version of the Aharonov-Bohm effect. The standard version will be discussed later on when we have discussed path integrals.↓