The Energy Spectrum of Bound states in the Vortex Core of Type II Superconductors

Question

In the calculation of the so-called Caroli-de Gennes energy states in a type II superconductor the following Bogoliubov-de Gennes Hamiltonian is used: $$ H=- (\frac{1}{2m}(\nabla+ie\mathbf{A}\sigma_z)^2+\mu)\sigma_z+\Delta(r)e^{i\theta} \sigma_+ + \Delta(r)e^{-i\theta}\sigma_- $$ to solve the eigenvalue problem $H\psi=E\psi$ where $\psi=(u,v)^T$ is the wavefunction in particle-hole space and $\Delta(r)$ is real. Then, to get rid of the phase in front of order parameter a transformation of the form $\psi\rightarrow e^{-i\sigma_z\theta/2}\psi$ is applied and one will have the following transformed Hamiltonian: $$H=-(\frac{1}{2m}(\nabla+ie\mathbf{A}\sigma_z+i\sigma_z \frac{\hat{\theta}}{2r})^2+\mu)\sigma_z+\Delta(r)\sigma_x$$ Both the original paper (Caroli,de Gennes,Matricon, 1964) and de Gennes' book proceed to neglect the $\mathbf{A}$ term since it is smaller than $\hat{\theta}$ term and they say that we can neglect all of the magnetic field effects and they solve this Hamiltonian: $$H=-(\frac{1}{2m}(\nabla+i\sigma_z \frac{\hat{\theta}}{2r})^2+\mu)\sigma_z+\Delta(r)\sigma_x$$

Now, my problem is that apparently we cannot neglect all of the effects of $\mathbf{A}$ as I describe below: The magnetic flux quantization condition says that: $$\oint d\varphi=-2e \oint \mathbf{A}.d\mathbf{r}$$ Where $\varphi$ is the phase of order parameter. At arbitrarily long distances from the vortex the magnetic field is zero and hence $\mathbf{A}$ is a pure gauge but this does not mean that it is zero. Using the flux condition we can see that for our vortex where $\varphi(\mathbf{r})=\theta$ we have: $$\mathbf{A}=-\frac{1}{2er}\hat{\theta}$$ And therefore the original Hamiltonian after the transformation $\psi\rightarrow e^{-i\sigma_z\theta/2}\psi$ takes the form: $$H=-(\frac{\nabla^2}{2m}+\mu)\sigma_z+\Delta(r)\sigma_x$$ which is much simpler than the one mentioned in the original paper. The problem now, is that without the $\hat{\theta}$ term the spectrum will be different if one follows the method which the authors use in the paper.

So, apparently I have made a mistake, but everything in my statement seems clear to me.

Answer 1

In a type-II superconductor the fraction of the magnetic flux in the vortex core is only a small fraction of the total flux because the magnetic penetration depth is usually much larger than that the coherence length (size of the core). It is only in the core that the bound state wavefunctions are substantial. Your formula $A_\theta \propto 1/r$ only holds for large $r$ and not in the core. In the core $A_\theta$ is small, so it's not unreasonable to neglect it.

If I remember correctely (it's some years since I worked this out) one can express the CdG energies as $E_l=\hbar \omega_0 l$ where $\omega_0$ is the rotation frequency of the superfluid at the edge of the core and $\hbar l$ is the angular momentum. When one includes the effect of the $B$ field, and assumes it roughly uniform over the size of the core, then there is a correction to the CdG spectrum of $\Delta E_l=\hbar \omega_c l$ where $\omega_c$ is the cyclotron frequency of the charge-$e$ core quasiparticles in the $B$ field.