We have Bogoliubov-De Gennes (BdG) equation, $$\left(\begin{array}{cc} \mathbf{p}\cdot\boldsymbol{\sigma} - V & \Delta_{0}e^{i\phi} \\ \Delta_{0}e^{-i\phi} & V - \mathbf{p}\cdot\boldsymbol{\sigma} \end{array} \right) \left( \begin{array}{c}u\\v\end{array}\right) = \mathcal{E}\left( \begin{array}{c}u\\v\end{array}\right)$$ with definition, $$\mathbf{p}\cdot\boldsymbol{\sigma} = -i\hbar v_{F}(\partial_{x}\sigma_x + \partial_y\sigma_y)$$ We solve the equation through ansatz, $$\Phi = {\rm Const}\cdot \exp(iqy + ik_{0}x + \kappa x)$$ for $k_{0} > 0$ and $\kappa > 0$. The question is: How to use the assumption that $V >> \Delta_{0}, \mathcal{E}$ in the calculations to get nice looking formulas like in Beenakker (A13)?

  • $\begingroup$ Did you at least tried to find $k_{0}$, $q$ and $\kappa$ ? The assumption come from their expressions ... $\endgroup$
    – FraSchelle
    Sep 2 '14 at 9:14
  • $\begingroup$ Yes I did, but it involves double square root (nested). You can find Yourself expressions for $k_{0}$ and $\kappa$ in Mathematica which depend on $q$. $\endgroup$
    – WoofDoggy
    Sep 2 '14 at 20:04

When you write down the formulas A15-A18, apply this assumption to these definitions and put it into wave functions, which are first calculated by solving BdG equation. You can reach the Beenakker results.


This is actually exactly the same question I was asking myself a while ago, and it took me quite some time to figure it out. What I ended up doing was:

  • diagonalise the BdG Hamiltonian in Mathematica
  • solve the expressions for the eigenvalues for $\kappa$
  • neglect terms proportional to $\kappa^2$, $\kappa \Delta$ in the expressions for the eigenvectors, expand them to first order in $\kappa$, insert the expression for $\kappa$ you got from the eigenvalues
  • simplify everything (using the expression for $\epsilon$ and approximate $\hbar v k \approx E_F+U$) and arrive at Beenakker's simple-looking expressions (tedious)

I hope that helps some, in case you haven't figured it out yourself in the meantime. If you need more info/details don't hesitate to ask, I can elaborate some more if needed!


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