

In [1]:

    
%pylab inline
from ipywidgets import interact
import scipy.sparse as scsp
import scipy.sparse.linalg as scspl
from topobp import *









    



Populating the interactive namespace from numpy and matplotlib

Topological insulators II/02: Proximity, effective p-wave and experiment

Model two site proximity effect

Build a model Hamiltonian where one site is characterised by an s-wave type superconducting orderparameter $\Delta$ and another site is coupled to it vie conventional hopping of sterngth $t$ and has onsite energy $\varepsilon$.
Calculate the spectrum using the $\mathcal{H}_{BdG}$ in the in the case of zero onsite potential and show that for small hoppingsterngth you recover the $t^2/\Delta$ law.
Calculate the spectrum for finite $\varepsilon$ onsite potential on the dot and calculate the average charge on the dot for the lowest 4 levels.

Calculate the phase diagram of the Lutchyn wire

Following a similar rute as we took for the HgTe edge investigate the phase diagram and finite size spectrum of the Lutchyn wire. Use the following assumptions

a magnetic field of strength $B$ in the z-direction
a spin dependent hopping $\alpha$ proportional to $i\sigma_y$
a spin independent hopping $\gamma$
s-wave superconductivity of strength $\Delta$

Calculate the single particle spectrum without superconductivity.
Calculate the winding and explore the phase diagram of the system.
Consider arbitrary direction magnetic field in a finite size sample and observe the behaviour of the zero modes.