Ground state wave function of Andreev bound states
Dear Kwant community, (I don't know if here is a good place to ask this question. Let me know if it is inadequate. ) I calculate the eigen vectors of a close system containing a large swave superconductor and a small nonSC region. The spin is conserved, i.e. eigen states can be grouped into pure spin up and down. Can I infer the ground state wavefunction by the eigen vectors I obtained from kwant? The nonSC region is like a quantum dot. As calculated in [1], if there's a single orbital level and it couples to SC lead, the state in the quantum dot will become ground state >=v*up,down>+u0>, excited states up> and down>, and the highest excited state +>=uup,down>+v*0>. However that is an effective model. I'm interested in how the ground state > looks like in tight binding model. Here is my procedure/reasoning: kwant can calculate BdG Hamiltonian on the lattice. The entire system has a ground state g>, which is a product state of: g> = (u1psi1,up>psi1,down> + v10>)*(u2psi2,up>psi2,down> + v20> )* ... . Here 0> is the state with no electron. The eigen vectors I obtained from kwant should be interpreted as excitations on g>. Now I find a eigen vector localized in nonSC region. It is spinup, with spinup electron part [e1, e2, e3, ...] and hole part [h1, h2, h3, ...] at lattice points 1,2,3,... . These coefficients {e1,h1, e2,h2, e3, h3, ...} can be used to construct a fermionics operator gamma_up. Its Kramer's partner gamma_down would have spindown part [e1, e2, e3, ...] and hole part [h1, h2, h3, ...] at each lattice point. By applying gamma_up and gamma_down on 0> I can get one of the states in g>. It turns out to be [sum_i {e_i h_i } + sum_i,j {e_i e_j a_i^dagger a_j^dagger}] 0>, where i,j are lattice point index, up to a normalization factor. Just as a reference, the onsite and hopping Hamiltonian are: def onsite(): (4 * p.t  p.mu + p.pot[x] ) * pauli.szs0 + p.delta[x]* pauli.sxs0 def hop(): p.t * pauli.szs0 p.pot[x] is potential. p.delta[x] is pairing potential. pauli.szs0 is the tensor product of sigma_z (eh basis) and identity (spinbasis). Thanks, Chien [1] "Selfconsistent description of Andreev bound states in Josephson quantum dot devices", Phys. Rev. B 79, 224521 (2009)
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ChienAn Wang