The one dimensional semi-classical Bogoliubov-de Gennes Hamiltonian with PT symmetry: generalized Bohr-Sommerfeld quantization rules

A Ifa,M Rouleux

doi:10.1088/1742-6596/1194/1/012049

Abstract

We present a method for computing first order asymptotics of semiclassical spectra for 1-D Bogoliubov-de Gennes (BdG) Hamiltonian from Supraconductivity, which models the electron/hole scattering through two SNS junctions. This involves: 1) reducing the system to Weber equation near the branching point at the junctions; 2) constructing local sections of the fibre bundle of microlocal solutions; 3) normalizing these solutions for the “flux norm” associated to the microlocal Wronskians; 4) finding the relative monodromy matrices in the gauge group that leaves invariant the flux norm; 5) from this we deduce Bohr-Sommerfeld (BS) quantization rules that hold precisely when the fibre bundle of microlocal solutions (depending on the energy parameter E) has trivial holonomy. Such a semi-classical treatement reveals interesting continuous symetries related to monodromy.

Highlights

The physical mechanism goes roughly as follows: An electron e− moving in the metallic lead, say, to the right, with energy 0 < E ≤ ∆ below the gap and kinetic energy K+(x) = μ(x) + E2 − ∆(x)2 is reflected back as a hole e+ from the supraconductor, injecting a Cooper pair into the superconducting contact
When inf[−L,L] K−(x) > 0 it bounces along the lead to the left and picks up a Cooper pair in the supraconductor, transforming again to the original electron state, a process known as Andreev reflection
The hole can propagate throughout the lead only if inf[−L,L] μ(x) ≥ E

Summary

Bohr-Sommerfeld quantization rules

{Gjε,,♭ω = Fεj,,ω♭,+ − Fεj,,ω♭,− : interpreted as a basis of j∈ the microlocal co-kernel of P near a, a′. Following [10], we introduce Gram matrix Gρ of vectors U1ρ and U2ρ in this basis, namely G =. The condition det(G(ρ)) = 0 means that U1 is colinear to U2, i.e. there is a global section of. Recall eρ12 = ρ dρ −; for ρ = +1 (electronic state) we get Imd+21 = 0, that is sin τ (+)(h) h. We eventually obtain BS by “surgery”: namely (ignoring tunneling) we cut and paste the half-bicharacteristic Λ>E,+ in the upper-half plane ξ > 0 with its symmetric part.

Relative monodromy matrices