Superconducting tight binding - bertdupe/Matjes GitHub Wiki
The implemented tight-binding formalism including the superconductivity follows mainly the formalism as described in the book Bogoliubov-de Gennes Method and Its Applications and PRB 76 014512 with the aim to describe skyrmion systems as in Commun. Phys. 2, 126.
An issue when comparing the above publications is the inconsistency of their notation, especially with respect to their choice of the Nambu-vector. Hence, it is necessary to lay out the here-used implementation choices and state the corresponding equations for the considered quantities:
The Nambu-vector is chosen as:
.
The order of the Basis in the Hamiltonian from inner to outer index is: Spin (up/dn), orbital(1..No), site(1..Ns), dagger(a/c) with up/dn corresponding to spin-up and spin-down, the number of orbitals No and the number of sites Ns, as well as the annihilation(a) and creation(c) operator.
Hence denoting the states as a(up/dn,1..No,1..Ns,a/c) the order for a system with magnetism, 2 orbital, 2 sites, and superconductivity would be: a(up,1,1,a) a(dn,1,1,a) a(up,2,1,a) a(dn,2,1,a) a(up,1,2,a) a(dn,1,2,a) a(up,2,2,a) a(dn,2,2,a) a(up,1,1,c) a(dn,1,1,c) a(up,2,1,c) a(dn,2,1,c) a(up,1,2,c) a(dn,1,2,c) a(up,2,2,c) a(dn,2,2,c)
With this choice of the Nambu-basis, the BdG equation takes the form
,
as derived in the Ref.1 Eq.2.22. Analogously describing each eigenstates as
,
where u corresponds to the solution in the annihilation sector and v corresponds to the creation sector. Here, the index i runs over both the orbitals and the sites.
The densities can be calculated according to
,
where the n-sum includes all energies above 0.