Relativistic Bogoliubov-de Gennes equations

In the nonrelativistic regime, the microscopic description of inhomogeneous singlet superconductors is based on the Bogoliubov-de Gennes (BdG) equations (de Gennes 

Originally, the effective potential Veff(r) and the effective pairing potential A eff(r, r ) were either treated as given functions of r or they were approximated by the standard mean-field expressions, i.e. the Hartree potential for Veff(r) and the BCS term for Aeff (r, r ). Mcxe recently, a density functional theory for superconductors was developed (Oliveira etal. 1988) where the effective potentials Veff(r) and Aeff(r, r ) are expressed as functionals of the density. 

i CT(r) and (r) represent the ordinary nonrelativistic electron annihilation and creation operators. An LDA-type approximation has recently been derived for the exchange-correlation free energy Fxc[n, xl leading to explicit expressions for the effective potentials Veg(r) and Aeff (r, r ) (Kurth et al. 1999). 

In second-quantized notation, the BdG Hamiltonian corresponding to Equations (5.8) and (5.9) reads 

Before proceeding with the construction of relativistic order parameters, we note that the singlet order parameter given in Equation (5.10) does not represent the only possible pair state in the nonrelativistic domain. Spin-triplet order parameters found, for example, in liquid 3He, can be formed as well. A convenient way of describing 

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