Meissner effect and gauge invariance in anisotropic narrow-band Bloch-electron and hole-type superconductors

Within the framework of the ladder-diagram approximation, the manifestly gauge-invariant expression for the static electromagnetic response kernel of the anisotropic narrow-band superconductor consisting of Bloch electrons is derived, on the basis of the charge-conserving current expression for them under the magnetic field. For the two-dimensional square lattice, the magnetic penetration depth ${\lambda }_{\mathit{L}}$(T) is calculated as a function of hole density ${\mathit{n}}_{\mathit{h}}$ and temperature T (${\mathit{T}}_{\mathit{c}}$). The resultant ${\lambda }_{\mathit{L}}$(0) has a symmetrical dependence on the hole and electron densities, and is almost unaffected by the anisotropy of the order parameter. The behavior of ${\lambda }_{\mathit{L}}$(T)/${\lambda }_{\mathit{L}}$(0) as a function of T/${\mathit{T}}_{\mathit{c}}$ for the d-wave state substantially deviates from those for the extended and usual s-wave states for all T${\mathit{T}}_{\mathit{c}}$. The obtained results are compared with those of the effective hole- and electron-mass approximations.