Quantum geometry of singlet superconductors

For s-wave and d-wave superconductors described by mean field theories, the overlap of two quasihole states at slightly different momenta defines a quantum metric. The integration of quantum metric over the momentum space represents an average distance between neighboring quasihole states, of which we call the fidelity number, and it can be further expressed as a fidelity marker defined locally on every lattice site. We investigate the momentum and spatial profiles of the quantum metric and fidelity marker, respectively, as an attempt to understand the quantum geometrical properties of singlet superconductors. For s-wave superconductors, we unveil that the quantum metric generally influences the charge and current responses of the superconductor against an electric field or electric potential that has a finite wave length, such as the infrared absorption and paramagnetic current. In addition, the dielectric response is directly proportional to the fidelity number. For d-wave superconductors, we will elaborate the singular behavior of the quantum metric near the nodal points, and how it renders a diverging fidelity number.