Geometric origin of supercurrents in Berry phase: Formula for computing currents from wavefunctions with correlation and particle number variation

The complexity of itinerant and many-body nature in Bardeen-Cooper-Schrieffer (BCS) wavefunctions has traditionally led to the use of coarse-grained order parameters for describing currents in superconductors (SC), rather than directly utilizing wavefunctions. In this work, we introduce a phase-based formula that enables the direct computation of currents from microscopic wavefunctions, accounting for correlation and particle number variations. Interestingly, the formulation draws parallels with insulators, suggesting a unified framework for understanding (intra-band) charge transport across two extremes of conductivity. A "group velocity" current J band ∝ 1 ℏ ∂ k E(k) is derived from Berry phase, independent of wave package dynamics, robust against correlation. Additionally, we identify a correlation-driven contribution, J_corr, which reveals that the pairing correlations ⟨c k c -k ⟩ among "dancing partners" provide a current component beyond the velocity operator.