Conserved current of nonconserved quantities

We provide a unified semiclassical theory for the conserved current of nonconserved quantities, and manifest it in two physical contexts: spin Bloch electrons charge mean-field Bogoliubov quasiparticles. Several longstanding problems that limit playground are solved. reveal hitherto overlooked torque quadrupole density Berry phase correction to dipole essential assure circulating with vanishing net flow at equilibrium. The band geometric origin bulk transport is ascertained be momentum space texture curvature instead curvature, paving way material related studies. In superconductors attained corresponds quasiparticle renormalized by condensate backflow. Its circulation equilibrium gives an orbital magnetization, which involves characteristics superconductivity, such as arising from unconventional pairing magnetic moment induced moving