Nonconserved density accumulations in orbital Hall transport: Insights from linear response theory

We present a linear response theory for stationary density accumulations in anomalous transport phenomena, such as the orbital Hall effect, where the transported density is odd under time reversal and the underlying charge is not conserved. Our framework applies to both metals and insulators, topologically trivial or nontrivial, and distinguishes between contributions from bulk and edge states, as well as undergap and dissipative currents. In time-reversal invariant systems, we prove a microscopic reciprocity theorem showing that only dissipative currents at the Fermi level contribute to density accumulation, while undergap currents do not. In contrast, in non-time-reversal invariant systems, nondissipative density accumulations, such as magnetoelectric polarization, can appear in both the bulk and edges. Importantly, we find that the net density accumulation does not always vanish, pointing to a global nonconservation that implies the existence of a nonvanishing integrated ``net torque'' in addition to a ``distributed torque,'' which has zero spatial average. We show that the distributed torque can be absorbed in the divergence of a redefined current that satisfies Onsager reciprocity, while the net torque must be explicitly accounted for. Finally, we apply our theory to two-dimensional models with edge terminations.