Quantum Geometric Entropy Production and Entropy Hall Effect

Quantum geometry has been extensively linked to charge, spin, orbital, and thermal transport in crystalline solids, but its imprint on entropy transport remains largely unexplored. Here we develop a quantum theory of entropy transport for Bloch electrons under an electric field. {Within the relaxation-time approximation, we show that the leading and subleading contributions to entropy production reproduce the Joule heating relation, with the former arising from the linear Drude current and the latter from the nonlinear Drude current and the intrinsic nonlinear Ohmic current. Microscopically, near topological Dirac band crossings and anticrossings, these contributions are controlled by the quantum metric and the fully symmetrized quantum metric dipole, respectively. We further predict a Berry-curvature-induced entropy Hall effect, which, together with the local heat Hall current, provides a simple way to recover the modern theory of orbital magnetization.} Finally, we derive general relations between entropy and charge currents, offering a practical route to probe entropy flow experimentally.