Topological gauge theory for mixed Dirac stationary states in all dimensions

We derive the universal real time $U(1)$ topological gauge field action for mixed quantum states in all dimensions, and demonstrate its independence of the underlying equilibrium or non-equilibrium nature of dynamics. The only prerequisites are charge quantization and charge conservation. The gauge action encodes non-quantized linear responses as expected for mixed states, but also quantized non-linear responses, associated with mixed state topology and accessible in experiment. Our construction furthermore demonstrates how the physical pictures of anomaly inflow and bulk-boundary correspondence extend to non-equilibrium systems.