Transport theory in non-Hermitian systems

Non-Hermitian systems have garnered significant attention due to the emergence of novel topology of complex spectra and skin modes. However, investigating transport phenomena in such systems faces obstacles stemming from the nonunitary nature of time evolution. Here, we establish the continuity equation for a general non-Hermitian Hamiltonian in the Schr & ouml;dinger picture. It attributes the universal nonconservativity to the anticommutation relationship between particle number and non-Hermitian terms. Our paper derives a comprehensive current formula for non-Hermitian systems using Green's function, applicable to both time-dependent and steady-state responses. To demonstrate the validity of our approach, we calculate the local current of models with one-dimensional and two-dimensional settings, incorporating scattering potentials. The spatial distribution of local current highlights the widespread non-Hermitian phenomena, including skin modes, nonreciprocal quantum dots, and corner states. Additionally, we revisit a recent experiment within the quantum Hall regime and propose a set-up for experimentally detecting non-Hermitian current. Our findings offer valuable insights for advancing theoretical and experimental research in the transport of non-Hermitian systems.