  Non-Hermitian materials can not only exhibit exotic energy band structures
but also an anomalous velocity induced by non-Hermitian anomalous Berry
connection as predicted by the semiclassical equations of motion for Bloch
electrons. However, it is not clear how the modified semiclassical dynamics
modifies transport phenomena. Here, we theoretically demonstrate the emergence
of anomalous oscillations driven by either an external dc or ac electric field,
which arise from non-Hermitian anomalous Berry connection. Moreover, it is a
well-known fact that geometric structures of electric wave functions can only
affect the Hall conductivity. However, we are surprised to find a non-Hermitian
anomalous Berry connection induced anomalous linear longitudinal conductivity
independent of the scattering time. We also show the emergence of a
second-order nonlinear longitudinal conductivity induced by non-Hermitian
anomalous Berry connection, violating a well-known fact of its absence in a
Hermitian system with symmetric energy spectra. These anomalous phenomena are
illustrated in a pseudo-Hermitian system with large non-Hermitian anomalous
Berry connection. Finally, we propose a practical scheme to realize the
anomalous oscillations in an optical system.
