Quantum--classical correspondence and dissipative to dissipationless crossover in magnetotransport phenomena.

The three-dimensional magneto-conductivity tensor was derived in a gauge invariant form based on the Kubo formula considering quantum effects under a magnetic field, such as the Landau quantization and quantum oscillations.
We analytically demonstrated that the quantum formula of the magneto-conductivity can be obtained by adding a quantum oscillation factor to the classical formula.
This result establishes the quantum--classical correspondence, which has long been missing in magnetotransport phenomena.
Moreover, we found dissipative-to-dissipationless crossover in the Hall conductivity by paying special attention to the analytic properties of the thermal Green's function.
Finally, by calculating the magnetoresistance of semimetals, we identified a phase shift in quantum oscillation originating from the dissipationless transport predominant at high fields.