Theory for the Charge-Density-Wave Mechanism of 3D Quantum Hall Effect.

The charge-density-wave (CDW) mechanism of the 3D quantum Hall effect has been observed recently in ZrTe_{5} [Tang et al., Nature 569, 537 (2019)10.1038/s41586-019-1180-9]. Different from previous cases, the CDW forms on a one-dimensional (1D) band of Landau levels, which strongly depends on the magnetic field. However, its theory is still lacking. We develop a theory for the CDW mechanism of 3D quantum Hall effect. The theory can capture the main features in the experiments. We find a magnetic field induced second-order phase transition to the CDW phase. We find that electron-phonon interactions, rather than electron-electron interactions, dominate the order parameter. We extract the electron-phonon coupling constant from the non-Ohmic I-V relation. We point out a commensurate-incommensurate CDW crossover in the experiment. More importantly, our theory explores a rare case, in which a magnetic field can induce an order-parameter phase transition in one direction but a topological phase transition in other two directions, both depend on one magnetic field.