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The charge-density-wave (CDW) mechanism of the 3D quantum Hall effect hasbeen observed recently in ZrTe$_5$ [Tang et al., Nature 569, 537 (2019)].Different from previous cases, the CDW forms on a one-dimensional (1D) band ofLandau levels, which strongly depends on the magnetic field. However, itstheory is still lacking. We develop a theory for the CDW mechanism of 3Dquantum Hall effect. The theory can capture the main features in theexperiments. We find a magnetic field induced second-order phase transition tothe CDW phase. We find that electron-phonon interactions, rather thanelectron-electron interactions, dominate the order parameter. We extract theelectron-phonon coupling constant from the non-Ohmic I-V relation. We point outa commensurate-incommensurate CDW crossover in the experiment. Moreimportantly, our theory explores a rare case, in which a magnetic field caninduce an order-parameter phase transition in one direction but a topologicalphase transition in other two directions, both depend on one magnetic field.

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