Theory for the charge-density-wave mechanism of the 3D quantum Hall effect

Fang Qin, Shuai Li, Z. Z. Du, C. M. Wang, Hai-Zhou Lu, X. C. Xie

Published 2020-03-05Version 1

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)]. Quite different from previous cases, the CDW forms on a 1D band of Landau levels, which strongly depends on the magnetic field. However, its theory is still lacking. To set up a theoretical framework to address the issues, we study the quantum limit of 3D Dirac fermions, focusing on the magnetic field dependence of the CDW order parameter. Our theory can capture the major features in the experiments, such as the non-Ohmic $I$-$V$ relation. 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 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. It will be useful and inspire further experiments and theories on this emergent phase of matter.