Quantum Hall Physics in Three-Dimensional Weyl Semimetals: A Perspective Beyond Two Dimensions

Abstract

The quantum Hall effect (QHE) discovered in 1980 has been understood as a quintessentially two-dimensional phenomenon: the quantization of Landau levels in a two-dimensional electron gas yields Hall conductance plateaus at integer multiples of e²/h . The advent of Weyl semimetals – three-dimensional topological materials characterized by linearly dispersing band crossings (Weyl nodes) and topologically protected Fermi arc surface states – has changed this picture fundamentally. In these systems, a true three-dimensional quantum Hall effect (3D QHE) is achieved via the Weyl orbit mechanism: electrons perform cyclotron orbits that embrace both surfaces of a macroscopic slab by tunneling through chiral bulk Landau levels, leading to quantized Hall conductance in a bona fide three-dimensional crystal. In this perspective review, we summarize theoretical basis and experimental evidence of 3D QHE. We develop the Landau quantization formalism for Weyl and Dirac semimetals, derive the Weyl orbit quantization condition and its thickness dependence, and review experimental observations in Cd₃As₂, ZrTe₅, and acoustic metamaterial analogs. We critically discuss the effect of disorder, Weyl node tilt and inter-node scattering on the robustness of the plateau. We highlight the major unsolved questions, including the separation of 3D bulk QHE from surface QHE contributions, the role of electron correlations in chiral Landau levels and the theoretical description of 3D QHE in amorphous topological systems.