The quantum Hall effect is usually observed in 2D systems. We show that the Fermi arcs can give rise to a distinctive 3D quantum Hall effect in topological semimetals. Because of the topological constraint, the Fermi arc at a single surface has an open Fermi surface, which cannot host the quantum Hall effect. Via a "wormhole" tunneling assisted by the Weyl nodes, the Fermi arcs at opposite surfaces can form a complete Fermi loop and support the quantum Hall effect. The edge states of the Fermi arcs show a unique 3D distribution, giving an example of (d-2)-dimensional boundary states. This is distinctly different from the surface-state quantum Hall effect from a single surface of topological insulator. As the Fermi energy sweeps through the Weyl nodes, the sheet Hall conductivity evolves from the 1/B dependence to quantized plateaus at the Weyl nodes. This behavior can be realized by tuning gate voltages in a slab of topological semimetal, such as the TaAs family, Cd_{3}As_{2}, or Na_{3}Bi. This work will be instructive not only for searching transport signatures of the Fermi arcs but also for exploring novel electron gases in other topological phases of matter.

Berry phase physics is closely related to a number of topological states of matter. Recently discovered topological semimetals are believed to host a nontrivial π Berry phase to induce a phase shift of ±1/8 in the quantum oscillation (+ for hole and − for electron carriers). We theoretically study the Shubnikov-de Haas oscillation of Weyl and Dirac semimetals, taking into account their topological nature and inter-Landau band scattering. For a Weyl semimetal with broken timereversal symmetry, the phase shift is found to change nonmonotonically and go beyond known values of ±1/8 and ±5/8. For a Dirac semimetal or paramagnetic Weyl semimetal, time-reversal symmetry leads to a discrete phase shift of ±1/8 or ±5/8, as a function of the Fermi energy. Different from the previous works, we find that the topological band inversion can lead to beating patterns in the absence of Zeeman splitting. We also find the resistivity peaks should be assigned integers in the Landau index plot. Our findings may account for recent experiments in Cd2As3 and should be helpful for exploring the Berry phase in various 3D systems. The Shubnikov-de Haas oscillation of resistance in a metal arises from the Landau quantization of electronic states under strong magnetic fields. The oscillation can be described by the Lifshitz-Kosevich formula [1] cos[2π(F/B + φ)], where B is the magnitude of magnetic field, and the oscillation frequency F and phase shift φ can provide valuable information about the Fermi surface topography of materials. It is widely believed that an energy band with linear dispersion carries an extra π Berry phase [2, 3], leading to phase shifts of φ = 0 and ±1/8 in 2D and 3D, respectively, compared with ±1/2 and ±5/8 for parabolic energy bands without the Berry phase (+ for hole and − for electron carriers). Topological semimetals [4][5][6][7][8] provide a new platform to study the nontrivial Berry phase in 3D. They have linear dispersion near the Weyl nodes at which the conduction and valence bands touch. The Weyl nodes host monopoles connected by Fermi arcs, and have been discovered in the Dirac semimetals Na 3 Bi [9-11] and Cd 3 As 2 [12][13][14][15][16][17][18][19], and the Weyl semimetals TaAs family [20][21][22][23][24][25][26][27][28][29] and YbMnBi 2 [30].Exploring the π Berry phase in 3D semimetals remains difficult [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. To extract the phase shift, the Landau indices, i.e., where F/B + φ takes integers n, need to be identified first from the magnetoresistivity. A plot of n vs 1/B then extrapolates to the phase shift on the n axis. However, the first step in 3D is highly nontrivial. In 3D, a magnetic field quantizes the energy spectrum into a set of 1D bands of Landau levels. There may be multiple Landau bands on the Fermi surface and scattering among them. This situation never occurs for discrete Landau levels in 2D. It is not intuitive to determine the Landau indices in 3D without a sophisticated theoretical analysis of the resistivity of the Landau ban...

The nonlinear Hall effect has opened the door towards deeper understanding of topological states of matter. Disorder plays indispensable roles in various linear Hall effects, such as the localization in the quantized Hall effects and the extrinsic mechanisms of the anomalous, spin, and valley Hall effects. Unlike in the linear Hall effects, disorder enters the nonlinear Hall effect even in the leading order. Here, we derive the formulas of the nonlinear Hall conductivity in the presence of disorder scattering. We apply the formulas to calculate the nonlinear Hall response of the tilted 2D Dirac model, which is the symmetry-allowed minimal model for the nonlinear Hall effect and can serve as a building block in realistic band structures. More importantly, we construct the general scaling law of the nonlinear Hall effect, which may help in experiments to distinguish disorder-induced contributions to the nonlinear Hall effect in the future.