nLab quantum anomalous Hall effect

Context

Quantum systems

quantum logic


quantum physics


quantum probability theoryobservables and states


quantum information


quantum technology


quantum computing

Topological physics

Contents

Idea

The quantum anomalous Hall effect (QAHE) is a joint variant of the quantum Hall effect and the anomalous Hall effect: Where a quantum Hall effect is induced by a strong external magnetic field, in the “anomalous” version — realized in crystalline topological phases of matter called Chern insulators — the effect of the external magnetic field on the electrons is instead mimicked by the latter’s spin-orbit coupling in the presence of magnetization, jointly reflected in a non-vanishing Berry curvature over the Brillouin torus which now plays the role of the external field’s flux density.

In analogy to how the ordinary quantum Hall effect has a fractional version, there is even a fractional version of the QAHE: the fractional quantum anomalous Hall effect (FQAHE).

Details

[Chang, Liu & MacDonald 2023 §II.A:] A common feature of all the QAH systems that are established as of this writing — magnetically doped TI films, films of the intrinsic magnetic TI MnBi 2Te 4Mn Bi_2 Te_4, magic-angle TBG, ABC trilayer graphene on h-BNh\text{-}BN, and TMD moirés — is adiabatic connection to a limit in which the band states close to the Fermi level can be described by 2D massive Dirac equations. […] The 2D Dirac model is not periodic in momentum and is therefore not a crystal Hamiltonian. When applied to crystalline electronic degrees of freedom, it is intended to apply only in small isolated portions of the Brillouin zone (BZ) with large Berry curvatures […] the Berry curvature in the 2D Dirac equation is concentrated within a momentum-space area proportional to (m/v D) 2(m/\hbar v_D)^2 (where v Dv D,xv D,yv_D \sim v_{D,x} \sim v_{D,y}), and that it decays as |k| 3{\vert \mathbf{k} \vert}^{-3} for large |k|{\vert \mathbf{k} \vert}. […] Each 2D Dirac Hamiltonian therefore contributes ±(e 2/2h)\pm (e^2/2 h) to the Hall conductivity

Moreover, for fractional quantum Hall systems the valence band:

  • is “almost flat”, meaning that its energy gradient with respect to momentum is small, so that the kinetic energy of electrons is small (“quenched”) and the electron-interaction/correlation is dominant

  • overlaps the Fermi energy, so that it is only partially (fractionally) filled, with holes at the peaks of the Dirac domes:

Types of Hall effects

References

Integer QAHE

The theoretical prediction of Hall conductance proportional to the first Chern number (integrated Berry curvature) of the valence band in a topological insulator:

  • D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs: Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett. 49 (1982) 405 [doi:10.1103/PhysRevLett.49.405]

The first theoretical lattice model, which came to be called the Haldane model:

Experimental realization of QAH systems:

Review:

See also:

Fractional QAHE

Theoretical prediction:

Experimental realization of FQAH systems:

Review:

See also:

Further discussion:

  • Nicolas Regnault et al.: Fractional topological states in rhombohedral multilayer graphene modulated by kagome superlattice [arXiv:2502.17320]

  • Sen Niu, Jason Alicea, D. N. Sheng, Yang Peng: Quantum anomalous Hall effects and Hall crystals at fractional filling of helical trilayer graphene [arXiv:2505.24146]

Last revised on June 4, 2025 at 10:27:24. See the history of this page for a list of all contributions to it.