nLab quantum anomalous Hall effect

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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

(from Chang, Liu, MacDonald 2023)

[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/ℏv D) 2(m/\hbar v_D)^2 (where v D∼v D,x∼v 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

General

The original theoretical prediction (published back-to-back in the same volume) of the fractional quantum anomalous Hall in fractional Chern insulators:

and further early theoretical development:

Relation to W-infinity algebra:

See also:

Experimental realization of FQAH systems:

Review:

See also:

Further discussion:

The case of crystalline topological insulators and symmetry protected topological order:

  • Yuan-Ming Lu, Ying Ran: Symmetry protected fractional Chern insulators and fractional topological insulators, Phys. Rev. B 85 (2012) 165134 [arXiv:1109.0226, doi:10.1103/PhysRevB.85.165134]

    “In fact, the recently discovered FCI states preserve all the lattice point group symmetry as well as translational symmetry. Here in this paper, we point out that as a consequence of the lattice symmetry, there exist many different quantum FCI phases, all respecting the full lattice symmetry, even at the same filling fraction with the same quantum Hall conductance […] These distinct FCI phases cannot be adiabatically connected with each other without a phase transition while the lattice symmetry is respected”

  • Ke Huang, Xiao Li, Sankar Das Sarma, Fan Zhang: Self-consistent theory of fractional quantum anomalous Hall states in rhombohedral graphene Phys. Rev. B 110 (2024) 115146 [doi:10.1103/PhysRevB.110.115146, arXiv:2407.08661]

  • Yuxuan Zhang, Maissam Barkeshli: Fractionally Quantized Electric Polarization and Discrete Shift of Crystalline Fractional Chern Insulators, Phys. Rev. B [arXiv:2411.04171, doi:10.1103/qslx-ybf6]

  • Naren Manjunath: Crystalline invariants of integer and fractional Chern insulators, talk at Recent Developments and Challenges in Topological Phases, Kyoto University (2024) [pdf, pdf]

See also:

On the corresponding Wannier functions:

Relation to superconductors:

  • Taige Wang, Michael P. Zaletel: Chiral superconductivity near a fractional Chern insulator [arXiv:2507.07921]

Derivation from a Hubbard model?:

  • Fabian J. Pauw, Ulrich Schollwöck, Nathan Goldman, Sebastian Paeckel, Felix A. Palm: From hidden order to skyrmions: Quantum Hall states in an extended Hofstadter-Fermi-Hubbard model, Phys. Rev. B 113 (2026) 035142 [arXiv:2509.12184, doi:10.1103/cqt2-b327]

See also:

  • Taige Wang: Topological superconductivity from Abelian fractional Chern insulators [arXiv:2605.29034]

  • Jennifer Cano, Jie Wang: Ideal Quantum Geometry for Fractional Chern Insulators [arXiv:2606.05496]

  • Siddhartha Sarkar, Yitong Zhang, Kai Sun: How Similar Can Fractional Chern Insulators Be to Fractional Quantum Hall States? Moiré-Enhanced Gaps and Excitation-Spectrum Correspondence [arXiv:2606.07323]

  • Haoyu Hu, Lei Chen: Quantum-Geometry-Induced Superconductivity near a Fractional Chern Insulator [arXiv:2606.21166]

  • Julian May-Mann, Tixuan Tan, Patrick J. Ledwith, Zhengyan Darius Shi, Trithep Devakul: Skyrmion Fractional Chern Insulator: An Intrinsically Multiband Route to Fractionalization in Rhombohedral Graphene [arXiv:2608.14535]

Anyonic states

On topological order and anyons in FQAH systems:

In view of TQC

In view of potential topological quantum computing hardware:

Collective excitations

On the chiral graviton mode in FCIs:

  • Min Long, Zeno Bacciconi, Hongyu Lu, Hernan B. Xavier, Zi Yang Meng, Marcello Dalmonte: Chiral Graviton Modes in Fermionic Fractional Chern Insulators [arXiv:2601.05196]

  • Yi-Hsien Du: Controlled Theory of Skyrmion Chern Bands in Moiré Quantum Materials: Quantum Geometry and Collective Dynamics [arXiv:2602.15016]

Last revised on October 4, 2026 at 09:22:09. See the history of this page for a list of all contributions to it.

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