nLab quantum anomalous Hall effect (changes)

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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

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[[Chang, Liu & MacDonald 2023 §II.A](#ChangLiuMacDonald23):] 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 $Mn Bi_2 Te_4$, magic-angle TBG, ABC trilayer graphene on $h\text{-}BN$, 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 $(m/\hbar v_D)^2$ (where $v_D \sim v_{D,x} \sim v_{D,y}$), and that it decays as ${\vert \mathbf{k} \vert}^{-3}$ for large ${\vert \mathbf{k} \vert}$. […] Each 2D Dirac Hamiltonian therefore contributes $\pm (e^2/2 h)$ 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:

\begin{imagefromfile} “file_name”: “FQAH-BandStructure.png”, “width”: 640, “unit”: “px”, “margin”: { “top”: -30, “bottom”: 20, “right”: 0, “left”: 30 } \end{imagefromfile}

Related entries

Types of Hall effects

• Hall effect, anomalous Hall effect

• quantum Hall effect, quantum anomalous Hall effect

• quantum$\;$spin Hall effect

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](https://doi.org/10.1103/PhysRevLett.49.405)]

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

• Duncan Haldane: Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the “Parity Anomaly”, Phys. Rev. Lett. 61 (1988) 2015 [[doi:10.1103/PhysRevLett.61.2015](https://doi.org/10.1103/PhysRevLett.61.2015)]

Experimental realization of QAH systems:

• Cui-Zu Chang et al.: Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator, Science 340 (2013) 167 [[doi:10.1126/science.1234414](https://doi.org/10.1126/science.1234414), arXiv:1605.08829]

Review:

• Jing Wang, Biao Lian, Shou-Cheng Zhang: Quantum anomalous Hall effect in magnetic topological insulators, Physica Scripta 2015 T164 (2015) 014003 [[arXiv:1409.6715](https://arxiv.org/abs/1409.6715), doi:10.1088/0031-8949/2015/T164/014003]

• Chao-Xing Liu, Shou-Cheng Zhang, Xiao-Liang Qi: The Quantum Anomalous Hall Effect: Theory and Experiment, Annual Review of Condensed Matter Physics 7 (2016) [[arXiv:1508.07106](https://arxiv.org/abs/1508.07106), doi:10.1146/annurev-conmatphys-031115-011417]

• Cui-Zu Chang, Chao-Xing Liu, Allan H. MacDonald: Colloquium: Quantum anomalous Hall effect, Rev. Mod. Phys. 95 (2023) 011002 [[arXiv:2202.13902](https://arxiv.org/abs/2202.13902), doi:10.1103/RevModPhys.95.011002]

See also:

• Wikipedia: Quantum anomalous Hall effect

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:

• Evelyn Tang, Jia-Wei Mei, Xiao-Gang Wen: High-Temperature Fractional Quantum Hall States, Phys. Rev. Lett. 106 (2011) 236802 [[arXiv:1101.1942](https://arxiv.org/abs/1101.1942), doi:10.1103/PhysRevLett.106.236802]

• Kai Sun, Zheng-Cheng Gu, Hosho Katsura, Sankar Das Sarma: Nearly Flatbands with Nontrivial Topology, Phys. Rev. Lett. 106 (2011) 236803 [[arXiv:1012.5864](https://arxiv.org/abs/1012.5864), doi:10.1103/PhysRevLett.106.236803]

• Titus Neupert, Luiz Santos, Claudio Chamon, Christopher Mudry: Fractional quantum Hall states at zero magnetic field, Phys. Rev. Lett. 106 (2011) 236804 [[arXiv:1012.4723](https://arxiv.org/abs/1012.4723), doi:10.1103/PhysRevLett.106.236804]

and further early theoretical development:

• Siddharth A. Parameswaran, Rahul Roy, Shivaji L. Sondhi: Fractional Quantum Hall Physics in Topological Flat Bands, Comptes Rendus. Physique, Topological insulators / Isolants topologiques, 14 9-10 (2013) 816-839 [[arXiv:1302.6606](https://arxiv.org/abs/1302.6606), doi:10.1016/j.crhy.2013.04.003]

  “The possibility of stabilizing exotic topological phases outside of a dilution fridge and without a superconducting magnet, and the avenues it opens for further experimental probes of these phases, is among the primary motivations of the activity in this field”

• Rahul Roy: Band geometry of fractional topological insulators, Phys. Rev. B 90 (2014) 165139 [[arXiv:1208.2055](https://arxiv.org/abs/1208.2055), doi:10.1103/PhysRevB.90.165139]

• Nicolas Regnault, B. Andrei Bernevig: Fractional Chern Insulator, Phys. Rev. X 1 (2011) 021014 [[arXiv:1105.4867](https://arxiv.org/abs/1105.4867), doi:10.1103/PhysRevX.1.021014]

Relation to W-infinity algebra:

• Siddharth A. Parameswaran, Rahul Roy, Shivaji L. Sondhi: Fractional Chern Insulators and the W-Infinity Algebra, Phys. Rev. B 85 (2012) 241308(R) [[arXiv:1106.4025](https://arxiv.org/abs/1106.4025), doi:10.1103/PhysRevB.85.241308]

See also:

• Peleg Emanuel, Anna Keselman, Yuval Oreg: A Unifying Framework for Fractional Chern Insulator Stabilization, Phys. Rev. B 112 (2025) 235133 [[arXiv:2505.07950](https://arxiv.org/abs/2505.07950), doi:10.1103/9hpw-kz7g]

Experimental realization of FQAH systems:

• Jiaqi Cai et al.: Signatures of Fractional Quantum Anomalous Hall States in Twisted $MoTe_2$ Bilayer, Nature 622 (2023) 63–68 [[arXiv:2304.08470](https://arxiv.org/abs/2304.08470), doi:10.1038/s41586-023-06289-w]

  (in twisted bilayer molybdenum ditelluride, $Mo Te_2$)

• Yihang Zeng et al.: Thermodynamic evidence of fractional Chern insulator in moiré $MoTe_2$, Nature 622 (2023) 69–73 [[doi:10.1038/s41586-023-06452-3](https://doi.org/10.1038/s41586-023-06452-3)]

• Heonjoon Park et al.: Observation of fractionally quantized anomalous Hall effect, Nature 622 (2023) 74–79 [[doi:10.1038/s41586-023-06536-0](https://doi.org/10.1038/s41586-023-06536-0)]

• Boran Zhou, Hui Yang, and Ya-Hui Zhang: Fractional Quantum Anomalous Hall Effect in Rhombohedral Multilayer Graphene in the Moiréless Limit, Phys. Rev. Lett. 133 (2024) 206504 [[doi:10.1103/PhysRevLett.133.206504](https://doi.org/10.1103/PhysRevLett.133.206504), arXiv:2311.04217]

• Zhengguang Lu et al.: Fractional quantum anomalous Hall effect in multilayer graphene, Nature 626 (2024) 759–764 [[doi:10.1038/s41586-023-07010-7](https://doi.org/10.1038/s41586-023-07010-7), arXiv:2309.17436]

• Jingwei Dong, et al.: Observation of Integer and Fractional Chern insulators in high Chern number flatbands [[arXiv:2507.09908](https://arxiv.org/abs/2507.09908)]

• Naitian Liu et al.: Diverse high-Chern-number quantum anomalous Hall insulators in twisted rhombohedral graphene [[arXiv:2507.11347](https://arxiv.org/abs/2507.11347)]

• Hongyun Zhang et al.: Moiré enhanced flat band in rhombohedral graphene, Nat. Mater. (2025) [[arXiv:2504.06251](https://arxiv.org/abs/2504.06251), doi:10.1038/s41563-025-02416-2]

• Yiping Wang et al.: Magnetic Signatures of a Putative Fractional Topological Insulator in Twisted $MoTe_2$ [[arXiv:2601.18508](https://arxiv.org/abs/2601.18508)]

• Jackson P. Butler: 1/3 Fractional and Gapless Integer Quantum Anomalous Hall States in Rhombohedral Graphene [[arXiv:2606.06450](https://arxiv.org/abs/2606.06450)]

• Zexu Li et al.: Fractional high-Chern insulator in twisted rhombohedral graphene, Nature (2026) [[arXiv:2512.21612](https://arxiv.org/abs/2512.21612), https://doi.org/10.1038/s41586-026-10762-7]

Commercial:

• FLEET: RMIT

Review:

• Rahul Roy, Shivaji L. Sondhi: Fractional quantum Hall effect without Landau levels, Physics 4 46 (June 2011) [[physics.aps:v4/46](https://physics.aps.org/articles/v4/46)]

• Emil J. Bergholtz, Zhao Liu: Topological Flat Band Models and Fractional Chern Insulators, Int. J. Mod. Phys. B 27 (2013) 1330017 [[doi:10.1142/S021797921330017X](https://doi.org/10.1142/S021797921330017X), arXiv:1308.0343]

• Titus Neupert, Claudio Chamon, Thomas Iadecola, Luiz H. Santos, Christopher Mudry: Fractional (Chern and topological) insulators Physica Scripta 2015 (2015) 014005 [[arXiv:1410.5828](https://arxiv.org/abs/1410.5828), doi:10.1088/0031-8949/2015/T164/014005]

• Long Yu, Allan H. MacDonald, Kin Fai Mak, Jie Shan, Xiaodong Xu: The fractional quantum anomalous Hall effect, Nature Reviews Materials 9 (2024) 455–-459 [[doi:10.1038/s41578-024-00694-x](https://doi.org/10.1038/s41578-024-00694-x)]

• Nicolás Morales-Durán, Jingtian Shi, Allan H. MacDonald: Fractionalized electrons in moiré materials, Nature Reviews Physics 6 (2024) 349–351 [[doi:10.1038/s42254-024-00718-z](https://doi.org/10.1038/s42254-024-00718-z)]

• Jian Zhao et al.: Exploring the Fractional Quantum Anomalous Hall Effect in Moiré Materials: Advances and Future Perspectives, ACS Nano (2025) [[doi:10.1021/acsnano.5c01598](https://doi.org/10.1021/acsnano.5c01598)]

See also:

• Nicolas Regnault: Fractional Topogical Insulators: numerical evidences [[pdf](https://www.ggi.infn.it/talkfiles/slides/talk2313.pdf), pdf]

• Wikipedia: Fractional Chern insulator

Further discussion:

• Adolfo G. Grushin, Titus Neupert, Claudio Chamon, Christopher Mudry: Enhancing the stability of a fractional Chern insulator against competing phases, Phys. Rev. B 86, 205125 (2012) [doi:10.1103/PhysRevB.86.205125, arXiv:1207.4097 cond-mat.str-el]

• Ahmed Abouelkomsan, Zhao Liu, Emil J. Bergholtz: Particle-Hole Duality, Emergent Fermi Liquids and Fractional Chern Insulators in Moiré Flatbands, Phys. Rev. Lett. 124 (2020) 106803 [doi:10.1103/PhysRevLett.124.106803, arXiv:1912.04907]

• V. Yu. Irkhin, Yu. N. Skryabin: Electronic States and Anomalous Hall Effect in Strongly Correlated Topological Systems, Journal of Experimental and Theoretical Physics, 133 1 (2021) 116–123 [doi:10.1134/S1063776121060030, arXiv:2108.13851 cond-mat.str-el]

• Valentin Crépel, Liang Fu: Anomalous Hall metal and fractional Chern insulator in twisted transition metal dichalcogenides, Phys. Rev. B 107 (2023) L201109 [[arXiv:2207.08895](https://arxiv.org/abs/2207.08895), doi:10.1103/PhysRevB.107.L201109]

• Gal Shavit, Yuval Oreg: Quantum Geometry and Stabilization of Fractional Chern Insulators Far from the Ideal Limit, Phys. Rev. Lett. 133 (2024) 156504 [[doi:10.1103/PhysRevLett.133.156504](https://doi.org/10.1103/PhysRevLett.133.156504), arXiv:2405.09627]

• Nicolas Regnault et al.: Fractional topological states in rhombohedral multilayer graphene modulated by kagome superlattice [[arXiv:2502.17320](https://arxiv.org/abs/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](https://arxiv.org/abs/2505.24146)]

• Hongyu Lu, Han-Qing Wu, Bin-Bin Chen, Wang Yao, Zi Yang Meng: Generic (fractional) quantum anomalous Hall crystals from interaction-driven band folding [[arXiv:2505.04138](https://arxiv.org/abs/2505.04138)]

• Raul Perea-Causin, Hui Liu, Emil J. Bergholtz: Quantum anomalous Hall crystals in moiré bands with higher Chern number, Nature Communications 16 6875 (2025) [[doi:10.1038/s41467-025-62224-9](https://doi.org/10.1038/s41467-025-62224-9)]

• Zhengyan Darius Shi, T. Senthil: Doping a fractional quantum anomalous Hall insulator [[arXiv:2409.20567](https://arxiv.org/abs/2409.20567)]

• Zhengyan Darius Shi, T. Senthil: Non-Abelian topological superconductivity from melting Abelian fractional Chern insulators [arXiv:2512.17996]

• Xi-Hang Feng et al.: Fractional Chern insulators in alternating twisted multilayer $MoTe_2$ [[arXiv:2607.13807](https://arxiv.org/abs/2607.13807)]

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](https://arxiv.org/abs/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](https://doi.org/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](https://arxiv.org/abs/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](https://www2.yukawa.kyoto-u.ac.jp/~yitpmolecule-topology-2024/slide/Naren.pdf), pdf]

See also:

• Kryštof Kolář, Kang Yang, Felix von Oppen, Christophe Mora: Robustness of real-space topology in moiré systems [[arXiv:2507.00130](https://arxiv.org/abs/2507.00130)]

On the corresponding Wannier functions:

• Xiao-Liang Qi: Generic Wavefunction Description of Fractional Quantum Anomalous Hall States and Fractional Topological Insulators, Phys. Rev. Lett. 107 (2011) 126803 [doi:10.1103/PhysRevLett.107.126803", arXiv:1105.4298]

• Maissam Barkeshli, Xiao-Liang Qi: Topological Nematic States and Non-Abelian Lattice Dislocations, Phys. Rev. X 2 (2012) 031013 [doi:10.1103/PhysRevX.2.031013, arXiv:1112.3311]

• Chao-Ming Jian, Xiao-Liang Qi: Crystal-symmetry preserving Wannier states for fractional chern insulators, Phys. Rev. B 88 (2013) 165134 [[arXiv:1303.1787](https://arxiv.org/abs/1303.1787), doi:10.1103/PhysRevB.88.165134]

Relation to superconductors:

• Taige Wang, Michael P. Zaletel: Chiral superconductivity near a fractional Chern insulator [[arXiv:2507.07921](https://arxiv.org/abs/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](https://arxiv.org/abs/2509.12184), doi:10.1103/cqt2-b327]

See also:

• Taige Wang: Topological superconductivity from Abelian fractional Chern insulators [[arXiv:2605.29034](https://arxiv.org/abs/2605.29034)]

• Jennifer Cano, Jie Wang: Ideal Quantum Geometry for Fractional Chern Insulators [[arXiv:2606.05496](https://arxiv.org/abs/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](https://arxiv.org/abs/2606.07323)]

• Haoyu Hu, Lei Chen: Quantum-Geometry-Induced Superconductivity near a Fractional Chern Insulator [[arXiv:2606.21166](https://arxiv.org/abs/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](https://arxiv.org/abs/2608.14535)]

Anyonic states

On topological order and anyons in FQAH systems:

• Fabian J. Pauw, Felix A. Palm, Ulrich Schollwöck, Annabelle Bohrdt, Sebastian Paeckel, Fabian Grusdt: Detecting Hidden Order in Fractional Chern Insulators, Phys. Rev. Research 6 (2024) 023180 [[doi:10.1103/physrevresearch.6.023180](https://doi.org/10.1103/physrevresearch.6.023180), arXiv:2309.03666]

• Aidan P. Reddy, Nisarga Paul, Ahmed Abouelkomsan, Liang Fu: Non-Abelian fractionalization in topological minibands, Phys. Rev. Lett. 133 (2024) 166503 [[arXiv:2403.00059](https://arxiv.org/abs/2403.00059), doi:10.1103/PhysRevLett.133.166503 ]

• Hui Liu, Zhao Liu, Emil J. Bergholtz: Non-Abelian Fractional Chern Insulators and Competing States in Flat Moiré Bands, Phys. Rev. Lett. 135 (2025) 106604 [[arXiv:2405.08887](https://arxiv.org/abs/2405.08887), doi:10.1103/43nq-ntqm]

• Ryohei Kobayashi, Yuxuan Zhang, Naren Manjunath, Maissam Barkeshli: Crystalline invariants of fractional Chern insulators, Phys. Rev. B (2025) [[arXiv:2405.17431](https://arxiv.org/abs/2405.17431), doi:10.1103/8bpm-qbzp]

• Hui Liu, Raul Perea-Causin, Emil J. Bergholtz: Parafermions in moiré minibands, Nature Communications 16 (2025) 1770 [[doi:10.1038/s41467-025-57035-x](https://doi.org/10.1038/s41467-025-57035-x)]

• Zhengyan Darius Shi, T. Senthil: Anyon delocalization transitions out of a disordered FQAH insulator, PNAS 122 51 (2025) e2520608122 [[arXiv:2506.02128](https://arxiv.org/abs/2506.02128), doi:10.1073/pnas.2520608122]

• Hisham Sati, Urs Schreiber: Identifying Anyonic Topological Order in Fractional Quantum Anomalous Hall Systems, Applied Physics Letters 128 023101 (2026) [[arXiv:2507.00138](https://arxiv.org/abs/2507.00138), doi:10.1063/5.0305441]

• Felix A. Palm, Nader Mostaan, Nathan Goldman, Fabian Grusdt: Interferometric Braiding of Anyons in Chern insulators [[arXiv:2511.09445](https://arxiv.org/abs/2511.09445)]

• Chuyi Tuo, Ming-Rui Li, Hong Yao: Fractional quantum anomalous Hall and anyon density-wave halo in a minimal interacting lattice model of twisted bilayer $MoTe_2$ [[arXiv:2512.23608](https://arxiv.org/abs/2512.23608)]

• Hisham Sati, Urs Schreiber: Fragile Topological Phases and Topological Order of 2D Crystalline Chern Insulators [[arXiv:2512.24709](https://arxiv.org/abs/2512.24709)]

• Tianhong Lu, Luiz H. Santos: Exciton-Anyon Binding in Fractional Chern Insulators: Spectral Fingerprints [[arXiv:2601.14365](https://arxiv.org/abs/2601.14365)]

• Kishore Iyer, Andreas Feuerpfeil, Valentin Crépel, Nicolas Regnault, Christophe Mora: Dispersion of Anyon Bloch Bands [[arXiv:2604.24859](https://arxiv.org/abs/2604.24859)]

• Zhengyan Darius Shi, T. Senthil: Superconductivity and non-Fermi liquid metals in a charge-$1/3$ anyon fluid [[arXiv:2606.20403](https://arxiv.org/abs/2606.20403)]

• Umang Mehta, Yuto Nakajima, Hart Goldman: Coloring in anyon superconductivity [[arXiv:2607.19470](https://arxiv.org/abs/2607.19470)]

  (relation to superconductivity)

• Botao Wang, Amit Vashisht, Felix A. Palm, Fabian Grusdt, Laurens Vanderstraeten, Nathan Goldman: Anyon-Impurity Bound States in Quantum-Engineered Fractional Chern Insulators [[arXiv:2608.06233](https://arxiv.org/abs/2608.06233)]

• Weijie Li et al.: Signatures of fractional charges via anyon–trions in twisted $MoTo_2$, Nature 651 (2026) 48–53 [doi:10.1038/s41586-026-10101-w]

In view of TQC

In view of potential topological quantum computing hardware:

• James Urton: Researchers make a quantum computing leap with a magnetic twist, UW News (June 2023)

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](https://arxiv.org/abs/2601.05196)]

• Yi-Hsien Du: Controlled Theory of Skyrmion Chern Bands in Moiré Quantum Materials: Quantum Geometry and Collective Dynamics [[arXiv:2602.15016](https://arxiv.org/abs/2602.15016)]