Physics:Fractional Chern insulator

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Fractional Chern insulators (FCIs) are lattice generalizations of the fractional quantum Hall effect that have been studied theoretically since the early 2010's.[1][2] They were first[3] predicted to exist in topological flat bands carrying Chern numbers. They can appear in topologically non-trivial band structures even in the absence of the large magnetic fields needed for the fractional quantum Hall effect. They promise physical realizations at lower magnetic fields, higher temperatures, and with shorter characteristic length scales compared to their continuum counterparts.[4][5] FCIs were initially studied by adding electron-electron interactions[3] to a fractionally filled Chern insulator, in one-body models where the Chern band is quasi-flat,[6][7] at zero magnetic field. The FCIs exhibit a fractional quantized Hall conductance.

Prior work and experiments with finite magnetic fields

In works predating the theoretical studies of FCIs, the analogue of the Laughlin state was demonstrated in Hofstadter-type models.[8][9] The essential features of the topology of single-particle states in such models still stems from the presence of a magnetic field. Chern Insulators - single-particle states exhibiting an integer anomalous quantized Hall effect at zero field - have been theoretically proposed.[10] Fractionally filling such states, in the presence of repulsive interactions, can lead to the zero-field Fractional Chern Insulator. These FCIs are sometimes not connected to the Fractional Quantum Hall Effect in Landau Levels. This is the case in bands with Chern number [math]\displaystyle{ |C|\gt 1 }[/math],[11] and are therefore a new type of states inherent to such lattice models. They have been explored with respect to their quasi-charge excitations, non-Abelian states and the physics of twist defects,[12] which may be conceptually interesting for topological quantum computing.

Experimentally, [math]\displaystyle{ |C|\gt 1 }[/math] Chern insulators have been realized without a magnetic field.[13] FCIs have been claimed to be realized experimentally in van der Waals heterostructures, but with an external magnetic field of order 10 – 30 T and, more recently, FCIs in a [math]\displaystyle{ |C|=1 }[/math] band have been claimed to be observed in twisted bilayer graphene close to the magic angle, yet again requiring a magnetic field, of order 5 T in order to "smoothen" out the Berry curvature of the bands.[14] These states have been called FCIs due to their link to lattice physics -- either in Hofstadter bands or in the moiré structure, but still required nonzero-magnetic field for their stabilization.

Zero field fractional Chern insulators

In 2023 a series of groups have reported an FCI at zero magnetic field[15] in twisted MoTe2 samples. These samples, where the moiré bands are valley-spin locked, undergo a spin-polarization transition which gives rise to a [math]\displaystyle{ C=1 }[/math] Chern insulator state at integer filling [math]\displaystyle{ -1 }[/math] of the moiré bands. Upon fractional filling at [math]\displaystyle{ -2/3 }[/math] and [math]\displaystyle{ -3/5 }[/math], a gapped state develops with a fractional slope in the Streda formula, a hallmark of an FCI. These fractional states are identical to the predicted zero magnetic field FCIs.[3] The full matching of FCI physics in MoTe2, using the single particle model proposed in,[16] to experiments still holds intriguing and unresolved mysteries. These were only partially theoretically addressed,[17] where the issues of model parameters, sample magnetization, and the appearance of some FCI states (at filling [math]\displaystyle{ -2/3 }[/math] and [math]\displaystyle{ -3/5 }[/math]) but the absence of others (so far at filling at [math]\displaystyle{ -1/3, -2/5 }[/math]) are partially addressed.

References

  1. ↑ Neupert, Titus; Chamon, Claudio; Iadecola, Thomas; Santos, Luiz H; Mudry, Christopher (2015-12-01). "Fractional (Chern and topological) insulators". Physica Scripta T164: 014005. doi:10.1088/0031-8949/2015/T164/014005. ISSN 0031-8949. Bibcode: 2015PhST..164a4005N. https://iopscience.iop.org/article/10.1088/0031-8949/2015/T164/014005. 
  2. ↑ Liu, Zhao; Bergholtz, Emil J. (2024-01-01), Chakraborty, Tapash, ed., Recent developments in fractional Chern insulators, Oxford: Academic Press, pp. 515–538, doi:10.1016/b978-0-323-90800-9.00136-0, ISBN 978-0-323-91408-6, https://www.sciencedirect.com/science/article/pii/B9780323908009001360, retrieved 2023-11-05 
  3. ↑ 3.0 3.1 3.2 T. Neupert, L. Santos, C. Chamon, and C. Mudry, Phys. Rev. Lett. 106, 236804 (2011); D. N. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Nature Communications 2, 389 (2011); N. Regnault and B. A. Bernevig, Phys. Rev. X 1, 021014 (2011)
  4. ↑ Bergholtz, Emil J.; Liu, Zhao (2013-09-30). "Topological Flat Band Models and Fractional Chern Insulators" (in en). International Journal of Modern Physics B 27 (24): 1330017. doi:10.1142/S021797921330017X. ISSN 0217-9792. Bibcode: 2013IJMPB..2730017B. https://www.worldscientific.com/doi/abs/10.1142/S021797921330017X. 
  5. ↑ Parameswaran, Siddharth A.; Roy, Rahul; Sondhi, Shivaji L. (2013-11-01). "Fractional quantum Hall physics in topological flat bands". Comptes Rendus Physique. Topological insulators / Isolants topologiques 14 (9): 816–839. doi:10.1016/j.crhy.2013.04.003. ISSN 1631-0705. Bibcode: 2013CRPhy..14..816P. https://www.sciencedirect.com/science/article/pii/S163107051300073X. 
  6. ↑ Tang, Evelyn; Mei, Jia-Wei; Wen, Xiao-Gang (6 June 2011). "High-Temperature Fractional Quantum Hall States". Physical Review Letters 106 (23): 236802. doi:10.1103/PhysRevLett.106.236802. ISSN 0031-9007. PMID 21770532. Bibcode: 2011PhRvL.106w6802T. 
  7. ↑ Sun, Kai; Gu, Zhengcheng; Katsura, Hosho; Das Sarma, S. (6 June 2011). "Nearly Flatbands with Nontrivial Topology". Physical Review Letters 106 (23): 236803. doi:10.1103/PhysRevLett.106.236803. ISSN 0031-9007. PMID 21770533. Bibcode: 2011PhRvL.106w6803S. 
  8. ↑ Hafezi, M.; Sørensen, A. S.; Demler, E.; Lukin, M. D. (28 August 2007). "Fractional quantum Hall effect in optical lattices". Physical Review A 76 (2): 023613. doi:10.1103/PhysRevA.76.023613. ISSN 1050-2947. Bibcode: 2007PhRvA..76b3613H. 
  9. ↑ Möller, G.; Cooper, N. R. (4 September 2009). "Composite Fermion Theory for Bosonic Quantum Hall States on Lattices". Physical Review Letters 103 (10): 105303. doi:10.1103/PhysRevLett.103.105303. ISSN 0031-9007. PMID 19792327. Bibcode: 2009PhRvL.103j5303M. 
  10. ↑ Haldane, F. D. M. (31 October 1988). "Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly"". Physical Review Letters 61 (18): 2015–2018. doi:10.1103/PhysRevLett.61.2015. ISSN 0031-9007. PMID 10038961. 
  11. ↑ Z. Liu, E. J. Bergholtz, H. Fan, and A. M. La ̈uchli, Phys. Rev. Lett. 109, 186805 (2012); A. Sterdyniak, C. Repellin, B. A. Bernevig, and N. Regnault Phys. Rev. B 87, 205137 (2013). M. Udagawa and E. J. Bergholtz, Journal of Statistical Mechanics: Theory and Experiment 2014, P10012 (2014); Y.-H. Wu, J. K. Jain, and K. Sun, Phys. Rev. B 91, 041119 (2015); G. Mo ̈ller and N. R. Cooper, Phys. Rev. Lett. 115, 126401(2015); B. Andrews and G. Mo ̈ller, Phys. Rev. B 97, 035159 (2018).
  12. ↑ B. d. z. Jaworowski, N. Regnault, and Z. Liu, Phys. Rev. B 99, 045136 (2019); M. Barkeshli and X.-L. Qi, Phys. Rev. X 2, 031013 (2012); E. J. Bergholtz, Z. Liu, M. Trescher, R. Moessner, and M. Udagawa, Phys. Rev. Lett. 114, 016806 (2015); Z. Liu, G. Möller, and E. J. Bergholtz, Phys. Rev. Lett. 119, 106801 (2017).
  13. ↑ G. Chen, A. L. Sharpe, E. J. Fox, Y.-H. Zhang, S. Wang, L. Jiang, B. Lyu, H. Li, K. Watanabe, T. Taniguchi, Z. Shi,T. Senthil, D. Goldhaber-Gordon, Y. Zhang, and F. Wang, Nature 579, 56 (2020)
  14. ↑ E. M. Spanton, A. A. Zibrov, H. Zhou, T. Taniguchi, K. Watanabe, M. P. Zaletel, and A. F. Young, Science 360, 62 (2018), https://science.sciencemag.org/content/360/6384/62.full.pdf; Y. Xie, A. T. Pierce, J. M. Park, D. E. Parker, E. Khalaf, P. Ledwith, Y. Cao, S. H. Lee, S. Chen, P. R. Forrester, K. Watanabe, T. Taniguchi, A. Vishwanath, P. Jarillo- Herrero, and A. Yacoby, Nature 600, 439 (2021).
  15. ↑ Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Knu ̈ppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, “Thermodynamic evidence of fractional Chern insulator in moiré MoTe2,” (2023), Nature volume 622, pages 69–73 (2023); J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe, Y. Ran, T. Cao, L. Fu, D. Xiao, W. Yao, and X. Xu, Nature (2023), 10.1038/s41586-023-06289-w; H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu, T. Taniguchi, K. Watanabe, J. haw Chu, T. Cao, L. Fu, W. Yao, C.-Z. Chang, D. Cobden, D. Xiao, and X. Xu, “Observation of fractionally quantized anomalous hall effect,” (2023), arXiv:2308.02657 [cond-mat.mes-hall]
  16. ↑ F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. H. Mac- Donald, Phys. Rev. Lett. 122, 086402 (2019)
  17. ↑ C. Wang, X.-W. Zhang, X. Liu, Y. He, X. Xu, Y. Ran, T. Cao, and D. Xiao, “Fractional chern insulator in twisted bilayer MoTe2,” (2023), arXiv:2304.11864 [cond-mat.str-el]; V. Cr ́epel and L. Fu, "Anomalous Hall metal and fractional Chern insulator in twisted transition metal dichalcogenides," Phys. Rev. B 107, L201109 (2023); N. Morales-Dura ́n, J. Wang, G. R. Schleder, M. Angeli, Z. Zhu, E. Kaxiras, C. Repellin, and J. Cano, “Pressure–enhanced fractional chern insulators in moiré transition metal dichalcogenides along a magic line,” (2023), arXiv:2304.06669 [cond-mat.str-el]; N. Morales-Dura ́n, N. Wei, and A. H. MacDonald, “Magic angles and fractional chern insulators in twisted homobilayer tmds,” (2023), arXiv:2308.03143 [cond-mat.str-el]; J. Yu, J. Herzog-Arbeitman, M. Wang, O. Vafek, B. A. Bernevig, and N. Regnault, “Fractional chern insulators vs. non-magnetic states in twisted bilayer MoTe2,” (2023), arXiv:2309.14429 [cond-mat.mes-hall]



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