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Tunable moiré materials for probing Berry physics and topology

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Abstract

Berry curvature physics and quantum geometric effects have been instrumental in advancing topological condensed matter physics in recent decades. Although Landau level-based flat bands and conventional 3D solids have been pivotal in exploring rich topological phenomena, they are constrained by their limited ability to undergo dynamic tuning. By stark contrast, moiré systems have risen as a versatile platform for engineering bands and manipulating the distribution of Berry curvature in momentum space. These moiré systems not only harbour tunable topological bands, modifiable through a plethora of parameters, but also provide unprecedented access to large length scales and low energy scales. Furthermore, they offer unique opportunities stemming from the symmetry-breaking mechanisms and electron correlations associated with the underlying flat bands that are beyond the reach of conventional crystalline solids. A diverse array of tools, encompassing quantum electron transport in both linear and nonlinear response regimes and optical excitation techniques, provide direct avenues for investigating Berry physics in these materials. This Review navigates the evolving landscape of tunable moiré materials, highlighting recent experimental breakthroughs in the field of topological physics. Additionally, we delineate the most pressing challenges and offer insights into promising avenues for future research.

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Fig. 1: Berry physics in moiré systems.
Fig. 2: Symmetry breaking in moiré superlattices.
Fig. 3: Valley Hall effect probed in electron transport.
Fig. 4: Quantum valley Hall effect.
Fig. 5: Quantum anomalous Hall effect.
Fig. 6: Nonlinear Hall effect.

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References

  1. Kennes, D. M. et al. Moiré heterostructures as a condensed-matter quantum simulator. Nat. Phys. 17, 155–163 (2021). This review discusses using the tunability of moiré heterostructures to experimentally simulate different fundamental many-body quantum models in condensed matter.

    Article  CAS  Google Scholar 

  2. He, F. et al. Moiré patterns in 2D materials: a review. ACS Nano 15, 5944–5958 (2021).

    Article  CAS  PubMed  Google Scholar 

  3. Mounet, N. et al. Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds. Nat. Nanotechnol. 13, 246–252 (2018). This theoretical study identifies possible 2D materials.

    Article  CAS  PubMed  Google Scholar 

  4. Geim, A. K. & Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 419–425 (2013). This review provides an early overview of 2D materials research.

    Article  CAS  PubMed  Google Scholar 

  5. Novoselov, K. S., Mishchenko, A., Carvalho, A. & Neto, A. H. C. 2D materials and van der Waals heterostructures. Science 353, aac9439 (2016). This review provides an overview of the emerging 2D materials research.

    Article  CAS  PubMed  Google Scholar 

  6. Liu, Y. et al. Van der Waals heterostructures and devices. Nat. Rev. Mater. 1, 16042 (2016).

    Article  CAS  Google Scholar 

  7. Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Nat. Mater. 19, 1265–1275 (2020).

    Article  CAS  PubMed  Google Scholar 

  8. Andrei, E. Y. et al. The marvels of moiré materials. Nat. Rev. Mater. 6, 201–206 (2021).

    Article  CAS  Google Scholar 

  9. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018). This paper provides the first experimental demonstration of a correlated insulator arising in a moiré superlattice.

    Article  CAS  PubMed  Google Scholar 

  10. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018). This paper reports the first observation of superconductivity in flat bands of a moiré superlattice.

    Article  CAS  PubMed  Google Scholar 

  11. Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 367, 900–903 (2020). This paper reports the observation of the quantized anomalous Hall effect in a moiré superlattice.

    Article  CAS  PubMed  Google Scholar 

  12. Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene. Science 365, 605–608 (2019). This paper reports the observation of the anomalous Hall effect in a moiré superlattice.

    Article  CAS  PubMed  Google Scholar 

  13. Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect. Rev. Mod. Phys. 82, 1539–1592 (2010). This review covers the different mechanisms responsible for the anomalous Hall effect observed in non-moiré materials.

    Article  Google Scholar 

  14. von Klitzing, K. The quantized Hall effect. Rev. Mod. Phys. 58, 519–531 (1986). This review discusses the quantum Hall effect.

    Article  Google Scholar 

  15. Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010). This review provides a pedagogical introduction to Berry physics and its experimental effects.

    Article  CAS  Google Scholar 

  16. Gao, A. et al. Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure. Science 381, 181–186 (2023).

    Article  CAS  PubMed  Google Scholar 

  17. Wang, N. et al. Quantum metric-induced nonlinear transport in a topological antiferromagnet. Nature 621, 487–492 (2023).

    Article  CAS  PubMed  Google Scholar 

  18. Liu, J., Ma, Z., Gao, J. & Dai, X. Quantum valley Hall effect, orbital magnetism, and anomalous Hall effect in twisted multilayer graphene systems. Phys. Rev. X 9, 031021 (2019).

    CAS  Google Scholar 

  19. Sodemann, I. & Fu, L. Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials. Phys. Rev. Lett. 115, 216806 (2015). This paper theoretically proposes second-harmonic Hall voltage generation owing to the Berry curvature dipole.

    Article  PubMed  Google Scholar 

  20. Sinha, S. et al. Berry curvature dipole senses topological transition in a moiré superlattice. Nat. Phys. 18, 765–770 (2022). This paper reports the experimental detection of a topological transition in a moiré material, probed using the Berry curvature dipole.

    Article  CAS  Google Scholar 

  21. Bhalla, P., Das, K., Culcer, D. & Agarwal, A. Resonant second-harmonic generation as a probe of quantum geometry. Phys. Rev. Lett. 129, 227401 (2022).

    Article  CAS  PubMed  Google Scholar 

  22. Ahn, J., Guo, G.-Y., Nagaosa, N. & Vishwanath, A. Riemannian geometry of resonant optical responses. Nat. Phys. 18, 290–295 (2022).

    Article  CAS  Google Scholar 

  23. Chang, C.-Z., Liu, C.-X. & MacDonald, A. H. Colloquium: quantum anomalous Hall effect. Rev. Mod. Phys. 95, 011002 (2023).

    Article  CAS  Google Scholar 

  24. Jungwirth, T., Niu, Q. & MacDonald, A. H. Anomalous Hall effect in ferromagnetic semiconductors. Phys. Rev. Lett. 88, 207208 (2002).

    Article  CAS  PubMed  Google Scholar 

  25. Ren, Y., Qiao, Z. & Niu, Q. Topological phases in two-dimensional materials: a review. Rep. Prog. Phys. 79, 066501 (2016).

    Article  PubMed  Google Scholar 

  26. Vanderbilt, D. Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators (Cambridge Univ. Press, 2018).

  27. Bernevig, B. A. Topological Insulators and Topological Superconductors (Princeton Univ. Press, 2013).

  28. Yankowitz, M. et al. Emergence of superlattice Dirac points in graphene on hexagonal boron nitride. Nat. Phys. 8, 382–386 (2012).

    Article  CAS  Google Scholar 

  29. Xiao, D., Yao, W. & Niu, Q. Valley-contrasting physics in graphene: magnetic moment and topological transport. Phys. Rev. Lett. 99, 236809 (2007). This paper presents a theoretical study of the Berry phase inducing a valley-dependent Hall transport.

    Article  PubMed  Google Scholar 

  30. Sinha, S. et al. Bulk valley transport and Berry curvature spreading at the edge of flat bands. Nat. Commun. 11, 5548 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  31. Song, J. C. W., Samutpraphoot, P. & Levitov, L. S. Topological Bloch bands in graphene superlattices. Proc. Natl Acad. Sci. USA 112, 10879–10883 (2015). This theoretical study highlights that moiré bands can be topological.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  32. Wolf, T. M. R., Zilberberg, O., Levkivskyi, I. & Blatter, G. Substrate-induced topological minibands in graphene. Phys. Rev. B 98, 125408 (2018).

    Article  CAS  Google Scholar 

  33. Zhang, Y.-H., Mao, D., Cao, Y., Jarillo-Herrero, P. & Senthil, T. Nearly flat Chern bands in moiré superlattices. Phys. Rev. B 99, 075127 (2019).

    Article  CAS  Google Scholar 

  34. Chittari, B. L., Chen, G., Zhang, Y., Wang, F. & Jung, J. Gate-tunable topological flat bands in trilayer graphene boron-nitride moiré superlattices. Phys. Rev. Lett. 122, 016401 (2019).

    Article  CAS  PubMed  Google Scholar 

  35. Zhang, S., Dai, X. & Liu, J. Spin-polarized nematic order, quantum valley Hall states, and field-tunable topological transitions in twisted multilayer graphene systems. Phys. Rev. Lett. 128, 026403 (2022).

    Article  CAS  PubMed  Google Scholar 

  36. Chen, G. et al. Tunable correlated Chern insulator and ferromagnetism in a moiré superlattice. Nature 579, 56–61 (2020).

    Article  CAS  PubMed  Google Scholar 

  37. Zhu, J., Su, J.-J. & MacDonald, A. Voltage-controlled magnetic reversal in orbital Chern insulators. Phys. Rev. Lett. 125, 227702 (2020).

    Article  CAS  PubMed  Google Scholar 

  38. Polshyn, H. et al. Electrical switching of magnetic order in an orbital Chern insulator. Nature 588, 66–70 (2020).

    Article  CAS  PubMed  Google Scholar 

  39. Adak, P. C. et al. Perpendicular electric field drives Chern transitions and layer polarization changes in Hofstadter bands. Nat. Commun. 13, 7781 (2022).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  40. Hao, Z. et al. Electric field-tunable superconductivity in alternating-twist magic-angle trilayer graphene. Science 371, 1133–1138 (2021).

    Article  CAS  PubMed  Google Scholar 

  41. Mannaï, M., Fuchs, J.-N., Piéchon, F. & Haddad, S. Stacking-induced Chern insulator. Phys. Rev. B 107, 045117 (2023).

    Article  Google Scholar 

  42. Stauber, T., Low, T. & Gómez-Santos, G. Chiral response of twisted bilayer graphene. Phys. Rev. Lett. 120, 046801 (2018).

    Article  CAS  PubMed  Google Scholar 

  43. Crosse, J. A., Nakatsuji, N., Koshino, M. & Moon, P. Hofstadter butterfly and the quantum Hall effect in twisted double bilayer graphene. Phys. Rev. B 102, 035421 (2020).

    Article  CAS  Google Scholar 

  44. Koshino, M. Band structure and topological properties of twisted double bilayer graphene. Phys. Rev. B 99, 235406 (2019).

    Article  CAS  Google Scholar 

  45. Chebrolu, N. R., Chittari, B. L. & Jung, J. Flat bands in twisted double bilayer graphene. Phys. Rev. B 99, 235417 (2019).

    Article  CAS  Google Scholar 

  46. Liu, B. et al. Higher-order band topology in twisted moiré superlattice. Phys. Rev. Lett. 126, 066401 (2021).

    Article  CAS  PubMed  Google Scholar 

  47. He, M. et al. Symmetry-broken Chern insulators in twisted double bilayer graphene. Nano Lett. 23, 11066–11072 (2023).

    Article  CAS  PubMed  Google Scholar 

  48. Lau, C. N., Bockrath, M. W., Mak, K. F. & Zhang, F. Reproducibility in the fabrication and physics of moiré materials. Nature 602, 41–50 (2022). This review highlights the reproducibility issues and challenges in the fabrication of moiré superlattice devices.

    Article  CAS  PubMed  Google Scholar 

  49. Hu, J.-X., Zhang, C.-P., Xie, Y.-M. & Law, K. T. Nonlinear Hall effects in strained twisted bilayer WSe2. Commun. Phys. 5, 255 (2022).

    Article  CAS  Google Scholar 

  50. Zhang, C.-P. et al. Giant nonlinear Hall effect in strained twisted bilayer graphene. Phys. Rev. B 106, L041111 (2022).

    Article  CAS  Google Scholar 

  51. Chakraborty, A. et al. Nonlinear anomalous Hall effects probe topological phase-transitions in twisted double bilayer graphene. 2D Mater. 9, 045020 (2022).

    Article  CAS  Google Scholar 

  52. Zhang, D., Schoenherr, P., Sharma, P. & Seidel, J. Ferroelectric order in van der Waals layered materials. Nat. Rev. Mater. 8, 25–40 (2023).

    Article  Google Scholar 

  53. McGilly, L. J. et al. Visualization of moiré superlattices. Nat. Nanotechnol. 15, 580–584 (2020).

    Article  CAS  PubMed  Google Scholar 

  54. Kalinin, S. V. & Meunier, V. Electronic flexoelectricity in low-dimensional systems. Phys. Rev. B 77, 033403 (2008).

    Article  Google Scholar 

  55. Wang, B., Gu, Y., Zhang, S. & Chen, L.-Q. Flexoelectricity in solids: progress, challenges, and perspectives. Prog. Mater. Sci. 106, 100570 (2019).

    Article  CAS  Google Scholar 

  56. Weston, A. et al. Interfacial ferroelectricity in marginally twisted 2D semiconductors. Nat. Nanotechnol. 17, 390–395 (2022).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  57. Kang, K. et al. Switchable moiré potentials in ferroelectric WTe2/WSe2 superlattices. Nat. Nanotechnol. 18, 861–866 (2023).

    Article  CAS  PubMed  Google Scholar 

  58. Zhong, D. et al. Van der Waals engineering of ferromagnetic semiconductor heterostructures for spin and valleytronics. Sci. Adv. 3, e1603113 (2017).

    Article  PubMed  PubMed Central  Google Scholar 

  59. Lin, J.-X. et al. Zero-field superconducting diode effect in small-twist-angle trilayer graphene. Nat. Phys. 18, 1221–1227 (2022).

    Article  CAS  Google Scholar 

  60. Arora, H. S. et al. Superconductivity in metallic twisted bilayer graphene stabilized by WSe2. Nature 583, 379–384 (2020).

    Article  CAS  PubMed  Google Scholar 

  61. Lin, J.-X. et al. Spin-orbit-driven ferromagnetism at half moiré filling in magic-angle twisted bilayer graphene. Science 375, 437–441 (2022).

    Article  CAS  PubMed  Google Scholar 

  62. Devakul, T. & Fu, L. Quantum anomalous Hall effect from inverted charge transfer gap. Phys. Rev. X 12, 021031 (2022).

    CAS  Google Scholar 

  63. Wu, F., Lovorn, T., Tutuc, E. & MacDonald, A. Hubbard model physics in transition metal dichalcogenide moiré bands. Phys. Rev. Lett. 121, 026402 (2018). This theoretical paper proposes TMDCs as a Hubbard model simulator.

    Article  CAS  PubMed  Google Scholar 

  64. Uri, A. et al. Superconductivity and strong interactions in a tunable moiré quasicrystal. Nature 620, 762–767 (2023).

    Article  CAS  PubMed  Google Scholar 

  65. Wu, F., Lovorn, T., Tutuc, E., Martin, I. & MacDonald, A. Topological insulators in twisted transition metal dichalcogenide homobilayers. Phys. Rev. Lett. 122, 086402 (2019). This theoretical study highlights that TMDC moiré materials are a platform to realize correlated and topological states.

    Article  CAS  PubMed  Google Scholar 

  66. Crépel, V. & Fu, L. Anomalous Hall metal and fractional Chern insulator in twisted transition metal dichalcogenides. Phys. Rev. B 107, L201109 (2023).

    Article  Google Scholar 

  67. Wu, F., Lovorn, T. & MacDonald, A. Topological exciton bands in moiré heterojunctions. Phys. Rev. Lett. 118, 147401 (2017).

    Article  PubMed  Google Scholar 

  68. Xiong, R. et al. Correlated insulator of excitons in WSe2/WS2 moiré superlattices. Science 380, 860–864 (2023).

    Article  CAS  PubMed  Google Scholar 

  69. Lu, X. et al. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene. Nature 574, 653–657 (2019).

    Article  CAS  PubMed  Google Scholar 

  70. Tang, Y. et al. Simulation of Hubbard model physics in WSe2/WS2 moiré superlattices. Nature 579, 353–358 (2020).

    Article  CAS  PubMed  Google Scholar 

  71. Xu, Y. et al. Correlated insulating states at fractional fillings of moiré superlattices. Nature 587, 214–218 (2020).

    Article  CAS  PubMed  Google Scholar 

  72. Shimazaki, Y. et al. Strongly correlated electrons and hybrid excitons in a moiré heterostructure. Nature 580, 472–477 (2020).

    Article  CAS  PubMed  Google Scholar 

  73. Choi, Y. et al. Correlation-driven topological phases in magic-angle twisted bilayer graphene. Nature 589, 536–541 (2021).

    Article  CAS  PubMed  Google Scholar 

  74. Liu, X. et al. Tuning electron correlation in magic-angle twisted bilayer graphene using Coulomb screening. Science 371, 1261–1265 (2021).

    Article  CAS  PubMed  Google Scholar 

  75. Burg, G. W. et al. Emergence of correlations in alternating twist quadrilayer graphene. Nat. Mater. 21, 884–889 (2022).

    Article  CAS  PubMed  Google Scholar 

  76. Balents, L., Dean, C. R., Efetov, D. K. & Young, A. F. Superconductivity and strong correlations in moiré flat bands. Nat. Phys. 16, 725–733 (2020).

    Article  CAS  Google Scholar 

  77. Mak, K. F. & Shan, J. Semiconductor moiré materials. Nat. Nanotechnol. 17, 686–695 (2022).

    Article  CAS  PubMed  Google Scholar 

  78. Bhowmik, S., Ghosh, A. & Chandni, U. Emergent phases in graphene flat bands. Preprint at http://arxiv.org/abs/2309.08938 (2023).

  79. Lee, J., Mak, K. F. & Shan, J. Electrical control of the valley Hall effect in bilayer MoS2 transistors. Nat. Nanotechnol. 11, 421–425 (2016).

    Article  CAS  PubMed  Google Scholar 

  80. Shimazaki, Y. et al. Generation and detection of pure valley current by electrically induced Berry curvature in bilayer graphene. Nat. Phys. 11, 1032–1036 (2015).

    Article  CAS  Google Scholar 

  81. Sui, M. et al. Gate-tunable topological valley transport in bilayer graphene. Nat. Phys. 11, 1027–1031 (2015).

    Article  CAS  Google Scholar 

  82. Yin, J. et al. Tunable and giant valley-selective Hall effect in gapped bilayer graphene. Science 375, 1398–1402 (2022).

    Article  CAS  PubMed  Google Scholar 

  83. Mak, K. F., McGill, K. L., Park, J. & McEuen, P. L. The valley Hall effect in MoS2 transistors. Science 344, 1489–1492 (2014). This experimental work is the first demonstration of the valley Hall effect.

    Article  CAS  PubMed  Google Scholar 

  84. Gorbachev, R. V. et al. Detecting topological currents in graphene superlattices. Science 346, 448–451 (2014).

    Article  CAS  PubMed  Google Scholar 

  85. Adak, P. C. et al. Tunable bandwidths and gaps in twisted double bilayer graphene on the verge of correlations. Phys. Rev. B 101, 125428 (2020).

    Article  CAS  Google Scholar 

  86. Wang, Y. et al. Bulk and edge properties of twisted double bilayer graphene. Nat. Phys. 18, 48–53 (2022).

    Article  CAS  Google Scholar 

  87. Ma, C. et al. Moiré band topology in twisted bilayer graphene. Nano Lett. 20, 6076–6083 (2020).

    Article  CAS  PubMed  Google Scholar 

  88. Aharon-Steinberg, A. et al. Long-range nontopological edge currents in charge-neutral graphene. Nature 593, 528–534 (2021).

    Article  CAS  PubMed  Google Scholar 

  89. Liu, C.-X., Zhang, S.-C. & Qi, X.-L. The quantum anomalous Hall effect: theory and experiment. Annu. Rev. Condens. Matter Phys. 7, 301–321 (2016).

    Article  Google Scholar 

  90. Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988). This theoretical paper outlines the Haldane model that demonstrates the possibility of having a quantum Hall effect in the absence of a magnetic field.

    Article  CAS  PubMed  Google Scholar 

  91. Chang, C.-Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013). This paper reports the first experimental observation of the quantum anomalous Hall effect in a topological insulator.

    Article  CAS  PubMed  Google Scholar 

  92. Deng, Y. et al. Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi2Te4. Science 367, 895–900 (2020).

    Article  CAS  PubMed  Google Scholar 

  93. Wang, Z., Tang, C., Sachs, R., Barlas, Y. & Shi, J. Proximity-induced ferromagnetism in graphene revealed by the anomalous Hall effect. Phys. Rev. Lett. 114, 016603 (2015).

    Article  CAS  PubMed  Google Scholar 

  94. Li, T. et al. Quantum anomalous Hall effect from intertwined moiré bands. Nature 600, 641–646 (2021).

    Article  CAS  PubMed  Google Scholar 

  95. Cai, J. et al. Signatures of fractional quantum anomalous Hall states in twisted MoTe2. Nature 622, 63–68 (2023).

    Article  CAS  PubMed  Google Scholar 

  96. Park, H. et al. Observation of fractionally quantized anomalous Hall effect. Nature 622, 74–79 (2023).

    Article  CAS  PubMed  Google Scholar 

  97. Zeng, Y. et al. Thermodynamic evidence of fractional Chern insulator in moiré MoTe2. Nature 622, 69–73 (2023).

    Article  CAS  PubMed  Google Scholar 

  98. Xu, F. et al. Observation of integer and fractional quantum anomalous Hall effects in twisted bilayer MoTe2. Phys. Rev. X 13, 031037 (2023). Together with Cai et al. (2023), Park et al. (2023) and Zeng et al. (2023), this paper experimentally demonstrates the fractional quantum anomalous Hall effect in moiré heterostructures.

    CAS  Google Scholar 

  99. Ledwith, P. J., Tarnopolsky, G., Khalaf, E. & Vishwanath, A. Fractional Chern insulator states in twisted bilayer graphene: an analytical approach. Phys. Rev. Res. 2, 023237 (2020).

    Article  CAS  Google Scholar 

  100. Repellin, C. & Senthil, T. Chern bands of twisted bilayer graphene: fractional Chern insulators and spin phase transition. Phys. Rev. Res. 2, 023238 (2020).

    Article  CAS  Google Scholar 

  101. Wilhelm, P., Lang, T. C. & Läuchli, A. M. Interplay of fractional Chern insulator and charge density wave phases in twisted bilayer graphene. Phys. Rev. B 103, 125406 (2021).

    Article  CAS  Google Scholar 

  102. Wang, C. et al. Fractional Chern insulator in twisted bilayer MoTe2. Phys. Rev. Lett. 132, 036501 (2024).

    Article  CAS  PubMed  Google Scholar 

  103. Reddy, A. P., Alsallom, F. F., Zhang, Y., Devakul, T. & Fu, L. Fractional quantum anomalous Hall states in twisted bilayer MoTe2 and WSe2. Phys. Rev. B 108, 085117 (2023).

    Article  CAS  Google Scholar 

  104. Reddy, A. P. & Fu, L. Toward a global phase diagram of the fractional quantum anomalous Hall effect. Phys. Rev. B 108, 245159 (2023).

    Article  CAS  Google Scholar 

  105. Devakul, T., Crépel, V., Zhang, Y. & Fu, L. Magic in twisted transition metal dichalcogenide bilayers. Nat. Commun. 12, 6730 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  106. Li, H., Kumar, U., Sun, K. & Lin, S.-Z. Spontaneous fractional Chern insulators in transition metal dichalcogenide moiré superlattices. Phys. Rev. Res. 3, L032070 (2021).

    Article  CAS  Google Scholar 

  107. Spanton, E. M. et al. Observation of fractional Chern insulators in a van der Waals heterostructure. Science 360, 62–66 (2018).

    Article  CAS  PubMed  Google Scholar 

  108. Xie, Y. et al. Fractional Chern insulators in magic-angle twisted bilayer graphene. Nature 600, 439–443 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  109. Lu, Z. et al. Fractional quantum anomalous Hall effect in multilayer graphene. Nature 626, 759–764 (2023).

    Article  Google Scholar 

  110. Dong, Z., Patri, A. S. & Senthil, T. Theory of fractional quantum anomalous Hall phases in pentalayer rhombohedral graphene moiré structures. Preprint at http://arxiv.org/abs/2311.03445 (2023).

  111. Zhou, B., Yang, H. & Zhang, Y.-H. Fractional quantum anomalous Hall effects in rhombohedral multilayer graphene in the moiréless limit and in Coulomb imprinted superlattice. Preprint at http://arxiv.org/abs/2311.04217 (2023).

  112. Dong, J. et al. Anomalous Hall crystals in rhombohedral multilayer graphene I: interaction-driven Chern bands and fractional quantum Hall states at zero magnetic field. Preprint at http://arxiv.org/abs/2311.05568 (2023).

  113. Polshyn, H. et al. Topological charge density waves at half-integer filling of a moiré superlattice. Nat. Phys. 18, 42–47 (2022).

    Article  CAS  Google Scholar 

  114. Bhowmik, S. et al. Broken-symmetry states at half-integer band fillings in twisted bilayer graphene. Nat. Phys. 18, 639–643 (2022).

    Article  CAS  Google Scholar 

  115. Rhodes, D. A. et al. Enhanced superconductivity in monolayer Td-MoTe2. Nano Lett. 21, 2505–2511 (2021).

    Article  CAS  PubMed  Google Scholar 

  116. Jindal, A. et al. Coupled ferroelectricity and superconductivity in bilayer Td-MoTe2. Nature 613, 48–52 (2023).

    Article  CAS  PubMed  Google Scholar 

  117. Kappera, R. et al. Phase-engineered low-resistance contacts for ultrathin MoS2 transistors. Nat. Mater. 13, 1128–1134 (2014).

    Article  CAS  PubMed  Google Scholar 

  118. Claassen, M., Xian, L., Kennes, D. M. & Rubio, A. Ultra-strong spin-orbit coupling and topological moiré engineering in twisted ZrS2 bilayers. Nat. Commun. 13, 4915 (2022).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  119. Dong, J., Wang, J., Ledwith, P. J., Vishwanath, A. & Parker, D. E. Composite fermi liquid at zero magnetic field in twisted MoTe2. Phys. Rev. Lett. 131, 136502 (2023).

    Article  CAS  PubMed  Google Scholar 

  120. Goldman, H., Reddy, A. P., Paul, N. & Fu, L. Zero-field composite fermi liquid in twisted semiconductor bilayers. Phys. Rev. Lett. 131, 136501 (2023).

    Article  CAS  PubMed  Google Scholar 

  121. Jain, J. In a twist, composite fermions form and flow without a magnetic field. Physics 16, 163 (2023).

    Article  Google Scholar 

  122. Ma, Q., Grushin, A. G. & Burch, K. S. Topology and geometry under the nonlinear electromagnetic spotlight. Nat. Mater. 20, 1601–1614 (2021).

    Article  CAS  PubMed  Google Scholar 

  123. Du, Z., Lu, H.-Z. & Xie, X. Nonlinear Hall effects. Nat. Rev. Phys. 3, 744–752 (2021).

    Article  Google Scholar 

  124. Ma, Q. et al. Observation of the nonlinear Hall effect under time-reversal-symmetric conditions. Nature 565, 337–342 (2019). This paper represents the first experimental report of the nonlinear Hall effect.

    Article  CAS  PubMed  Google Scholar 

  125. Kang, K., Li, T., Sohn, E., Shan, J. & Mak, K. F. Nonlinear anomalous Hall effect in few-layer WTe2. Nat. Mater. 18, 324–328 (2019).

    Article  CAS  PubMed  Google Scholar 

  126. Son, J., Kim, K.-H., Ahn, Y., Lee, H.-W. & Lee, J. Strain engineering of the Berry curvature dipole and valley magnetization in monolayer MoS2. Phys. Rev. Lett. 123, 036806 (2019).

    Article  CAS  PubMed  Google Scholar 

  127. Qin, M.-S. et al. Strain tunable Berry curvature dipole, orbital magnetization and nonlinear Hall effect in WSe2 monolayer. Chin. Phys. Lett. 38, 017301 (2021).

    Article  CAS  Google Scholar 

  128. Huang, M. et al. Giant nonlinear Hall effect in twisted bilayer WSe2. Natl Sci. Rev. 10, nwac232 (2023).

    Article  CAS  PubMed  Google Scholar 

  129. Huang, M. et al. Intrinsic nonlinear Hall effect and gate-switchable Berry curvature sliding in twisted bilayer graphene. Phys. Rev. Lett. 131, 066301 (2023).

    Article  CAS  PubMed  Google Scholar 

  130. Kazmierczak, N. P. et al. Strain fields in twisted bilayer graphene. Nat. Mater. 20, 956–963 (2021).

    Article  CAS  PubMed  Google Scholar 

  131. Turkel, S. et al. Orderly disorder in magic-angle twisted trilayer graphene. Science 376, 193–199 (2022).

    Article  CAS  PubMed  Google Scholar 

  132. Pantaleón, P. A., Low, T. & Guinea, F. Tunable large Berry dipole in strained twisted bilayer graphene. Phys. Rev. B 103, 205403 (2021).

    Article  Google Scholar 

  133. He, Z. & Weng, H. Giant nonlinear Hall effect in twisted bilayer WTe2. npj Quantum Mater. 6, 101 (2021).

    Article  CAS  Google Scholar 

  134. Mannaï, M. & Haddad, S. Twistronics versus straintronics in twisted bilayers of graphene and transition metal dichalcogenides. Phys. Rev. B 103, L201112 (2021).

    Article  Google Scholar 

  135. Facio, J. I. et al. Strongly enhanced Berry dipole at topological phase transitions in BiTeI. Phys. Rev. Lett. 121, 246403 (2018).

    Article  CAS  PubMed  Google Scholar 

  136. He, P. et al. Graphene moiré superlattices with giant quantum nonlinearity of chiral Bloch electrons. Nat. Nanotechnol. 17, 378–383 (2022).

    Article  CAS  PubMed  Google Scholar 

  137. Duan, J. et al. Giant second-order nonlinear Hall effect in twisted bilayer graphene. Phys. Rev. Lett. 129, 186801 (2022).

    Article  CAS  PubMed  Google Scholar 

  138. Du, Z. Z., Wang, C. M., Li, S., Lu, H.-Z. & Xie, X. C. Disorder-induced nonlinear Hall effect with time-reversal symmetry. Nat. Commun. 10, 3047 (2019).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  139. Lahiri, S., Bhore, T., Das, K. & Agarwal, A. Nonlinear magnetoresistivity in two-dimensional systems induced by Berry curvature. Phys. Rev. B 105, 045421 (2022).

    Article  CAS  Google Scholar 

  140. Xiao, J. et al. Berry curvature memory through electrically driven stacking transitions. Nat. Phys. 16, 1028–1034 (2020).

    Article  CAS  Google Scholar 

  141. Nishijima, T. et al. Ferroic Berry curvature dipole in a topological crystalline insulator at room temperature. Nano Lett. 23, 2247–2252 (2023).

    Article  CAS  PubMed  Google Scholar 

  142. Song, J. C. W. & Rudner, M. S. Chiral plasmons without magnetic field. Proc. Natl Acad. Sci. USA 113, 4658–4663 (2016).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  143. Kumar, A. et al. Chiral plasmon in gapped Dirac systems. Phys. Rev. B 93, 041413 (2016).

    Article  Google Scholar 

  144. Huang, T. et al. Observation of chiral and slow plasmons in twisted bilayer graphene. Nature 605, 63–68 (2022).

    Article  CAS  PubMed  Google Scholar 

  145. Chakraborty, A., Dutta, D. & Agarwal, A. Tunable interband and intraband plasmons in twisted double bilayer graphene. Phys. Rev. B 106, 155422 (2022).

    Article  CAS  Google Scholar 

  146. Hesp, N. C. H. et al. Observation of interband collective excitations in twisted bilayer graphene. Nat. Phys. 17, 1162–1168 (2021).

    Article  CAS  Google Scholar 

  147. Zhao, W. et al. Efficient Fizeau drag from Dirac electrons in monolayer graphene. Nature 594, 517–521 (2021).

    Article  CAS  PubMed  Google Scholar 

  148. Arora, A., Rudner, M. S. & Song, J. C. W. Quantum plasmonic nonreciprocity in parity-violating magnets. Nano Lett. 22, 9351–9357 (2022).

    Article  CAS  PubMed  Google Scholar 

  149. Papaj, M. & Lewandowski, C. Plasmonic nonreciprocity driven by band hybridization in moiré materials. Phys. Rev. Lett. 125, 066801 (2020).

    Article  CAS  PubMed  Google Scholar 

  150. Dutta, D., Chakraborty, A. & Agarwal, A. Intrinsic nonreciprocal bulk plasmons in noncentrosymmetric magnetic systems. Phys. Rev. B 107, 165404 (2023).

    Article  CAS  Google Scholar 

  151. Rossi, E. Quantum metric and correlated states in two-dimensional systems. Curr. Opin. Solid State Mater. Sci. 25, 100952 (2021).

    Article  CAS  Google Scholar 

  152. Mandal, D., Sarkar, S., Das, K. & Agarwal, A. Quantum geometry induced third order nonlinear transport responses. Preprint at https://doi.org/10.48550/arXiv.2310.19092 (2023).

  153. Das, K., Ghorai, K., Culcer, D. & Agarwal, A. Nonlinear valley Hall effect. Phys. Rev. Lett. 132, 096302 (2024).

    Article  CAS  PubMed  Google Scholar 

  154. Das, K., Lahiri, S., Atencia, R. B., Culcer, D. & Agarwal, A. Intrinsic nonlinear conductivities induced by the quantum metric. Phys. Rev. B 108, L201405 (2023).

    Article  CAS  Google Scholar 

  155. Komissarov, I., Holder, T. & Queiroz, R. The quantum geometric origin of capacitance in insulators. Preprint at https://doi.org/10.48550/arXiv.2306.08035 (2023).

  156. Wei, M. et al. Quantum fluctuation of the quantum geometric tensor and its manifestation as intrinsic Hall signatures in time-reversal invariant systems. Phys. Rev. Lett. 130, 036202 (2023).

    Article  CAS  PubMed  Google Scholar 

  157. Peotta, S. & Törmä, P. Superfluidity in topologically nontrivial flat bands. Nat. Commun. 6, 8944 (2015). These theoretical papers connect the idea of quantum metric to that of flat-band superconductivity.

    Article  CAS  PubMed  Google Scholar 

  158. Julku, A., Peotta, S., Vanhala, T. I., Kim, D.-H. & Törmä, P. Geometric origin of superfluidity in the Lieb-lattice flat band. Phys. Rev. Lett. 117, 045303 (2016). Together with Peotta and Törmä (2015), this theoretical paper connects the idea of quantum metric to that of flat-band superconductivity.

    Article  PubMed  Google Scholar 

  159. Liang, L. et al. Band geometry, Berry curvature, and superfluid weight. Phys. Rev. B 95, 024515 (2017).

    Article  Google Scholar 

  160. Törmä, P., Peotta, S. & Bernevig, B. A. Superconductivity, superfluidity and quantum geometry in twisted multilayer systems. Nat. Rev. Phys. 4, 528–542 (2022).

    Article  Google Scholar 

  161. Tian, H. et al. Evidence for Dirac flat band superconductivity enabled by quantum geometry. Nature 614, 440–444 (2023).

    Article  CAS  PubMed  Google Scholar 

  162. Hazra, T., Verma, N. & Randeria, M. Bounds on the superconducting transition temperature: applications to twisted bilayer graphene and cold atoms. Phys. Rev. X 9, 031049 (2019).

    CAS  Google Scholar 

  163. Mao, D. & Chowdhury, D. Diamagnetic response and phase stiffness for interacting isolated narrow bands. Proc. Natl Acad. Sci. USA 120, e2217816120 (2023).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  164. Arora, A., Kong, J. F. & Song, J. C. W. Strain-induced large injection current in twisted bilayer graphene. Phys. Rev. B 104, L241404 (2021).

    Article  CAS  Google Scholar 

  165. Du, L. et al. Moiré photonics and optoelectronics. Science 379, eadg0014 (2023).

    Article  CAS  PubMed  Google Scholar 

  166. Ma, C. et al. Intelligent infrared sensing enabled by tunable moiré quantum geometry. Nature 604, 266–272 (2022).

    Article  CAS  PubMed  Google Scholar 

  167. Mak, K. F., He, K., Shan, J. & Heinz, T. F. Control of valley polarization in monolayer MoS2 by optical helicity. Nat. Nanotechnol. 7, 494–498 (2012).

    Article  CAS  PubMed  Google Scholar 

  168. Hu, J.-X., Xie, Y.-M. & Law, K. T. Berry curvature, spin Hall effect, and nonlinear optical response in moiré transition metal dichalcogenide heterobilayers. Phys. Rev. B 107, 075424 (2023).

    Article  CAS  Google Scholar 

  169. Tschirhart, C. L. et al. Intrinsic spin Hall torque in a moiré Chern magnet. Nat. Phys. 19, 807–813 (2023).

    Article  CAS  Google Scholar 

  170. Tao, Z. et al. Giant spin Hall effect in AB-stacked MoTe2/WSe2 bilayers. Nat. Nanotechnol. 19, 28–33 (2023).

    Article  PubMed  Google Scholar 

  171. Kaplan, D., Holder, T. & Yan, B. Twisted photovoltaics at terahertz frequencies from momentum shift current. Phys. Rev. Res. 4, 013209 (2022).

    Article  CAS  Google Scholar 

  172. Chaudhary, S., Lewandowski, C. & Refael, G. Shift-current response as a probe of quantum geometry and electron-electron interactions in twisted bilayer graphene. Phys. Rev. Res. 4, 013164 (2022).

    Article  CAS  Google Scholar 

  173. Khalifa, A., Murthy, G. & Kaul, R. K. Absence of edge states in the valley Chern insulator in moiré graphene. Phys. Rev. B 107, 085138 (2023).

    Article  CAS  Google Scholar 

  174. Parameswaran, S. A., Roy, R. & Sondhi, S. L. Fractional quantum Hall physics in topological flat bands. C. R. Phys. 14, 816–839 (2013).

    Article  CAS  Google Scholar 

  175. Liu, Z. & Bergholtz, E. J. in Encyclopedia of Condensed Matter Physics 2nd edn, Vol. 1 (eds Chakraborty, T. et al.) 515–538 (Elsevier, 2023).

  176. Park, M. J., Kim, Y., Cho, G. Y. & Lee, S. Higher-order topological insulator in twisted bilayer graphene. Phys. Rev. Lett. 123, 216803 (2019).

    Article  CAS  PubMed  Google Scholar 

  177. Liu, B.-B., Zeng, X.-T., Chen, C., Chen, Z. & Sheng, X.-L. Second-order and real Chern topological insulator in twisted bilayer α-graphyne. Phys. Rev. B 106, 035153 (2022).

    Article  CAS  Google Scholar 

  178. Rodriguez-Vega, M., Vogl, M. & Fiete, G. A. Low-frequency and Moiré-Floquet engineering: a review. Ann. Phys. 435, 168434 (2021).

    Article  CAS  Google Scholar 

  179. Topp, G. E. et al. Topological Floquet engineering of twisted bilayer graphene. Phys. Rev. Res. 1, 023031 (2019).

    Article  CAS  Google Scholar 

  180. Li, Y., Fertig, H. A. & Seradjeh, B. Floquet-engineered topological flat bands in irradiated twisted bilayer graphene. Phys. Rev. Res. 2, 043275 (2020).

    Article  CAS  Google Scholar 

  181. Lu, M., Zeng, J., Liu, H., Gao, J.-H. & Xie, X. C. Valley-selective Floquet Chern flat bands in twisted multilayer graphene. Phys. Rev. B 103, 195146 (2021).

    Article  CAS  Google Scholar 

  182. Vogl, M., Rodriguez-Vega, M., Flebus, B., MacDonald, A. H. & Fiete, G. A. Floquet engineering of topological transitions in a twisted transition metal dichalcogenide homobilayer. Phys. Rev. B 103, 014310 (2021).

    Article  CAS  Google Scholar 

  183. Brem, S., Linderälv, C., Erhart, P. & Malic, E. Tunable phases of moiré excitons in van der Waals heterostructures. Nano Lett. 20, 8534–8540 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  184. Zhao, S. et al. Excitons in mesoscopically reconstructed moiré heterostructures. Nat. Nanotechnol. 18, 572–579 (2023).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  185. Suri, N., Wang, C., Zhang, Y. & Xiao, D. Chiral phonons in moiré superlattices. Nano Lett. 21, 10026–10031 (2021).

    Article  CAS  PubMed  Google Scholar 

  186. Liu, X., Peng, R., Sun, Z. & Liu, J. Moiré phonons in magic-angle twisted bilayer graphene. Nano Lett. 22, 7791–7797 (2022).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  187. Lin, M.-L. et al. Moiré phonons in twisted bilayer MoS2. ACS Nano 12, 8770–8780 (2018).

    Article  CAS  PubMed  Google Scholar 

  188. Li, Z., Lai, J.-M. & Zhang, J. Review of phonons in moiré superlattices. J. Semicond. 44, 011902 (2023).

    Article  Google Scholar 

  189. Li, Y.-H. & Cheng, R. Moiré magnons in twisted bilayer magnets with collinear order. Phys. Rev. B 102, 094404 (2020).

    Article  CAS  Google Scholar 

  190. Ganguli, S. C. et al. Visualization of moiré magnons in monolayer ferromagnet. Nano Lett. 23, 3412–3417 (2023).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  191. Fujimoto, M., Koschke, H. & Koshino, M. Topological charge pumping by a sliding moiré pattern. Phys. Rev. B 101, 041112 (2020).

    Article  CAS  Google Scholar 

  192. Zhang, Y., Gao, Y. & Xiao, D. Topological charge pumping in twisted bilayer graphene. Phys. Rev. B 101, 041410 (2020).

    Article  CAS  Google Scholar 

  193. Su, Y. & Lin, S.-Z. Topological sliding moiré heterostructure. Phys. Rev. B 101, 041113 (2020).

    Article  CAS  Google Scholar 

  194. Zhai, D. & Yao, W. Theory of tunable flux lattices in the homobilayer moiré of twisted and uniformly strained transition metal dichalcogenides. Phys. Rev. Mater. 4, 094002 (2020).

    Article  CAS  Google Scholar 

  195. Classen, L., Honerkamp, C. & Scherer, M. M. Competing phases of interacting electrons on triangular lattices in moiré heterostructures. Phys. Rev. B 99, 195120 (2019).

    Article  CAS  Google Scholar 

  196. Wu, F. Topological chiral superconductivity with spontaneous vortices and supercurrent in twisted bilayer graphene. Phys. Rev. B 99, 195114 (2019).

    Article  CAS  Google Scholar 

  197. Pershoguba, S. S. & Yakovenko, V. M. Optical control of topological memory based on orbital magnetization. Phys. Rev. B 105, 064423 (2022).

    Article  CAS  Google Scholar 

  198. Yuan, S. et al. Geometric deep optical sensing. Science 379, eade1220 (2023).

    Article  CAS  PubMed  Google Scholar 

  199. Onishi, Y. & Fu, L. High-efficiency energy harvesting based on nonlinear Hall rectifier. Preprint at http://arxiv.org/abs/2211.17219 (2023).

  200. Montblanch, A. R.-P., Barbone, M., Aharonovich, I., Atatüre, M. & Ferrari, A. C. Layered materials as a platform for quantum technologies. Nat. Nanotechnol. 18, 555–571 (2023).

    Article  CAS  PubMed  Google Scholar 

  201. He, P. et al. Quantum frequency doubling in the topological insulator Bi2Se3. Nat. Commun. 12, 698 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  202. Ho, S.-C. et al. Hall effects in artificially corrugated bilayer graphene without breaking time-reversal symmetry. Nat. Electron. 4, 116–125 (2021).

    Article  CAS  Google Scholar 

  203. Valenzuela, S. O. & Tinkham, M. Direct electronic measurement of the spin Hall effect. Nature 442, 176–179 (2006).

    Article  CAS  PubMed  Google Scholar 

  204. Chen, S. et al. Electrically tunable correlated and topological states in twisted monolayer–bilayer graphene. Nat. Phys. 17, 374–380 (2021).

    Article  CAS  Google Scholar 

  205. Tseng, C.-C. et al. Anomalous Hall effect at half filling in twisted bilayer graphene. Nat. Phys. 18, 1038–1042 (2022).

    Article  CAS  Google Scholar 

  206. Kuiri, M. et al. Spontaneous time-reversal symmetry breaking in twisted double bilayer graphene. Nat. Commun. 13, 6468 (2022).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  207. Sundaram, G. & Niu, Q. Wave-packet dynamics in slowly perturbed crystals: gradient corrections and Berry-phase effects. Phys. Rev. B 59, 14915–14925 (1999).

    Article  CAS  Google Scholar 

  208. Ong, N. & Lee, W. Geometry and the anomalous Hall effect in ferromagnets. In Proc. 8th International Symposium on Foundations of Quantum Mechanics in the Light of New Technology, ISQM-Tokyo 2005 121–126 (World Scientific, 2006).

  209. Zhou, J., Shan, W.-Y., Yao, W. & Xiao, D. Berry phase modification to the energy spectrum of excitons. Phys. Rev. Lett. 115, 166803 (2015).

    Article  PubMed  Google Scholar 

  210. Karplus, R. & Luttinger, J. M. Hall effect in ferromagnetics. Phys. Rev. 95, 1154–1160 (1954).

    Article  Google Scholar 

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Acknowledgements

The authors thank J. Sarkar, A. Mukherjee, K. Maji, S. Layek, A. Basu, U. Chandni, B. Datta, J. Song, J. K. Jain, A. Chakraborty, K. Das and P. Yadav for the discussions. M.M.D. acknowledges the Department of Science and Technology (DST) of India for the J.C. Bose fellowship JCB/2022/000045, Nanomission grant SR/NM/NS45/2016, and DST SUPRA grant SPR/2019/001247 along with the Department of Atomic Energy of the Government of India 12-R&D-TFR-5.10-0100 for the support. A.A. thanks the Department of Science and Technology of the Government of India for project number DST/NM/TUE/QM-6/2019(G)-IIT Kanpur, for the financial support. P.C.A. acknowledges support by the National Science Foundation under grant number OMA-2328993.

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  1. These authors contributed equally: Pratap Chandra Adak, Subhajit Sinha.

Authors and Affiliations

  1. Department of Condensed Matter Physics and Materials Science, Tata Institute of Fundamental Research, Mumbai, India

    Pratap Chandra Adak, Subhajit Sinha & Mandar M. Deshmukh

  2. Department of Physics, City College of New York, New York, NY, USA

    Pratap Chandra Adak

  3. Department of Physics, Indian Institute of Technology, Kanpur, India

    Amit Agarwal

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Adak, P.C., Sinha, S., Agarwal, A. et al. Tunable moiré materials for probing Berry physics and topology. Nat Rev Mater (2024). https://doi.org/10.1038/s41578-024-00671-4

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