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Superconductivity, superfluidity and quantum geometry in twisted multilayer systems

Nature Reviews Physics volume 4, pages 528–542 (2022)Cite this article

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Abstract

Superconductivity has been observed in moiré systems such as twisted bilayer graphene, which host flat, dispersionless electronic bands. In parallel, theory work has discovered that superconductivity and superfluidity of flat-band systems can be made possible by the quantum geometry and topology of the band structure. These recent key developments are merging into a flourishing research topic: understanding the possible connection and ramifications of quantum geometry on the induced superconductivity and superfluidity in moiré multilayer and other flat-band systems. This article presents an introduction to how quantum geometry governs superconductivity and superfluidity in platforms including, and beyond, graphene. Ultracold gases are introduced as a complementary platform for quantum geometric effects and a comparison is made to moiré materials. An outlook sketches the prospects of twisted multilayer systems in providing the route to room-temperature superconductivity.

Key points

  • Bands of (quasi-)flat dispersion dramatically enhance Cooper pairing as their (nearly) vanishing kinetic energy allows interaction effects to dominate.

  • Superfluidity and stable supercurrents are possible in a flat band if the band has non-trivial quantum geometry: the related overlap of the Wannier functions facilitates movement of interacting particles even when non-interacting particles would be localized.

  • Twisted bilayer graphene exhibits nearly flat bands at its Fermi energy for small twist angles. Theory work suggests that quantum geometry is essential for the experimentally observed twisted bilayer graphene superconductivity and that a topological invariant called Euler class provides a lower bound for its superfluid weight.

  • Ultracold gases offer another promising platform for highly controllable studies of superfluidity in moiré geometries. The first experiments are on the way.

  • A flat-band dispersion together with a quantum geometry that guarantees superfluidity are powerful guidelines for the search of superconductivity at elevated temperatures.

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Fig. 1: Formation of flat bands.
Fig. 2: Role of Wannier function overlap in flat-band superfluidity.
Fig. 3: Twisted bilayer graphene lattice, experimental phase diagram and band structures.
Fig. 4: Geometric contribution in twisted bilayer graphene superfluid weight.
Fig. 5: Wilson loops/Wannier centres: fragile and stable topology of twisted bilayer graphene.
Fig. 6: Ultracold gas analogues of twisted bilayer graphene.

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References

  1. Zhou, X. et al. High-temperature superconductivity. Nat. Rev. Phys. 3, 462–465 (2021).

    Article  Google Scholar 

  2. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).

    Article  ADS  Google Scholar 

  3. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

    Article  ADS  Google Scholar 

  4. MacDonald, A. H. Bilayer graphene’s wicked, twisted road. Physics 12, 12 (2019).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  6. 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  Google Scholar 

  7. Kennes, D. M. et al. Moiré heterostructures as a condensed-matter quantum simulator. Nat. Phys. 17, 155–163 (2021).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  9. Kopnin, N., Heikkilä, T. & Volovik, G. High-temeprature surface superconductivity in topological flat-band systems. Phys. Rev. B 83, 220503 (2011).

    Article  ADS  Google Scholar 

  10. Heikkilä, T., Kopnin, N. & Volovik, G. Flat bands in topological media. JETP Lett. 94, 233 (2011).

    Article  ADS  Google Scholar 

  11. Khodel, V. A. & Shaginyan, V. R. New approach in the microscopic Fermi systems theory. Phys. Rep. 249, 1–134 (1994).

    Article  ADS  Google Scholar 

  12. Schrieffer, J. R. Theory of Superconductivity. Frontiers in Physics (Benjamin, 1964).

  13. Peotta, S. & Törmä, P. Superfluidity in topologically nontrivial flat bands. Nat. Commun. 6, 8944 (2015).

    Article  ADS  Google Scholar 

  14. 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).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  16. Törmä, P., Liang, L. & Peotta, S. Quantum metric and effective mass of a two-body bound state in a flat band. Phys. Rev. B 98, 220511 (2018).

    Article  ADS  Google Scholar 

  17. Huhtinen, K.-E., Herzog-Arbeitman, J., Chew, A., Bernevig, B. A. & Törmä, P. Revisiting flat band superconductivity: dependence on minimal quantum metric and band touchings. Preprint at arXiv 2203.11133 (2022).

  18. Provost, J. P. & Vallee, G. Riemannian structure on manifolds of quantum states. Commun. Math. Phys. 76, 289–301 (1980).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Resta, R. The insulating state of matter: a geometrical theory. Eur. Phys. J. B 79, 121–137 (2011).

    Article  ADS  Google Scholar 

  20. Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    Article  ADS  Google Scholar 

  21. Bernevig, B. A. & Hughes, T. L. Topological Insulators and Topological Superconductors (Princeton Univ. Press, 2013).

  22. Mielke, A. Ferromagnetic ground states for the Hubbard model on line graphs. J. Phys. A Math. Gen. 24, 2 (1991).

    Article  MathSciNet  Google Scholar 

  23. Lieb, E. H. Two theorems on the Hubbard model. Phys. Rev. Lett. 62, 1201 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  24. Leykam, D., Andreanov, A. & Flach, S. Artificial flat band systems: from lattice models to experiments. Adv. Phys. X 3, 1473052 (2018).

    Google Scholar 

  25. Călugăru, D. et al. General construction and topological classification of crystalline flat bands. Nat. Phys. 18, 185–189 (2022).

    Article  Google Scholar 

  26. Tinkham, M. Introduction to Superconductivity 2nd edn (Dover Publications, 2004).

  27. Scalapino, D., White, S. R. & Zhang, S. C. Superfluid density and the Drude weight of the Hubbard model. Phys. Rev. Lett. 68, 2830 (1992).

    Article  ADS  Google Scholar 

  28. Scalapino, D., White, S. R. & Zhang, S. C. Insulator, metal, or superconductor: The criteria. Phys. Rev. B 47, 7995 (1993).

    Article  ADS  Google Scholar 

  29. Chandrasekhar, B. S. & Einzel, D. The superconducting penetration depth from the semiclassical model. Ann. Phys. 505, 535–546 (1993).

    Article  Google Scholar 

  30. Leggett, A. J. On the superfluid fraction of an arbitrary many-body system at T=0. J. Stat. Phys. 93, 927–941 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. London, F. & London, H. The electromagnetic equations of the supraconductor. Proc. R. Soc. A Math. Phys. Eng. 149, 71–88 (1935).

    ADS  MATH  Google Scholar 

  32. Basov, D. N. & Chubukov, A. V. Manifesto for a higher Tc. Nat. Phys. 7, 272–276 (2011).

    Article  Google Scholar 

  33. Tovmasyan, M., Peotta, S., Törmä, P. & Huber, S. D. Effective theory and emergent SU(2) symmetry in the flat bands of attractive Hubbard models. Phys. Rev. B 94, 245149 (2016).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  35. Iskin, M. Two-body problem in a multiband lattice and the role of quantum geometry. Phys. Rev. A 103, 053311 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  36. Oh, M. et al. Evidence for unconventional superconductivity in twisted bilayer graphene. Nature 600, 240–245 (2021).

    Article  ADS  Google Scholar 

  37. Moon, K. et al. Spontaneous interlayer coherence in double-layer quantum Hall systems: Charged vortices and Kosterlitz-Thouless phase transitions. Phys. Rev. B 51, 5138–5170 (1995).

    Article  ADS  Google Scholar 

  38. Kopnin, N. B. Surface superconductivity in multilayered rhombohedral graphene: Supercurrent. JETP Lett. 94, 81 (2011).

    Article  ADS  Google Scholar 

  39. Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I. & Vanderbilt, D. Maximally localized Wannier functions: Theory and applications. Rev. Mod. Phys. 84, 1419 (2012).

    Article  ADS  Google Scholar 

  40. Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).

    Article  ADS  Google Scholar 

  41. Brouder, C., Panati, G., Calandra, M., Mourougane, C. & Marzari, N. Exponential localization of Wannier functions in insulators. Phys. Rev. Lett. 98, 046402 (2007).

    Article  ADS  Google Scholar 

  42. Panati, G. Triviality of Bloch and Bloch–Dirac bundles. Ann. Henri Poincaré 8, 995–1011 (2007).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Monaco, D., Panati, G., Pisante, A. & Teufel, S. Optimal decay of Wannier functions in Chern and quantum Hall insulators. Commun. Math. Phys. 359, 61–100 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  44. Verma, N., Hazra, T. & Randeria, M. Optical spectral weight, phase stiffness, and Tc bounds for trivial and topological flat band superconductors. Proc. Natl Acad. Sci. USA 118, e2106744118 (2021).

    Article  Google Scholar 

  45. Xie, F., Song, Z., Lian, B. & Bernevig, B. A. Topology-bounded superfluid weight in twisted bilayer graphene. Phys. Rev. Lett. 124, 167002 (2020).

    Article  ADS  Google Scholar 

  46. Herzog-Arbeitman, J., Peri, V., Schindler, F., Huber, S. D. & Bernevig, B. A. Superfluid weight bounds from symmetry and quantum geometry in flat bands. Phys. Rev. Lett. 128, 087002 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  47. Nelson, D. R. & Kosterlitz, J. M. Universal jump in the superfluid density of two-dimensional superfluids. Phys. Rev. Lett. 39, 1201–1205 (1977).

    Article  ADS  Google Scholar 

  48. Hofmann, J. S., Berg, E. & Chowdhury, D. Superconductivity, pseudogap, and phase separation in topological flat bands. Phys. Rev. B 102, 201112 (2020).

    Article  ADS  Google Scholar 

  49. Peri, V., Song, Z.-D., Bernevig, B. A. & Huber, S. D. Fragile topology and flat-band superconductivity in the strong-coupling regime. Phys. Rev. Lett. 126, 027002 (2021).

    Article  ADS  Google Scholar 

  50. Tovmasyan, M., Peotta, S., Liang, L., Törmä, P. & Huber, S. D. Preformed pairs in flat Bloch bands. Phys. Rev. B 98, 134513 (2018).

    Article  ADS  Google Scholar 

  51. Mondaini, R., Batrouni, G. G. & Grémaud, B. Pairing and superconductivity in the flat band: Creutz lattice. Phys. Rev. B 98, 155142 (2018).

    Article  ADS  Google Scholar 

  52. Chan, S. M., Grémaud, B. & Batrouni, G. G. Pairing and superconductivity in quasi-one-dimensional flat-band systems: Creutz and sawtooth lattices. Phys. Rev. B 105, 024502 (2022).

    Article  ADS  Google Scholar 

  53. Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233–12237 (2011).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  55. Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059–1064 (2019).

    Article  ADS  Google Scholar 

  56. Saito, Y., Ge, J., Watanabe, K., Taniguchi, T. & Young, A. F. Independent superconductors and correlated insulators in twisted bilayer graphene. Nat. Phys. 16, 926–930 (2020).

    Article  Google Scholar 

  57. Stepanov, P. et al. Untying the insulating and superconducting orders in magic-angle graphene. Nature 583, 375–378 (2020).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  59. Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 367, 900–903 (2019).

    Article  ADS  Google Scholar 

  60. Jiang, Y. et al. Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene. Nature 573, 91–95 (2019).

    Article  ADS  Google Scholar 

  61. Xie, Y. et al. Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene. Nature 572, 101–105 (2019).

    Article  ADS  Google Scholar 

  62. Choi, Y. et al. Electronic correlations in twisted bilayer graphene near the magic angle. Nat. Phys. 15, 1174–1180 (2019).

    Article  Google Scholar 

  63. Nuckolls, K. P. et al. Strongly correlated Chern insulators in magic-angle twisted bilayer graphene. Nature 588, 610–615 (2020).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  65. Das, I. et al. Symmetry broken Chern insulators and magic series of Rashba-like Landau level crossings in magic angle bilayer graphene. Nat. Phys. 17, 710–714 (2021).

    Article  Google Scholar 

  66. Wu, S., Zhang, Z., Watanabe, K., Taniguchi, T. & Andrei, E. Y. Chern insulators, van Hove singularities and topological flat bands in magic-angle twisted bilayer graphene. Nat. Mater. 20, 488–494 (2021).

    Article  ADS  Google Scholar 

  67. Lu, X. et al. Multiple flat bands and topological Hofstadter butterfly in twisted bilayer graphene close to the second magic angle. Proc. Natl Acad. Sci. USA 118, e2100006118 (2021).

    Article  Google Scholar 

  68. Burg, G. W. et al. Evidence of emergent symmetry and valley Chern number in twisted double-bilayer graphene. Preprint at arXiv 2006.14000 (2020).

  69. Zou, L., Po, H. C., Vishwanath, A. & Senthil, T. Band structure of twisted bilayer graphene: Emergent symmetries, commensurate approximants, and Wannier obstructions. Phys. Rev. B 98, 085435 (2018).

    Article  ADS  Google Scholar 

  70. Fu, Y., König, E. J., Wilson, J. H., Chou, Y.-Z. & Pixley, J. H. Magic-angle semimetals. NPJ Quantum Mater. 5, 71 (2020).

    Article  ADS  Google Scholar 

  71. Liu, J., Liu, J. & Dai, X. Pseudo Landau level representation of twisted bilayer graphene: Band topology and implications on the correlated insulating phase. Phys. Rev. B 99, 155415 (2019).

    Article  ADS  Google Scholar 

  72. Kang, J. & Vafek, O. Symmetry, maximally localized Wannier states, and a low-energy model for twisted bilayer graphene narrow bands. Phys. Rev. X 8, 031088 (2018).

    Google Scholar 

  73. Song, Z. et al. All magic angles in twisted bilayer graphene are topological. Phys. Rev. Lett. 123, 036401 (2019).

    Article  ADS  Google Scholar 

  74. Po, H. C., Zou, L., Senthil, T. & Vishwanath, A. Faithful tight-binding models and fragile topology of magic-angle bilayer graphene. Phys. Rev. B 99, 195455 (2019).

    Article  ADS  Google Scholar 

  75. Ahn, J., Park, S. & Yang, B.-J. Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle. Phys. Rev. X 9, 021013 (2019).

    Google Scholar 

  76. Bouhon, A., Black-Schaffer, A. M. & Slager, R.-J. Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry. Phys. Rev. B 100, 195135 (2019).

    Article  ADS  Google Scholar 

  77. Lian, B., Xie, F. & Bernevig, B. A. Landau level of fragile topology. Phys. Rev. B 102, 041402 (2020).

    Article  ADS  Google Scholar 

  78. Hejazi, K., Liu, C. & Balents, L. Landau levels in twisted bilayer graphene and semiclassical orbits. Phys. Rev. B 100, 035115 (2019).

    Article  ADS  Google Scholar 

  79. Kang, J. & Vafek, O. Strong coupling phases of partially filled twisted bilayer graphene narrow bands. Phys. Rev. Lett. 122, 246401 (2019).

    Article  ADS  Google Scholar 

  80. Bultinck, N. et al. Ground state and hidden symmetry of magic-angle graphene at even integer filling. Phys. Rev. X 10, 031034 (2020).

    Google Scholar 

  81. Po, H. C., Zou, L., Vishwanath, A. & Senthil, T. Origin of Mott insulating behavior and superconductivity in twisted bilayer graphene. Phys. Rev. X 8, 031089 (2018).

    Google Scholar 

  82. Julku, A., Peltonen, T., Liang, L., Heikkilä, T. & Törmä, P. Superfluid weight and Berezinskii-Kosterlitz-Thouless transition temperature of twisted bilayer graphene. Phys. Rev. B 101, 060505 (2020).

    Article  ADS  Google Scholar 

  83. Hu, X., Hyart, T., Pikulin, D. I. & Rossi, E. Geometric and conventional contribution to the superfluid weight in twisted bilayer graphene. Phys. Rev. Lett. 123, 237002 (2019).

    Article  ADS  Google Scholar 

  84. Kang, J. & Vafek, O. Non-Abelian Dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic-angle twisted bilayer graphene. Phys. Rev. B 102, 035161 (2020).

    Article  ADS  Google Scholar 

  85. Soejima, T., Parker, D. E., Bultinck, N., Hauschild, J. & Zaletel, M. P. Efficient simulation of moiré materials using the density matrix renormalization group. Phys. Rev. B 102, 205111 (2020).

    Article  ADS  Google Scholar 

  86. Pixley, J. H. & Andrei, E. Y. Ferromagnetism in magic-angle graphene. Science 365, 543 (2019).

    Article  ADS  Google Scholar 

  87. Xie, M. & MacDonald, A. H. Nature of the correlated insulator states in twisted bilayer graphene. Phys. Rev. Lett. 124, 097601 (2020).

    Article  ADS  Google Scholar 

  88. Liu, J. & Dai, X. Theories for the correlated insulating states and quantum anomalous Hall effect phenomena in twisted bilayer graphene. Phys. Rev. B 103, 035427 (2021).

    Article  ADS  Google Scholar 

  89. Cea, T. & Guinea, F. Band structure and insulating states driven by Coulomb interaction in twisted bilayer graphene. Phys. Rev. B 102, 045107 (2020).

    Article  ADS  Google Scholar 

  90. Da Liao, Y. et al. Correlation-induced insulating topological phases at charge neutrality in twisted bilayer graphene. Phys. Rev. X 11, 011014 (2021).

    Google Scholar 

  91. Abouelkomsan, A., Liu, Z. & Bergholtz, E. J. Particle-hole duality, emergent Fermi liquids, and fractional Chern insulators in moiré flatbands. Phys. Rev. Lett. 124, 106803 (2020).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  93. Vafek, O. & Kang, J. Renormalization group study of hidden symmetry in twisted bilayer graphene with Coulomb interactions. Phys. Rev. Lett. 125, 257602 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  94. Fernandes, R. M. & Venderbos, J. W. F. Nematicity with a twist: Rotational symmetry breaking in a moiré superlattice. Sci. Adv. 6, eaba8834 (2020).

    Article  ADS  Google Scholar 

  95. Wilson, J. H., Fu, Y., Das Sarma, S. & Pixley, J. H. Disorder in twisted bilayer graphene. Phys. Rev. Res. 2, 023325 (2020).

    Article  Google Scholar 

  96. Wang, J., Zheng, Y., Millis, A. J. & Cano, J. Chiral approximation to twisted bilayer graphene: Exact intravalley inversion symmetry, nodal structure, and implications for higher magic angles. Phys. Rev. Res. 3, 023155 (2021).

    Article  Google Scholar 

  97. Song, Z.-D., Lian, B., Regnault, N. & Bernevig, B. A. Twisted bilayer graphene. II. Stable symmetry anomaly. Phys. Rev. B 103, 205412 (2021).

    Article  ADS  Google Scholar 

  98. Bernevig, B. A. et al. Twisted bilayer graphene. V. Exact analytic many-body excitations in Coulomb Hamiltonians: Charge gap, Goldstone modes, and absence of Cooper pairing. Phys. Rev. B 103, 205415 (2021).

    Article  ADS  Google Scholar 

  99. Codecido, E. et al. Correlated insulating and superconducting states in twisted bilayer graphene below the magic angle. Sci. Adv. 5, eaaw9770 (2019).

    Article  ADS  Google Scholar 

  100. Roy, B. & Juričić, V. Unconventional superconductivity in nearly flat bands in twisted bilayer graphene. Phys. Rev. B 99, 121407 (2019).

    Article  ADS  Google Scholar 

  101. Wang, J., Cano, J., Millis, A. J., Liu, Z. & Yang, B. Exact Landau level description of geometry and interaction in a flatband. Phys. Rev. Lett. 127, 246403 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  102. Lian, B. et al. Twisted bilayer graphene. IV. Exact insulator ground states and phase diagram. Phys. Rev. B 103, 205414 (2021).

    Article  ADS  Google Scholar 

  103. Zhang, X. et al. Correlated insulating states and transport signature of superconductivity in twisted trilayer graphene superlattices. Phys. Rev. Lett. 127, 166802 (2021).

    Article  ADS  Google Scholar 

  104. Chen, G. et al. Signatures of tunable superconductivity in a trilayer graphene moiré superlattice. Nature 572, 215–219 (2019).

    Article  Google Scholar 

  105. Park, J., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene. Nature 590, 249 (2021).

    Article  ADS  Google Scholar 

  106. Classen, L. Geometry rescues superconductivity in twisted graphene. Physics 13, 23 (2020).

    Article  Google Scholar 

  107. Su, Y. & Lin, S.-Z. Pairing symmetry and spontaneous vortex-antivortex lattice in superconducting twisted-bilayer graphene: Bogoliubov-de Gennes approach. Phys. Rev. B 98, 195101 (2018).

    Article  ADS  Google Scholar 

  108. Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H. Graphene bilayer with a twist: Electronic structure. Phys. Rev. Lett. 99, 256802 (2007).

    Article  ADS  Google Scholar 

  109. Wang, Z., Chaudhary, G., Chen, Q. & Levin, K. Quantum geometric contributions to the BKT transition: Beyond mean field theory. Phys. Rev. B 102, 184504 (2020).

    Article  ADS  Google Scholar 

  110. Kitamura, T., Yamashita, T., Ishizuka, J., Daido, A. & Yanase, Y. Superconductivity in monolayer FeSe enhanced by quantum geometry. Preprint at https://arxiv.org/abs/2108.10002 (2021).

  111. Lee, D.-H. Hunting down unconventional superconductors. Science 357, 32–33 (2017).

    Article  ADS  Google Scholar 

  112. Gallego, S. V., Tasci, E. S., Flor, G., Perez-Mato, J. M. & Aroyo, M. I. Magnetic symmetry in the Bilbao Crystallographic Server: a computer program to provide systematic absences of magnetic neutron diffraction. J. Appl. Crystallogr. 45, 1236–1247 (2012).

    Article  Google Scholar 

  113. Alexandradinata, A., Dai, X. & Bernevig, B. A. Wilson-loop characterization of inversion-symmetric topological insulators. Phys. Rev. B 89, 155114 (2014).

    Article  ADS  Google Scholar 

  114. Ahn, J., Kim, D., Kim, Y. & Yang, B.-J. Band topology and linking structure of nodal line semimetals with Z2 monopole charges. Phys. Rev. Lett. 121, 106403 (2018).

    Article  ADS  Google Scholar 

  115. Ünal, F. N., Bouhon, A. & Slager, R.-J. Topological euler class as a dynamical observable in optical lattices. Phys. Rev. Lett. 125, 053601 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  116. Po, H. C., Watanabe, H. & Vishwanath, A. Fragile topology and Wannier obstructions. Phys. Rev. Lett. 121, 126402 (2018).

    Article  ADS  Google Scholar 

  117. Cano, J. et al. Topology of disconnected elementary band representations. Phys. Rev. Lett. 120, 266401 (2018).

    Article  ADS  Google Scholar 

  118. Yu, R., Qi, X. L., Bernevig, A., Fang, Z. & Dai, X. Equivalent expression of \({{\mathbb{Z}}}_{2}\) topological invariant for band insulators using the non-Abelian Berry connection. Phys. Rev. B 84, 075119 (2011).

  119. Bultinck, N., Chatterjee, S. & Zaletel, M. P. Mechanism for anomalous Hall ferromagnetism in twisted bilayer graphene. Phys. Rev. Lett. 124, 166601 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  120. Bernevig, B. A., Song, Z. D., Regnault, N. & Lian, B. Twisted bilayer graphene. III. Interacting Hamiltonian and exact symmetries. Phys. Rev. B 103, 205413 (2021).

    Article  ADS  Google Scholar 

  121. Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008).

    Article  ADS  Google Scholar 

  122. Giorgini, S., Pitaevskii, L. P. & Stringari, S. Theory of ultracold atomic Fermi gases. Rev. Mod. Phys. 80, 1215–1274 (2008).

    Article  ADS  Google Scholar 

  123. Törmä, P. & Sengstock, K. (eds) Quantum Gas Experiments: Exploring Many-Body States (Imperial College Press, 2015).

  124. Lewenstein, M., Sanpera, A. & Ahufinger, V. Ultracold Atoms in Optical Lattices: Simulating Quantum Many-Body Systems 1st edn. (Oxford Univ. Press, 2012).

  125. Cooper, N. R., Dalibard, J. & Spielman, I. B. Topological bands for ultracold atoms. Rev. Mod. Phys. 91, 015005 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  126. Soltan-Panahi, P. et al. Multi-component quantum gases in spin-dependent hexagonal lattices. Nat. Phys. 7, 434–440 (2011).

    Article  Google Scholar 

  127. Tarruell, L., Greif, D., Uehlinger, T., Jotzu, G. & Esslinger, T. Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice. Nature 483, 302–305 (2012).

    Article  ADS  Google Scholar 

  128. Jo, G.-B. et al. Ultracold atoms in a tunable optical kagome lattice. Phys. Rev. Lett. 108, 045305 (2012).

    Article  ADS  Google Scholar 

  129. Taie, S. et al. Coherent driving and freezing of bosonic matter wave in an optical Lieb lattice. Sci. Adv. 1, e1500854 (2015).

    Article  ADS  Google Scholar 

  130. Gall, M., Wurz, N., Samland, J., Chan, C. F. & Köhl, M. Competing magnetic orders in a bilayer Hubbard model with ultracold atoms. Nature 589, 40–43 (2021).

    Article  ADS  Google Scholar 

  131. Sbroscia, M. et al. Observing localization in a 2D quasicrystalline optical lattice. Phys. Rev. Lett. 125, 200604 (2020).

    Article  ADS  Google Scholar 

  132. Mancini, M. et al. Observation of chiral edge states with neutral fermions in synthetic Hall ribbons. Science 349, 1510–1513 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  133. Stuhl, B. K., Lu, H.-I., Aycock, L. M., Genkina, D. & Spielman, I. B. Visualizing edge states with an atomic Bose gas in the quantum Hall regime. Science 349, 1514–1518 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  134. Ozawa, T. & Price, H. M. Topological quantum matter in synthetic dimensions. Nat. Rev. Phys. 1, 349–357 (2019).

    Article  Google Scholar 

  135. O’Riordan, L. J., White, A. C. & Busch, T. Moiré superlattice structures in kicked Bose-Einstein condensates. Phys. Rev. A 93, 023609 (2016).

    Article  ADS  Google Scholar 

  136. González-Tudela, A. & Cirac, J. I. Cold atoms in twisted-bilayer optical potentials. Phys. Rev. A 100, 053604 (2019).

    Article  ADS  Google Scholar 

  137. Salamon, T. et al. Simulating twistronics without a twist. Phys. Rev. Lett. 125, 030504 (2020).

    Article  ADS  Google Scholar 

  138. Salamon, T., Chhajlany, R. W., Dauphin, A., Lewenstein, M. & Rakshit, D. Quantum anomalous Hall phase in synthetic bilayers via twistronics without a twist. Phys. Rev. B 102, 235126 (2020).

    Article  ADS  Google Scholar 

  139. Luo, X.-W. & Zhang, C. Spin-twisted optical lattices: Tunable flat bands and Larkin-Ovchinnikov superfluids. Phys. Rev. Lett. 126, 103201 (2021).

    Article  ADS  Google Scholar 

  140. Meng, Z. et al. Atomic Bose-Einstein condensate in a twisted-bilayer optical lattice. Preprint at https://arxiv.org/abs/2110.00149 (2021).

  141. Carusotto, I. & Castin, Y. Nonequilibrium and local detection of the normal fraction of a trapped two-dimensional Bose gas. Phys. Rev. A 84, 053637 (2011).

    Article  ADS  Google Scholar 

  142. Sidorenkov, L. A. et al. Second sound and the superfluid fraction in a Fermi gas with resonant interactions. Nature 498, 78–81 (2013).

    Article  ADS  Google Scholar 

  143. Ho, T.-L. & Zhou, Q. Obtaining the phase diagram and thermodynamic quantities of bulk systems from the densities of trapped gases. Nat. Phys. 6, 131–134 (2010).

    Article  Google Scholar 

  144. John, S. T., Hadzibabic, Z. & Cooper, N. R. Spectroscopic method to measure the superfluid fraction of an ultracold atomic gas. Phys. Rev. A 83, 023610 (2011).

    Article  ADS  Google Scholar 

  145. Edge, J. M. & Cooper, N. R. Probing ultracold Fermi gases with light-induced gauge potentials. Phys. Rev. A 83, 053619 (2011).

    Article  ADS  Google Scholar 

  146. Peotta, S., Chien, C.-C. & Di Ventra, M. Phase-induced transport in atomic gases: From superfluid to Mott insulator. Phys. Rev. A 90, 053615 (2014).

    Article  ADS  Google Scholar 

  147. Rossini, D., Fazio, R., Giovannetti, V. & Silva, A. Quantum quenches, linear response and superfluidity out of equilibrium. EPL 107, 30002 (2014).

    Article  ADS  Google Scholar 

  148. Krinner, S., Esslinger, T. & Brantut, J.-P. Two-terminal transport measurements with cold atoms. J. Phys. Condens. Matter 29, 343003 (2017).

    Article  Google Scholar 

  149. Krinner, S., Stadler, D., Husmann, D., Brantut, J.-P. & Esslinger, T. Observation of quantized conductance in neutral matter. Nature 517, 64–67 (2015).

    Article  ADS  Google Scholar 

  150. Krinner, S., Stadler, D., Meineke, J., Brantut, J.-P. & Esslinger, T. Superfluidity with disorder in a thin film of quantum gas. Phys. Rev. Lett. 110, 100601 (2013).

    Article  ADS  Google Scholar 

  151. Pyykkönen, V. A. J. et al. Flat-band transport and Josephson effect through a finite-size sawtooth lattice. Phys. Rev. B 103, 144519 (2021).

    Article  ADS  Google Scholar 

  152. Huhtinen, K.-E. & Törmä, P. Possible insulator-pseudogap crossover in the attractive Hubbard model on the Lieb lattice. Phys. Rev. B 103, L220502 (2021).

    Article  ADS  Google Scholar 

  153. Chin, C., Grimm, R., Julienne, P. & Tiesinga, E. Feshbach resonances in ultracold gases. Rev. Mod. Phys. 82, 1225–1286 (2010).

    Article  ADS  Google Scholar 

  154. Mazurenko, A. et al. A cold-atom Fermi–Hubbard antiferromagnet. Nature 545, 462–466 (2017).

    Article  ADS  Google Scholar 

  155. Dolgirev, P. E. et al. Characterizing two-dimensional superconductivity via nanoscale noise magnetometry with single-spin qubits. Phys. Rev. B 105, 024507 (2022).

    Article  ADS  Google Scholar 

  156. Tian, H. et al. Evidence for flat band Dirac superconductor originating from quantum geometry. Preprint at https://arxiv.org/abs/2112.13401 (2021).

  157. Jia, Y. et al. Evidence for a monolayer excitonic insulator. Nat. Phys. 18, 87–93 (2022).

    Article  Google Scholar 

  158. Wang, P. et al. One-dimensional Luttinger liquids in a two-dimensional moiré lattice. Nature 605, 57–62 (2022).

    Article  ADS  Google Scholar 

  159. Zhou, H., Xie, T., Taniguchi, T., Watanabe, K. & Young, A. F. Superconductivity in rhombohedral trilayer graphene. Nature 598, 434–438 (2021).

    Article  ADS  Google Scholar 

  160. 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).

    Google Scholar 

  161. Hofmann, J. S., Chowdhury, D., Kivelson, S. A. & Berg, E. Heuristic bounds on superconductivity and how to exceed them. Preprint at https://arxiv.org/abs/2105.09322 (2021).

  162. Topp, G. E., Eckhardt, C. J., Kennes, D. M., Sentef, M. A. & Törmä, P. Light-matter coupling and quantum geometry in moiré materials. Phys. Rev. B 104, 064306 (2021).

    Article  ADS  Google Scholar 

  163. Chaudhary, S., Lewandowski, G. & 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  Google Scholar 

  164. Hu, X., Hyart, T., Pikulin, D. I. & Rossi, E. Quantum-metric-enabled exciton condensate in double twisted bilayer graphene. Phys. Rev. B 105, L140506 (2022).

    Article  ADS  Google Scholar 

  165. Julku, A., Bruun, G. M. & Törmä, P. Quantum geometry and flat band Bose-Einstein condensation. Phys. Rev. Lett. 127, 170404 (2021).

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

S.P. and P.T. acknowledge support by the Academy of Finland under project numbers 330384, 336369, 303351 and 327293. B.A.B. acknowledges support from the Office of Naval Research grant no. N00014-20-1-2303 and from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 101020833).

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Authors and Affiliations

  1. Department of Applied Physics, Aalto University School of Science, Aalto, Finland

    Päivi Törmä & Sebastiano Peotta

  2. Department of Physics, Princeton University, Princeton, NJ, USA

    Bogdan A. Bernevig

  3. Donostia International Physics Center, Donostia–San Sebastián, Spain

    Bogdan A. Bernevig

  4. Ikerbasque, Basque Foundation for Science, Bilbao, Spain

    Bogdan A. Bernevig

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  1. Päivi Törmä
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Törmä, P., Peotta, S. & Bernevig, B.A. Superconductivity, superfluidity and quantum geometry in twisted multilayer systems. Nat Rev Phys 4, 528–542 (2022). https://doi.org/10.1038/s42254-022-00466-y

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