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  • Review Article
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The physics of quantum materials

Abstract

The physical description of all materials is rooted in quantum mechanics, which describes how atoms bond and electrons interact at a fundamental level. Although these quantum effects can in many cases be approximated by a classical description at the macroscopic level, in recent years there has been growing interest in material systems where quantum effects remain manifest over a wider range of energy and length scales. Such quantum materials include superconductors, graphene, topological insulators, Weyl semimetals, quantum spin liquids, and spin ices. Many of them derive their properties from reduced dimensionality, in particular from confinement of electrons to two-dimensional sheets. Moreover, they tend to be materials in which electrons cannot be considered as independent particles but interact strongly and give rise to collective excitations known as quasiparticles. In all cases, however, quantum-mechanical effects fundamentally alter properties of the material. This Review surveys the electronic properties of quantum materials through the prism of the electron wavefunction, and examines how its entanglement and topology give rise to a rich variety of quantum states and phases; these are less classically describable than conventional ordered states also driven by quantum mechanics, such as ferromagnetism.

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Figure 1: Collective order of electrons on a two-dimensional square lattice.
Figure 2: Resonant X-ray scattering from a Kitaev magnet.
Figure 3: Neutron scattering from Higgs modes.
Figure 4: Quantum oscillations and quantum criticality in high-temperature superconductors.
Figure 5: Examples of massless fermions in quantum materials.

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References

  1. Parkin, S. S. P. et al. Giant tunnelling magnetoresistance at room temperature with MgO (100) tunnel barriers. Nat. Mater. 3, 862–867 (2004).

    Article  ADS  Google Scholar 

  2. Yuasa, S., Nagahama, T., Fukushima, A., Suzuki, Y. & Ando, K. Giant room-temperature magnetoresistance in single-crystal Fe/MgO/Fe magnetic tunnel junctions. Nat. Mater. 3, 868–871 (2004).

    Article  ADS  Google Scholar 

  3. Yin, J. et al. Satellite-based entanglement distribution over 1200 kilometers. Science 356, 1140–1144 (2017).

    Article  Google Scholar 

  4. Schrödinger, E. Discussion of probability relations between separated systems. Math. Proc. Cambr. 31, 555–563 (1935).

    Article  MATH  Google Scholar 

  5. Shoenberg, D. Magnetic Oscillations in Metals (Cambridge Univ. Press, 1984).

    Book  Google Scholar 

  6. Bardeen, J., Cooper, L. N. & Schrieffer, J. R. Theory of superconductivity. Phys. Rev. 108, 1175–1204 (1957).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Bednorz, J. G. & Müller, K. A. Possible high Tc superconductivity in the Ba–La–Cu–O system. Z. Phys. B 64, 189–193 (1986).

    Article  ADS  Google Scholar 

  8. Drozdov, A. P., Eremets, M. I., Troyan, I. A., Ksenofontov, V. & Shylin, S. I. Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system. Nature 525, 73–76 (2015).

    Article  ADS  Google Scholar 

  9. Balasubramanian, G. et al. Ultralong spin coherence time in isotopically engineered diamond. Nat. Mater. 8, 383–387 (2009).

    Article  ADS  Google Scholar 

  10. Maurer, P. C. et al. Room-temperature quantum bit memory exceeding one second. Science 336, 1283–1286 (2012).

    Article  ADS  Google Scholar 

  11. Savary, L. & Balents, L. Quantum spin liquids: a review. Rep. Prog. Phys. 80, 016502 (2017). This paper provides a timely and accessible review of theoretical work on quantum spin liquids and the challenges facing materials research in this field.

    Article  ADS  Google Scholar 

  12. A longer overview under the auspices of the US Department of Energy is “Basic Research Needs Workshop on Quantum Materials for Energy Relevant Technology”, Broholm, C., Fisher, I., Moore, J. & Murnane, M. (Office of Science, US Department of Energy, 2016); http://science.energy.gov/bes/efrc/research/bes-reports

  13. Basov, D. N., Averitt, R. D. & Hsieh, D. Controlling the properties of quantum materials. Nat. Mater. 16, 1077–1088 (2017).

    Article  ADS  Google Scholar 

  14. Tokura, Y., Kawasaki, M. & Nagaosa, N. Emergent functions of quantum materials. Nat. Phys. 13, 1056–1068 (2017).

    Article  Google Scholar 

  15. James, H. M. & Coolidge, A. S. The ground state of the hydrogen molecule. J. Chem. Phys. 1, 825–835 (1933).

    Article  ADS  Google Scholar 

  16. Blunt, N. S. et al. Semi-stochastic full configuration interaction quantum Monte Carlo: developments and application. J. Chem. Phys. 142, 184107 (2015).

    Article  ADS  Google Scholar 

  17. Goodenough, J. B. Theory of the role of covalence in the perovskite-type manganites [La, M(II)]MnO3 . Phys. Rev. 100, 564–573 (1955).

    Article  ADS  Google Scholar 

  18. Kanamori, J. Superexchange interaction and symmetry properties of electron orbitals. J. Phys. Chem. Solids 10, 87–98 (1959).

    Article  ADS  Google Scholar 

  19. Franchini, C. et al. Maximally localized Wannier functions in LaMnO3 within PBE + U, hybrid functionals and partially self-consistent GW: an efficient route to construct ab initio tight-binding parameters for eg perovskites. J. Phys. Condens. Matter 24, 235602 (2012).

    Article  ADS  Google Scholar 

  20. Marques, M. A. L. et al. Ab initio theory of superconductivity. II. Application to elemental metals. Phys. Rev. B 72, 024546 (2005).

    Article  ADS  Google Scholar 

  21. Giustino, F. Electron–phonon interactions from first principles. Rev. Mod. Phys. 89, 015003 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  22. Jackeli, G. & Khaliullin, G. Mott insulators in the strong spin–orbit coupling limit: from Heisenberg to a quantum compass and Kitaev models. Phys. Rev. Lett. 102, 017205 (2009). This theoretical work showed how bond-directional exchange interactions and Kitaev models can be realized in magnetic insulators with strong spin–orbit coupling.

    Article  ADS  Google Scholar 

  23. Plumb, K. W. et al. α-RuCl3: a spin–orbit assisted Mott insulator on a honeycomb lattice. Phys. Rev. B 90, 041112(R) (2014).

    Article  ADS  Google Scholar 

  24. Takayama, T. et al. Hyperhoneycomb iridate β-Li2IrO3 as a platform for Kitaev magnetism. Phys. Rev. Lett. 114, 077202 (2015).

    Article  ADS  Google Scholar 

  25. Chun, S. H. et al. Direct evidence for dominant bond-directional interactions in a honeycomb lattice iridate Na2IrO3 . Nat. Phys. 11, 462–466 (2015).

    Article  Google Scholar 

  26. Rüegg, Ch. et al. Pressure-induced quantum phase transition in the spin-liquid TlCuCl3 . Phys. Rev. Lett. 93, 257201 (2004). These authors established a model system that allows detailed, quantitative neutron scattering experiments on the evolution of spin correlations across a quantum phase transition.

    Article  ADS  Google Scholar 

  27. Jain, A. et al. Higgs mode and its decay in a two-dimensional antiferromagnet. Nat. Phys. 13, 633–637 (2017).

    Article  Google Scholar 

  28. Hong, T. et al. Higgs amplitude mode in a two-dimensional quantum antiferromagnet near the quantum critical point. Nat. Phys. 13, 638–643 (2017).

    Article  Google Scholar 

  29. Pekker, D. & Varma, C. M. Amplitude/Higgs modes in condensed matter physics. Annu. Rev. Condens. Matter Phys. 6, 269–297 (2014).

    Article  ADS  Google Scholar 

  30. Cercellier, H. et al. Evidence for an excitonic insulator phase in 1T-TiSe2 . Phys. Rev. Lett. 99, 146403 (2007). The photoemission data presented in this paper revived interest in excitonic insulators. The data were taken on a material system that is now being studied intensely in the form of exfoliated monolayers.

    Article  ADS  Google Scholar 

  31. Wakisaka, Y. et al. Excitonic insulator state in Ta2NiSe5 probed by photoemission spectroscopy. Phys. Rev. Lett. 103, 026402 (2009).

    Article  ADS  Google Scholar 

  32. Lu, Y. F. et al. Zero-gap semiconductor to excitonic insulator transition in Ta2NiSe5 . Nat. Commun. 8, 14408 (2017).

    Article  ADS  Google Scholar 

  33. Jérome, D., Rice, T. M. & Kohn, W. Excitonic insulator. Phys. Rev. 158, 462–475 (1967).

    Article  ADS  Google Scholar 

  34. Morosan, E. et al. Superconductivity in CuxTiSe2 . Nat. Phys. 2, 544–550 (2006).

    Article  Google Scholar 

  35. Kusmartseva, A. F., Sipos, B., Berger, H., Forró, L. & Tutiš, E. Pressure induced superconductivity in pristine 1T-TiSe2 . Phys. Rev. Lett. 103, 236401 (2009).

    Article  ADS  Google Scholar 

  36. Joe, Y. I. et al. Emergence of charge density wave domain walls above the superconducting dome in 1T-TiSe2 . Nat. Phys. 10, 421–425 (2014).

    Article  Google Scholar 

  37. Edge, J. M., Kedem, Y., Aschauer, U., Spaldin, N. A. & Balatsky, A. V. Quantum critical origin of the superconducting dome in SrTiO3 . Phys. Rev. Lett. 115, 247002 (2016).

    Article  ADS  Google Scholar 

  38. Ruhman, J. & Lee, P. A. Superconductivity at very low density: the case of strontium titanate. Phys. Rev. B 94, 224515 (2016).

    Article  ADS  Google Scholar 

  39. Ohtomo, A. & Hwang, H. Y. A high-mobility electron gas at the LaAlO3/SrTiO3 heterointerface. Nature 427, 423–426 (2004). A variety of quantum states have now been observed to emerge at atomic-scale interfaces between correlated materials. The creation of such interfaces was enabled by progress in materials synthesis highlighted in this paper.

    Article  ADS  Google Scholar 

  40. Boschker, H. & Mannhart, J. Quantum-matter heterostructures. Annu. Rev. Condens. Matter Phys. 8, 145–164 (2017).

    Article  ADS  Google Scholar 

  41. Novoselov, K. S., Mishchenko, A., Carvalho, A. & Castro Neto, A. H. 2D materials and van der Waals heterostructures. Science 353, aac9439 (2016).

    Article  Google Scholar 

  42. Kamihara, Y. et al. Iron-based layered superconductor: LaOFeP. J. Am. Chem. Soc. 128, 10012–10013 (2006).

    Article  Google Scholar 

  43. Hsu, F. C. et al. Superconductivity in the PbO-type structure α-FeSe. Proc. Natl Acad. Sci. USA 105, 14262–14264 (2008).

    Article  ADS  Google Scholar 

  44. Si, Q., Yu, R. & Abrahams, E. High-temperature superconductivity in iron pnictides and chalcogenides. Nat. Rev. Mater. 1, 16017 (2016).

    Article  ADS  Google Scholar 

  45. Dai, P. C. Antiferromagnetic order and spin dynamics in iron-based superconductors. Rev. Mod. Phys. 87, 855–896 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  46. Wang, F., Zhai, H., Ran, Y., Vishwanath, A. & Lee, D.-H. Functional renormalization-group study of the pairing symmetry and pairing mechanism of the FeAs-based high-temperature superconductor. Phys. Rev. Lett. 102, 047005 (2009). The functional renormalization group approach gives a way to understand how various electron–electron interactions and Fermi surfaces combine to lead to the different superconducting ordering parameters observed in iron-based superconductors.

    Article  ADS  Google Scholar 

  47. Metzner, W., Salmhofer, M., Honerkamp, C., Meden, V. & Schönhammer, K. Functional renormalization group approach to correlated fermion systems. Rev. Mod. Phys. 84, 299–352 (2012).

    Article  ADS  Google Scholar 

  48. Imada, M., Fujimori, A. & Tokura, Y. Metal–insulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998).

    Article  ADS  Google Scholar 

  49. LeBlanc, J. P. F. et al. Solutions of the two-dimensional Hubbard model: benchmarks and results from a wide range of numerical algorithms. Phys. Rev. X 5, 041041 (2015).

    Google Scholar 

  50. Fradkin, E., Kivelson, S. A. & Tranquada, J. M. Theory of intertwined orders in high temperature superconductors. Rev. Mod. Phys. 87, 457–482 (2015). This paper provides a lively yet in-depth review of current research on intertwined order and electronic analogues of liquid crystals in copper oxide superconductors.

    Article  ADS  Google Scholar 

  51. Tranquada, J. M. et al. Evidence for unusual superconducting correlations coexisting with stripe order in La1.875Ba0.125CuO4 . Phys. Rev. B 78, 174529 (2008).

    Article  ADS  Google Scholar 

  52. Dagotto, E. Complexity in strongly correlated electronic systems. Science 309, 257–262 (2005).

    Article  ADS  Google Scholar 

  53. Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature 518, 179–186 (2015).

    Article  ADS  Google Scholar 

  54. Stewart, G. R. Unconventional superconductivity. Adv. Phys. 6, 75–196 (2017).

    Article  ADS  Google Scholar 

  55. Scalapino, D. J. A common thread: the pairing interaction for unconventional superconductors. Rev. Mod. Phys. 84, 1383–1417 (2012).

    Article  ADS  Google Scholar 

  56. Sachdev, S. & Keimer, B. Quantum criticality. Phys. Today 64, 29–35 (February, 2011).

    Article  Google Scholar 

  57. Sebastian, S. E. & Proust, C. Quantum oscillations in hole-doped cuprates. Annu. Rev. Condens. Matter Phys. 6, 411–430 (2015).

    Article  ADS  Google Scholar 

  58. Sebastian, S. E. et al. Normal-state nodal electronic structure in underdoped high-Tc copper oxides. Nature 511, 61–64 (2014).

    Article  ADS  Google Scholar 

  59. Ramshaw, B. J. et al. Quasiparticle mass enhancement approaching optimal doping in a high-Tc superconductor. Science 348, 317–320 (2015).

    Article  ADS  Google Scholar 

  60. Braicovich, L. et al. Dispersion of magnetic excitations in the cuprate La2CuO4 and CaCuO2 compounds measured using resonant X-ray scattering. Phys. Rev. Lett. 102, 167401 (2009). This paper introduced resonant inelastic X-ray scattering (RIXS) as an energy- and momentum-resolved probe of collective spin excitations in metal oxides. The sensitivity of RIXS now allows detection of dispersive magnons in thin films and at interfaces.

    Article  ADS  Google Scholar 

  61. Hu, W. et al. Optically enhanced coherent transport in YBa2Cu3O6.5 by ultrafast redistribution of interlayer coupling. Nat. Mater. 13, 705–711 (2014).

    Article  ADS  Google Scholar 

  62. Carrasquilla, J. & Melko, R. G. Machine learning phases of matter. Nat. Phys. 13, 431–434 (2017).

    Article  Google Scholar 

  63. Booth, G. H., Grüneis, A., Kresse, G. & Alavi, A. Towards an exact description of electronic wavefunctions in real solids. Nature 493, 365–370 (2013).

    Article  ADS  Google Scholar 

  64. Schneider, U. et al. Fermionic transport and out-of-equilibrium dynamics in a homogeneous Hubbard model with ultracold atoms. Nat. Phys. 8, 213–218 (2012).

    Article  Google Scholar 

  65. Cheuk, L. W. et al. Observation of spatial charge and spin correlations in the 2D Fermi–Hubbard model. Science 353, 1260–1264 (2016).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  66. Norman, M. R. Materials design for new superconductors. Rep. Prog. Phys. 79, 074502 (2016).

    Article  ADS  Google Scholar 

  67. von Klitzing, K., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).

    Article  ADS  Google Scholar 

  68. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982). The importance of this classic paper on how topology of wavefunctions (specifically, the Berry phase of Bloch electrons) underlies the integer quantum Hall effect only became fully clear two decades later.

    Article  ADS  Google Scholar 

  69. 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).

    Article  ADS  MathSciNet  Google Scholar 

  70. Murakami, S., Nagaosa, N. & Zhang, S.-C. Spin-Hall insulator. Phys. Rev. Lett. 93, 156804 (2004).

    Article  ADS  Google Scholar 

  71. Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005). The possibility of a new type of topological invariant enabled by time-reversal symmetry in systems with spin–orbit coupling was spelled out clearly in this paper on two-dimensional electron systems, which inspired a great deal of subsequent theoretical work.

    Article  ADS  Google Scholar 

  72. König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).

    ADS  Google Scholar 

  73. Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

    Article  ADS  Google Scholar 

  74. Moore, J. E. & Balents, L. Topological invariants of time-reversal-invariant band structures. Phys. Rev. B 75, 121306(R) (2007).

    Article  ADS  Google Scholar 

  75. Fu, L., Kane, C. L. & Mele, E. J. Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007).

    Article  ADS  Google Scholar 

  76. Roy, R. Topological phases and the quantum spin Hall effect in three dimensions. Phys. Rev. B 79, 195322 (2009).

    Article  ADS  Google Scholar 

  77. Xia, Y. et al. Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nat. Phys. 5, 398–402 (2009). Previous topological states were discovered primarily by transport measurements. Many of the discoveries in the last ten years were enabled by angle-resolved photoemission, as in this work, which found the Dirac cone electronic structure at the surface of bismuth selenide.

    Article  Google Scholar 

  78. Zhang, H. et al. Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface. Nat. Phys. 5, 438–442 (2009).

    Article  Google Scholar 

  79. Qi, X.-L., Hughes, T. L. & Zhang, S.-C. Topological field theory of time-reversal invariant insulators. Phys. Rev. B 78, 195424 (2008). This paper gave a more abstract/emergent picture of what topological insulators are, by making a profound connection between topological insulators and previous work on possible electromagnetic responses in solids (‘axion electrodynamics’).

    Article  ADS  Google Scholar 

  80. Essin, A. M., Moore, J. E. & Vanderbilt, D. Magnetoelectric polarizability and axion electrodynamics in crystalline insulators. Phys. Rev. Lett. 102, 146805 (2009).

    Article  ADS  Google Scholar 

  81. Wu, L. et al. Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator. Science 354, 1124–1127 (2016).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  82. Chang, C. Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).

    Article  ADS  Google Scholar 

  83. Liu, Z. K. et al. A stable three-dimensional topological Dirac semimetal Cd3As2 . Nat. Mater. 13, 677–681 (2014).

    Article  ADS  Google Scholar 

  84. Neupane, M. et al. Observation of a topological 3D Dirac semimetal phase in high-mobility Cd3As2 . Nat. Commun. 5, 3786 (2014).

    Article  ADS  Google Scholar 

  85. Borisenko, S. et al. Experimental realization of a three-dimensional Dirac semimetal. Phys. Rev. Lett. 113, 027603 (2014).

    Article  ADS  Google Scholar 

  86. Murakami, S. Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase. New J. Phys. 9, 356 (2007).

    Article  ADS  Google Scholar 

  87. Burkov, A. A. & Balents, L. Weyl semimetal in a topological insulator multilayer. Phys. Rev. Lett. 107, 127205 (2011).

    Article  ADS  Google Scholar 

  88. Nielsen, H. B. & Ninomiya, M. A no-go theorem for regularizing chiral fermions. Phys. Lett. B 105, 219–223 (1981).

    Article  ADS  Google Scholar 

  89. Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).

    Article  ADS  Google Scholar 

  90. Vafek, O. & Vishwanath, A. Dirac fermions in solids: from high-Tc cuprates and graphene to topological insulators and Weyl semimetals. Annu. Rev. Condens. Matter Phys. 5, 83–112 (2014).

    Article  ADS  Google Scholar 

  91. Son, D. T. & Spivak, B. Z. Chiral anomaly and classical negative magnetoresistance of Weyl metals. Phys. Rev. B 88, 104412 (2013).

    Article  ADS  Google Scholar 

  92. Xiong, J. et al. Evidence for the chiral anomaly in the Dirac semimetal Na3Bi. Science 350, 413–416 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  93. Huang, S.-M. et al. Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class. Nat. Commun. 6, 7373 (2015).

    Article  ADS  Google Scholar 

  94. Weng, H., Fang, C., Fang, Z., Bernevig, A. & Dai, X. Weyl semimetal phase in non-centrosymmetric transition metal monophosphides. Phys. Rev. X 5, 011029 (2015).

    Google Scholar 

  95. Xu, S.-Y. et al. Discovery of a Weyl Fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015).

    Article  ADS  Google Scholar 

  96. Cho, G. Y., Bardarson, J. H., Lu, Y. M. & Moore, J. E. Superconductivity of doped Weyl semimetals: finite-momentum pairing and electronic analog of the 3He-A phase. Phys. Rev. B 86, 214514 (2012).

    Article  ADS  Google Scholar 

  97. Dzero, M., Xia, J., Galitski, V. & Coleman, P. Topological Kondo insulators. Annu. Rev. Condens. Matter Phys. 7, 249–280 (2016).

    Article  ADS  Google Scholar 

  98. Ando, Y. & Fu, L. Topological crystalline insulators and topological superconductors: from concepts to materials. Annu. Rev. Condens. Matter Phys. 6, 361–381 (2015).

    Article  ADS  Google Scholar 

  99. Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).

    Article  ADS  Google Scholar 

  100. Laughlin, R. B. Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398 (1983).

    Article  ADS  Google Scholar 

  101. Leinaas, J. M. & Myrheim, J. On the theory of identical particles. Nuovo Cimento B 37, 1–23 (1977).

    Article  ADS  Google Scholar 

  102. Willett, R. et al. Observation of an even-denominator quantum number in the fractional quantum Hall effect. Phys. Rev. Lett. 59, 1776–1779 (1987).

    Article  ADS  Google Scholar 

  103. Moore, G. & Read, N. Non-Abelions in the fractional quantum Hall effect. Nucl. Phys. B 360, 362–396 (1991).

    Article  ADS  Google Scholar 

  104. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Sarma, S. D. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  105. Read, N. & Green, D. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect. Phys. Rev. B 61, 10267–10297 (2000). This paper greatly clarified the relationship between the seemingly different problems of superconductivity and exotic non-Abelian fractional quantum Hall phases. It turns out that even a simple superconducting wavefunction of BCS type contains within it non-Abelian order.

    Article  ADS  Google Scholar 

  106. Maeno, Y., Kittaka, S., Nomura, T., Yonezawa, S. & Ishida, K. Evaluation of spin-triplet superconductivity in Sr2RuO4 . J. Phys. Soc. Jpn 81, 011009 (2012).

    Article  ADS  Google Scholar 

  107. Mourik, V. et al. Signatures of Majorana fermions in hybrid superconductor–semiconductor nanowire devices. Science 336, 1003–1007 (2012).

    ADS  Google Scholar 

  108. Nadj-Perge, S. et al. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346, 602–607 (2014).

    Article  ADS  Google Scholar 

  109. Albrecht, S. M. et al. Exponential protection of zero modes in Majorana islands. Nature 531, 206–209 (2016).

    Article  ADS  Google Scholar 

  110. Volovik, G. E. The Universe in a Helium Droplet (International Series of Monographs on Physics, Oxford Univ. Press, 2009).

    Book  Google Scholar 

  111. Moessner, R. & Sondhi, S. Resonating valence bond phase in the triangular lattice quantum dimer model. Phys. Rev. Lett. 86, 1881–1884 (2001).

    Article  ADS  Google Scholar 

  112. Han, T. H. et al. Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet. Nature 492, 406–410 (2012).

    Article  ADS  Google Scholar 

  113. Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  114. Banerjee, A. et al. Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet. Nat. Mater. 15, 733–740 (2016).

    Article  ADS  Google Scholar 

  115. Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).

    Article  ADS  Google Scholar 

  116. Sachdev, S. Emergent gauge fields and the high temperature superconductors. Phil. Trans. R. Soc. A 374, 20150248 (2016).

    Article  ADS  Google Scholar 

  117. Senthil, T., Vishwanath, A., Balents, L., Sachdev, S. & Fisher, M. P. A. Deconfined quantum critical points. Science 303, 1490–1494 (2004).

    Article  ADS  Google Scholar 

  118. Hwang, H. Y. et al. Emergent phenomena at oxide interfaces. Nat. Mater. 11, 103–113 (2012).

    Article  ADS  Google Scholar 

  119. Hoffmann, A. & Bader, S. D. Opportunities at the frontiers of spintronics. Phys. Rev. Appl. 4, 047001 (2015).

    Article  ADS  Google Scholar 

  120. Bozhko, D. A. et al. Supercurrent in a room-temperature Bose–Einstein magnon condensate. Nat. Phys. 12, 1057–1062 (2016).

    Article  Google Scholar 

  121. Schulz, T. et al. Emergent electrodynamics of skyrmions in a chiral magnet. Nat. Phys. 8, 301–304 (2012). The experimental discovery of emergent magnetic skyrmion phases in MnSi and other materials led to an outpouring of experiment and theory, with significant potential for new kinds of magnetic storage and spintronic devices.

    Article  Google Scholar 

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Acknowledgements

B.K. acknowledges financial support by the European Research Council under Advanced Grant No. 669550 (Com4Com) and by the German Science Foundation (DFG) in the Collaborative Research Center TRR80. J.E.M. was supported by the Office of Basic Energy Sciences, Materials Sciences and Engineering Division, of the US Department of Energy under Contract No. DE-AC02-05CH11231, and the National Science Foundation under grant DMR-1507141.

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Keimer, B., Moore, J. The physics of quantum materials. Nature Phys 13, 1045–1055 (2017). https://doi.org/10.1038/nphys4302

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