An understanding of the missing antinodal electronic excitations in the pseudogap state is essential for uncovering the physics of the underdoped cuprate high-temperature superconductors1,2,3,4,5,6. The majority of high-temperature experiments performed thus far, however, have been unable to discern whether the antinodal states are rendered unobservable due to their damping or whether they vanish due to their gapping7,8,9,10,11,12,13,14,15,16,17,18. Here, we distinguish between these two scenarios by using quantum oscillations to examine whether the small Fermi surface pocket, found to occupy only 2% of the Brillouin zone in the underdoped cuprates19,20,21,22,23,24, exists in isolation against a majority of completely gapped density of states spanning the antinodes, or whether it is thermodynamically coupled to a background of ungapped antinodal states. We find that quantum oscillations associated with the small Fermi surface pocket exhibit a signature sawtooth waveform characteristic of an isolated two-dimensional Fermi surface pocket25,26,27,28,29,30,31,32. This finding reveals that the antinodal states are destroyed by a hard gap that extends over the majority of the Brillouin zone, placing strong constraints on a drastic underlying origin of quasiparticle disappearance over almost the entire Brillouin zone in the pseudogap regime7,8,9,10,11,12,13,14,15,16,17,18.
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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).
Timusk, T. & Statt, B. The pseudogap in high-temperature superconductors: an experimental survey. Rep. Prog. Phys. 62, 61–122 (1999).
Norman, M. R., Pines, D. & Kallin, C. The pseudogap: friend or foe of high T c? Adv. Phys. 54, 715–733 (2005).
Ding, H. et al. Spectroscopic evidence for a pseudogap in the normal state of underdoped high-T c superconductors. Nature 382, 51–54 (1996).
Damascelli, A., Hussain, Z. & Shen, Z.-X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473–541 (2003).
Basov, D. N. & Timusk, T. Electrodynamics of high-T c superconductors. Rev. Mod. Phys. 77, 721–779 (2005).
Chakravarty, S., Laughlin, R. B., Morr, D. K. & Nayak, C. Hidden order in the cuprates. Phys. Rev. B 63, 094503 (2001).
Shen, K. M. et al. Nodal quasiparticles and antinodal charge ordering in Ca2 − xNaxCuO2Cl2. Science 307, 901–904 (2005).
Wise, W. D. et al. Charge-density-wave origin of cuprate checkerboard visualized by scanning tunnelling microscopy. Nat. Phys. 4, 696–699 (2008).
Senthil, T., Sachdev, S. & Vojta, M. Fractionalized Fermi liquids. Phys. Rev. Lett. 90, 216403 (2003).
Lee, P. A., Nagaosa, N. & Wen, X. G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).
Yang, K.-Y., Rice, T. M. & Zhang, F.-C. Phenomenological theory of the pseudogap state. Phys. Rev. B 73, 174501 (2006).
Kaul, R. K., Kolezhuk, A., Levin, M., Sachdev, S. & Senthil, T. Hole dynamics in an antiferromagnet across a deconfined quantum critical point. Phys. Rev. B 75, 235122 (2007).
Kohsaka, Y. et al. How Cooper pairs vanish approaching the Mott insulator in Bi2Sr2CaCu2O8 + δ. Nature 454, 1072–1078 (2008).
Emery, V. J. & Kivelson, S. A. Importance of phase fluctuations in superconductors with small superfluid density. Nature 374, 434–437 (1995).
Randeria, M., Trivedi, N., Moreo, A. & Scalettar, R. T. Pairing and spin gap in the normal state of short coherence length superconductors. Phys. Rev. Lett. 69, 2001–2004 (1992).
Valla, T., Fedorov, A. V., Lee, J. & Davis, J. C. The ground state of the pseudogap in cuprate superconductors. Science 314, 1914–1916 (2006).
Gomes, K. K. et al. Visualizing pair formation on the atomic scale in the high-T c superconductor Bi2Sr2CaCu2O8 + δ. Nature 447, 569–572 (2007).
Doiron-Leyraud, N. et al. Quantum oscillations and the Fermi surface in an underdoped high-T c superconductor. Nature 447, 565–568 (2007).
Yelland, E. A. et al. Quantum oscillations in the underdoped cuprate YBa2Cu4O8. Phys. Rev. Lett. 100, 047003 (2008).
Bangura, A. F. et al. Small Fermi surface pockets in underdoped high temperature superconductors: observation of Shubnikov–de Haas oscillations in YBa2Cu4O8. Phys. Rev. Lett. 100, 047004 (2008).
Barišić, N. et al. Universal quantum oscillations in the underdoped cuprate superconductors. Nat. Phys. 9, 761–764 (2013).
Sebastian, S. E., Harrison, N. & Lonzarich, G. G. Towards resolution of the Fermi surface in underdoped high-T c superconductors. Rep. Prog. Phys. 75, 102501 (2012).
Chan, M. K. et al. Single reconstructed Fermi surface pocket in an underdoped single-layer cuprate superconductor. Nat. Commun. 7, 12244 (2016).
Shoenberg, D. Magnetization of a two-dimensional electron-gas. J. Low Temp. Phys. 56, 417–440 (1984).
Eisenstein, J. P. et al. Density of states and de Haas–van Alphen effect in two-dimensional electron systems. Phys. Rev. Lett. 55, 875–878 (1985).
Jauregui, K., Marchenko, V. I. & Vagner, I. D. Magnetization of a two-dimensional electron gas. Phys. Rev. B 41, 12922–12925 (1990).
Harrison, N. et al. Numerical model of quantum oscillations in quasi-two-dimensional organic metals in high magnetic fields. Phys. Rev. B 54, 9977–9987 (1996).
Potts, A. et al. Magnetization studies of Landau level broadening in two-dimensional electron systems. J. Phys. Condens. Matter 8, 5189–5207 (1996).
Itskovsky, M. A., Maniv, T. & Vagner, I. D. Wave form of de Haas–van Alphen oscillations in a two-dimensional metal. Phys. Rev. B 61, 14616–14627 (2000).
Champel, T. Chemical potential oscillations and de Haas–van Alphen effect. Phys. Rev. B 64, 054407 (2001).
Wilde, M. A. et al. Experimental evidence of the ideal de Haas–van Alphen effect in a two-dimensional system. Phys. Rev. B 73, 125325 (2006).
Millis, A. J. & Norman, M. R. Antiphase stripe order as the origin of electron pockets observed in 1/8-hole-doped cuprates. Phys. Rev. B 76, 220503 (2007).
Chakravarty, S. & Kee, H.-Y. Fermi pockets and quantum oscillations of the Hall coefficient in high-temperature superconductors. Proc. Natl Acad. Sci. USA 105, 8835–8839 (2008).
Allais, A., Chowdhury, D. & Sachdev, S. Connecting high-field quantum oscillations to zero-field electron spectral functions in the underdoped cuprates. Nat. Commun. 5, 5771 (2014).
Maharaj, A. V., Hosur, P. & Raghu, S. Crisscrossed stripe order from interlayer tunneling in hole-doped cuprates. Phys. Rev. B 90, 125108 (2014).
Shoenberg, D. Magnetic Oscillations in Metals (Cambridge Univ. Press, 1984).
Wosnitza, J. et al. Two-dimensional Fermi liquid with fixed chemical potential. Phys. Rev. B 61, 7383–7387 (2000).
Sebastian, S. E. et al. Normal-state nodal electronic structure in underdoped high-T c copper oxides. Nature 511, 61–64 (2014).
Alexandrov, A. S. & Bratkovsky, A. M. de Haas–van Alphen effect in canonical and grand canonical multiband Fermi liquid. Phys. Rev. Lett. 76, 1308–1311 (1996).
Champel, T. & Mineev, V. P. de Haas–van Alphen effect in two- and quasi-two-dimensional metals and superconductors. Phil. Mag. B 81, 55–74 (2001).
Riggs, S. C. et al. Heat capacity through the magnetic-field-induced resistive transition in an underdoped high-temperature superconductor. Nat. Phys. 7, 332–335 (2011).
Marcenat, C. et al. Calorimetric determination of the magnetic phase diagram of underdoped ortho-II YBa2Cu3O6.54 single crystals. Nat. Commun. 6, 7927 (2015).
Grissonnanche, G. et al. Wiedemann–Franz law in the underdoped cuprate superconductor YBa2Cu3Oy. Phys. Rev. B 93, 064513 (2016).
Harrison, N. & Sebastian, S. E. Fermi surface reconstruction from bilayer charge ordering in the underdoped high temperature superconductor YBa2Cu3O6 + x. New J. Phys. 14, 095023 (2012).
Harrison, N. & Sebastian, S. E. Magnetotransport signatures of a single nodal electron pocket constructed from Fermi arcs. Phys. Rev. B 92, 224505 (2015).
Doiron-Leyraud, N. et al. Evidence for a small hole pocket in the Fermi surface of underdoped YBa2Cu3Oy. Nat. Commun. 6, 6034 (2015).
Proust, C., Vignolle, B., Levallois, J., Adachi, S. & Hussey, N. E. Fermi liquid behavior of the in-plane resistivity in the pseudogap state of YBa2Cu4O8. Proc. Natl Acad. Sci. USA 113, 13654–13659 (2016).
Kawasaki, S., Lin, C., Kuhns, P. L., Reyes, A. P. & Zheng, G.-Q. Carrier-concentration dependence of the pseudogap ground state of superconducting Bi2Sr2 − xLaxCuO6 + δ revealed by 63,65Cu-nuclear magnetic resonance. Phys. Rev. Lett 105, 137002 (2010).
Zhou, R. et al. Spin susceptibility of charge-ordered YBa2Cu3Oy across the upper critical field. Proc. Natl Acad. Sci. USA 114, 13148–13153 (2017).
Harrison, N. & Sebastian, S. E. Protected nodal electron pocket from multiple-Q ordering in underdoped high temperature superconductors. Phys. Rev. Lett. 106, 226402 (2011).
Atkinson, W. A., Kampf, A. P. & Bulut, S. Charge order in the pseudogap phase of cuprate superconductors. New J. Phys. 17, 013025 (2015).
Harrison, N. Robustness of the biaxial charge density wave reconstructed electron pocket against short-range spatial antiferromagnetic fluctuations. Phys. Rev. B 97, 245150 (2018).
Read, N. & Sachdev, S. Spin-Peierls, valence-bond solid and Néel ground states of low-dimensional quantum antiferromagnets. Phys. Rev. B 42, 4568–4589 (1990).
Yu, F. et al. Magnetic phase diagram of underdoped YBa2Cu3Oy inferred from torque magnetization and thermal conductivity. Proc. Natl Acad. Sci. USA 113, 12667–12672 (2016).
Norman, M. R. & Davis, J. C. S. Quantum oscillations in a pair density wave state. Proc. Natl Acad. Sci. USA 115, 5389–5391 (2018).
Dai, Z., Zhang, Y.-H., Senthil, T. & Lee, P. A. Pair density wave, charge density wave and vortex in high-T c cuprates. Phys. Rev. B 97, 174511 (2018).
Lee, P. A. Amperean pairing and the pseudogap phase of cuprate superconductors. Phys. Rev. X 4, 031017 (2014).
Rajasekaran, S. et al. Probing optically silent superfluid stripes in cuprates. Science 359, 575–579 (2018).
Edkins, S. D. et al. Magnetic-field induced pair density wave state in the cuprate vortex halo. Science 364, 976–980 (2019).
Liang, R., Bonn, D. A. & Hardy, W. N. Evaluation of CuO2 plane hole doping in YBa2Cu3O6 + x single crystals. Phys. Rev. B 73, 180505 (2006).
Hinkov, V. et al. Two-dimensional geometry of spin excitations in the high-transition-temperature superconductor YBa2Cu3O6 + x. Nature 430, 650–654 (2004).
M.H., Y.-T.H. and S.E.S. acknowledge support from the Royal Society, the Winton Programme for the Physics of Sustainability, EPSRC (studentship, grant no. EP/P024947/1 and EPSRC Strategic Equipment grant no. EP/M000524/1) and the European Research Council (grant no. 772891). S.E.S. acknowledges support from the Leverhulme Trust by way of the award of a Philip Leverhulme Prize. H.Z., J.W. and Z.Z. acknowledge support from the National Key Research and Development Program of China (grant no. 2016YFA0401704). A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by the National Science Foundation Cooperative Agreement no. DMR-1644779, the state of Florida and the US Department of Energy. Work performed by M.K.C., R.D.M. and N.H. was supported by the US DOE BES ‘Science of 100 T’ programme.
The authors declare no competing interests.
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Hartstein, M., Hsu, Y., Modic, K.A. et al. Hard antinodal gap revealed by quantum oscillations in the pseudogap regime of underdoped high-Tc superconductors. Nat. Phys. 16, 841–847 (2020). https://doi.org/10.1038/s41567-020-0910-0