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Angle-resolved photoemission spectroscopy of a Fermi–Hubbard system


Angle-resolved photoemission spectroscopy (ARPES) measures the single-particle excitations of a many-body quantum system with energy and momentum resolution, providing detailed information about strongly interacting materials1. ARPES directly probes fermion pairing, and hence is a natural technique to study the development of superconductivity in systems ranging from high-temperature superconductors to unitary Fermi gases. In these systems, a remnant gap-like feature persists in the normal state2. Developing a quantitative understanding of these so-called pseudogap regimes may elucidate details about the pairing mechanisms that lead to superconductivity, but this is difficult in real materials partly because the microscopic Hamiltonian is not known. Here, we report on the development of ARPES to study strongly interacting fermions in an optical lattice using a quantum gas microscope. We benchmark the technique by measuring the occupied single-particle spectral function of an attractive Fermi–Hubbard system across the BCS–BEC crossover and comparing the results to those of quantum Monte Carlo calculations. We find evidence for a pseudogap that opens well above the expected critical temperature for superfluidity. This technique may also be applied to the doped repulsive Hubbard model, which is expected to exhibit a pseudogap at temperatures close to those achieved in recent experiments3.

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Fig. 1: ARPES technique and raw data.
Fig. 2: Trap-averaged spectral function.
Fig. 3: Occupied spectral function versus interaction.
Fig. 4: Occupied spectral function versus temperature at strong coupling.

Data availability

The data displayed in Figs. 14 are available online at Supporting data generated during the current study are available from the corresponding author on reasonable request.

Code availability

The code to reproduce the analysis in this study is available from the corresponding author on reasonable request.


  1. Damascelli, A., Hussain, Z. & Shen, Z.-X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473–541 (2003).

    ADS  Article  Google Scholar 

  2. Mueller, E. J. Review of pseudogaps in strongly interacting Fermi gases. Rep. Prog. Phys. 80, 104401 (2017).

    ADS  MathSciNet  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  4. Chen, Q., He, Y., Chien, C.-C. & Levin, K. Theory of radio frequency spectroscopy experiments in ultracold Fermi gases and their relation to photoemission in the cuprates. Rep. Prog. Phys. 72, 122501 (2009).

    ADS  Article  Google Scholar 

  5. Törmä, P. Physics of ultracold Fermi gases revealed by spectroscopies. Phys. Scr. 91, 043006 (2016).

    ADS  Article  Google Scholar 

  6. Vishik, I. M. et al. ARPES studies of cuprate fermiology: superconductivity, pseudogap and quasiparticle dynamics. New J. Phys. 12, 105008 (2010).

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  8. Chin, C. et al. Observation of the pairing gap in a strongly interacting Fermi gas. Science 305, 1128–1130 (2004).

    ADS  Article  Google Scholar 

  9. Schunck, C. H., Shin, Y., Schirotzek, A., Zwierlein, M. W. & Ketterle, W. Pairing without superfluidity: the ground state of an imbalanced Fermi mixture. Science 316, 867–870 (2007).

    ADS  Article  Google Scholar 

  10. Gaebler, J. P. et al. Observation of pseudogap behaviour in a strongly interacting Fermi gas. Nat. Phys. 6, 569–573 (2010).

    Article  Google Scholar 

  11. Nascimbène, S. et al. Fermi-liquid behavior of the normal phase of a strongly interacting gas of cold atoms. Phys. Rev. Lett. 106, 215303 (2011).

    ADS  Article  Google Scholar 

  12. Feld, M., Fröhlich, B., Vogt, E., Koschorreck, M. & Köhl, M. Observation of a pairing pseudogap in a two-dimensional Fermi gas. Nature 480, 75–78 (2011).

    ADS  Article  Google Scholar 

  13. Sommer, A. T., Cheuk, L. W., Ku, M. J. H., Bakr, W. S. & Zwierlein, M. W. Evolution of fermion pairing from three to two dimensions. Phys. Rev. Lett. 108, 045302 (2012).

    ADS  Article  Google Scholar 

  14. Murthy, P. A. et al. High-temperature pairing in a strongly interacting two-dimensional Fermi gas. Science 359, 452–455 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  15. Dao, T.-L., Georges, A., Dalibard, J., Salomon, C. & Carusotto, I. Measuring the one-particle excitations of ultracold fermionic atoms by stimulated Raman spectroscopy. Phys. Rev. Lett. 98, 240402 (2007).

    ADS  Article  Google Scholar 

  16. Stewart, J. T., Gaebler, J. P. & Jin, D. S. Using photoemission spectroscopy to probe a strongly interacting Fermi gas. Nature 454, 744–747 (2008).

    ADS  Article  Google Scholar 

  17. Sagi, Y., Drake, T. E., Paudel, R., Chapurin, R. & Jin, D. S. Breakdown of the Fermi liquid description for strongly interacting fermions. Phys. Rev. Lett. 114, 075301 (2015).

    ADS  Article  Google Scholar 

  18. Fröhlich, B. et al. Two-dimensional Fermi liquid with attractive interactions. Phys. Rev. Lett. 109, 130403 (2012).

    ADS  Article  Google Scholar 

  19. Schneider, W. & Randeria, M. Universal short-distance structure of the single-particle spectral function of dilute Fermi gases. Phys. Rev. A 81, 021601 (2010).

    ADS  Article  Google Scholar 

  20. Loktev, V. M., Quick, R. M. & Sharapov, S. G. Phase fluctuations and pseudogap phenomena. Phys. Rep. 349, 1–123 (2001).

    ADS  Article  Google Scholar 

  21. Perali, A., Pieri, P., Strinati, G. C. & Castellani, C. Pseudogap and spectral function from superconducting fluctuations to the bosonic limit. Phys. Rev. B 66, 024510 (2002).

    ADS  Article  Google Scholar 

  22. Perali, A. et al. Evolution of the normal state of a strongly interacting Fermi gas from a pseudogap phase to a molecular Bose gas. Phys. Rev. Lett. 106, 060402 (2011).

    ADS  Article  Google Scholar 

  23. Wurz, N. et al. Coherent manipulation of spin correlations in the Hubbard model. Phys. Rev. A 97, 051602 (2018).

    ADS  Article  Google Scholar 

  24. Murthy, P. A. et al. Matter-wave Fourier optics with a strongly interacting two-dimensional Fermi gas. Phys. Rev. A 90, 043611 (2014).

    ADS  Article  Google Scholar 

  25. Bohrdt, A., Greif, D., Demler, E., Knap, M. & Grusdt, F. Angle-resolved photoemission spectroscopy with quantum gas microscopes. Phys. Rev. B 97, 125117 (2018).

    ADS  Article  Google Scholar 

  26. Strohmaier, N. et al. Interaction-controlled transport of an ultracold Fermi gas. Phys. Rev. Lett. 99, 220601 (2007).

    ADS  Article  Google Scholar 

  27. Hackermuller, L. et al. Anomalous expansion of attractively interacting fermionic atoms in an optical lattice. Science 327, 1621–1624 (2010).

    ADS  Article  Google Scholar 

  28. 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 

  29. Mitra, D. et al. Quantum gas microscopy of an attractive Fermi–Hubbard system. Nat. Phys. 14, 173–177 (2017).

    Article  Google Scholar 

  30. Paiva, T., dos Santos, R. R., Scalettar, R. T. & Denteneer, P. J. H. Critical temperature for the two-dimensional attractive Hubbard model. Phys. Rev. B 69, 184501 (2004).

    ADS  Article  Google Scholar 

  31. Singer, J. M., Pedersen, M. H., Schneider, T., Beck, H. & Matuttis, H.-G. From BCS-like superconductivity to condensation of local pairs: a numerical study of the attractive Hubbard model. Phys. Rev. B 54, 1286–1301 (1996).

    ADS  Article  Google Scholar 

  32. Singer, J. M., Schneider, T. & Pedersen, M. H. On the phase diagram of the attractive Hubbard model: crossover and quantum critical phenomena. Eur. Phys. J. B 2, 17–30 (1998).

    ADS  Article  Google Scholar 

  33. Singer, J. M., Schneider, T. & Meier, P. F. Spectral properties of the attractive Hubbard model. Eur. Phys. J. B 7, 37–51 (1999).

    ADS  Article  Google Scholar 

  34. Brown, P. T. et al. Bad metallic transport in a cold atom Fermi–Hubbard system. Science 363, 379–382 (2019).

    ADS  Article  Google Scholar 

  35. Nichols, M. A. et al. Spin transport in a Mott insulator of ultracold fermions. Science 363, 383–387 (2019).

    ADS  Article  Google Scholar 

  36. Xu, W., Haule, K. & Kotliar, G. Hidden Fermi liquid, scattering rate saturation, and Nernst effect: a dynamical mean-field theory perspective. Phys. Rev. Lett. 111, 036401 (2013).

    ADS  Article  Google Scholar 

  37. Deng, X. et al. How bad metals turn good: spectroscopic signatures of resilient quasiparticles. Phys. Rev. Lett. 110, 086401 (2013).

    ADS  Article  Google Scholar 

  38. Grusdt, F. et al. Parton theory of magnetic polarons: mesonic resonances and signatures in dynamics. Phys. Rev. X 8, 011046 (2018).

    Google Scholar 

  39. O’Hara, K. M., Gehm, M. E., Granade, S. R., Bali, S. & Thomas, J. E. Stable, strongly attractive, two-state mixture of lithium fermions in an optical trap. Phys. Rev. Lett. 85, 2092–2095 (2000).

    ADS  Article  Google Scholar 

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This work was supported by the NSF (grant no. DMR-1607277), the David and Lucile Packard Foundation (grant no. 2016-65128) and the AFOSR Young Investigator Research Program (grant no. FA9550-16-1-0269). T.P.D. and E.W.H. acknowledge support from the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract no. DE-AC02-76SF00515. Computational work was performed on the Sherlock cluster at Stanford University.

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Authors and Affiliations



P.T.B. and W.S.B. conceived the experiment. P.T.B., E.G.-S. and B.M.S. collected the experimental data and performed the data analysis. E.W.H. and T.P.D. performed the quantum Monte Carlo calculations. T.P.D. and W.S.B. supervised the project. All authors contributed to the writing of the manuscript.

Corresponding author

Correspondence to Waseem S. Bakr.

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The authors declare no competing interests.

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Supplementary information

Supplementary Information

Supplementary text, Figs. 1–7 and references.

Supplementary Data

Text files containing the data plotted in Figs. 1–4.

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Brown, P.T., Guardado-Sanchez, E., Spar, B.M. et al. Angle-resolved photoemission spectroscopy of a Fermi–Hubbard system. Nat. Phys. 16, 26–31 (2020).

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