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Direct observation of incommensurate magnetism in Hubbard chains

An Author Correction to this article was published on 22 January 2019

This article has been updated


The interplay between magnetism and doping is at the origin of exotic strongly correlated electronic phases and can lead to novel forms of magnetic ordering. One example is the emergence of incommensurate spin-density waves, which have wavevectors that do not belong to the reciprocal lattice. In one dimension this effect is a hallmark of Luttinger liquid theory, which also describes the low-energy physics of the Hubbard model1. Here we use a quantum simulator that uses ultracold fermions in an optical lattice2,3,4,5,6,7,8 to directly observe such incommensurate spin correlations in doped and spin-imbalanced Hubbard chains using fully spin- and density-resolved quantum gas microscopy. Doping is found to induce a linear change in the spin-density wavevector, in excellent agreement with predictions from Luttinger theory. For non-zero polarization we observe a reduction in the wavevector with magnetization, as expected from the antiferromagnetic Heisenberg model in a magnetic field. We trace the microscopic-scale origin of these incommensurate correlations to holes, doublons (double occupancies) and excess spins, which act as delocalized domain walls for the antiferromagnetic order. In addition, by inducing interchain coupling we observe fundamentally different spin correlations around doublons and suppression of incommensurate magnetism at finite (low) temperature in the two-dimensional regime9. Our results demonstrate how access to the full counting statistics of all local degrees of freedom can be used to study fundamental phenomena in strongly correlated many-body physics.

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Fig. 1: Probing incommensurate spin correlations in Hubbard chains.
Fig. 2: Incommensurate spin correlations versus doping.
Fig. 3: Incommensurate spin correlations versus polarization.
Fig. 4: Spin correlations in the 1D–2D crossover.

Data availability

The datasets generated and analysed during this study are available from the corresponding author upon reasonable request.

Change history

  • 22 January 2019

    In this Letter, the affiliation for Christian Gross should have been ‘Max-Planck-Institut für Quantenoptik, Garching, Germany’ instead of ‘Fakultät für Physik, Ludwig-Maximilians-Universität, Munich, Germany’; this has been corrected online.


  1. Giamarchi, T. Quantum Physics in One Dimension (Clarendon Press, Oxford, 2003).

    Book  Google Scholar 

  2. Greif, D., Uehlinger, T., Jotzu, G., Tarruell, L. & Esslinger, T. Short-range quantum magnetism of ultracold fermions in an optical lattice. Science 340, 1307–1310 (2013).

    Article  ADS  CAS  Google Scholar 

  3. Hart, R. A. et al. Observation of antiferromagnetic correlations in the Hubbard model with ultracold atoms. Nature 519, 211–214 (2015).

    Article  ADS  CAS  Google Scholar 

  4. Parsons, M. F. et al. Site-resolved measurement of the spin-correlation function in the Fermi–Hubbard model. Science 353, 1253–1256 (2016).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  5. Boll, M. et al. Spin- and density-resolved microscopy of antiferromagnetic correlations in Fermi–Hubbard chains. Science 353, 1257–1260 (2016).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  6. 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  CAS  Google Scholar 

  7. Drewes, J. H. et al. Antiferromagnetic correlations in two-dimensional fermionic Mott-insulating and metallic phases. Phys. Rev. Lett. 118, 170401 (2017).

    Article  ADS  CAS  Google Scholar 

  8. Brown, P. T. et al. Spin-imbalance in a 2D Fermi–Hubbard system. Science 357, 1385–1388 (2017).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  9. Dagotto, E. Correlated electrons in high-temperature superconductors. Rev. Mod. Phys. 66, 763–840 (1994).

    Article  ADS  CAS  Google Scholar 

  10. Haldane, F. D. M. ‘Luttinger liquid theory’ of one-dimensional quantum fluids. I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas. J. Phys. C Solid State Phys. 14, 2585–2609 (1981).

    Article  ADS  Google Scholar 

  11. Wen, X. G. Chiral Luttinger liquid and the edge excitations in the fractional quantum Hall states. Phys. Rev. B 41, 12838–12844 (1990).

    Article  ADS  CAS  Google Scholar 

  12. Frahm, H. & Korepin, V. E. Correlation functions of the one-dimensional Hubbard model in a magnetic field. Phys. Rev. B 43, 5653–5662 (1991).

    Article  ADS  CAS  Google Scholar 

  13. Cardy, J. Scaling and Renormalization in Statistical Physics (Cambridge Univ. Press, Cambridge, 1996).

    Book  Google Scholar 

  14. Bockrath, M. et al. Luttinger-liquid behaviour in carbon nanotubes. Nature 397, 598–601 (1999).

    Article  ADS  CAS  Google Scholar 

  15. Lee, J. et al. Real space imaging of one-dimensional standing waves: direct evidence for a Luttinger liquid. Phys. Rev. Lett. 93, 166403 (2004).

    Article  ADS  Google Scholar 

  16. Stone, M. B. et al. Extended quantum critical phase in a magnetized spin-1/2 antiferromagnetic chain. Phys. Rev. Lett. 91, 037205 (2003).

    Article  ADS  CAS  Google Scholar 

  17. Lake, B., Tennant, D. A., Frost, C. D. & Nagler, S. E. Quantum criticality and universal scaling of a quantum antiferromagnet. Nat. Mater. 4, 329–334 (2005).

    Article  ADS  CAS  Google Scholar 

  18. Klanjšek, M. et al. Controlling Luttinger liquid physics in spin ladders under a magnetic field. Phys. Rev. Lett. 101, 137207 (2008).

    Article  ADS  Google Scholar 

  19. Tranquada, J. M., Sternlieb, B. J., Axe, J. D., Nakamura, Y. & Uchida, S. Evidence for stripe correlations of spins and holes in copper oxide superconductors. Nature 375, 561–563 (1995).

    Article  ADS  Google Scholar 

  20. Omran, A. et al. Microscopic observation of Pauli blocking in degenerate fermionic lattice gases. Phys. Rev. Lett. 115, 263001 (2015).

    Article  ADS  Google Scholar 

  21. Hilker, T. A. et al. Revealing hidden antiferromagnetic correlations in doped Hubbard chains via string correlators. Science 357, 484–487 (2017).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  22. Kruis, H. V., McCulloch, I. P., Nussinov, Z. & Zaanen, J. Geometry and the hidden order of Luttinger liquids: the universality of squeezed space. Phys. Rev. B 70, 075109 (2004).

    Article  ADS  Google Scholar 

  23. Ogata, M. & Shiba, H. Bethe–Ansatz wave function, momentum distribution, and spin correlation in the one-dimensional strongly correlated Hubbard model. Phys. Rev. B 41, 2326–2338 (1990).

    Article  ADS  CAS  Google Scholar 

  24. Woynarovich, F. Excitations with complex wavenumbers in a Hubbard chain. I. States with one pair of complex wavenumbers. J. Phys. C 15, 85–96 (1982).

    Article  ADS  CAS  Google Scholar 

  25. Bogoliubov, N. M., Izergin, A. G. & Korepin, V. E. Critical exponents for integrable models. Nucl. Phys. B 275, 687–705 (1986).

    Article  ADS  MathSciNet  Google Scholar 

  26. Brinkman, W. F. & Rice, T. M. Single-particle excitations in magnetic insulators. Phys. Rev. B 2, 1324–1338 (1970).

    Article  ADS  Google Scholar 

  27. Greif, D., Jotzu, G., Messer, M., Desbuquois, R. & Esslinger, T. Formation and dynamics of antiferromagnetic correlations in tunable optical lattices. Phys. Rev. Lett. 115, 260401 (2015).

    Article  ADS  Google Scholar 

  28. White, S. R. & Affleck, I. Density matrix renormalization group analysis of the Nagaoka polaron in the two-dimensional tJ model. Phys. Rev. B 64, 024411 (2001).

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  30. Grusdt, F. et al. Parton theory of magnetic polarons: mesonic resonances and signatures in dynamics. Phys. Rev. X 8, 011046 (2017).

    Google Scholar 

  31. Dagotto, E. & Rice, T. M. Surprises on the way from one- to two-dimensional quantum magnets: the ladder materials. Science 271, 618–623 (1996).

    Article  ADS  CAS  Google Scholar 

  32. White, S. R. & Scalapino, D. J. Hole and pair structures in the tJ model. Phys. Rev. B 55, 6504–6517 (1997).

    Article  ADS  CAS  Google Scholar 

  33. White, S. R. & Scalapino, D. J. Density matrix renormalization group study of the striped phase in the 2D tJ model. Phys. Rev. Lett. 80, 1272–1275 (1998).

    Article  ADS  CAS  Google Scholar 

  34. Büchler, H. P. Microscopic derivation of Hubbard parameters for cold atomic gases. Phys. Rev. Lett. 104, 090402 (2010).

    Article  ADS  Google Scholar 

  35. Jordan, P. & Wigner, E. Über das Paulische Äquivalenzverbot. Z. Phys. 47, 631–651 (1928).

    Article  ADS  CAS  Google Scholar 

  36. Prokof’ev, N. V., Svistunov, B. V. & Tupitsyn, I. S. Exact, complete, and universal continuous-time worldline Monte Carlo approach to the statistics of discrete quantum systems. J. Exp. Theor. Phys. 87, 310–321 (1998).

    Article  ADS  Google Scholar 

  37. Pollet, L., Houcke, K. V. & Rombouts, S. M. Engineering local optimality in quantum Monte Carlo algorithms. J. Comput. Phys. 225, 2249–2266 (2007).

    Article  ADS  Google Scholar 

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We thank T. Giamarchi for exchanges on incommensurate magnetism, D. Huse, A. Recati, E. Demler and F. Grusdt for discussions and P. Sompet for reading the manuscript. Financial support was provided by the Max Planck Society (MPG) and the European Union (UQUAM, QSIMGAS, MIR-BOSE), and J.K. acknowledges funding from the Hector Fellow Academy.

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Nature thanks M. Lewenstein, C. de Morais Smith and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Authors and Affiliations



G.S., T.A.H., J.K., J.V., I.B. and C.G. planned the experiment and analysed and discussed the data. J.N. and L.P. performed the QMC simulations. All authors contributed to the interpretation of the data and the writing of the manuscript.

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Correspondence to Guillaume Salomon.

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Extended data figures and tables

Extended Data Fig. 1 Chain statistics.

Hubbard chain statistics are shown for a typical dataset containing 5,240 shots. The total spin Sz and total atom number N of individual Hubbard chains are conserved quantities of the Hamiltonian for each experimental run. However, they fluctuate for different experimental realizations, allowing us to explore the effects of doping and polarization individually through data grouping.

Extended Data Fig. 2 Density properties of the 1D clouds.

a, Density profiles n0(iN) of the chain located at the centre of the cloud in the y direction (j = 0). b, Antiferromagnetic spin correlations across a hole fixed at x = 1 (green) or a doublon (blue), measured through Cdw(x). The correlation signal is shifted by the hole, which is the microscopic-scale origin of the incommensurate spin correlations away from half-filling in the spin-balanced case. Error bars denote one standard error of the mean.

Extended Data Fig. 3 Spin correlations in squeezed space.

a, Doublon–hole correlations, measured by g2(x). The strong bunching at |x| = 1 reveals neighbouring doublon–hole pairs as mostly stemming from quantum fluctuations. This justifies our extension of the squeezed-space concept away from U → ∞. b, Spin correlations in the zero-magnetization sector at the centre of the cloud. Averaging over different polarizations (blue) results in a faster decay of the spin correlations with distance x in squeezed space compared to the Sz = 0 sector (green). Exponential fits of the correlation envelope for distances x = 2, …, 6 yield ξavg = 1.3(1) without magnetization post-selection and ξ0 = 2(1) in the Sz = 0 sector. Error bars denote one standard error of the mean.

Extended Data Fig. 4 Squeezed-space spin correlations at fixed distance.

Experimental spin correlations in squeezed space are shown as a function of polarization m for distances \(\mathop{x}\limits^{ \sim }=1,\ldots ,9\) (blue circles), along with exact diagonalization results for spin correlations in the Heisenberg chain at T = 0.7J averaged over the experimental {SzNs} distribution (grey squares). Error bars denote one standard error of the mean.

Extended Data Fig. 5 Chain statistics for the polarization study.

a, Experimental distribution {SzNs} used for studying the effects of polarization on the SDW vector. b, Histograms of pairs of parallel spins for {Ns = 10, Sz = 0} and {Ns = 11, |Sz| = 0.5}. The upper row shows, as expected, an upward shift of the distribution towards larger number of domain walls away from Sz = 0. By using the convention that spins pointing in the same direction at the edges contribute as one pair of parallel spins (lower row), we find that the parity of the number of domain walls is even in the integer-spin sectors and odd in the half-integer case. In the Sz = 0 sector, domain walls appear in pairs of opposite quantum numbers, which do not affect the SDW wavevector. In the |Sz| = 0.5 case on the other hand, we find a minimum of one domain wall owing to the excess spin and higher numbers of domain walls corresponding to pairs of additional excited parallel-spin pairs with opposite quantum numbers.

Extended Data Fig. 6 Properties of the prepared 2D clouds.

a, Density distribution for ty/tx = 1. b, Doublon–hole correlations g2(r). The strong bunching of the doubon–hole correlations g2(r) at |r| = 1 justifies the rejection of outcomes in which holes and doublons are found nearby when studying the effects of doping.

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Salomon, G., Koepsell, J., Vijayan, J. et al. Direct observation of incommensurate magnetism in Hubbard chains. Nature 565, 56–60 (2019).

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