# Direct observation of incommensurate magnetism in Hubbard chains

## Abstract

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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## 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.

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## Acknowledgements

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.

### Reviewer information

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

## Author information

Authors

### Contributions

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.

### Corresponding author

Correspondence to Guillaume Salomon.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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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). https://doi.org/10.1038/s41586-018-0778-7

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