Revealing hidden antiferromagnetic correlations in doped Hubbard chains via string correlators

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Science  04 Aug 2017:
Vol. 357, Issue 6350, pp. 484-487
DOI: 10.1126/science.aam8990
  • Fig. 1 Analysis of a doped Hubbard chain.

    (A) Experimental spin and density-resolved image of a single, slightly doped Hubbard chain after a local Stern-Gerlach-like detection. The reconstructed chain is shown below the picture. (B) Illustration of the magnetic environment around a hole. For aligned spins (top) the hole cannot freely delocalize because of the magnetic energy cost J, which is absent for anti-aligned spins (bottom). (C) Illustration of hole-induced AFM parity flips, squeezed space, and string correlator. Hole doping leads to AFM parity flips highlighted by the color mismatch between the spins and the background (top). Squeezed space is constructed by removing all sites with holes from the chain (bottom left). In the string correlator analysis, the flip in the AFM parity is canceled by a multiplication of –1 for each hole (bottom right). Comparing either of these analyses to the conventional two-point correlator reveals the hidden finite-range AFM order in the system.

  • Fig. 2 Revealing the magnetic environment around holes.

    (A) Connected two-point spin correlation function Embedded Image analyzed on occupied sites only (blue). The finite-range AFM order without holes asymptotically falls off with an exponential decay length of Embedded Image sites. The spin correlations at a distance of two sites switch sign in the presence of a hole as measured by Embedded Image (red diamond) demonstrating an AFM environment surrounding the hole. The solid black line indicates the finite-size offset (27), the blue line is a guide to the eye, and statistical uncertainties are smaller than the symbol sizes. (Inset) Comparison of experimental values (red lines) of Embedded Image (top) and Embedded Image (bottom) with finite temperature results from exact diagonalization (gray curves). The systematic error originating from a finite atom loss rate of up to 3% during imaging is negligible. (B) Amplitude of the correlation function Embedded Image as a function of distance d and the number of holes Nh between the two spins with the finite-size offset subtracted. The parity of the AFM order flips with every hole.

  • Fig. 3 Effect of hole doping on spin order.

    (A) Comparison of the spin correlation function (blue) Embedded Image and the spin-string correlation function (red) Embedded Image averaged over all local densities in the trap. The spin order is not visible with the conventional two-point spin correlator, but can be revealed by disentangling spin and charge sector with the string correlator. The extracted exponential decay length of Embedded Image sites matches the one extracted at unity filling (compare Fig. 1). The insets show the data binned by density (bin widths Embedded Image) for Embedded Image (blue), Embedded Image (red), and Embedded Image (green). Finite-range AFM order in the conventional correlator Embedded Image is present at Embedded Image, whereas it quickly gets suppressed when the system is doped away from half filling. At the same time, we observe a decreasing wave vector of the oscillations shown by the two-point spin correlations with decreasing density (left). By contrast, string correlations Embedded Image only marginally depend on density (right). Solid lines are guides to the eye. (B) Spin correlation measured directly in squeezed space for Embedded Image (blue), Embedded Image (red), and Embedded Image (green) as a function of density Embedded Image (bin widths Embedded Image). Dotted lines represent spin correlations Embedded Image and Embedded Image in the Heisenberg model for temperatures Embedded Image obtained by exact diagonalization with a coupling constant Embedded Image. The correlation decreases with increasing ratio Embedded Image. All correlations shown are corrected for the constant finite-size offset (27).

  • Fig. 4 Single holes as domain walls for the AFM order.

    (A) Tailored string correlator Embedded Image measuring the effect of a single hole on the doped Hubbard-chain. As expected for separated spin and charge sectors, the correlations are independent of the distance s between the hole and the spin, except for the opposite sign when the hole sits in between the two spins at relative distance d. In addition, there is a dynamic picture to the measurements shown here. Interpreting the vertical axis as time, one obtains the picture of a delocalized hole freely propagating through an antiferromagnetic background. The correlator Embedded Image is set to zero whenever two operators are evaluated at the same site. (B) Rectified correlator Embedded Image with hole position referenced to the string center. The hole-associated AFM parity flips are directly visible by the different domains. The expected parity is observed consistently for spin-spin distances of up to eight sites. The point symmetry around the origin is by construction of the correlator.

Supplementary Materials

  • Revealing hidden antiferromagnetic correlations in doped Hubbard chains via string correlators

    Timon A. Hilker, Guillaume Salomon, Fabian Grusdt, Ahmed Omran, Martin Boll, Eugene Demler, Immanuel Bloch, Christian Gross

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    • Supplementary Text
    • Figs. S1 to S5
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