Fig. 4: Hole-doping cold Fermi Hubbard systems.

a,b, To coherently introduce hole dopants into the half-filled state after splitting the BI, we increase the kinetic energy and decrease the interaction strength by reducing the lattice depths out of the tight-binding regime. This prepares cold weakly interacting Fermi liquid that inherits the low initial entropy, which is adiabatically expanded to reach a given target density. Reloading the lattices and ramping up interactions create cold, hole-doped Hubbard systems at U/t = 8.0(3) (III). Schemes of splitting the BI into a half-filled antiferromagnetic Mott insulator at strong interaction of U/t = 18.6(8) (I) and intermediate interaction of U/t = 8.3(2) (II) are also marked. c, Doping scheme in quasimomentum space. Splitting the BI, which completely fills the Brillouin zone, into the square lattice doubles the number of sites and the size of the Brillouin zone. The population of quasimomentum states remain nearly unchanged. Expanding in real space then decreases the size of the Fermi surface and coherently introduces doping. d, Spin correlations as a function of bond displacements \({C}_{{\bf{d}}}^{zz}\) measured in an ROI of r = 3. The range and magnitude of the antiferromagnetic correlation decreases with increased doping. e, Azimuthal average of the sign-corrected spin correlations shown in d. At small dopings of δ ≤ 10%, the antiferromagnetism still remains long-ranged over the ROI. As we increase the number of hole dopants, the strengths of spin correlations at all distances are reduced by similar amounts. f, Short-range spin correlations \({C}_{d}^{zz}\) at different distances \(d=1,\sqrt{2},2,\sqrt{5}\) as we dope the system. The nearest-neighbour correlations at d = 1 show quantitative agreement between experimental data and CP-AFQMC simulations at U/t = 8 and T/t ∈ [0, 0.1] (solid line). At longer bond distances d > 1, experimental data show stronger correlations than CP-AFQMC simulations. The error bars denote 1 s.e.m.

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