  Understanding quantum many-body states of correlated electrons is one main
theme in modern condensed matter physics. Given that the Fermi-Hubbard model,
the prototype of correlated electrons, has been recently realized in ultracold
optical lattices, it is highly desirable to have controlled numerical
methodology to provide precise finite-temperature results upon doping, to
directly compare with experiments. Here, we demonstrate the exponential tensor
renormalization group (XTRG) algorithm [Phys. Rev. X 8, 031082 (2018)],
complemented with independent determinant quantum Monte Carlo (DQMC) offer a
powerful combination of tools for this purpose. XTRG provides full and accurate
access to the density matrix and thus various spin and charge correlations,
down to unprecedented low temperature of few percents of the fermion tunneling
energy scale. We observe excellent agreement with ultracold fermion
measurements at both half-filling and finite-doping, including the
sign-reversal behavior in spin correlations due to formation of magnetic
polarons, and the attractive hole-doublon and repulsive hole-hole pairs that
are responsible for the peculiar bunching and antibunching behavior of the
antimoments.
