Quantum many-body simulations of the two-dimensional Fermi-Hubbard model in ultracold optical lattices

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, was 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 [Chen et al., Phys. Rev. X 8, 031082 (2018)], complemented by independent determinant quantum Monte Carlo, offers 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 an unprecedented low temperature of a few percent of the tunneling energy. 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 behaviors of the antimoments.

Figure 1

(a) Bilayer calculation of the spin-spin $\langle {\widehat{S}}_i·{\widehat{S}}_j\rangle$ and hole-doublon $\langle {\widehat{h}}_i·{\widehat{d}}_j\rangle$ correlators by sandwiching corresponding operators in between $\widehat{\rho }(\beta /2)$ and ${\widehat{\rho }}^{†}(\beta /2)$, where the snakelike ordering of sites for the XTRG is indicated by thick gray lines. (b) In the low-temperature AF background (blue down and red up spins), a magnetic polaron (gray shaded region) emerges around a moving hole, where the spins around the hole can be in a superposition of spin-up and -down states. The blue ellipse represents a hole-doublon pair showing a strong bunching effect. (c) A hole moves in the system along the path indicated by the gray string, leading to a sign reversal of the diagonal spin correlation. The red and blue shaded regions illustrate the deformed magnetic background due to the interplay between the hole and spins. Diagonal correlations are indicated red (aligned) or blue (antialigned).

Figure 2

Half-filled FHM with $U=7.2$ and $L=4,6,8$. (a) The finite-size AF order pattern is determined from the spin correlation $C_S\left(d\right)$ versus $(d_x,d_y)$, which melts gradually as $T$ increases. We show in (b) the spin correlation function $|C_S\left(d\right)|$ of various $d=1,\sqrt{2},2$ and in (c) the finite-size spontaneous magnetization $m_s$. Excellent agreement between the calculated ($L=8$) data and the experimental data [17] can be observed.

Figure 3

Doped FHM with $U=7.2$ and $L=6,8$. (a) The spin correlation pattern $C_S\left(d\right)$ versus $\delta$, plotted at $T=0.06$, where the finite-size AF order fades out for $\delta \gtrsim 0.15$. The computed (b) spin correlations $|C_S(d=1)|$ and (c) staggered magnetization $m_s$ are compared to the experimental data [17]. The XTRG data in (b) and (c) are obtained via extrapolation $1/D^{*}\to 0$ [53]. In the inset in (c), we show how $\delta$, computed by both XTRG and DQMC, varies with $T$ and on an $L=6$ lattice and at a fixed chemical potential $\mu =1.5$.

Figure 4

Diagonal and NNN $C_S\left(d\right)$ correlations versus $\delta$ for an $L\times L$ system with $U=7.2$ and $L=6,8$ for (a) $d=\sqrt{2}$ and (b) $d=2$. The inset in (b) zooms in on small $C_S\left(d\right)$ values. The sign reversal of $C_d$ is in good agreement with experimental data [20].

Figure 5

Various $g_2$ correlators for an $8\times 8$ system with $U=7.2$. The antimoment correlators (a) ${\overline{g}}_2(d=1)$ and (b) ${\tilde{g}}_2(d=2)$ are shown versus $\delta$. Experimental data with $d=1$, $T/t\simeq 1.0$ [13] and $d=2$, $T/t\simeq 0.25$ [20] are included for comparison. (c) and (d) The two-site hole-doublon ($g_2^{hd}$), hole-hole ($g_2^{hh}$), and full-density ($g_2^{nn}$) correlations for (c) $d=1$ and (d) $d=2$. The $d=1$ hole-doublon correlation $g_2^{hd}$ is compared with experiment in (c), with a nice agreement despite a separate $U/t\simeq 11.8$ in experiment [25].