Numerical results on the short-range spin correlation functions in the ground state of the two-dimensional Hubbard model

Optical lattice experiments with ultracold fermion atoms and quantum gas microscopy have recently realized direct measurements of magnetic correlations at the site-resolved level. We calculate the short-range spin-correlation functions in the ground state of the two-dimensional repulsive Hubbard model with the auxiliary-field quantum Monte Carlo (AFQMC) method. The results are numerically exact at half filling where the fermion sign problem is absent. Away from half filling, we employ the constrained path AFQMC approach to eliminate the exponential computational scaling from the sign problem. The constraint employs unrestricted Hartree-Fock trial wave functions with an effective interaction strength $U$, which is optimized self-consistently within AFQMC. Large supercells are studied, with twist averaged boundary conditions as needed, to reach the thermodynamic limit. We find that the nearest-neighbor spin correlation always increases with the interaction strength $U$, contrary to the finite-temperature behavior where a maximum is reached at a finite $U$ value. We also observe a change of sign in the next-nearest-neighbor spin correlation with increasing density, which is a consequence of the buildup of the long-range antiferromagnetic correlation. We expect the results presented in this paper to serve as a benchmark as lower temperatures are reached in ultracold atom experiments.

Figure 1

Nearest-neighbor $[\mathbf{r}=(0,1\left)\right]$ correlation function vs $1/L^3$ for $U/t=2,4$, and 8. The largest system size included is $22\times 22$. Finite-size scaling fits using Eq. (12) are also shown.

Figure 2

The magnitude of short-range spin-correlation functions at half filling. The correlation function $c\left(\mathbf{r}\right)$ is shown vs interaction strength, for several values of $\mathbf{r}$: $(0,1),(1,1),(0,2)$, and $(1,2)$. The upper horizontal line represents the infinite-$U$ value of the NN spin-correlation function from the Heisenberg model [43]. The lower horizontal line represents the infinite distance value at the infinite-$U$ limit (the square of the magnetization in the Heisenberg model [43]). The dotted red curve is the NN spin-correlation function at $T=0.31$, taken from Ref. [18].

Figure 3

Optimal value of $U_{\mathrm{eff}}^{\mathrm{UHF}}$. The relative difference ${\delta }_{\text{UHF}}$ is plotted vs the effective interaction used in the UHF calculation, for an $8\times 8$ system with $N_{\uparrow }=N_{\downarrow }=28$ $\left(n=0.875\right)$ and $U/t=6,8,12$.

Figure 4

Comparison of CP-AFQMC and exact diagonalization results of the NN spin-correlation function for a $4\times 4$ system with $N_{\uparrow }=N_{\downarrow }=7$ and $U=12$. A series of UHF trial wave functions, generated with different $U_{\mathrm{eff}}^{\mathrm{UHF}}$ values, is used in CP-AFQMC to evaluate the systematic errors. CP-free indicates CP-AFQMC using the free-electron trial wave function. In the inset, we also plot the relative error of the twist averaged results vs $U_{\mathrm{eff}}^{\mathrm{UHF}}$ of the UHF in the trial wave function $[{\delta }_{\text{CP}}$ in Eq. (15)]. The minimum error corresponds to the optimal $U_{\mathrm{eff}}^{\mathrm{UHF}}\sim 3.5$ in this case.

Figure 5

Illustration of the dependence of NN (upper panel) and NNN (lower panel) spin-correlation functions on supercell size. Two interaction strengths, $U=4$ and 8, are shown in a system with $n=0.75$. Linear fits in $1/L$ are also plotted.

Figure 6

NN (upper panel) and NNN (lower panel) spin-correlation functions vs filling factors at different interaction strengths. The half-filling results in Fig. 2 are included here for completeness. The value of the NN spin-correlation function at half filling for $U=\infty$ is from Ref. [43].