Spatially inhomogeneous phase in the two-dimensional repulsive Hubbard model

Using recent advances in auxiliary-field quantum Monte Carlo techniques and the phaseless approximation to control the sign/phase problem, we determine the equation of state in the ground state of the two-dimensional repulsive single-band Hubbard model at intermediate interactions. Shell effects are eliminated and finite-size effects are greatly reduced by boundary-condition integration. Spin-spin correlation functions and structure factors are also calculated. In lattice sizes up to $16\times 16$, the results show signals for phase separation. Upon doping, the system separates into one phase of density $n=1$ (hole free) and the other at density $n_c$ $\left(\sim 0.9\right)$. The long-range antiferromagnetic order is coupled to this process and is lost below $n_c$.

Figure 1

(Color online) Upper panel: Ground-state energy per site $e\left(n\right)$, versus density, of the $3\times 3$ Hubbard lattice at $U=4$ (blue) and 8 (red) calculated by ED (empty symbols) and our QMC method (filled symbols). At each density, the result is the average from 1000 random $\mathbf{\Theta }$ values and the statistical error is estimated from their distribution. Bottom panel: Relative error (see text) of QMC ground-state energy compared to the exact result (percentage).

Figure 2

(Color online) Ground-state energy per site of the 2D Hubbard model vs density for several interaction strengths and lattice sizes. Error bars are combined QMC and $\mathbf{\Theta }$-integration statistical errors. As a result of TABC, curves are smooth and different lattice sizes are indistinguishable. The inset shows convergence to the thermodynamic limit with a magnified view. (To reduce clutter, only every fifth density is shown for each size.) It also illustrates the accuracy of the fit $e_{\text{fit}}\left(n\right)$ across the density range for the phase below $n_c$.

Figure 3

(Color online) The hole energy $e_h\left(h\right)$ vs hole density $h$ for interacting systems derived from Fig. 2. A clear minimum is seen for $U\ge 4$ at finite hole density $h_c$. The inset shows $e_h\left(h\right)$ for noninteracting Hubbard model calculated for lattices up to $40\times 40$ with PBC. Note the kinks and the flat part of the curves near half filling. The magenta curve is for a $12\times 12$ lattice (and the dashed line for a $40\times 40$) using TABC, which effectively eliminates the finite-size and shell effects.

Figure 4

(Color online) Spin-spin correlation function $C\left(\mathbf{r}\right)$ for a $12\times 12$ Hubbard lattice with $U=4$. Within the PS region, the system exhibits long-range AF correlation. The strength of the long-range correlation decreases with doping and vanishes at smaller densities. The inset shows a $16\times 16$ lattice at $U=4$, at a few selected densities near $n_c$; $n=0.9688$ (blue squares), 0.9453 (green diamonds), 0.9297 (red empty diamonds), and 0.9063 (black empty squares). To aid the eye, the absolute value $\left|C\left(\mathbf{r}\right)\right|$ is shown along two separate directions. The behavior of the curves indicates the finite sizes of the AF phase in the periodic lattice.

Figure 5

(Color online) Spin structure factor at $\mathbf{q}=\left(\pi ,\pi \right)$ for three system sizes calculated at $U=4$. The lines are guides to the eye. The inset shows $S\left(\pi ,\pi \right)$ vs lattice size at several densities (obtained by linear interpolation if the exact $n$ is not available in the particular lattice).