Phase separation and competition of superconductivity and magnetism in the two-dimensional Hubbard model: From strong to weak coupling

Cooperation and competition between the antiferromagnetic, $d$-wave superconducting, and Mott-insulating states are explored for the two-dimensional Hubbard model including nearest and next-nearest-neighbor hoppings at zero temperature. Using the variational cluster approach with clusters of different shapes and sizes up to 10 sites, it is found that the doping-driven transition from a phase with microscopic coexistence of antiferromagnetism and superconductivity to a purely superconducting phase is discontinuous for strong interaction and accompanied by phase separation. At half-filling the system is in an antiferromagnetic Mott-insulating state with vanishing charge compressibility. Upon decreasing the interaction strength $U$ below a certain critical value of roughly $U_c\sim 4$ (in units of the nearest-neighbor hopping), however, the filling-dependent magnetic transition changes its character and becomes continuous. Phase separation or, more carefully, the tendency towards the formation of inhomogeneous states disappears. This critical value is in contrast to previous studies, where a much larger value was obtained. Moreover, we find that the system at half-filling undergoes the Mott transition from an insulator to a state with a finite charge compressibility at essentially the same value. The weakly correlated state at half-filling exhibits superconductivity microscopically admixed to the antiferromagnetic order. This scenario suggests a close relation between phase separation and the Mott-insulator physics.

Figure 1

(Color online) Chemical potential $\mu$ as function of the average density $n$, calculated with a $2\times 2$ cluster as reference system for different values of $U$. From top to bottom: $U=12$, 8, and 4, respectively. $n_1$ and $n_2$ denote the boundaries of the instability (PS) region. Solid lines: Pure SC phase. Dashed lines: $\mathrm{AF}+\mathrm{SC}$ mixed phase. Dotted horizontal line: Critical chemical potential (where applicable).

Figure 2

Clusters used as reference systems in this study. The “$\sqrt{10}\times \sqrt{10}$” and “$\sqrt{8}\times \sqrt{8}$” clusters are marked explicitly in the figure.

Figure 3

(Color online) Characteristic energy $\Delta \mu ={\mu }^{*}-{\mu }_c$ (top) and critical densities $n_1$ and $n_2$ (bottom) as functions of $U$, calculated using the clusters shown in Fig. 2 as reference systems, $2\times 2$ (circles), $\sqrt{8}\times \sqrt{8}$ (squares), $4\times 2$ (stars), $5\times 2$ (triangles down), and $\sqrt{10}\times \sqrt{10}$ (triangles up). Lower plot: Solid lines correspond to the critical density $n_1$ and dashed lines to the critical density $n_2$.

Figure 4

(Color online) Ground-state energy per site $E_0∕L$ as function of electron density $n$, for $U=4$ (top), $U=8$ (middle), and $U=12$ (bottom). Results are shown for the $2\times 2$ (solid), $4\times 2$ (dashed-dotted), and $5\times 2$ (dashed) reference systems.

Figure 5

(Color online) Order parameters as a function of density for $U=4$ (left), $U=8$ (middle), and $U=12$ (right). Top: AF order parameter. Bottom: $d$-wave SC order parameter. Results are shown for the $2\times 2$ (solid), $4\times 2$ (dashed-dotted), and $5\times 2$ (dashed) reference systems.

Figure 6

(Color online) Closing of the Mott gap as function of $U$. From top to bottom: $L_c=2\times 2$, $L_c=4\times 2$, and $L_c=\sqrt{10}\times \sqrt{10}$ site clusters. From left to right: $U=3$ (green), 3.5 (blue), 4 (black), and 5 (red).

Figure 7

Representation of the Fermi surface for an interaction $U=3.5<U_c$ and $n=1$. Calculation using the $\sqrt{8}\times \sqrt{8}$ cluster. Dark regions denote large spectral weight integrated over a small frequency window of $\Delta \omega =\pm 0.05$ around $\omega =0$.

Figure 8

Spectral function for $U=8$ (top) and $U=4$ (bottom). Calculation using a $\sqrt{8}\times \sqrt{8}$ cluster. The horizontal dashed line marks the chemical potential.