Magnetic Correlations and Pairing in the $1/5$-Depleted Square Lattice Hubbard Model

We study the single-orbital Hubbard model on the $1/5$-depleted square-lattice geometry, which arises in such diverse systems as the spin-gap magnetic insulator ${\mathrm{CaV}}_4{\mathrm{O}}_9$ and ordered-vacancy iron selenides, presenting new issues regarding the origin of both magnetic ordering and superconductivity in these materials. We find a rich phase diagram that includes a plaquette singlet phase, a dimer singlet phase, a Néel and a block-spin antiferromagnetic phase, and stripe phases. Quantum Monte Carlo simulations show that the dominant pairing correlations at half filling change character from $d$ wave in the plaquette phase to extended $s$ wave upon transition to the Néel phase. These findings have intriguing connections to iron-based superconductors, and suggest that some physics of multiorbital systems can be captured by a single-orbital model at different dopings.

Figure 1

The geometry of the $1/5$-depleted square lattice. $2\times 2$ plaquettes have intersite hopping $t$. Different plaquettes are linked by hopping $t^{\prime }$. The two primitive vectors are shown by red arrows and the unit cell is shown by the tilted square.

Figure 2

Ground state phase diagram at half filling. The thick horizontal lines indicate the region with long-range Néel order. At $U=\infty$, the AF region is obtained from Ref. [19] (Heisenberg model study on the same geometry) by considering $t^{\prime }/t=\sqrt{J^{\prime }/J}$ ($J$ and $J^{\prime }$ are the intra- and interplaquette spin exchange interactions). Blue circles track the high symmetry point (HSP) inside the AF region where the NN spin correlations are equal on all bonds [see Fig. 3]. Similar results for the HSP are not available for the Heisenberg limit. So instead, the empty blue circle indicates the location of the maximum of the AF moment in that limit. The dashed line shows the AF–band insulator (BI) phase boundary as predicted by the RPA. The inset shows the AF ordering. Filled (empty) circles denote up (down) spins.

Figure 3

(a) The absolute value of the difference in the NN spin correlations on $t$ and $t^{\prime }$ bonds at $\beta =20$ from DQMC vs $t^{\prime }/t$ for several values of the interaction strength ($m(\mathbf{r})$ is the spin-spin correlation function at distance $\mathbf{r}$ and “a” denotes the lattice constant between NNs [29]). The shaded region on the horizontal axis is added to show the boundaries of the Néel phase in the $U\to \infty$ limit. The lattice is a $4\times 4$ arrangement of $2\times 2$ plaquettes ($N=64$), except for $U=1$ and 2 where the $8\times 8$ arrangement ($N=256$) is used. We have also simulated a 576-site lattice for the latter interactions and found no significant changes in the location of the HSP. (b) The AF structure factor vs $t^{\prime }/t$ at $\beta =20$ from DQMC. Except for $U=1$, for which $N=256$, the results are obtained for the $N=64$ lattice.

Figure 4

(a) Pairing structure factor [29] at $\beta =20$ and $t^{\prime }/t=0.3$ vs the interaction strength. For $U=1$ and 2, two different system sizes are shown. (b) Pairing structure factor at $\beta =20$ and $U=4$ vs the ratio $t^{\prime }/t$. Full (empty) symbols are for the $d$-wave (extended $s$-wave) symmetry. The error bars are smaller than the symbols. The arrow indicates the location of the AF phase transition.

Figure 5

(Top) Four of the magnetic orderings considered in this study. The up (down) spins are denoted by filled (empty) circles. $Q$ indicates the phase between unit cells and the FM, AF, and stripe denote the ordering of spins within each unit cell. (Bottom) The ground state RPA phase diagram of the model at $t^{\prime }=t$ away from half filling. The inset shows the density of states (DOS).