Doped valence-bond solid and superconductivity on the Shastry-Sutherland lattice

Motivated by recent experiments on $\mathrm{Sr}{\mathrm{Cu}}_2{\left(\mathrm{B}{\mathrm{O}}_3\right)}_2$, we investigate the ground states of the doped Mott insulator on the Shastry-Sutherland lattice. To provide a unified theoretical framework for both the valence-bond solid state found in undoped $\mathrm{Sr}{\mathrm{Cu}}_2{\left(\mathrm{B}{\mathrm{O}}_3\right)}_2$ and the doped counterpart being pursued in on-going experiments, we analyze the $t\text{−}J\text{−}V$ model via the bond-operator formulation. It is found that superconducting states emerge upon doping with their properties crucially depending on the underlying valence-bond order. Implications to future experiments are discussed.

Figure 1

(Color online) Schematic diagram for the ground states on the Shastry-Sutherland lattice. The valence-bond solid state is the ground state at half-filling, as depicted in the left figure where ellipses represent spin-singlet pairs. When doped with holes, the system exhibits superconductivity in addition to the coexisting valence-bond solid order. Furthermore, when the nearest-neighbor repulsive interaction $V$ is larger than a critical value $V_c$, the plaquette $D$-wave superconductivity appears in a range of doping concentration $x$, with a peculiar spatial pattern, as shown in the right figure. In this situation, the pairing amplitudes residing in the four links encircling the horizontal dimers have the opposite sign to those for the vertical dimers. Different colors are used to emphasize the plaquette pattern of the pairing amplitude.

Figure 2

(Color online) Phase diagram of the $t\text{−}J\text{−}V$ model on the Shastry-Sutherland lattice as a function of hole concentration $x$ and the nearest-neighbor repulsive interaction $V$. The thick line separating the normal metal and the $S$-wave superconducting phase is a first-order phase boundary, while other boundaries are all second order. The undoped Mott-insulating state is represented by the thick solid line at $x=0$.

Figure 3

(Color online) Superconducting order parameter $\chi$ as well as condensation densities of the spin-Peierls singlets $\overline{s}$ and double-hole bosons $\overline{d}$ as a function of hole concentration $x$. When $V=0.4t>V_c$ (top panel), $S$-wave $\chi$ and $\overline{d}$ simultaneously vanish at $x=0.431$. Before the full plaquette $D$-wave superconductivity sets in at larger $x$, there is a region, $0.389<x<0.431$, where the $S$- and $D$-wave superconductivities coexist in the form of $S+iD$ wave. Finally, at $x=0.5$, the plaquette $D$-wave superconducting order parameter becomes zero and so does $\overline{s}$. No valence-bond solid correlation exists for $x⩾0.5$. When $V=0.1t<V_c$ (bottom panel), $\overline{d}$ increases monotonically as a function of $x$ while $\overline{s}$ remains finite all the way to $x=1$. For clarity, ${\overline{d}}^2$ is magnified 20 times in the top panel.

Figure 4

(Color online) $h$-fermion pairing amplitude $\Delta$ as a function of hole concentration $x$. In the plot, ${\Delta }_S$ and ${\Delta }_D$ indicate the $S$- and $D$-wave components of $\Delta$, respectively. Note that there is a region, $0.389<x<0.431$, where the $S$- and $D$-wave superconductivities coexist in the form of the $S+iD$ wave.

Figure 5

(Color online) Schematic diagram showing the origin of different behaviors of the $d$-boson condensate density below and above $V_c\simeq 0.23t$. The bottom figure plots ${\overline{d}}^2$ as a function of hole concentration $x$ for $V=0.4t$ $(\Delta E>0)$ and $V=0.1t$ $(\Delta E<0)$. Note that for clarity, ${\overline{d}}^2$ is magnified 30 times in the case of $V=0.4t$.

Figure 6

(Color online) Band structure of the $h$ fermion at half-filling. (a) The full band structure of the mean-field Hamiltonian, $H_{\text{fermion}}=H_{\text{fermion}}^{\left(0\right)}+H_{+,-}+H_{++--,-+}$. (b) The band structure obtained if $H_{\text{fermion}}$ is approximated to be $H_{\text{fermion}}^{\left(0\right)}+H_{++,--}$. (c) The additional band structure due to the remaining term, $H_{+-,-+}$. For convenience, we plot the band structure of $H_{\text{fermion}}^{\left(0\right)}+H_{+-,-+}$.