Supersolid Phases of a Doped Valence-Bond Quantum Antiferromagnet: Evidence for a Coexisting Superconducting Order Parameter

Motivated by numerical evidence of the valence-bond ground state of the two-dimensional Heisenberg pyrochlore lattice, we argue using a $t\mathrm{\text{−}}J$ model that it evolves under doping into novel phases characterized by superconductivity coexisting with the underlying valence-bond solid order. A fermionic mean-field theory supplemented by exact diagonalization results provide strong arguments in favor of the stability of such supersolid phases. The resemblance with modulated superconducting patterns in high-$T_c$ cuprates as well as possible relevance to frustrated noncuprate superconductors such as spinels and pyrochlores is discussed.

Figure 1

Schematic pattern of $d_{(\pi ,\pi )}$-VBSS (a) and $s_{(\pi ,\pi )}$-VBSS (b) phases. In the checkerboard lattice, diagonal bonds are on every second plaquette. Rotation symmetry of the pairing amplitude ${\Delta }_{ij}$ reduces the number of inequivalent bonds down to three. The linewidths of these bonds labeled by $a$ ($\equiv$ “$ij$” running from 1 to 3) are proportional to the magnitude of the corresponding spin-spin correlations (for $t^{\prime }/t=-1$). Different types of those lines (solid and dashed) correspond to opposite signs of ${\Delta }_a$.

Figure 2

RMFT phase diagram of the generalized $t\mathrm{\text{−}}J$ model on the checkerboard lattice with $t^{\prime }>0$ (left) $t^{\prime }<0$ (right) while $J^{\prime }/J=(t^{\prime }/t{)}^2$. Solid (dashed) lines show first- (second-order) phase transition, respectively; the striped area corresponds to phase separation.

Figure 3

(a),(b) Spin-spin correlations $S_a$ and (c),(d) bond charge hoppings $T_a$ on the three inequivalent bonds of the $d_{(\pi ,\pi )}$-VBSS (left) and $s_{(\pi ,\pi )}$-VBSS (right) phase for $t^{\prime }/t=-1$. Panels (e),(f) show the SC order parameters ${\Delta }_a^{\mathrm{SC}}$ found to be enhanced with respect to ${\Delta }_0^{\mathrm{SC}}={\Delta }_1^{\mathrm{SC}}={\Delta }_2^{\mathrm{SC}}$ on a simple square lattice ($t^{\prime }=0$); see also Ref. 15. Shaded areas denote the region(s) of relative stability of each phase.

Figure 4

Quasiparticle band structure found for $t^{\prime }/t=-1$ at $x=1/4$: (a) $d_{(\pi ,\pi )}$-VBSS and (b) $s_{(\pi ,\pi )}$-VBSS.

Figure 5

Maxwell construction (see text) for $t^{\prime }<0$ (a) and $t^{\prime }>0$ (b). From top to bottom: $|t^{\prime }/t|=0.7$, 0.8, 0.9, 1.0, 1.05, and 1.1. The arrows indicate small kinks due to the first-order transition expected if PS (cf. Fig. 2) is not considered.