Figure 1

Phase diagram of the Hubbard-Heisenberg model on the half-filled ATL. As the frustration, $J^{\prime }/J=(t^{\prime }/t{)}^2$, is varied the spin correlations change from commensurate $(\pi ,\pi )$ characteristic of the square lattice for $t^{\prime }/t<0.93$, to incommensurate $(q,q)$ in the highly frustrated regime, to commensurate $(\pi /2,\pi /2)$ for $t^{\prime }/t\gtrsim 1.3$ characteristic of weakly coupled chains (see Fig. 2). The spin correlations mediate superconductivity, and the changes in the spin correlations cause changes in the symmetry of the superconducting state, which changes from “$d_{x^2-y^2}$” ($A_2$) for small $t^{\prime }/t$ to “$d+id$” ($A_1+i{A}_2$) for $t^{\prime }\sim t$ to “$s+d_{xy}$” ($A_1$) for large $t^{\prime }/t$.Reuse & Permissions

Figure 2

The wave vector $(q,q)$, where the spin correlations, which mediate superconductivity, are strongest. RVB theory shows that quantum effects increase the region where commensurate correlations are found relative to the classical theory. The lower inset shows the angle $\theta$, which determines the symmetry of the superconducting order parameter. $\theta =-\pi /2$ implies $A_2$ (“$d_{x^2-y^2}$”) superconductivity; $\theta =-\pi$ implies $A_1$ (“$s+d_{xy}$”) superconductivity; and $-\pi /2<\theta <-\pi$ implies an “$A_1+i{A}_2$” (“$d+id$”) state. Note, in particular, that $\theta =-2\pi /3$ for $t^{\prime }=t$, independent of $U$. The upper inset is a sketch of the ATL indicating the relevant hopping integrals (exchange parameters) to nearest neighbors (solid lines) and across one diagonal (dashed line).Reuse & Permissions

Figure 3

Variation of the RVB mean fields as the frustration is varied for $U=10t$. ${\chi }_x$, ${\chi }_y$, and ${\chi }_d$ all vary smoothly, but two phase transitions occur between superconducting states. The first occurs when ${\Delta }_d$ becomes finite, but the second involves the angle $\theta$ (inset of Fig. 2). ${\Delta }_x$, ${\Delta }_y$, and ${\Delta }_d$ become small at large $t^{\prime }/t$ due because the bandwidth $W\propto t^{\prime }$ in this regime and so we are moving away from the Mott transition as $t^{\prime }/t$ increases for $t^{\prime }\gg t$ (cf. Fig. 1).Reuse & Permissions

Figure 4

The $k$ dependence of superconducting order parameter, $\Delta (\mathbf{k})$, as the frustration ($t^{\prime }/t$) is varied. Each plot covers the first Brillouin zone, and the solid lines denote the noninteracting Fermi surfaces. For $t^{\prime }/t<0.92$ we have a “$d_{x^2-y^2}$” superconductor where ${\Delta }_{\mathbf{k}}$ transforms as the $A_2$ representation of $C_{2v}$ and has symmetry required nodes along the lines $k_x=\pm k_y$. This superconducting state is driven by the strong spin correlations at wave vector $(\pi ,\pi )$. For $t^{\prime }/t>1.06$ we find an “$s+d_{xy}$” order parameter which transforms like the $A_1$ representation of $C_{2v}$. This superconducting state is favored by the strong spin correlations near $(\pi /2,\pi /2)$. We find that this state has nodes, although they are not required by symmetry and therefore their location is dependent on $t^{\prime }/t$ and $U/t$. For $t^{\prime }=t$ the lattice maps onto the hexagonal lattice and the ground state is a “$d+id$” state which transforms as the $E_2$ representation of $C_{6v}$. Intermediate states such as those found at $t^{\prime }/t=0.95$ and 1.04 still show the effects of the strongly frustrated triangular spin correlations and form “$A_1+i{A}_2$” states. The “$A_1+i{A}_2$” states found for $0.92<t^{\prime }/t<1.06$ all break time reversal symmetry and are fully gapped. The results shown are for $U=10t$.Reuse & Permissions