Figure 1

(Color online) (a) Zero-temperature phase diagram of the checkerboard Hubbard model for small $x(=0.025)$ and (b) of $H^{eff}$ on the checkerboard lattice. The two superconducting phases both have $d$-wave symmetry and no nodal quasi-particles, but one is BCS-like ($d$-BCS) and the other is a Bose condensate of preformed two-particle bound states ($d$-BEC). The dashed lines indicate a crossover between them rather than a sharp phase transition. The two Fermi liquid phases (FL and ${\mathrm{FL}}^{\prime }$) are distinguished by the quantum numbers of the quasi-particles. The dotted curve with arrows in (b) represents the effective trajectory corresponding to the Hubbard model with fixed $t^{\prime }∕t$ $(=0.005)$ as a function of increasing $U$.Reuse & Permissions

Figure 2

Schematic finite temperature phase diagram of the checkerboard model. Notice that this figure is not drawn to scale and the x-axis has been offset from zero. As in Fig. 1a, there is a crossover within the superconducting phase from a $d$-BEC to a $d$-BCS limit. For large enough $U$ there is a FL phase. Above $T^{*}\sim \mathcal{O}\left({t^{\prime }}^0\right)$, indicated by the top dashed curve, the effective theory breaks down. The lower dashed curve, $T_p$, signifies the crossover between a normal Bose liquid of preformed pairs for $T<T_p$ and a high temperature state with little electron pairing for $T>T_p$. The range $T_c<T<T_p$ is a pseudo-gap regime. This can be a broad regime since the energy scale of coherence between preformed pairs would be down to $\mathcal{O}\left({t^{\prime }}^2\right)$ as indicated by the solid curve.Reuse & Permissions