Unidirectional $d$-wave superconducting domains in the two-dimensional $t\text{−}J$ model

Motivated by the recently observed pattern of unidirectional domains in high-$T_c$ superconductors [Y. Kohsaka et al., Science 315, 1380 (2007)], we investigate the emergence of spontaneous modulations in the $d$-wave superconducting resonating valence bond phase using the $t$-$J$ model at $x=1∕8$ doping. Half-filled charge domains separated by four lattice spacings are found to form along one of the crystal axes, leading to modulated superconductivity with out-of-phase $d$-wave order parameters in neighboring domains. Both renormalized mean-field theory and variational Monte Carlo calculations reveal that the energies of modulated and uniform phases are very close to each other.

Figure 1

(Color online) Spatial modulation of the hole density $n_{hi}$ (labeled from 0 to 7) found at hole doping $x=1∕8$ and $t∕J=3$ for (a) bond-, and (b) site-centered $\pi$DRVB configurations. Circle diameters are proportional to the hole densities while widths of bond lines connecting them are proportional to the magnitudes of the pair-order parameter ${\Delta }_{ij}$.

Figure 2

(Color online) Same as in Fig. 1 but for the $\pi$DSF patterns. Bond lines with arrows indicate the direction of charge currents, and their widths are proportional to the magnitudes of the bond-order parameter ${\chi }_{ij}$; $+$ $(-)$ symbols refer to positive (negative) flux flowing through each plaquette, the magnitude of which is represented by the size of the $+$ $(-)$ symbol.

Figure 3

(Color online) (a), (b) Hole density $n_{hi}$, (c), (d) SC order parameter ${\Delta }_{i\alpha }^{\mathrm{SC}}$ (modulated flux ${\Phi }_{\pi i}$) indicated by left (right) arrows, (e), (f) spin correlation $S_i^x$, and (g), (h) bond charge $T_i^y$, found in the $\pi$DSF (open symbols) and $\pi$DRVB (solid symbols) phase. Left (right) panels depict the bond- (site-centered) patterns; circles (triangles) in panels (c)–(h) correspond to the $x\left(y\right)$ direction, respectively.

Figure 4

(Color online) MF spectrum $E_{\mathbf{k}}$ of the (a) $\pi$DSF phase and (b) $\pi$DRVB phase along the nodal $\Gamma \text{−}M$ path near the $S$ point. Dashed line shows $E_{\mathbf{k}}$ of the $d$-wave RVB phase.

Figure 5

(Color) Real part of the SC order parameter ${\Delta }_{\mathbf{k}}^{\mathrm{SC}}\left(\mathbf{Q}\right)$ in the (a) $d$-wave RVB phase with $\mathbf{Q}=(0,0)$ as well as in the bond-centered $\pi$DRVB phase with (b) $\mathbf{Q}=\pm (\pi ∕4,0)$ and (c) $\mathbf{Q}=\pm (3\pi ∕4,0)$ (scaled, respectively, by factors 4.5 and 8 for clarity).