$d$-wave superconductivity in coupled ladders

We study the one-band Hubbard model on the trellis lattice, a two-dimensional frustrated lattice of coupled two-leg ladders, with hopping amplitude $t$ within ladders and $t^{\prime }$ between ladders. For large $U/t$ this is a model for the cuprate ${\mathrm{Sr}}_{14-x}{\mathrm{Ca}}_x{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$. We investigate the phase diagram as a function of doping for $U=10t$ using two quantum cluster methods: The variational cluster approximation (VCA), with clusters of sizes 8 and 12, and cellular dynamical mean field theory (CDMFT), both at zero temperature. Both methods predict a superconducting dome, ending at roughly 20% doping in VCA and 15% in CDMFT. In VCA, the superconducting order parameter is complex in a range of doping centered around 10%, corresponding to bulk chiral, $T$-violating superconductivity. However, the CDMFT solution is not chiral. We find evidence for a migration of the Cooper pairs from the interladder region towards the plaquettes as doping is increased.

Figure 1

Model for the ladder cuprate ${\mathrm{Sr}}_{14-x}{\mathrm{Ca}}_x{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ on the trellis lattice. The dots stand for copper atoms. The open circles represent oxygen atoms (not shown in the middle of the figure), which are not explicitly included in the model. The shaded areas are the clusters used in VCA, labeled C8 and C12. $I_p$ and $I_c$ are the supercurrent loops circulating around the plaquette and between the plaquettes, respectively.

Figure 2

Superconducting order parameter ${\mathrm{\Delta }}_{ab}\left(\mathbf{k}\right)$ for the BCS Hamiltonian (3), as a function of wave vector, with parameters $D_x=0.09,D_y=0.03$ and $D_1=D_2=-0.01+0.008i$. Doping is set at 10%. The Brillouin zone is indicated. Panels (a) and (b) show the real and imaginary parts of the rung component (different sublattices), whereas panels (c) and (d) show the leg component (same sublattice). Red (gray) means negative, blue (dark gray) means positive, and the range is indicated on top of each panel. The normal-state Fermi surface is shown in red (gray) on panel (a).

Figure 3

Pairing order parameters $({\mathrm{\Delta }}^x,{\mathrm{\Delta }}^y,{\mathrm{\Delta }}^{\perp })$ as a function of doping $\delta$ for the two clusters used in VCA, at $U=10$. The complex phase of ${\mathrm{\Delta }}^{\perp }$ can be read on the right axis. The $T$-breaking solution $\left(arg{\mathrm{\Delta }}_{\perp }\ne 0\right)$ exists in a finite range of doping.

Figure 4

Superconducting order parameter ${\mathrm{\Delta }}_{ab}\left(\mathbf{k}\right)$ computed from the VCA solution, as a function of wave vector. $U=10$ and doping is 10%. This is to be compared with the BCS order parameter of Fig. 2.

Figure 5

Loop supercurrents $I_p$ and $I_c$ circulating respectively around a plaquette and around the center parallelogram of the C8 cluster, as a function of doping, for $U=10$. The loops are illustrated on Fig. 1.

Figure 6

Value of the Potthoff functional at the solution as a function of chemical potential $\mu$ for $U=10$ and different sets of variational parameters: the normal solution (no variational parameters), two $T$-preserving solutions, with $(D_x,D_y)$ and $(D_x,D_y,\mathrm{Re}{D}_{\perp })$ used as variational parameters, and the $T$-violating solution with variational set $(D_x,R_y,\mathrm{Re}{D}_{\perp },\mathrm{Im}{D}_{\perp })$. The latter has the lowest value of $\mathrm{\Omega }$. Two values of doping are indicated for that solution. A multiple of $\mu$ is added in order to better separate the different curves.

Figure 7

CDMFT cluster-bath system used in this work. The trellis lattice was deformed into a square lattice for simplicity. Right panel: arrangement of the two clusters A and B needed to tile the lattice. Intra- and intercluster hopping terms are represented by green (dark gray) and light gray lines, respectively (full for $t$, dashed for $t^{\prime })$. Left panel: pictorial representation of the two clusters: lattice sites are gray spheres and bath orbitals are represented by smaller, red (dark gray) spheres. Dashed (blue) lines are bath-cluster hybridization terms and dotted (red) lines anomalous terms between bath sites, forming the anomalous part of the matrix $\mathbf{\epsilon }$.

Figure 8

Order parameters computed from the CDMFT solutions found with the cluster-bath system illustrated in Fig. 7, as a function of doping. The CDMFT solutions do not break time-reversal invariance.

Figure 9

Order parameters ${\mathrm{\Delta }}^x$ (top) and $|{\mathrm{\Delta }}_{\perp }|$ (bottom) as a function of doping $\delta$ for cluster C8 and several values of on-site repulsion $U$.