Fragmented Cooper Pair Condensation in Striped Superconductors

Condensation of bosons in Bose-Einstein condensates or Cooper pairs in superconductors refers to a macroscopic occupation of a few single- or two-particle states. A condensate is called “fragmented” if not a single, but multiple states are macroscopically occupied. While fragmentation is known to occur in particular Bose-Einstein condensates, we propose that fragmentation naturally takes place in striped superconductors. To this end, we investigate the nature of the superconducting ground state realized in the two-dimensional $t\text{−}{t}^{\prime }\text{−}J$ model. In the presence of charge density modulations, the condensate is shown to be fragmented and composed of partial condensates located on the stripes. The fragments of the condensates hybridize to form an extended macroscopic wave function across the system. The results are obtained from evaluating the singlet-pairing two-particle density matrix of the ground state on finite cylinders computed via the density matrix renormalization group method. Our results shed light on the intricate relation between stripe order and superconductivity in systems of strongly correlated electrons.

Figure 1

Spectrum ${ϵ}_n$ of the singlet density matrix ${\widehat{\rho }}_S$ of the ground state on the width $W=4$ cylinder at $t^{\prime }=0.2$ and $J=0.4$. We compare system lengths $L=8$, 16, 24, 32 and show results for hole doping $p=1/16$ (a) and $p=1/8$ (b). The number of dominant eigenvalues above the residual continuum exactly matches the number of stripes in the system. The insets show enlarged views of the largest eigenvalues. The condensate fractions ${ϵ}_n$ increase with system size.

Figure 2

(a) Condensate wave functions ${\chi }_n({\mathit{r}}_i,\alpha )$ for the dominant four eigenvalues of the two-body density matrix on a $32\times 4$ cylinder at doping $p=1/16$ and $t^{\prime }=0.2$. For $\alpha =\widehat{\mathit{x}}$ we show the value of ${\chi }_n({\mathit{r}}_i,\alpha )$ as the color and line width right to the site ${\mathit{r}}_i$, for $\alpha =\widehat{\mathit{y}}$ it is shown on the link on top of site ${\mathit{r}}_i$. Blue (red) indicates a positive (negative) value of ${\chi }_n({\mathit{r}}_i,\alpha )$. The hole density $1-⟨n_i⟩$ is shown as the area of the gray circles. We observe a uniform $d$-wave pattern, where vertical and horizontal bonds have opposite signs in the most dominant condensate wave function, while the other dominant condensates exhibit modulation of the $d$-wave orientation concomitant with the stripes. (b) Rung-averaged $d$-wave condensate wave function ${\overline{\chi }}_i^d$ and the rung-averaged hole density $1-⟨{\overline{n}}_i⟩$.

Figure 3

Condensate wave functions ${\chi }_n({\mathit{r}}_i,\alpha )$ for the dominant four eigenvalues of ${\widehat{\rho }}_S$ on a $32\times 4$ cylinder at doping $p=1/16$ and $t^{\prime }=0$. The hole density $1-⟨n_i⟩$ is shown as the area of the gray circles.

Figure 4

Spectrum ${ϵ}_n$ of ${\widehat{\rho }}_S$ on the $W=6$ cylinder for (a) the $d$-wave superconducting state at $t^{\prime }=0.2$, $p=1/16$ and (b) a nonsuperconducting stripe state at $t^{\prime }=-0.2$, $p=1/8$. No dominant eigenvalues are observed in the nonsuperconducting case in (b). (c) Spectrum of ${\rho }_{\mathrm{loc}}$ [Eq. (11), diagonal elements have been set to zero] for the $s$-wave superconducting state realized in the attractive Hubbard model on a $W=4$ cylinder for $U/t=-2$, $p=1/2$, and $t^{\prime }/t=0$. Only one dominant eigenvalue is observed for this uniform condensate.

Figure 5

Condensate wave functions ${\chi }_n({\mathit{r}}_i,\alpha )$ for the dominant three eigenvalues of ${\widehat{\rho }}_S$ on a $16\times 6$ cylinder at doping $p=1/16$ and $t^{\prime }=0.2$. The hole density $1-⟨n_i⟩$ is shown as the area of the gray circles.

Figure 6

Temperature dependence of the $d$-wave pairing susceptibility $\mathcal{D}$ (a) and the antiferromagnetic spin structure factor $S_m(\pi ,\pi )$ (b) for the $t\text{−}{t}^{\prime }\text{−}J$ model on the $32\times 4$ cylinder at $p=1/16$ and $t^{\prime }=0.2$. Strong pairing correlations develop below a temperature $T/t\approx 0.05$, which suppresses antiferromagnetism.