Superconductivity on a quasiperiodic lattice: Extended-to-localized crossover of Cooper pairs

We study a possible superconductivity in quasiperiodic systems by portraying the issue within the attractive Hubbard model on a Penrose lattice. Applying a real-space dynamical mean-field theory to the model consisting of 4181 sites, we find a superconducting phase at low temperatures. Reflecting the nonperiodicity of the Penrose lattice, the superconducting state exhibits an inhomogeneity. According to the type of the inhomogeneity, the superconducting phase is categorized into three different regions which cross over each other. Among them, the weak-coupling region exhibits spatially extended Cooper pairs, which are nevertheless distinct from the conventional pairing of two electrons with opposite momenta.

Figure 1

Two-dimensional Penrose lattice of 4181 sites. The sites are located at vertices of rhombuses. Top-right panel is an enlarged view of a part of the lattice.

Figure 2

(a) $T$ dependence of ${\text{OP}}_i$ (red crosses) and $\overline{\text{OP}}$ (black dots) for $\overline{n}=0.5$ and $U=-4$. Blue curve plots the DMFT results for the Bethe lattice with the bandwidth $8t$ at quarter filling. (b) The same for $U$ dependence for $\overline{n}=0.5$ and $T=0.01$.

Figure 3

Spatial patterns of ${\text{OP}}_i$ and $n_i$ at $T=0.01$ for three different sets of $U$ and $\overline{n}$. The sites with $Q_i>\overline{Q}\left(Q_i<\overline{Q}\right)$ with $Q=\text{OP}$ and $n$ are colored by red (yellow).

Figure 4

${\text{OP}}_i$ plotted against $n_i/2(1-n_i/2)$ for various sets of $\overline{n}$ and $U$ at $T=0.01$. In order of (a)-(b)-(f)-(c)-(d), $\left|U\right|$ increases at a fixed $\overline{n}=0.5$. In order of (e)-(f)-(g)-(h), $\overline{n}$ increases at a fixed $U=-8$. Thick gray dashed curve in (d) shows the result calculated for the infinite-dimensional Bethe lattice with the bandwidth $8t$.

Figure 5

(a) Off-site pair amplitude ${\text{OP}}_{ij}\equiv \langle c_{i\uparrow }{c}_{j\downarrow }\rangle$ (normalized by $\overline{\text{OP}}$) plotted against the Euclidean distance $\left|\right|{\mathbf{r}}_i-{\mathbf{r}}_j\left|\right|$ for the three states shown in Fig. 3. Inset: Enlarged view for the short-distance part of red triangles and blue crosses. (b) and (c) Intensity map of $|{\text{OP}}_{\mathbf{k}{\mathbf{k}}^{\prime }}|$ calculated for a square lattice with the bandwidth $8t$ at quarter filling for $U=-2$ and $-16$, respectively. Just for plotting purpose, we set $k_x=k_y=k$ and $k_x^{\prime }=k_y^{\prime }=k^{\prime }$. (d)–(f) The same quantity for the three states in (a), plotted with a cutoff at $\left|k\right|,|k^{\prime }|=10$.

Figure 6

Phase diagram on the $\overline{n}\text{−}U$ plane at $T=0.01$. SC denotes the superconducting phase. The yellow, red, and blue regions, which are judged from the characteristics seen in the ${\text{OP}}_i\text{−}{n}_i/2(1-n_i/2)$ plots like Fig. 4, denote the superconducting states represented by Figs. 3, 3, and 3, respectively.