Exotic pairing state in quasicrystalline superconductors under a magnetic field

We theoretically study the effect of a magnetic field on quasicrystalline superconductors, by modeling them as an attractive Hubbard model on a Penrose-tiling structure. We find that at low temperatures and under a high magnetic field there appears an exotic superconducting state with the order parameter changing its sign in real space. We discuss the state in comparison with the Fulde-Ferrell-Larkin-Ovchinnikov state proposed many years ago for periodic systems, clarifying commonalities and differences. It is remarkable that, even in the absence of periodicity, the electronic system finds a way to keep a coherent superconducting state with a spatially sign-changing order parameter compatible with the underlying quasiperiodic structure.

Figure 1

(a) Superconducting order parameter ${\mathrm{\Delta }}_i/U$ plotted against the magnetic field strength $h$ at $T=0.01$ (red circles) and 0.3 (blue crosses). $U=-3, \overline{n}=0.5$, and $N=11006$ are used. Black triangles and squares denote the value averaged over the sites at $T=0.01$ and 0.3, respectively. (b) Corresponding plot of the magnetization $m_i$. The inset shows an example of Penrose tiling.

Figure 2

Spatial pattern of (a)–(d) the superconducting order parameter ${\mathrm{\Delta }}_i$ and (e)–(h) magnetization $m_i$ for $T=0.01, U=-3$, and $N=11006$. (a), (e) $\overline{n}=0.5, h=0$; (b), (f) $\overline{n}=0.5, h=0.52$; (c), (g) $\overline{n}=0.3, h=0.32$; and (d), (h) $\overline{n}=0.7, h=0.62$.

Figure 3

Phase diagram against temperature $T$ and magnetic field $h$, calculated for $\overline{n}=0.5, U=-3$, and $N=11006$. Squares, circles, and triangles denote the normal state, alternating-sign superconducting state, and uniform-sign superconducting state, respectively, where we regard the results to be superconducting when ${max}_i{\mathrm{\Delta }}_i/U>0.001$.

Figure 4

(a), (b) On-site order parameter ${\mathrm{\Delta }}_i$ plotted against the Euclidean distance $|{\mathbf{r}}_i|=\sqrt{x_i^2+y_i^2}$ measured from the central site for $N=4181$ and $11006$, respectively. $\overline{n}=0.5, T=0.01$, and $h=0.52$ are used. Green arrows denote the “period” of the oscillation. (c), (d) Corresponding ${\mathrm{\Gamma }}_i$ for the normal-state solution, plotted against $|{\mathbf{r}}_i|$.