Edge insulating topological phases in a two-dimensional superconductor with long-range pairing

We study the zero-temperature phase diagram of a two-dimensional square lattice loaded by spinless fermions, with nearest-neighbor hopping and algebraically decaying pairing. We find that for sufficiently long-range pairing, new phases occur, not continuously connected with any short-range phase and not belonging to the standard families of topological insulators/superconductors. These phases are signaled by the violation of the area law for the von Neumann entropy, by semi-integer Chern numbers, and by edge modes with nonzero mass. The latter feature results in the absence of single-fermion edge conductivity, present instead in the short-range limit. The definition of a bulk-topology and the presence of a bulk-boundary correspondence is suggested also for the long-range phases. Recent experimental proposals and advances open the possibility to probe the described long-range effects in near-future realistic setups.

Figure 1

Phase diagram at $T=0$ for $\mathrm{\Delta }=2w=1$.

Figure 2

Decay exponents $\beta$ of $\langle c_i^{†}{c}_j\rangle$ (red) and $\langle c_i^{†}{c}_j^{†}\rangle$ (blue), for $L=1500,\mu =0.5$, and $\alpha$ varying. The regime close to $\alpha =1$ is not plotted, due to the presence of large finite-size effects.

Figure 3

Behavior of the von Neumann entropy $S$ for $L=18$ and $\mu =0.5$. Top: $S\left(l\right)$ for $\alpha =10$ (red) and for $\alpha =0.5$ (blue). Notice the difference in the two cases for the variation around $l=\frac{L}{2}$. Bottom: Same quantity for fixed $l=\frac{L}{2}$ and $\alpha$ varying.

Figure 4

Main panel: Square of the wave function, $w\left(x\right)$, of the massive edge modes at $L=24,\mu =0.5$, and $\alpha =0.5$, for $y=\frac{L}{2}$ and varying $x$. Inset: Finite-size scaling of the mass for the edge modes at $\mu =0.5$ and $\alpha =0.5$, for $L=12\text{–}48$ varied in steps of 12. A nonvanishing mass in the thermodynamic limit is obtained.