Bogoliubov flat bands in twisted layered materials

Abstract

Flat bands have attracted considerable interest in condensed matter physics because they provide a fertile platform for realizing strongly correlated and topological quantum phases. To date, however, most studies have focused on flat bands in normal-state electronic structures, such as those found in graphene and transition metal dichalcogenides. In this work, we investigate the emergence of flat bands in the superconducting Bogoliubov quasiparticle spectrum of twisted layered $d$-wave superconductors. We show that when the superconducting order parameter is odd under the in-plane $\mathrm{C}_2$ rotation, Bogoliubov flat bands can be engineered in the vicinity of the rotation axis. By analyzing a low-energy effective Hamiltonian, we demonstrate that the Berry connection of single layer system provides a clear criterion for the formation of the Bogoliubov flat bands. Our results establish a new paradigm of superconducting twistronics, in which the twist angle acts as a powerful tuning parameter for designing gapless flat-band superconductors.

Figures (6)

• Figure 1: Structure of twisted bilayer square lattice. Regular and inverted triangles denote lattice points on the top and bottom layers, respectively. Squares with a solid and dotted outlines represent the unit cell of bilayer and single-layer systems, respectively. There are five atoms per unit cell in each layer.
• Figure 2: (a) FSs of the lowest (red) and the second-lowest bands (blue) at $\mu=-0.25t$ and $t_z=0.5t$. Angle dependence of the energy gaps $E_{g}$ of the lowest (red) and the second-lowest bands (blue) at (b) $t_z=0$, (c) $t_z=0.02t$, and (d) $t_z=0.5t$.
• Figure 3: (a) $t_z$-dependence of the energy gap $E_g$ at $\varphi=\pi/4$. (b) The magnitude of the derivative of $E_g$ with respect to $\varphi$ at $\varphi=\pi/4$. It is proportional to the group velocity along the FS.
• Figure 4: (a) The quasiparticle density of states $\rho_{S}(E)$ at $t_z=0$ (dashed line) and $t_z=0.6t$ (solid line). (b) $\rho_{S}(E)$ at $E=0$ is plotted as a function of $t_z$.
• Figure 5: Illustration of a nodal creation with zero group velocity in ${\bm k}$-space. The blue solid and dotted lines denote the FSs with and without interlayer hopping, respectively. The arrows show the direction of the Berry connection of the top layer, $\mathbf{A}({\bm k})/|\mathbf{A}({\bm k})|$. At the momentum on the Red line, the direction of the Berry connection is $-3/4\pi$.