Higher-order topological state induced by $d$-wave competing orders in high-$T_c$ superconductor based heterostructures

We introduce a two-dimensional Chern insulator in proximity to a $d$-wave pseudogap state of the high-$T_c$ superconducting material as an effective platform to realize the higher-order topological system. The proximity-induced $d$-density-wave (DDW) order in the Chern insulator layer serves as an effective mass. The edge states will be fully gapped by this DDW order. In the real space, the sign of the DDW order parameter changes at the system corners due to the $d$-wave factor, leading to the gapless corner states, indicating that this system may be in a higher-order topological state. The higher-order topology in this coupled system is confirmed based on the calculation of the edge polarization and the quadrupole moment. In the superconducting state where the superconducting order and the DDW order coexist, the Majorana corner states emerge.

Figure 1

(a) The energy bands considering the open boundary condition along the $x$ direction with ${\mathrm{\Delta }}_0=0$. (b) The energy bands in the presence of the DDW modulation. (c) The zero-energy LDOS in the presence of the DDW order in a $20\times 20$ lattice with open boundaries along both the $x$ and the $y$ directions. (d) The corresponding eigenvalues of the Hamiltonian in the $20\times 20$ lattice.

Figure 2

(a) The Wannier spectral ${\nu }_x$ as a function of the effective Zeeman field $m$. (b) The edge polarization ($p_y^{\mathrm{edge},x},p_x^{\mathrm{edge},y}$) and quadrupole moments $q_{xy}$ as a function of $m$.

Figure 3

Similar to Fig. 1, but in presence of an additional $d$-wave pairing order with ${\mathrm{\Delta }}_d=0.5$.