Fig. 1. Floquet potentials in a checkerboard lattice (left panel). A unit cell (marked by the dotted box) has five sites (labeled as a, b, c, d, e). The four sites, with loss, for the on-site potential ΔV are modulated in sinusoidal form (right panel). Such Floquet NHSE systems offer rich non- Hermitian topological phase and states in a geometrically static arrangement.

Fig. 2. The topological phase of the non-Hermitian effective model. (a), (b) The real-space Chern numbers calculated for gaps I/IV and gaps II/III as functions of g1(g3) and g2(g4) under loss γ∈(0,1), respectively. The Chern numbers are distinguished by different colors. The other parameters are set as θm=(m−1)π/2, where m=1,2,3,4_.

Fig. 3. The non-Hermitian properties of this effective model under the C4 rotation symmetry. (a) The real part of the projected band structure. The Chern numbers for band gaps II and III are labeled. The edge states within band gaps II and III are shown in green and orange, respectively, depending on their localization at the upper and lower edges. (b) The complex energy spectra with xPBC/yPBC, xPBC/yOBC, and xOBC/yOBC, respectively, from top to bottom. The olive dots represent the degenerate edge states. The hybrid skin-topological corner modes are indicated in red. (c) The schematic depiction of the quadrilateral supercell with xOBC/yOBC. The black dots represent omitted unit cells. The number of sites for xPBC/yOBC and xOBC/yOBC is 60 and 3380, respectively. The parameters are t1=1, t2=2, g1,2,3,4=0.5, and γ=0.6.

Fig. 4. Non-Hermitian C4 symmetric effective model with corner skin modes and configurations. (a) The top of the schematic for corner skin modes existing in the configurations of region A. Blue (red) arrows represent chiral edge states exhibiting relative attenuation (amplification), accumulating waves in their propagation (opposite) directions. Glowing purple spheres indicate potential localized corner skin modes. The bottom of (a) shows corner skin modes with higher imaginary energy in the parallelogram supercell. The height of the columns in the supercell is proportional to the amplitude at each atomic position. (b)–(d) Similar to the explanation for (a).

Fig. 5. The non-Hermitian properties of the effective model under C2 symmetry. (a) The real part of the projected band structure with xPBC/yOBC. The edge states within band gaps II and III are labeled the same as in Fig. 3(a). (b) The complex energy spectra with xPBC/yPBC, xPBC/yOBC, and xOBC/yOBC, respectively, from the top to bottom. The olive dots represent the degenerate edge states. The hybrid skin-topological corner modes are indicated in red. The number of sites for xPBC/yOBC and xOBC/yOBC is 60 and 3380, respectively. The parameters are t1=1, t2=2, g1,3=1, g2,4=2 and γ=1.

Fig. 6. (a), (b) Configuration diagram of the supercell, with the left side showing the predicted corner skin modes at all corners, and the right side showing the predicted corner skin modes in regions A, D, and B, C under the C2 symmetry.

Fig. 7. Non-Hermitian C2 symmetric effective model with corner skin modes and configurations. (a) The top of the schematic for corner skin modes existing in the configurations of region A. The bottom of (a) shows corner skin modes with higher imaginary energy in the parallelogram supercell. The bottom right of (a) shows the P values of the corner skin modes at the four corners, where Pb1, Pb2, Pb3, and Pb4 correspond to the positions shown at the top. (b)–(h) Similar to the explanation for (a).

Fig. 8. (a), (b) The real-space Chern number of non-Hermitian checkerboard lattices as a function of the Fermi energy, where (a) represents C4 symmetry and (b) represents C2 symmetry. The inset shows the schematic diagram of the supercell used to calculate the real-space Chern number.