Non-Hermitian physics provides an effective framework for investigating non-equilibrium and open quantum systems as well as their classical counterparts, wherein dissipation, energy gain, and nonreciprocal couplings give rise to physical effects that are unattainable in conventional Hermitian systems. Rich non-Hermitian phenomena such as unidirectional invisibility [1], non-Hermitian skin effects (NHSEs) [2–6], exceptional points, lines, and surfaces [7–15], non-Bloch bands [16–19], non-Hermitian topology [20–23], and non-Hermitian dynamics [24–27] were found in the past decades. A particularly interesting frontier lies at the interface between topological physics and non-Hermitian physics. For instance, the NHSEs can be characterized by the spectral topology in the complex plane for finite non-Hermitian systems [2,28]. Moreover, in non-Hermitian topological systems, the non-Hermitian skin modes coexist with topological edge modes, which thus extends the bulk-edge correspondence to a more general picture [29]. The interplay between the NHSEs and the topological edge modes can also bring unconventional effects such as the skin-topological effect [30–33] wherein the corner skin modes originate from the NHSEs of the topological edge states.

On the other hand, Floquet lattice systems provide an efficient pathway toward engineering desired Hamiltonians and realizing the underlying physics [34–38]. In particular, the periodic modulation in Floquet systems introduces artificial gauge fields and enables access to various topological phases and the NHSEs without nonreciprocal couplings. Thus, Floquet non-Hermitian systems provide a rich realm for exploring dynamic and topological phenomena [39–43]. With experimental feasibility in photonic and acoustic systems, skin-topological effects are especially accessible in Floquet non-Hermitian systems [30,32,44]. However, the existing studies on skin-topological effects focus on the relatively simple regime where the spatial symmetry dictates the phenomenon and there is only one branch of edge states [33]. Much remains unknown when the Chern number (and hence the number of the branches of edge states) is varied and spatial symmetry is controllable, which may offer a valuable platform and unconventional mechanisms for steering light by the interplay between topology and non-Hermicity [45].

In this work, we study the interplay between the NHSE and the topological physics in checkerboard lattices where we find controllable Floquet skin-topological effects with varying Chern numbers. At the heart of our theory is the creation of a Floquet non-Hermitian checkerboard lattice model wherein rich, controllable non-Hermitian topological phases induced only by gain and loss can be realized. For concreteness, we design an array of evanescently coupled helical optical waveguides in three dimensions to realize the model. Nevertheless, the underlying physics is more general and is not restricted to photonic systems. For instance, the same physics can be realized also in cold atom systems where the synthetic gauge flux can be realized via other mechanisms [46–48]. We find that our checkerboard lattices can efficiently generate non-Hermitian bands with large Chern numbers. The induced multiple chiral edge states have rich non-Hermitian skin effects due to the fact that each branch of the chiral edge states has its own non-Hermitian properties. The total skin-topological effect comes from the NHSEs of the multiple chiral edge states as well as their interaction. We find that the scattering between these non-Hermitian topological edge channels at the corner boundaries can induce an unconventional mechanism for the corner skin effect, which is distinct from the previous mechanisms solely associated with the nonreciprocal properties of the non-Hermitian edge states. With a simple and controllable design, our system is found to support non-Hermitian hybrid skin-topological effects, exhibiting rich corner skin effects that can be engineered by varying the lattice symmetry.

The structure of this paper is organized as follows. In Section 2.A, we introduce the non-Hermitian effective model of photonic checkerboard lattices and the topological phases, which provide the groundwork for realizing the NHSE. In Section 2.B, we investigate the mixed skin-topological effects in the non-Hermitian effective model with $C_4$ symmetry. We analyze the origin of corner skin modes from a dynamical perspective. The locations of different corner skin modes are influenced by various non-equilibrium interactions of topological edge states. In Section 2.C, we explore the non-Hermitian effective model with $C_2$ symmetry. Corner skin modes still exist, and we present the dynamic analysis process, finding that the locations of corner skin modes depend not only on the signs of group velocity and the gain/loss of topological edge states, but also on the contributions from the topological edge states. Finally, we conclude in Section 3.

We introduce interstitial sites between the main sites of the two-dimensional square potential or Su-Schrieffer-Heeger (SSH) lattice, as shown in Fig. 1. In each unit cell (marked by the dotted-dashed box), an interstitial site (e) is added in the intermediate region between the four sublattice sites (a, b, c, d), forming a coupling path between each pair of sublattice sites. There is an inhomogeneous dissipation distribution and an imbalanced on-site potential in each unit cell. In this case, the $z$-dependent Hamiltonian in this model is given by$$H(z)=H_{\mathrm{SSH}}+H_{\mathrm{loss}\text{-}\text{potential}}(z)+H_{\text{inter}\text{-}\mathrm{SSH}}(z),$$(1)where $H_{\mathrm{SSH}}$ and $H_{\mathrm{loss}\text{-}\text{potential}}(z)$ respectively describe the main sites and their dissipation and on-site potential distributions, whereas $H_{\text{inter}\text{-}\mathrm{SSH}}$ is the coupling between the interstitial sites and main sites of sublattice. We take $$\begin{gathered} H_{\mathrm{SSH}}=\sum _{i,j}{t}_1(a_{i,j}^{†}{b}_{i,j}+d_{i,j}^{†}{c}_{i,j}+b_{i,j}^{†}{d}_{i,j}+a_{i,j}^{†}{c}_{i,j}) \\ +\sum _{i,j}{t}_2(a_{i+1,j}^{†}{b}_{i,j}+d_{i+1,j}^{†}{c}_{i,j}+b_{i,j+1}^{†}{d}_{i,j}+a_{i,j+1}^{†}{c}_{i,j})+\mathrm{h}\text{.c.}, \end{gathered}$$(2)where $t_1$ and $t_2$ represent the intra-cell and inter-cell coupling strengths in the 2D SSH lattice, respectively. ${\xi }_{i,j}^{†}({\xi }_{i,j})(\xi \in \{a,b,c,d\})$ creates (annihilates) the operator at the site $(i,j)$ and h.c. denotes the Hermitian conjugate. The coupling interaction between the two sub-systems is characterized by $H_{\text{inter}\text{-}\mathrm{SSH}}(z)={\sum }_{i,j}{\sum }_{m=1}^4{g}_m(z)(e_{i,j}^{†}{\xi }_{i,j})$. The clockwise rotation involves time-dependent nearest-neighbor interactions [with couplings $g_1(z),g_2(z),g_3(z),g_4(z)$], as shown in Fig. 1. The on-site potential at the sublattice sites (a, b, c, d), with dissipation $\gamma$, is given by $$H_{\mathrm{loss}\text{-}\text{potential}}(z)=\sum _{i,j}(V_{i,j}(z)-i\gamma ){\xi }_{i,j}^{†}{\xi }_{i,j}.$$(3)We consider a sinusoidal modulation $V_{i,j}(z)=\sin (2\pi z/T+2\pi m/4)$, with a chiral phase shift of $2\pi /4$ between the interstitial sites in the intermediate region of each main site, where $m\in 1,2,3,4$ and $T$ denotes the Floquet period along the $z$ direction. In the periodical Floquet systems, the static effective Hamiltonian $H_{\mathrm{eff}}$ is defined as $$H_{\mathrm{eff}}=i/T\log (U),$$(4)where $U=\mathcal{T}\exp (-i{\int }_0^{T}H(z)\mathrm{d}z)$ describes the evolution operator $U$ for one period $T$, and $\mathcal{T}$ is the time-ordering. In the high driving frequency regime, $H_{\mathrm{eff}}$ can be a perturbation expansion in powers of $1/T$ (see details in Appendix A).

Next, we analyze whether the edge states are topological in the effective model. Generally, topological invariants are used to characterize them. Notably, unlike the Hermitian case, for complex non-Hermitian systems in two dimensions, topological invariants can be calculated in real space. Detailed calculations can be found in Appendix B. To verify the experimental feasibility of the results, and to ensure that the model exhibits a large Chern number at a specific coupling strength, we consider the loss coefficient to be at the value of (0,1). Using this method, we compute the real-space topological invariants of the loss $\gamma \in (\mathrm{0,1})$ as a function of $g_1(g_3)$ and $g_2(g_4)$, resulting in the phase diagrams for gaps I/IV and II/III, as illustrated in Figs. 2(a) and 2(b), respectively. For coupling strengths $g_m\in (\mathrm{0,1.5})$, the system is gapless, making the calculation of the Chern number infeasible. As the coupling strength increases, topological phases with a Chern number $C=1$ emerge in gaps I and IV. However, with further increases in coupling strength, these phases eventually disappear. In gaps II and III, a larger nontrivial Chern number $C=2$ is observed, corresponding to two pairs of topological chiral edge states. The results for both gaps are consistent with the projected band structure.

The non-Hermitian properties of a kagome lattice under both $C_3$ symmetry and broken $C_3$ symmetry conditions are analyzed in Ref. [33], revealing corner modes induced by non-Hermitian skin effects arising from topological edge states. Furthermore, by analyzing the dynamic behavior and interactions of edge states at the corners, the study demonstrates how the skin effect localizes these states, accurately predicting hybrid topological-skin modes. This analytical approach is also applicable to our upcoming investigation of the non-Hermitian properties of the model under $C_4$ symmetry. However, unlike the previous research, which only analyzes a pair of edge states at the corners, we extend our study to multiple pairs of edge states at the corners, and similarly, accurately predict the emergence of hybrid topological-skin modes.

The model possesses $C_4$ symmetry and exhibits non-Hermitian properties, allowing easy engineering of the coupling strengths as $g_{\mathrm{1,2,3,4}}=0.5$, with the loss coefficient set to $\gamma =0.6$. Then, based on the phase diagrams (Fig. 2), one infers that two pairs of chiral edge states exist within band gaps II and III, as clearly evidenced in the real part of projected band structure depicted in Fig. 3(a). Moreover, the projected band structure exhibits as gapless, with edge states marked as dark purple and light purple in Fig. 3(a). This situation is not further analyzed in this paper. We use green (orange) to represent the eigenstates localized at the upper (lower) edge. This labeling facilitates the subsequent analysis of the hybrid skin-topological modes. The corresponding complex energy spectra for the non-Hermitian effective model are shown in Fig. 3(b), from top to bottom, corresponding to three different boundary conditions (xPBC/yPBC, xPBC/yOBC, and xOBC/yOBC). PBC (OBC) represents the periodic (open) boundary condition in the $x$ or $y$ direction. A schematic of the regular rectangular supercell with xOBC/yOBC is presented in Fig. 3(c). The cancellation of nonreciprocity arising from the symmetry of the local flux, along with a balance between gain and loss within the bulk, as explained in Ref. [30], leads to an almost unchanged complex energy spectrum of the bulk continuum. This invariance is illustrated in Fig. 3(b) under different boundary conditions.

Before specifically analyzing the hybrid skin-topological modes, we need to focus on the group velocity of the edge states and the imaginary part of the energy. The group velocity of the chiral edge states can be calculated from the specific wave vector $k$ in the projected band structure, i.e., $v=\frac{\partial \mathrm{Re}(E(k))}{\partial k}$. The sign of the group velocity indicates the direction in which the wave packet moves along the boundary. Furthermore, the magnitude of the imaginary part of the energy of the edge states causes an imbalance in the lifetime of the wave packet, meaning the wave packet will either amplify or decay as it propagates along the boundary.

Let us analyze a specific region, A. In Fig. 3(b), for the same real energy, the degenerate states with higher imaginary energy, marked in olive color, are composed of the upper edge state ${\psi }_{\text{green},\mathrm{I}}$ and the lower edge state ${\psi }_{\text{orange},\mathrm{I}}$, while the degenerate states with lower imaginary energy are composed of the upper edge state ${\psi }_{\text{green},\mathrm{II}}$ and the lower edge state ${\psi }_{\text{orange},\mathrm{II}}$. Meanwhile, the upper (lower) edge states marked in green (orange) have a negative (positive) group velocity, indicating that the wave packet propagates in the negative direction along that boundary, with the rightward direction along the lower boundary defined as the positive direction. Furthermore, applying OBC to the rectangular supercell, the upper and left boundaries correspond to the upper edge states, while the lower and right boundaries correspond to the lower edge states [see Fig. 3(c)]. It is worth noting that, when comparing the complex energy spectra of xPBC/yOBC and xOBC/yOBC in Fig. 3(b), at the same real energy, the imaginary energy in region A satisfies the following relation: $\mathrm{Im}(E_{\text{green/orange},\mathrm{II},\text{xPBC/yOBC}})<$. $\mathrm{Im}(E_{\text{red-up/-down,xOBC/yOBC}})<\mathrm{Im}(E_{\text{green/orange},I,\text{xPBC/yOBC}})$. Moreover, all the gap states marked in red are localized at the four corners of the system, as shown in Fig. 4.

Based on the above analysis, we can derive the directions of wave packet propagation on the four boundaries and their relative amplification or attenuation during propagation. As shown at the top of Fig. 4, the configurations of the hybrid skin-topological modes for regions A and B are displayed, where arrows indicate the direction of wave packet propagation, with blue (red) representing relative amplification (attenuation) along their respective directions. Further analyzing the configuration of the hybrid skin-topological modes with higher imaginary energy in region A, at the top of Fig. 4(a), we observe two types of wave packets exhibiting directional amplification or attenuation along the boundaries. For example, one dynamic type is as follows: the wave packet ${\psi }_{\text{green},\mathrm{I}}$ propagates leftward along the upper edge, has a longer lifetime, and eventually collapses to the upper-left corner over time. In contrast, the wave packet ${\psi }_{\text{green},\mathrm{II}}$ moves downward along the left edge, with a shorter lifetime and exponential decay. This dynamic type will localize at the corner, labeled as 2 (for ${\psi }_{\text{green},\mathrm{I}}$ and ${\psi }_{\text{green},\mathrm{II}}$), and this type of dynamics will eventually result in localized corner skin modes at all four corners, as shown by the glowing purple spheres in the top of Fig. 4(a). In contrast, another dynamic type involves the wave packet ${\psi }_{\text{green},\mathrm{II}}$ propagating leftward along the upper edge, with a shorter lifetime and faster decay. Meanwhile, ${\psi }_{\text{green},\mathrm{I}}$ moves downward along the left edge, has a longer lifetime, and eventually collapses to the lower-left corner during dynamic evolution. This shows that the interaction between these two wave packets at the upper-left corner does not lead to localization, so we do not consider this dynamic type.

A similar analysis has been conducted for the other corner skin modes, as shown at the top of Figs. 4(b)–4(d), where the corresponding corner skin mode configurations are displayed. It can be observed that the configurations are the same because the Chern number in gaps II and III is $+2$ for both. The bottoms of Figs. 4(a) and 4(b) [4(c) and 4(d)] show the hybrid topological skin modes with higher (lower) imaginary energy, respectively, where the height of the columns on the plot is proportional to the amplitude strength at each atomic site.

To demonstrate the feasibility of the system and to extend the discussion to a richer hybrid skin effect, we will investigate the non-Hermitian effective model with $C_2$ symmetry, where the coupling strengths are $g_{\mathrm{1,3}}=1$ and $g_{\mathrm{2,4}}=2$, and the dissipation coefficient is $\gamma =1$. As shown in Fig. 5(a), the real part of the projected band structure reveals that two pairs of edge states still exist in band gaps II and III, marked in the same way as before, while there is one pair of edge states in band gaps I and IV. Figure 5(b) corresponds to the complex energy spectra under three boundary conditions: xPBC/yPBC, xPBC/yOBC, and xOBC/yOBC, from the top to bottom. Notably, under the xPBC/yOBC and xOBC/yOBC boundary conditions, distinct crossing points are observed within both band gaps II and III. At these crossing points, the imaginary part of the energy reverses on either side. Consequently, these crossings divide band gap II into regions A and B, and band gap III into regions C and D. In regions A and D, the real energy is the same, and the imaginary energy of both the upper and lower supercell states lies between the chiral edge states. In regions B and C, only the imaginary energy of the upper supercell state is positioned between the chiral edge states, while the imaginary energy of the lower supercell state is located below the edge states. This slight shift in the edge states may be attributed to the $C_2$ symmetry of the structure. However, it does not affect the validity of our subsequent analysis. And this clear distinction in imaginary energy distribution across different regions aids in understanding the properties of the corner skin modes that will be discussed later.

More importantly, the dynamical behavior of chiral edge states is influenced not only by the imaginary energy and the associated group velocity but also by the weight of the chiral edge states in specific corner skin modes, denoted as $c_j$ ($j=\mathrm{1,2,3,4}$), where $j$ represents the $j$-th edge state. This factor must be considered, as in the coupling interaction $g_{\mathrm{1,3}}=1$ and $g_{\mathrm{2,4}}=2$, the Chern number $C=2$ emerges in gaps II and III, indicating the presence of two pairs of topological edge states rather than a single pair. Moreover, the $C_2$ symmetry of the model alters the three factors that affect the dynamical behavior of the chiral edge states, resulting in a more diverse set of hybrid skin-topological modes. Thus, we need to redefine the previous analysis and provide a detailed derivation (see details in Appendix C).

We redefine $P_j\equiv c_j\exp (E_j{t}_j)$ ($j=\mathrm{1,2,3,4}$), where the imaginary part of the energy $E_j$, time $t_j$, and coefficient $c_j$ are used to calculate the impact of wave functions during the dynamic process along different edges. In this definition, the imaginary part of $E_j$ represents the difference between the edge state and the imaginary energy of the corner skin mode (as reference energy). The time $t_j$ is determined by the group velocity of the edge state, representing the time required for a wave packet to propagate a unit lattice distance along the edge. Notably, for a wave packet with a longer lifetime, it will gradually collapse to the corner when interacting with another wave packet of a shorter lifetime, and the product operation of $P_j$ gives the magnitude of the wave function distribution at that corner. Assuming the initial wave function distribution at a corner is $P_j=c_j$, it will amplify or decay relative to the motion along their respective boundaries. By comparing the wave function distributions at different corners, we can assess the manifestation of the hybrid topological skin effect at each corner.

Based on the existing analytical methods, we also need to consider the group velocity of the chiral edge states, which is the same as in the case with $C_4$ symmetry considered previously. For convenience in subsequent analysis, the complex energy spectrum of xPBC/yOBC in Fig. 5(b) shows that, for the same real energy, the degenerate states (marked in olive) with higher imaginary energy in region A and lower imaginary energy in region B are composed of the upper edge state ${\psi }_{\text{green},\mathrm{I}}$ and the lower edge state ${\psi }_{\text{orange},\mathrm{I}}$, while the degenerate states with lower imaginary energy in region A and higher imaginary energy in region B are composed of the upper edge state ${\psi }_{\text{green},\mathrm{II}}$ and the lower edge state ${\psi }_{\text{orange},\mathrm{II}}$. The degenerate states in regions C and D are composed of edge states with opposite labels compared to those in regions A and B.

Figure 6 shows the corresponding configurations of corner skin modes in different regions. The configurations on the left are derived from the analysis on the right, reflecting the transition from $C_4$ symmetry to $C_2$ symmetry. We will now focus on the analysis of region A, with the corresponding configuration shown in Fig. 6(a). For instance, we choose the corner skin mode with higher imaginary energy [$\mathrm{Im}(E)=-0.8332$]. In this case, the configuration could either be Type 1 or Type 2. First, we consider the Type 1 configuration. Suppose the initial wave packet ${\psi }_{\text{green},\mathrm{I}}$ has a longer lifetime and propagates leftward along the upper edge. At this point, $P_{b3}=1.0616$, and after some time, the wave packet localizes at the upper-left corner, yielding $P_{b1}=c_{\text{green},\mathrm{I}}\exp ((E_{\text{green},\mathrm{I}}-E_{\text{xOBC/yOBC}})t_{\text{green},\mathrm{I}})=1.0801$. Simultaneously, the wave packet ${\psi }_{\text{green},\mathrm{II}}$, with a shorter lifetime, propagates downward along the left edge and decays, yielding $P_{b4}=P_{b1}\times P_{\text{green},\mathrm{II}}=1.0552$. Next, the wave packet ${\psi }_{\text{orange},\mathrm{I}}$, with a longer lifetime, moves rightward along the lower edge and eventually localizes at the lower-right corner, leading to $P_{b2}=P_{b1}\times P_{b4}=1.1402$. Clearly, the Type 1 configuration corresponds to the corner skin mode. Similarly, we can calculate the Type 2 configuration, and find that it also fits the corner skin mode. Finally, by comparing the $P$-values of the wave packets at each corner for both configurations [as shown at the bottom of Fig. 7(a)], we find that $P_{b1}$ and $P_{b2}$ for Type 1 are greater than $P_{b3}$ and $P_{b4}$ for Type 2. Therefore, the configuration in this mode is Type 1, as shown at the top of Fig. 7(a), with the corresponding corner skin mode at the bottom. The configuration analysis for other regions follows a similar procedure (for more details, see Fig. 6).

Through similar analysis, Figs. 7(a)–7(d) and 7(e)–7(h) show the corresponding configurations and corner skin effects for regions A, D, and B, C, respectively. The top of Fig. 7 presents the corresponding configuration diagrams, while the bottom displays the observed corner skin effect patterns. Notably, for the same real energy, the imaginary energy of the corner skin modes in Figs. 7(f) and 7(h) is lower than that of the edge states under xPBC/yOBC boundary conditions. In this case, we still need to refer to the imaginary energy of the supercell, but this does not affect the value of $P_j$. Similarly, we can calculate $P_{b1}$, $P_{b2}$, $P_{b3}$, and $P_{b4}$, and derive the corresponding configurations, as shown in the top of Figs. 7(f) and 7(h). Here, the introduced concept $P_j\equiv c_j\exp (E_j{t}_j)$ provides a more universally applicable method for analyzing multiple hybrid skin-topological modes, enriching the study of symmetry models.

We establish a non-Hermitian Chern insulator model based on checkerboard lattices. Despite its simplicity, the model encompasses rich non-Hermitian topological phases. Of special interest here are the non-Hermitian topological band gaps with large Chern numbers that support multiple chiral edge states. In finite systems, a corner boundary can trigger the scattering among these chiral edge states without causing back scattering, making the bulk-boundary correspondence more versatile. Interestingly, we find that the corner-induced scattering leads to a higher-order non-Hermitian skin effect where the waves are trapped at the corner boundaries. Depending on the system’s symmetry and parameters, the corner skin modes can appear at all the four corners or only at a pair of diagonal corners. This phenomenon is different from the previously found hybrid skin-topological effect where the system has only one branch of non-Hermitian chiral edge states and the corner skin effect is solely due to the gain and loss of the chiral edge states as determined often by the symmetry of the system. Instead, here, we reveal through the dynamical analysis that the corner-induced scattering plays an important role in the higher order non-Hermitian skin effect, which is unique for non-Hermitian topological phases with large Chern numbers.

We emphasize that the checkerboard lattice model proposed in this work can be realized in a number of systems. First, as studied here, a photonic Floquet lattice consisting of coupled optical waveguides can realize the model, which has been realized in experiments [35]. Second, in cold atom systems where the artificial gauge fields can be generated through the scheme of optical flux lattices [46], the checkerboard lattice model can also be realized if the dissipation is engineered as in a recent work [48].

We remark that this work not only adds to the fundamental research in non-Hermitian topological physics by enriching the mechanisms for non-Hermitian topological bulk-boundary correspondence, but also facilitates the development of non-Hermitian topological photonics, which may enable the realization of unconventional photonic materials and applications.

Acknowledgment. We are grateful to Dr. Weiwei Zhu for valuable discussions.