Hybrid Skin-Topological Effect

• Hybrid skin–topological effect is a boundary-localization phenomenon combining the non-Hermitian skin effect with higher-order topological modes to yield corner or hinge states.
• It is realized in models such as the 2D non-Hermitian SSH lattice where tuning of nonreciprocal hoppings and topological invariants enables precise control over edge and corner localization.
• The mechanism breaks conventional bulk–boundary correspondence, offering versatile applications in photonics, electronics, and metamaterials for robust state engineering.

The hybrid skin-topological effect (HSTE) is a boundary-localization phenomenon in non-Hermitian tight-binding systems where the non-Hermitian skin effect and higher-order topological modes interplay, giving rise to sharply localized corner or hinge states. In HSTE, the system’s bulk remains extended and free of conventional skin effect, but topological edge or surface states, protected by a bulk crystalline/topological invariant, themselves become exponentially localized at lower-dimensional boundaries due to the non-Hermitian pumping mechanism. HSTE distinctly generalizes both the traditional higher-order topological insulator (HOTI) physics and the non-Hermitian skin effect (NHSE), establishing a new paradigm for boundary phenomena in non-Hermitian condensed matter, photonics, electronics, and acoustics.

1. Hamiltonians and Minimal Models Realizing HSTE

The prototypical HSTE is realized in the 2D non-Hermitian Su–Schrieffer–Heeger (SSH) lattice with nonreciprocal (directional) hoppings. The Bloch Hamiltonian for a square lattice with a four-site basis is

$$H(\mathbf k) = \sum_{k} \vec c^\dagger(k) h(k) \vec c(k)$$

with $c(k) = (c_1, c_2, c_3, c_4)^T$ and

$$h(k) = \begin{pmatrix} q_{2\times 2}(-k_y, k_y) & q_{2\times 2}(k_x, k_x) \ q_{2\times 2}(-k_x, -k_x) & q_{2\times 2}(k_y, -k_y) \end{pmatrix}$$

where the %%%%1%%%% block is

$$q_{2\times2}(k_1, k_2) = \begin{pmatrix} 0 & t_1 - \gamma_1 + (t_2 - \gamma_2) e^{ik_1} \ t_1 + \gamma_1 + (t_2 + \gamma_2) e^{ik_2} & 0 \end{pmatrix}$$

Here, $t_1, t_2$ are intracell/intercell hoppings, and $\gamma_1, \gamma_2$ are their respective nonreciprocal (non-Hermitian) parts. Parameterizations like $$t_1 = 1 + \delta t,\, t_2 = 1 - \delta t,\, \gamma_1 = 1/2 + \delta\gamma,\, \gamma_2 = 1/2 - \delta\gamma$$ allow for control of both Hermitian and non-Hermitian regimes (Wakao, 2023).

Variants are realized in checkerboard Chern-insulator lattices with non-reciprocal couplings, non-Hermitian Haldane and SSH models with sublattice gain/loss, and quasicrystals implementing multi-site cells with nonreciprocal intracell hoppings (Zhang et al., 2024, Zhu et al., 2022, Chen et al., 2024). Floquet HSTE phases are realized in time-periodically modulated photonic honeycomb lattices with structured loss (Sun et al., 2023).

2. Topological Invariants Governing Hybrid Skin–Topological Phases

HSTE requires the coexistence of two complementary topological invariants:

• Crystalline/Higher-order Topological Invariant: For the 2D non-Hermitian SSH lattice, the real-line gap topology is characterized by the $\mathbb{Z}_4$ Berry phase, protected by a generalized fourfold rotational symmetry:

$$U_4 h(\mathbf k) U_4^{-1} = h^\dagger(R_4\mathbf k)$$

with $R_4: (k_x, k_y) \mapsto (-k_y, k_x)$. The Berry phase is computed biorthogonally as

$$\gamma_{\mathbb{Z}_4} = \mathrm{Re}\,\nu\in\{0,1/4,1/2,3/4\}$$

Only $\gamma_{\mathbb{Z}_4}=0$ and $1/2$ are realized in this model, and the value dictates the orientation of edge states (bottom/right for $0$, top/left for $1/2$).

• Point-gap Topology (Skin Effect Invariant): Edge bands under open boundary conditions in one direction may exhibit a point gap. The winding number around a reference energy $E_\mathrm{ref}$,

$$w = \frac{1}{2\pi i}\int_{-\pi}^{\pi} dk\, \partial_k \log \det [h_{1D}(k) - E_\mathrm{ref} \, I]$$

encodes the spectral winding in the complex plane. A nonzero $w$ indicates a non-Hermitian skin effect for that edge-state subspace (Wakao, 2023).

Therefore, the position and nature of corner-localized HSTE modes is uniquely dictated by the pair $(\gamma_{\mathbb{Z}_4}, w)$, implementing a bulk–boundary–boundary correspondence.

3. Boundary Localization and the Mechanism of HSTE

The HSTE emerges through a two-step bulk-to-boundary localization:

1. First-order (edge) topological localization: The crystalline $\mathbb{Z}_4$ Berry phase demands gapless edge states localized to specific edges, depending on the value of $\gamma_{\mathbb{Z}_4}$.
2. Second-order (skin) localization: When the edge subspace develops a nonzero point-gap winding $w$, the edge states themselves are pumped unidirectionally toward a corner—the unique intersection of those edges prescribed by the topological crystalline invariant.

Explicitly, for $(\gamma_{\mathbb{Z}_4}, w) = (1/2,-1)$, edge states reside on top/left and pile up at the upper-left corner. For $(0,+1)$, they reside on bottom/right and accumulate at the lower-right corner.

Under adiabatic variation of parameters, these corner skin modes are robust so long as neither the real-line gap nor the edge point gap closes (Wakao, 2023, Zou et al., 2021, Fu et al., 2020). In Floquet systems, the mechanism is analogous, with the Chern number and the Floquet spectral winding number pairing to localize edge-band states at a designated corner (Sun et al., 2023).

4. Phase Diagrams and Parameter Control

The HSTE manifests within precisely delineated regions of parameter space. For the 2D non-Hermitian SSH model, the $(\delta t, \delta\gamma)$ plane displays domains separated by real-line-gap-closing lines $|\delta\gamma| = |\delta t|$:

• $|\delta\gamma| < |\delta t|$: $\gamma_{\mathbb{Z}_4}=0$
• $|\delta\gamma| > |\delta t|$: $\gamma_{\mathbb{Z}_4}=1/2$

Within each domain, the sign and magnitude of $w$ for relevant edge bands further selects the localization site of HSTE corner modes (Wakao, 2023). In checkerboard Chern-insulator models, the emergence and relocation of HSTE corner states are governed by competing gain/loss, synthetic fluxes, and chiral edge-band lifetimes (Zhang et al., 2024).

In multi-cell quasicrystals, arbitrary control over the pattern of corner localization is enabled via the tuning of intracell nonreciprocity vectors and the shape/orientation of the outer boundary (Chen et al., 2024).

5. Breakdown of Conventional Bulk–Boundary Correspondence

In HSTE systems, the standard bulk–boundary correspondence—in which the bulk topological invariant dictates the presence of boundary states—breaks down in a distinctive manner. Here, only the boundary (or edge) Hamiltonian develops a nontrivial spectral winding, while the extended bulk remains free of point-gap topology. The topological edge bands thus act as an effective 1D non-Hermitian system, acquiring a nonzero point-gap winding that instantiates the skin effect for these edge states alone. The bulk bands remain extended under OBC; only the edge states become exponentially localized at corners (Wakao, 2023, Zhu et al., 2023, Fu et al., 2020).

6. Physical Intuition: Nonreciprocity Meets Higher-Order Topology

Nonreciprocal hopping, implemented via asymmetric couplings or engineered gain/loss, tilts the complex energy spectrum in a non-Hermitian lattice, creating conditions for the skin effect along open boundaries. In combination with higher-order topological band structure, the interplay causes a selective skin effect restricted to topologically protected boundary modes.

In the adiabatic connection picture, the value of the crystalline invariant ($\mathbb{Z}_4$ Berry phase in the SSH case) can be traced to limiting dimerized configurations in which the edge-state positions are explicit; non-Hermiticity then pumps those edge modes to unique corners. The mechanism is robust against disorder and insensitive to boundary geometry, provided the topological and skin invariants do not vanish (Wakao, 2023, Sun et al., 2023, Zhu et al., 2023, Chen et al., 2024).

7. Experimental Observations and Generalizations

The HSTE has been demonstrated in a range of physical platforms:

• Topolectrical circuits: Realization of HSTE in 2D and 3D nonreciprocal circuit lattices, with voltage mappings directly visualizing skin-corner and skin-hinge modes (Zou et al., 2021).
• Photonic Floquet lattices: Time-modulated optical waveguide arrays show Floquet HSTE with reconfigurable corner light funneling; the corner positions are tunable by spectral winding and topological switches are activated by gap closings (Sun et al., 2023).
• Acoustic metamaterials: Higher-order (hinge) skin–topological modes can be dynamically steered by gain/loss patterns, enabling multidimensional sound routing and robust energy harvest at hinges (Fang et al., 9 May 2025).
• Quasicrystals: Hybrid skin–topological effect and its localization pattern can be controlled in 2D Ammann–Beenker tilings with nonreciprocal 8-site cells (Chen et al., 2024).
• Checkerboard and Chern lattices: The interplay of large Chern number, multi-channel chiral edge states, and non-Hermitian effects produces a diversity of HSTE regimes (Zhang et al., 2024).

8. Impact of Fragile Topology and Suppression of HSTE

Recent evidence demonstrates that fragile topology—characterized by gapped Wannier bands—can suppress the HSTE even in the presence of robust edge modes. In a bilayer breathing honeycomb lattice with spiral interlayer couplings, increasing the Wannier gap (by tuning the interlayer coupling $\lambda$) monotonically suppresses the spectral difference between open and periodic boundary conditions, thereby “turning off” the skin–topological conversion. The HSTE is maximized when the Wannier bands are gapless (strongly nontrivial topology) and vanishes as the Wannier gap widens, providing a quantitative lever for controlling the effect in real systems (Liu, 1 Jan 2026).

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Key References:

Main Physical Setting	Reference	Key Features
2D SSH model, $\mathbb{Z}_4$ Berry phase	(Wakao, 2023)	Bulk–boundary–boundary correspondence, adiabatic connection
Floquet photonic topological system	(Sun et al., 2023)	Corner funneling, phase diagram, topological switch
Topolectrical circuit experiments	(Zou et al., 2021)	First experimental demonstration, hierarchy of NH topological effects
Fragile topology and HSTE suppression	(Liu, 1 Jan 2026)	Wannier gap controls HSTE amplitude in bilayer honeycomb
Checkerboard Chern lattice (large $C$)	(Zhang et al., 2024)	Multi-channel/degenerate edge-state induced HSTE
Quasicrystal with eight-site cell	(Chen et al., 2024)	Complete localization control via nonreciprocal intracell hopping

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The hybrid skin–topological effect synthesizes higher-order topology and non-Hermitian spectral flow into a robust localization mechanism, enriching the taxonomy of non-Hermitian phases, and establishing a versatile theoretical and experimental platform with broad applicability in future topological and open-system device engineering.