Hybrid Skin-Topological States

• Hybrid skin-topological states are non-Hermitian boundary states that merge topological edge/surface modes with the non-Hermitian skin effect to yield localized corner, hinge, or defect states.
• They are characterized by layered diagnostic methods combining line-gap (topological) and point-gap (nonreciprocal) invariants, including specific winding numbers and Berry phases.
• Experimental realizations in transmission-line networks, acoustics, and topolectrical circuits illustrate robust, higher-dimensional nonreciprocal behavior and selective boundary localization.

Hybrid skin-topological states are non-Hermitian boundary states whose localization results simultaneously from band topology and the non-Hermitian skin effect. In the standard formulation, topology first creates edge or surface modes, and non-Hermitian pumping then localizes those modes further along the boundary, producing corner, hinge, or defect-bound states. In two and three dimensions this yields a boundary-of-boundary response that is distinct from both ordinary topological boundary states and ordinary bulk skin modes, and it can occur even when the bulk remains extended (Lee et al., 2018, Lin et al., 2023).

1. Definition, scope, and classification

Hybrid skin-topological states occupy an intermediate category between three better-known classes of states. Ordinary topological boundary states are localized because of band topology alone; they are typically extended along the boundary on which they live. Ordinary skin modes arise from nonreciprocal or non-Hermitian pumping and usually involve a macroscopic fraction of bulk states. Conventional higher-order topological corner or hinge states are localized by higher-order bulk topology rather than by boundary spectral pumping. Hybrid skin-topological states instead combine these mechanisms: a state is first localized to an edge or surface by topology and then further accumulated by a skin effect into a corner, hinge, or defect (Lin et al., 2023).

This distinction becomes especially sharp in higher dimensions. In one dimension, topology and NHSE mainly compete because both act along the same spatial direction. In two and three dimensions they can act in different directions, so topology can localize a mode onto a codimension-1 boundary while skin accumulation acts within that boundary. This is why the original higher-dimensional classification introduced mixed channels such as $ST$ in two dimensions and $ST0$, $STT$, and $SST$ in three dimensions, where $S$ denotes skin localization, $T$ topological localization, and $0$ no localization in a given direction (Lee et al., 2018).

A central consequence is that hybridization need not imply a bulk skin effect. In the foundational nonreciprocal constructions, bulk states remain extended because opposite nonreciprocities cancel within each unit cell, while topological boundary states experience a “spontaneous breaking of reciprocity” because their sublattice polarization prevents the cancellation from operating in the boundary subspace (Lee et al., 2018). This selective pumping mechanism remains a recurring theme across later Chern-insulator, kagome, quasicrystal, and defect-based realizations.

Hybrid skin-topological states are also not identical to all higher-order NHSE phenomena. The review literature explicitly treats the hybrid skin-topological effect as a special class of higher-order NHSE in which boundary-of-boundary localization is additionally tied to conventional topological boundary states, whereas later work shows that higher-order skin accumulation can exist without conventional topological protection (Lin et al., 2023).

2. Mechanisms and diagnostics

The general non-Hermitian framework distinguishes line-gap topology, which stabilizes topological boundary states, from point-gap topology, which diagnoses NHSE. In one-dimensional non-Hermitian band theory the point-gap winding number is

$$w(E_r)=\int_0^{2\pi}\frac{dk}{2\pi i}\frac{d}{dk}\log\det[H(k)-E_r],$$

and non-Bloch bulk-boundary correspondence replaces the ordinary Brillouin zone by a generalized Brillouin zone defined by equal-modulus roots of the characteristic equation. In the broad hybrid setting, the usual logic is layered: topology establishes the relevant boundary sector, and spectral winding or effective nonreciprocal pumping in that sector determines whether a skin effect acts on it (Lin et al., 2023).

One explicit realization of this layered diagnosis appears in the two-dimensional nonreciprocal SSH model. There the bulk real-line-gap topology is characterized by a generalized-$C_4$-protected $\mathbb{Z}_4$ Berry phase $ST0$0, while the edge point-gap topology is characterized by a winding number $ST0$1 of the cylinder spectrum. The corner position of the hybrid mode is not fixed by either invariant alone but by the pair $ST0$2: $ST0$3 selects the left-top corner, $ST0$4 the right-bottom corner, $ST0$5 the right-top corner, and $ST0$6 the left-bottom corner (Wakao, 2023).

A second diagnostic route uses auxiliary Hermitian Hamiltonians. For the static gain/loss Haldane model,

$ST0$7

and for Floquet systems the corresponding construction is made directly from the Floquet operator $ST0$8. In this formulation, hybrid skin-topological modes in the original non-Hermitian system are diagnosed by zero-energy corner states of the auxiliary Hermitian system. The static auxiliary model can belong to either an intrinsic or an extrinsic second-order topological insulator phase, and the same construction extends to anomalous Floquet phases whose edge topology is not captured by band Chern numbers (Zhu et al., 2022).

A distinct mechanism arises in non-Hermitian Chern systems with gain/loss rather than explicit asymmetric hopping. The “non-Hermitian chiral skin effect” recasts topological edge states as chiral one-dimensional modes on a closed boundary loop with inhomogeneous dissipation. For a chiral mode with Hamiltonian

$ST0$9

the periodic-boundary-condition solution is

$STT$0

so the spectrum depends only on the average dissipation $STT$1, بينما the wavefunction depends on

$STT$2

Localization then occurs at “global dissipation’s domain walls” where $STT$3 changes sign; the defect type selected depends on the chirality sign $STT$4. In this framework the relevant spectrum is line-like rather than loop-like, so the standard point-gap winding diagnosis of conventional NHSE does not apply (Ma et al., 2023).

3. Principal model families

The earliest explicit higher-dimensional constructions used nonreciprocal four-band lattices in which topological boundary modes feel effective pumping even when the bulk has vanishing net reciprocity. In two dimensions this produces $STT$5 corner modes, and in three dimensions it generalizes to $STT$6, $STT$7, and $STT$8 boundary classes. These models established the now-standard picture that hybrid skin-topological modes can scale with system length rather than system area because they originate from skin accumulation of a lower-dimensional topological boundary band (Lee et al., 2018).

Chern-insulator realizations broadened the phenomenon beyond explicit asymmetric hopping. In non-Hermitian Haldane systems with staggered gain/loss, all topological edge states can localize at one corner under OBC while the bulk remains extended, defining a second-order NHSE without asymmetric couplings. The same gain/loss mechanism extends to anomalous Floquet topological systems, where hybrid skin-topological modes survive even though the relevant chiral edge states are not characterized by static band Chern numbers (Zhu et al., 2022).

Kagome-lattice Chern insulators provide a complementary setting with multiple gaps and tunable symmetry. In the $STT$9-symmetric non-Hermitian kagome model with pure-imaginary non-reciprocal NNN hoppings, both gap I and gap II are topological, with real-space Chern numbers $SST$0. The bulk continuum changes negligibly between PBC and OBC, so first-order bulk NHSE is nearly absent, yet the chiral edge states develop a second-order, edge-originated NHSE that produces corner skin modes. Breaking $SST$1 by reversing one of the six pure-imaginary NNN hoppings preserves the Chern phases but yields richer localization patterns, including one-corner and two-corner hybrid states (Wang et al., 2024).

Large-Chern-number systems introduce a multi-channel version of the effect. In the Floquet non-Hermitian checkerboard lattice, gap I carries $SST$2, gap II carries $SST$3, gap III carries $SST$4, and gap IV carries $SST$5. The key mechanism is no longer a single chiral boundary channel but corner-induced scattering among multiple chiral edge channels with different group velocities and imaginary energies. Under full OBC this produces corner-localized states derived from the topological edge sector, with patterns controlled by whether the lattice retains $SST$6 or only $SST$7 symmetry (Zhang et al., 2024).

Exciton-polariton honeycomb lattices extend hybrid skin-topological physics into spinful bosonic platforms. With sublattice-dependent gain and loss, non-Hermiticity acts selectively on pre-existing topological edge modes rather than on the bulk spectrum. Two regimes are identified: hybrid skin-Chern states, whose localization direction switches when the TE-TM splitting drives the Hermitian parent from $SST$8 to $SST$9, and hybrid skin-antichiral states, which preserve the spin-polarized property of the parent antichiral edge modes (Bao et al., 1 Dec 2025).

4. Geometry, symmetry, and defect selectivity

Boundary geometry is not an ancillary detail; it is often the variable that decides whether topological edge states remain edge-extended or collapse into corners. In gain/loss Haldane systems, zigzag edges acquire nonzero effective dissipation while armchair edges can have zero effective dissipation because of sublattice cancellation. In rectangular OBC geometries this produces a multi-defect boundary profile: chiral edge states remain extended on armchair segments but become directionally localized on zigzag segments, so the apparent corner localization is a boundary manifestation of non-Hermitian chiral skin physics rather than a conventional higher-order corner phase (Ma et al., 2023).

The same sensitivity appears in quasicrystals. In the Ammann–Beenker construction with eight-site cells and nonreciprocal intracell hopping, the real-space Chern numbers are approximately $S$0 and $S$1 for one parameter set, yet whether a topological edge state becomes a hybrid corner state depends strongly on the outer boundary. Regular octagonal, kite, and parallelogram boundaries support corner localization verified by zero-energy corner states of the auxiliary Hamiltonian, whereas the square boundary can leave the corresponding topological state as an ordinary edge state. The eight-site cell further permits four-corner or two-corner localization patterns and motivates an effective-dissipation domain-wall interpretation of the selected corners (Chen et al., 2024).

Negative curvature introduces another geometric route. In the non-Hermitian Haldane model on hyperbolic $S$2 and $S$3 lattices, topological edge states inside the bulk gap are diagnosed by a biorthogonal real-space Chern number, while suitable open hyperbolic truncations generate an effective net boundary flux in the mapped zigzag chain. The resulting skin-topological modes localize at four selected corners in the $S$4 case and six corners in the $S$5 case, disappearing when a smoother boundary removes the flux imbalance (Sun et al., 2023).

Defect topology extends the phenomenon beyond sample boundaries. In the non-Hermitian $S$6-symmetric 2D SSH disclination model, skin-topological disclination states are governed by the combined invariant

$S$7

where $S$8 is the biorthogonal polarization and $S$9 the real-space disclination index. For $T$0, one obtains $T$1 and a pair of complex-conjugate disclination modes plus a purely real disclination mode; for $T$2, $T$3 and no skin-topological disclination state appears (Li et al., 6 Jan 2026).

Antichiral systems reveal a different hybridization channel. In the modified Haldane zigzag ribbon with gain/loss applied only at the zigzag edges, antichiral edge states on opposite boundaries propagate in the same direction, so the relevant competition is between an edge state and counter-propagating bulk states rather than between opposite chiral edge channels. The effective-gain/loss imbalance

$T$4

controls the localization direction. In narrow ribbons, gain or loss applied to only one edge can induce a nonlocal antichiral skin effect on states localized at the opposite edge, while bulk skin modes remain forbidden in the $T$5-unbroken regime $T$6 and appear only after $T$7 breaking (Ito et al., 2 Mar 2026).

Fragile topology supplies an additional suppression mechanism. In the bilayer breathing honeycomb lattice with spiral intracell interlayer couplings and a gain-loss domain wall, increasing the interlayer coupling $T$8 monotonically increases the Wannier gap $T$9, while the spectral difference between PBC and OBC decreases overall and some corner states revert to edge states. The reported trend follows $0$0 more closely than the ordinary bulk gap, which the paper presents as evidence that fragile-topological Wannier structure suppresses the hybrid skin-topological effect (Liu, 1 Jan 2026).

5. Experimental realizations and observables

A direct experimental observation was achieved in a transmission-line network by using synthetic complex-frequency excitation to access modes with genuinely complex eigenfrequencies. In that experiment, a lower-left HSTE corner state at

$0$1

was reconstructed from real-frequency measurements through the synthesized response

$0$2

The same device also displayed asymmetric anti-diagonal transmission inside the topological band gap and separated genuine HSTE corner states from corner states generated by non-chiral edge bands with boundary-induced onsite imbalance (Jiang et al., 2024).

Three-dimensional acoustics offers a higher-order analogue. In the stacked sonic crystal implementing a non-Hermitian acoustic Haldane/Weyl structure, zigzag surfaces support gain or loss surface states once on-site gain and loss are introduced through

$0$3

At the representative point $0$4, $0$5, and excitation frequency $0$6, adjacent gain and loss zigzag boundaries produce hinge skin states at their intersections. Reversing $0$7 relocates the hinge states to a complementary set of hinges, and adding internal holes allows simultaneous inner and outer hinge localization (Fang et al., 9 May 2025).

Topolectrical circuits have also realized defect-based hybrid states. In the nonreciprocal $0$8-symmetric disclination circuit, the Hamiltonian is encoded in the circuit Laplacian, and complex eigenfrequencies are probed by adding a compensating conductance $0$9. The measured resonances include a complex skin-topological disclination mode at $$w(E_r)=\int_0^{2\pi}\frac{dk}{2\pi i}\frac{d}{dk}\log\det[H(k)-E_r],$$0 and a purely real disclination mode at $$w(E_r)=\int_0^{2\pi}\frac{dk}{2\pi i}\frac{d}{dk}\log\det[H(k)-E_r],$$1, with impedance profiles concentrated near the disclination core (Li et al., 6 Jan 2026).

Beyond direct observations, several platforms are identified as experimentally favorable. Non-Hermitian kagome models are proposed for acoustics, photonics, and mechanical systems (Wang et al., 2024); checkerboard large-Chern-number lattices are designed for integrated silicon photonic nanocircuits, microresonators, and helical optical waveguides (Zhang et al., 2024); and exciton-polariton honeycomb lattices are proposed as optical-frequency platforms in which sublattice-dependent gain and loss, TE-TM splitting, and polarization-resolved imaging can reveal hybrid skin-Chern and hybrid skin-antichiral states (Bao et al., 1 Dec 2025).

6. Open questions and current directions

A persistent conceptual issue has been how to distinguish a genuinely new hybrid mechanism from a conventional bulk NHSE acting indirectly on boundary modes. The non-Hermitian chiral skin effect addresses this directly for gain/loss Chern insulators by attributing localization to chiral boundary loops with inhomogeneous dissipation under periodic boundary conditions, not to bulk point-gap winding. In that framework the relevant spectrum is line-like rather than loop-like, and localization is controlled by real-space dissipation defects rather than by standard point-gap topology (Ma et al., 2023).

Several settings still lack a fully general bulk-boundary theory. The kagome large-gap constructions and the checkerboard large-Chern-number models demonstrate modified non-Hermitian bulk-boundary responses numerically and dynamically, but do not develop a full non-Bloch bulk-boundary correspondence for their multi-channel edge sectors (Wang et al., 2024, Zhang et al., 2024). A plausible implication is that large-Chern-number hybrid skin-topological phases will require boundary theories that track channel mixing, corner scattering, and imaginary-energy ordering simultaneously.

Boundary-selective non-Hermiticity also complicates taxonomy. The transmission-line-network experiment shows that corner-localized modes can arise not only from chiral topological edge states but also from non-chiral edge states created by unbalanced effective boundary onsite energies. This sharpens the need to separate genuine hybrid skin-topological states from more general boundary-induced corner skin states (Jiang et al., 2024).

Some phenomena remain explicitly unresolved. The non-Hermitian Haldane analysis reports a “non-local NHCSE” in narrow strips, where both edges localize near the same dissipation defect, and leaves its mechanism for future study (Ma et al., 2023). The fragile-topology study presents evidence, rather than a complete theory, that increasing the Wannier gap suppresses HSTE (Liu, 1 Jan 2026). In quasicrystals, the auxiliary-Hamiltonian and real-space-Chern approach works effectively, but a more complete real-space non-Bloch framework remains open (Chen et al., 2024).

Across these directions, the common conclusion is stable: hybrid skin-topological states are not a single model-specific anomaly but a broad non-Hermitian boundary phenomenon in which topology creates the relevant low-dimensional channel and non-Hermiticity reshapes its spatial profile into a higher-codimension state. The precise mechanism, however, depends on whether the channel is chiral, antichiral, multi-channel, defect-bound, symmetry-constrained, or boundary-induced, and current work increasingly treats those possibilities as distinct subclasses rather than as one universal effect.