Non-Hermitian Kagome Lattice: Topology & Dynamics

• Non-Hermitian Kagome lattices are 2D structures with broken Hermiticity via nonreciprocal hopping, balanced gain/loss, and engineered synthetic flux.
• They exhibit unique phenomena such as higher-order skin effects, hybrid corner modes, and tunable flat bands across acoustic, photonic, and metamaterial platforms.
• Experimental and theoretical studies show that asymmetric couplings and point-gap topology induce directional energy localization and novel Chern phases.

The non-Hermitian Kagome lattice denotes a class of two-dimensional kagome-lattice systems—kagome lattices consist of corner-sharing triangles—in which Hermiticity is broken by mechanisms such as nonreciprocal hopping, balanced gain and loss, purely imaginary next-nearest-neighbor hopping, or nonlinear gain/loss terms. In recent work, these systems have been used to study the non-Hermitian skin effect (NHSE), higher-order NHSE, non-Hermitian Chern insulator phases, hybrid topological-skin corner modes, bipolar NHSE, second-order topological corner lasing, and real-energy flat bands in acoustic, photonic, optical, and metamaterial settings (Zhong et al., 2024, Wang et al., 2024, Yang et al., 19 Jan 2026, Ezawa, 2021, Zhang et al., 2019).

1. Model classes and defining ingredients

The cited literature studies several distinct kagome constructions rather than a single canonical Hamiltonian. One class is obtained from a Hermitian higher-order topological insulator (HOTI) through a Hermitian–non-Hermitian correspondence, producing a kagome lattice with nonreciprocal nearest-neighbor hoppings. A second class adds non-reciprocal, purely imaginary next-nearest-neighbor hoppings to a kagome lattice with Hermitian nearest-neighbor couplings, generating non-Hermitian Chern phases. A third class is a photonic kagome crystal with balanced gain and loss and gyromagnetic time-reversal-symmetry breaking. A fourth class is the breathing Kagome lattice with linear loss and nonlinear gain saturation, used for topological lasing. A fifth class engineers real-energy flat bands by combining balanced gain/loss with nonreciprocal coupling and synthetic magnetic flux (Zhong et al., 2024, Wang et al., 2024, Yang et al., 19 Jan 2026, Ezawa, 2021, Zhang et al., 2019).

Construction	Non-Hermitian ingredient	Principal phenomenon
Kagome child of a HOTI	Unidirectional nearest-neighbor hopping	Higher-order NHSE
Kagome NN–NNN model	Purely imaginary, non-reciprocal NNN hopping	Chern insulator phases and corner skin modes
Photonic kagome crystal	Balanced gain and loss	Bipolar NHSE and lifted corner-mode degeneracy
Breathing Kagome laser	Linear loss and nonlinear gain terms	Second-order topological corner lasing
Engineered optical kagome flat-band model	Balanced gain/loss and synthetic flux	Real, tunable flat band

This diversity is essential for interpreting the phrase “non-Hermitian Kagome lattice.” In the current literature, it refers not only to non-Hermitian perturbations of the standard kagome tight-binding model, but also to kagome realizations derived from higher-order topology, active photonic media, and flat-band engineering protocols.

2. Hermitian–non-Hermitian correspondence and higher-order skin physics

A central construction begins from a Hermitian HOTI protected by chiral symmetry and rotational crystalline symmetry, and maps it to a non-Hermitian kagome Hamiltonian by isolating the off-diagonal block in the chiral basis. The Hermitian parent Hamiltonian is written as

$$h(\mathbf{k})= \begin{pmatrix} 0 & q(\mathbf{k})\ q^\dagger(\mathbf{k}) & 0 \end{pmatrix},$$

with chiral symmetry

$\Gamma h(\mathbf{k})\Gamma^{-1}=-h(\mathbf{k}),$

and the non-Hermitian child Hamiltonian is identified with the block $q(\mathbf{k})$. In the specific realization discussed for the kagome lattice, the Hermitian parent is the breathing honeycomb, or Kekulé-distorted honeycomb, lattice, while the child is a kagome lattice with nonreciprocal nearest-neighbor hoppings in which all hoppings are unidirectional (Zhong et al., 2024).

Within this framework, the non-Hermitian kagome model inherits chiral and crystalline structure. The crystalline relation is expressed as

$r_3 q(\mathbf{k}) r_3^\dagger = q(R_3 \mathbf{k}),$

corresponding to a $C_3$ rotation operator, and the model also satisfies

$\Theta q(\mathbf{k}) \Theta^{-1} = q(-\mathbf{k})$

for spinless time reversal. The physical role of nonreciprocal hopping is explicit: energy transport between two sites occurs only in one direction, so the hopping matrix elements satisfy $t_{ij}\neq t_{ji}^*$, and the system is non-Hermitian.

The associated higher-order skin effect is second order in two dimensions. First-order NHSE localizes eigenstates on an edge; higher-order NHSE localizes skin modes on a boundary of codimension two, namely a corner. In the kagome case, the model is non-separable, unlike previous square- or cubic-lattice demonstrations, so the effect does not reduce to a tensor product of one-dimensional chains. In the topologically nontrivial phase, defined by $$\kappa \equiv \kappa_{\text{intra}}/\kappa_{\text{inter}} < 1$$, skin modes derived from edge bands with nonzero winding in the complex energy plane collapse into one corner of a finite sample, whereas the bulk states remain delocalized. This reproduces, in the non-Hermitian setting, the corner-localized zero modes of the parent HOTI.

A common misconception is that higher-order NHSE requires separability. The kagome construction shows the opposite: higher-order skin accumulation can arise in a non-separable Bloch Hamiltonian when non-Hermiticity and crystalline symmetry act together.

3. Non-Hermitian Chern phases and hybrid corner skin modes

A different line of work studies kagome lattices with nearest-neighbor and next-nearest-neighbor hoppings, where the non-Hermitian ingredient is purely imaginary, non-reciprocal NNN hopping. The Bloch Hamiltonian is decomposed as

$H = H_A + H_B,$

with Hermitian nearest-neighbor terms and non-Hermitian NNN terms. Time-reversal symmetry breaking through purely imaginary, non-reciprocal NNN hoppings opens band gaps and induces Chern insulator phases with real-space Chern number $C=\pm 1$, computed באמצעות the Kitaev formula in a biorthogonal framework. Both principal gaps host counter-propagating, chiral, topological edge states (Wang et al., 2024).

In this setting, corner skin modes are hybrid modes with both topological and skin-effect character. They appear in distinct energy regions within the two principal gaps. Each gap is split into two regions by crossings of complex-energy edge states, and corner skin modes occur between distinct chiral edge branches but vanish at the crossing point, where the imaginary parts of the edge-state energies are equal. With $\Gamma h(\mathbf{k})\Gamma^{-1}=-h(\mathbf{k}),$0 symmetry, second-order hybrid skin-topological modes localize at three out of six corners of a hexagonal finite kagome lattice. Without $\Gamma h(\mathbf{k})\Gamma^{-1}=-h(\mathbf{k}),$1 symmetry, reversing one NNN hopping makes edges and corners inequivalent, and the corner skin modes can localize at only one or two out of six corners, depending on boundary type and parameter choice.

The dynamical analysis provides the physical criterion for corner accumulation. The amplification or attenuation factor along an edge segment of length $\Gamma h(\mathbf{k})\Gamma^{-1}=-h(\mathbf{k}),$2 is

$\Gamma h(\mathbf{k})\Gamma^{-1}=-h(\mathbf{k}),$3

where $\Gamma h(\mathbf{k})\Gamma^{-1}=-h(\mathbf{k}),$4 is the difference in imaginary part between edge state and corner mode, and $\Gamma h(\mathbf{k})\Gamma^{-1}=-h(\mathbf{k}),$5 is the group velocity. Chiral edge states with gain transport probability in a preferred direction; if the termination geometry causes multiple edge flows to converge, a corner skin mode forms. This picture explains why the skin modes vanish at edge-state crossings and why the set of skin corners changes when $\Gamma h(\mathbf{k})\Gamma^{-1}=-h(\mathbf{k}),$6 symmetry is broken.

This dynamical account also addresses a second misconception: in these chiral kagome Chern phases, spectral winding numbers are not the only useful language, and in the formulation of the cited work they are ill-defined for the relevant edge dynamics. Corner localization can instead be predicted from group velocity and gain/loss bias on the edges.

4. Point-gap topology and the breakdown of conventional bulk–boundary correspondence

In photonic kagome crystals, non-Hermiticity is introduced through balanced gain and loss by setting the relative permittivity of selected dielectric cylinders to

$\Gamma h(\mathbf{k})\Gamma^{-1}=-h(\mathbf{k}),$7

while time-reversal symmetry breaking is produced by an external magnetic field along $\Gamma h(\mathbf{k})\Gamma^{-1}=-h(\mathbf{k}),$8 through a gyromagnetic permeability tensor. In this platform, non-Hermiticity lifts the degeneracy of topological corner modes and also drives bulk states to accumulate at corners, producing a bipolar non-Hermitian skin effect (Yang et al., 19 Jan 2026).

The bipolar NHSE means that bulk states localize at opposite corners of the finite system. In the reported configurations, this behavior is robust for both rhombic and triangular geometries. The corner-mode spectrum changes qualitatively when gain and loss are switched on: one obtains one real-frequency mode at one corner and a pair of complex-conjugate modes at another corner. Edge modes remain present, but the non-Hermitian skin effect imposes preferred localization directions.

The topological characterization shifts from conventional Bloch-band invariants to point-gap topology. The relevant winding number is

$\Gamma h(\mathbf{k})\Gamma^{-1}=-h(\mathbf{k}),$9

A nonzero $q(\mathbf{k})$0 signals the possibility of the skin effect upon opening boundaries, and the direction and magnitude of the winding number correspond to the direction and type of skin localization. By contrast, conventional bulk polarization and Wilson-loop-based invariants can fail in the non-Hermitian setting: the cited work states that trivial bulk polarization does not prevent the emergence of localized corner modes under open boundary conditions.

This is one of the clearest points of controversy and clarification in the subject. In Hermitian higher-order topology, Bloch-band bulk invariants are expected to predict boundary modes. In the non-Hermitian kagome setting, the spectral reality under periodic and open boundary conditions diverges, gap closing can occur at different points for PBC and OBC, and conventional bulk–boundary correspondence based on the Bloch Hamiltonian breaks down. The kagome platform therefore functions as a concrete setting in which point-gap topology and dynamical accumulation replace standard Hermitian intuition (Wang et al., 2024).

5. Experimental realizations in acoustic and photonic platforms

A direct experimental observation of higher-order NHSE has been reported in an acoustic kagome lattice composed of 27 3D-printed hexagonal prism-shaped acoustic cavities, each resonant at 1034.5 Hz, arranged in a two-dimensional kagome configuration with $q(\mathbf{k})$1. Unidirectional hopping is engineered by pairs of microphones and loudspeakers between cavities: the signal from a microphone in one cavity is phase-shifted and amplified, then sent to a loudspeaker in the neighboring cavity, thereby realizing a unidirectional, complex-valued coupling $q(\mathbf{k})$2. Digital phase control is implemented in real time using digital fractional delay filters, and spectrum rotation by $q(\mathbf{k})$3 moves all eigenfrequencies below the real axis for stability (Zhong et al., 2024).

Loss mitigation is crucial in this experiment. The method used is complex-frequency excitation, an exponentially decaying sinusoidal excitation that provides virtual gain and compensates unavoidable losses without destabilizing the system. Acoustic response is measured at each cavity, and the spatial-temporal evolution of the acoustic energy density is tracked for different source positions and excitation frequencies. In the topologically nontrivial phase, $q(\mathbf{k})$4, acoustic energy accumulates at a specific corner of the sample even when the source is located far from that corner, provided that the excitation frequency lies in the skin-mode band. In the topologically trivial phase, $q(\mathbf{k})$5, no corner localization appears and the energy spreads into the bulk. The reported localization behavior constitutes unequivocal evidence of higher-order NHSE.

The broader experimental program is not restricted to acoustics. The kagome Chern models with non-reciprocal, purely imaginary NNN hoppings are proposed as feasible in acoustic, optical, and mechanical metamaterials, with purely imaginary non-reciprocal hopping realizable through engineered gain/loss, synthetic dimensions, or effective gauge fields (Wang et al., 2024). The photonic kagome crystal with balanced gain and loss is simulated in COMSOL and is described as compatible with mature photonic crystal fabrication and selective gain/loss introduction, for example by optical pumping or doping, together with an applied magnetic field to realize gyromagnetism (Yang et al., 19 Jan 2026).

Taken together, these implementations show that the non-Hermitian kagome lattice is not merely a formal tight-binding motif. It is a cross-platform experimental architecture in which nonreciprocity, balanced gain/loss, and spectral engineering can be directly programmed.

6. Related directions: nonlinear corner lasing and flat-band engineering

The breathing Kagome lattice provides a second-order topological insulator geometry with three corner states and a natural entry point for nonlinear non-Hermitian dynamics. Its Bloch Hamiltonian is

$q(\mathbf{k})$6

with hopping amplitudes parameterized by

$q(\mathbf{k})$7

The cited phase structure is: $q(\mathbf{k})$8 topological with $q(\mathbf{k})$9, $r_3 q(\mathbf{k}) r_3^\dagger = q(R_3 \mathbf{k}),$0 trivial with $r_3 q(\mathbf{k}) r_3^\dagger = q(R_3 \mathbf{k}),$1, and $r_3 q(\mathbf{k}) r_3^\dagger = q(R_3 \mathbf{k}),$2 metallic. In the nonlinear non-Hermitian model,

$r_3 q(\mathbf{k}) r_3^\dagger = q(R_3 \mathbf{k}),$3

uniform loss suppresses bulk and edge excitations while nonlinear gain with saturation is applied at the corners. When one corner is initially excited, all three corner sites begin to emit stable laser light after a delay, although no wave propagation is observed from the stimulated site. In the isolated-corner limit, the lasing state satisfies $r_3 q(\mathbf{k}) r_3^\dagger = q(R_3 \mathbf{k}),$4 for $r_3 q(\mathbf{k}) r_3^\dagger = q(R_3 \mathbf{k}),$5 (Ezawa, 2021).

A separate direction concerns non-Hermitian flat-band engineering on kagome lattices. The construction proceeds in two steps. First, the flat-band eigenstate is made momentum-independent by eliminating the couplings $r_3 q(\mathbf{k}) r_3^\dagger = q(R_3 \mathbf{k}),$6 between sublattices $r_3 q(\mathbf{k}) r_3^\dagger = q(R_3 \mathbf{k}),$7 and $r_3 q(\mathbf{k}) r_3^\dagger = q(R_3 \mathbf{k}),$8. Second, balanced gain and loss $r_3 q(\mathbf{k}) r_3^\dagger = q(R_3 \mathbf{k}),$9 are introduced together with a nonreciprocal coupling $C_3$0 between $C_3$1 and $C_3$2. Under the matching condition

$C_3$3

the flat-band energy is real and tunable:

$C_3$4

The compact localized state has amplitudes $C_3$5 on the $C_3$6, $C_3$7, and $C_3$8 sublattices, with the $C_3$9 site unoccupied, and the Bloch eigenstate is momentum-independent, $\Theta q(\mathbf{k}) \Theta^{-1} = q(-\mathbf{k})$0, or the analogous triplet for the kagome geometry. The cited work emphasizes that, in the original Hermitian kagome model, the flat band is the lowest energy band with all couplings positive and has nonzero flat-band energy; the non-Hermitian extension makes that flat-band energy tunable and real (Zhang et al., 2019).

These developments suggest a broader unification. The non-Hermitian kagome lattice is not only a host for skin effects and non-Hermitian Chern phases; it is also a setting in which higher-order topology, gain/loss engineering, nonlinearity, and flat-band compact localization can be co-designed within the same geometric motif.