Non-Hermitian Skyrmions in Complex Systems

• Non-Hermitian skyrmions are topological structures defined in complex, non-Hermitian systems that exhibit charge splitting, exceptional points, and modified state-space invariants.
• They manifest across multiple platforms—including magnetic textures, momentum-space pseudospin fields, dynamic quench systems, and scalar optical microcavities—with each setting featuring distinct invariants.
• Advanced analytical and numerical methods are employed to reveal how gain/loss, biorthogonality, and nonlocal damping affect topological protection and spectral winding.

Searching arXiv for papers on non-Hermitian skyrmions and closely related formulations. In recent arXiv literature, the term non-Hermitian skyrmion denotes several distinct constructions in which a skyrmionic map or skyrmion number is defined for a non-Hermitian generator, open-wave system, driven–dissipative field, or complexified Skyrme theory. The phrase is used for a real-space magnetic skyrmion with balanced gain and loss or a PT-symmetric anisotropy, for momentum-space pseudo-spin textures in leaky photonic bands, for dynamical skyrmions on generalized-Brillouin-zone–time manifolds, for scalar-field textures in optical microcavities, and for complex BPS Skyrmion configurations with real energies. Across these settings, the common problem is not merely whether a topological charge exists, but which state space it is built from, whether it remains quantized, and how non-Hermitian singularities, biorthogonality, damping, or antilinear symmetry modify its protection (Liu, 26 Jun 2026, Bouteyre et al., 2022, Li et al., 2021, Wingenbach et al., 1 Jun 2026, Koyama et al., 8 May 2026, Correa et al., 2021).

1. Real-space magnetic formulation

The most direct use of the term concerns a magnetic $CP^1$ texture $z$ with balanced gain/loss along the easy axis, introduced by the anti-Hermitian term $i\gamma \sigma_z$. The canonical PT-symmetric deformation is

$i\,\partial_t z = (H_0 + i\gamma\,\sigma_z)\,z ,$

and, because this evolution does not preserve the norm of $z$, the physical degree of freedom is the normalized spinor $\hat z=z/|z|$. Writing the projector $P=\hat z\hat z^\dagger$ and $\Gamma=\gamma\sigma_z$, the trace-preserving projective evolution is

$$\dot P = \{\Gamma,P\} - 2\,\mathrm{Tr}(\Gamma P)\,P .$$

To probe the left–right structure spatially, the local non-Hermitian generator is taken pointwise as

$$H(\mathbf{r}) = \mathbf{d}(\mathbf{r})\cdot\boldsymbol{\sigma} + i\gamma\,\sigma_z ,$$

where $z$0 encodes the Hermitian texture field and $z$1 implements PT-symmetric balanced gain/loss or anisotropy (Liu, 26 Jun 2026).

In the Hermitian limit $z$2, the magnetization is a unit Bloch vector $z$3. A standard axisymmetric skyrmion is parameterized by

$z$4

with $z$5 and $z$6. The equator ring is the locus $z$7 where $z$8, equivalently $z$9. Its Hermitian topological charge is the Pontryagin index

$i\gamma \sigma_z$0

The non-Hermitian construction begins from this standard skyrmion but alters the state space in which the charge is defined.

2. Charge splitting, biorthogonality, and exceptional-point breakdown

A central result of the magnetic non-Hermitian skyrmion construction is that the Hermitian skyrmion charge splits into two inequivalent quantities. The first is the right-state Bloch vector $i\gamma \sigma_z$1, with right-state charge

$i\gamma \sigma_z$2

The second is the biorthogonal Bloch vector

$i\gamma \sigma_z$3

which is generally complex and satisfies $i\gamma \sigma_z$4, so that it lives on the complex quadric $i\gamma \sigma_z$5. The corresponding biorthogonal charge is

$i\gamma \sigma_z$6

These two charges coincide in the Hermitian limit and part ways for $i\gamma \sigma_z$7: $i\gamma \sigma_z$8 stays integer, while $i\gamma \sigma_z$9 is complex and nonquantized once gain/loss is turned on (Liu, 26 Jun 2026).

The right-state sector remains homotopy protected because the PT flow reduces exactly to a Gilbert-type relaxation on the target sphere. In the paper’s formulation, the PT deformation viewed through the right state is indistinguishable from a relaxation of the texture toward the $i\,\partial_t z = (H_0 + i\gamma\,\sigma_z)\,z ,$0 axis, and the right Bloch vector remains a smooth map into $i\,\partial_t z = (H_0 + i\gamma\,\sigma_z)\,z ,$1. Under smooth evolution, $i\,\partial_t z = (H_0 + i\gamma\,\sigma_z)\,z ,$2 therefore cannot change. The only stated way it could change is a true singularity such as $i\,\partial_t z = (H_0 + i\gamma\,\sigma_z)\,z ,$3, where the projective class becomes undefined.

The biorthogonal sector behaves differently because its target manifold is noncompact and because the local generator can reach an exceptional point. The squared eigenvalues of the local generator are

$i\,\partial_t z = (H_0 + i\gamma\,\sigma_z)\,z ,$4

and the exceptional-point condition is

$i\,\partial_t z = (H_0 + i\gamma\,\sigma_z)\,z ,$5

On a skyrmion, $i\,\partial_t z = (H_0 + i\gamma\,\sigma_z)\,z ,$6 occurs on the equator ring. For a hedgehog of constant modulus $i\,\partial_t z = (H_0 + i\gamma\,\sigma_z)\,z ,$7, the exceptional-point criterion becomes $i\,\partial_t z = (H_0 + i\gamma\,\sigma_z)\,z ,$8. At that ring the left and right eigenvectors coalesce, the phase rigidity

$i\,\partial_t z = (H_0 + i\gamma\,\sigma_z)\,z ,$9

tends to zero, and the biorthogonal Bloch field diverges. In the equatorial two-level caricature,

$z$0

yields

$z$1

so $z$2 diverges as $z$3. The paper identifies this loss of quantization as the real-space topological counterpart of the analyticity breakdown that a causal response function suffers at an exceptional point. In this sense, topological protection is no longer a single statement in the non-Hermitian magnetic problem: it splits into a right-state homotopy invariant and a biorthogonal quantity that becomes complex, nonquantized, and singular at the exceptional point.

3. Momentum-space and dynamical skyrmions

A second usage of the term appears in open photonic band structures. In a 1D optical lattice with a symmetry-protected bound state in the continuum at $z$4, the familiar polarization-vortex winding number of the dark branch is $z$5 and flips when a symmetry-preserving band inversion changes the sign of $z$6. To obtain a conserved invariant, the band identity is incorporated into a momentum-space pseudo-spin

$z$7

The non-Hermitian skyrmion number is then

$z$8

with $z$9. The resulting texture is a half-skyrmion: the paper describes a transition from an antimeron to a meron-like texture through band inversion, while the half-charge skyrmion number remains conserved even though the BIC polarization-vortex winding does not (Bouteyre et al., 2022).

A third usage appears in non-Bloch quench dynamics with non-Hermitian skin effects. There the relevant manifold is not real space or momentum space alone, but the generalized Brillouin zone times time. Writing $\hat z=z/|z|$0 on the generalized Brillouin zone, the dynamical pseudospin is defined biorthogonally with a metric operator $\hat z=z/|z|$1, and one constructs a real unit vector $\hat z=z/|z|$2. The dynamic skyrmion charge is

$\hat z=z/|z|$3

or, on submanifolds between fixed points $\hat z=z/|z|$4 and $\hat z=z/|z|$5,

$\hat z=z/|z|$6

When the generalized Brillouin zones of the pre- and post-quench Hamiltonians coincide and both spectra are real, fixed points occur at $\hat z=z/|z|$7. If the non-Bloch winding numbers differ, $\hat z=z/|z|$8, the fixed points are of different types and dynamic skyrmions with $\hat z=z/|z|$9 appear; when $P=\hat z\hat z^\dagger$0, the dynamic Chern numbers vanish. Even when the generalized Brillouin zones do not coincide, global signatures persist through the time-averaged polarization $P=\hat z\hat z^\dagger$1, so the skyrmion construction remains a detection scheme for non-Bloch topology in the presence of skin effects (Li et al., 2021).

4. Scalar-field skyrmions in driven–dissipative microcavities

In optical microcavities the term is extended to a scalar complex field rather than a spin texture. The coherent exciton–polariton condensate is described by a single complex order parameter $P=\hat z\hat z^\dagger$2, and the skyrmionic texture is built from amplitude and phase gradients of that scalar field. For a stationary state $P=\hat z\hat z^\dagger$3, the construction introduces

$P=\hat z\hat z^\dagger$4

and then the unit vector

$P=\hat z\hat z^\dagger$5

Over a single Skyrmion cell, the charge is

$P=\hat z\hat z^\dagger$6

with the cell boundary chosen by critical lines of $P=\hat z\hat z^\dagger$7. In this formulation the Skyrmion charge is a bulk area integral mapping $P=\hat z\hat z^\dagger$8 and is explicitly distinct from vortex winding $P=\hat z\hat z^\dagger$9 (Wingenbach et al., 1 Jun 2026).

The driven–dissipative dynamics is governed by the open-dissipative Gross–Pitaevskii equation with an incoherent reservoir,

$\Gamma=\gamma\sigma_z$0

$\Gamma=\gamma\sigma_z$1

For nonresonant pumping, the continuity equation

$\Gamma=\gamma\sigma_z$2

shows that spatially inhomogeneous gain/loss requires a compensating current divergence in steady state. Near threshold, a Gaussian pump reduces the problem to a non-Hermitian Schrödinger equation with a complex harmonic frequency $\Gamma=\gamma\sigma_z$3, so the threshold mode necessarily carries nonzero phase curvature through $\Gamma=\gamma\sigma_z$4. That phase curvature produces radial sign reversals in $\Gamma=\gamma\sigma_z$5 and nodal rings in $\Gamma=\gamma\sigma_z$6, yielding an isolated Skyrmion of charge $\Gamma=\gamma\sigma_z$7 inside the first Skyrmion cell. The paper emphasizes that nonlinear interactions are not required for existence: Skyrmions persist even with $\Gamma=\gamma\sigma_z$8, and the essential ingredients are localized gain and finite lifetime.

This scalar non-Hermitian skyrmion framework supports isolated Skyrmions, self-organized Skyrmion lattices, and Skyrmion bags in moiré lattices. Resonant interference can generate hexagonal lattices with $\Gamma=\gamma\sigma_z$9 per unit cell, whereas structured incoherent pumping can induce spontaneous phase synchronization and nonresonant Skyrmion bags on a $$\dot P = \{\Gamma,P\} - 2\,\mathrm{Tr}(\Gamma P)\,P .$$0 domain. The same platform provides nonlinear all-optical control: an additional nonresonant ring pump changes the propagation phase through the reservoir-induced blueshift and switches the central Skyrmion number between distinct integers. The derivative field $$\dot P = \{\Gamma,P\} - 2\,\mathrm{Tr}(\Gamma P)\,P .$$1 also determines two force densities, $$\dot P = \{\Gamma,P\} - 2\,\mathrm{Tr}(\Gamma P)\,P .$$2 and $$\dot P = \{\Gamma,P\} - 2\,\mathrm{Tr}(\Gamma P)\,P .$$3, so isolated scalar Skyrmions generate inward intensity-gradient forces and outward phase-gradient forces.

5. Damped magnons and non-Hermitian topology on skyrmion strings

A different but related direction studies a skyrmion texture as the background that generates non-Hermitian topology for its excitations. In a three-dimensional chiral magnet hosting a skyrmion-string lattice, the static texture is a periodic array of skyrmion tubes extended along $$\dot P = \{\Gamma,P\} - 2\,\mathrm{Tr}(\Gamma P)\,P .$$4. The spin dynamics is described by a generalized Landau–Lifshitz–Gilbert equation with a real symmetric positive-semidefinite damping kernel $$\dot P = \{\Gamma,P\} - 2\,\mathrm{Tr}(\Gamma P)\,P .$$5, allowing both local Gilbert damping and nonlocal damping between different sites. Linearization about the stationary texture produces a bosonic Bogoliubov–de Gennes problem for the Nambu spinor $$\dot P = \{\Gamma,P\} - 2\,\mathrm{Tr}(\Gamma P)\,P .$$6,

$$\dot P = \{\Gamma,P\} - 2\,\mathrm{Tr}(\Gamma P)\,P .$$7

with complex magnon frequencies under damping (Koyama et al., 8 May 2026).

To leading order in small damping,

$$\dot P = \{\Gamma,P\} - 2\,\mathrm{Tr}(\Gamma P)\,P .$$8

where $$\dot P = \{\Gamma,P\} - 2\,\mathrm{Tr}(\Gamma P)\,P .$$9 quantifies particle–hole mixing and elliptical precession. The point-gap winding number is

$$H(\mathbf{r}) = \mathbf{d}(\mathbf{r})\cdot\boldsymbol{\sigma} + i\gamma\,\sigma_z ,$$0

A nontrivial $$H(\mathbf{r}) = \mathbf{d}(\mathbf{r})\cdot\boldsymbol{\sigma} + i\gamma\,\sigma_z ,$$1-dependence of $$H(\mathbf{r}) = \mathbf{d}(\mathbf{r})\cdot\boldsymbol{\sigma} + i\gamma\,\sigma_z ,$$2 produces loops in the complex spectrum and enables the non-Hermitian skin effect. The paper stresses that this can occur even without nonlocal damping: with purely local damping $$H(\mathbf{r}) = \mathbf{d}(\mathbf{r})\cdot\boldsymbol{\sigma} + i\gamma\,\sigma_z ,$$3, the texture-induced $$H(\mathbf{r}) = \mathbf{d}(\mathbf{r})\cdot\boldsymbol{\sigma} + i\gamma\,\sigma_z ,$$4-dependence of $$H(\mathbf{r}) = \mathbf{d}(\mathbf{r})\cdot\boldsymbol{\sigma} + i\gamma\,\sigma_z ,$$5 alone can give $$H(\mathbf{r}) = \mathbf{d}(\mathbf{r})\cdot\boldsymbol{\sigma} + i\gamma\,\sigma_z ,$$6. With nearest-neighbor nonlocal damping along one direction,

$$H(\mathbf{r}) = \mathbf{d}(\mathbf{r})\cdot\boldsymbol{\sigma} + i\gamma\,\sigma_z ,$$7

and weak variation of $$H(\mathbf{r}) = \mathbf{d}(\mathbf{r})\cdot\boldsymbol{\sigma} + i\gamma\,\sigma_z ,$$8, the sign rule becomes

$$H(\mathbf{r}) = \mathbf{d}(\mathbf{r})\cdot\boldsymbol{\sigma} + i\gamma\,\sigma_z ,$$9

for a stable particle band with minimum at $z$00. Under open boundary conditions, $z$01 and $z$02 correspond to top-boundary and bottom-boundary skin localization, respectively.

In the skyrmion-string lattice used in the demonstrations, the counterclockwise gyration mode shows strong $z$03-dependence of $z$04 and, with local damping only, yields a single prominent spectral loop with $z$05. With nonlocal damping, the localization side becomes band selective: at $z$06, low-energy bands with $z$07 localize at the top boundary, whereas at $z$08 the seventh band has $z$09 and localizes at the bottom boundary. The resulting directional propagation after a magnetic-field pulse is a dynamical signature of point-gap topology. This framework sharply separates two notions of topology: the skyrmion number of the static texture and the spectral winding number of damped magnons. The paper also states explicitly that exceptional points are not required for this point-gap topology or for the non-Hermitian skin effect in the perturbative regime considered.

6. Complex BPS Skyrmions and conceptual distinctions

In Skyrme-type field theory, non-Hermitian skyrmions appear as complex-valued $z$10-based or complexified field configurations whose energies remain real because of an antilinear CPT symmetry. The general setting uses

$z$11

with Skyrme-type Lagrangian

$z$12

topological charge

$z$13

and static energy $z$14. In the BPS sector, the Hamiltonian density can be written as $z$15, so the Bogomolny equation $z$16 reduces the problem to first-order radial equations. Non-Hermiticity is introduced by complex field shifts, complex potentials, or complex couplings, while energy reality follows when a modified antilinear CPT symmetry maps the Hamiltonian functional to its parity time-reversed complex conjugate and maps a solution either to itself or to a degenerate-energy partner (Correa et al., 2021).

The paper develops several concrete families. In the complex-boosted BPS model, $z$17 with $z$18 yields a non-Hermitian Hamiltonian density that remains pseudo-Hermitian and is related by a Dyson map to a Hermitian counterpart. This sector contains complex compactons, step solutions, cusp solutions, shell solutions, and purely imaginary compactons, all with real energies. Other constructions include semi-kink solutions, massless shell solutions with $z$19, Bender–Boettcher-type potentials $z$20 with real-energy sectors on discrete sets of $z$21, and a solvable submodel with purely imaginary coupling $z$22 and real energy

$z$23

In this field-theoretic usage, the skyrmion is non-Hermitian because the fields, couplings, or potentials are complex, not because of gain/loss or an exceptional spectrum.

Taken together, these formulations show that non-Hermitian skyrmion is a genuinely plural concept. It does not always mean a real-space magnetic spin texture, and its “charge” is not universally the integer Pontryagin index of a Hermitian map. Depending on the setting, the relevant invariant may remain integer and homotopy protected, become complex and nonquantized, stay fixed as a half-charge meron across band inversion, appear as a dynamic Chern number on generalized-Brillouin-zone–time space, or arise from a scalar derivative field rather than a spin field. Likewise, exceptional points are central to the breakdown of the biorthogonal magnetic charge, but they are not the organizing mechanism in every non-Hermitian skyrmion construction. Current literature therefore treats the subject less as a single topological object than as a family of skyrmionic structures whose definition is inseparable from the non-Hermitian state space in which they are embedded (Liu, 26 Jun 2026, Bouteyre et al., 2022, Li et al., 2021, Wingenbach et al., 1 Jun 2026, Koyama et al., 8 May 2026).