Non-Hermitian Skin Pumping: Mechanisms & Applications

• Non-Hermitian skin pumping is the mechanism driving the directional accumulation of eigenstates at system boundaries through asymmetric hopping, gain/loss, and complex momentum deformation.
• It encompasses multiple regimes—selective, hybrid, higher-order, and relative effects—each defined by distinct topological and spatial localization characteristics.
• Experimental and dynamical studies validate these phenomena, with applications spanning photonics, metamaterials, and bosonic quantum systems.

Non-Hermitian skin pumping denotes the directional accumulation of eigenmodes, wave packets, or steady-state excitations at system boundaries caused by non-Hermitian spectral structure, most commonly through non-reciprocal couplings, asymmetric gain/loss, or effective non-Hermitian terms generated in auxiliary descriptions. In its prototypical form, it is the pumping mechanism underlying the non-Hermitian skin effect (NHSE), where a macroscopic number of states condense at an edge and the conventional Bloch description fails; in more structured settings it can act selectively on topological boundary modes, higher-codimension boundary modes, or specific dynamical sectors (Lee et al., 2019, Zou et al., 2021). The topic now spans static spectral topology, wave-packet dynamics, adiabatic pumping, parametric bosonic systems, and experimental platforms ranging from topolectrical circuits and mechanical metamaterials to laser arrays, cold atoms, and driven quantum circuits.

1. Foundational mechanism

The basic mechanism of skin pumping is non-reciprocal spectral bias under open boundary conditions. In non-Hermitian lattices, asymmetric hopping or asymmetry in gain/loss can drive a macroscopic number of eigenstates toward one boundary, producing the NHSE and invalidating the usual bulk-boundary correspondence formulated in terms of Bloch waves (Li et al., 2022). In this sense, “pumping” is not an externally imposed transport cycle but an intrinsic directional flow encoded in the complex spectrum and boundary-sensitive eigenstates.

A complementary viewpoint is provided by the general framework of non-Hermitian pumping developed through complex momentum deformation. There, open-boundary spectra are described by a quasi-reciprocal surrogate Hamiltonian

$\bar H(k)=H(k+i\kappa(k)),$

with the deformation $\kappa(k)$ encoding the net effect of non-Hermitian pumping (Lee et al., 2019). This formulation places Hermitian and non-Hermitian lattice Hamiltonians on equal footing, while emphasizing that the physically relevant bulk data for open systems are not given by the original Bloch Hamiltonian alone.

Several works also show that skin pumping need not originate from explicitly asymmetric physical couplings. In bosonic BdG systems, reciprocal Hermitian Hamiltonians with parametric terms can generate an auxiliary non-Hermitian, nonreciprocal single-particle problem after BdG mapping, and this auxiliary problem can host the NHSE (Wang et al., 2022). In the continuum, Hermitian, nonlocal parametric pairing combined with uniform local dissipation produces exceptional points, a tilted diabolical line, and open-boundary skin modes, again without relying on conventional asymmetric hopping in the microscopic Hamiltonian (Bestler et al., 5 May 2025). A plausible implication is that skin pumping is best understood as a property of the effective spectral problem rather than of a single microscopic implementation.

2. Selective, hybrid, and higher-order forms

Non-Hermitian skin pumping is not restricted to the conventional case in which all bulk states accumulate at the same boundary. The literature now distinguishes several regimes in which pumping is selective with respect to boundary topology, mode type, band sector, or energy.

In topolectrical circuits, the hybrid higher-order skin-topological effect realizes a particularly sharp form of selectivity: the skin effect acts only on pre-existing topological boundary modes, not on bulk modes (Zou et al., 2021). In the 2D “ST” configuration, local non-reciprocity persists among sublattices while the net non-reciprocity across the lattice cancels, so only topological edge or corner modes are further pumped into corners. In 3D, the corresponding “STT” effect combines topological localization in some directions with non-reciprocal pumping in another, producing corner modes with no Hermitian nor lower-dimensional analog.

Higher-order skin effects generalize this idea further. In 2D systems of size $L\times L$, conventional first-order NHSE gives $\mathcal{O}(L^2)$ skin modes, whereas the second-order skin effect gives $\mathcal{O}(L)$ corner skin modes; in 3D, the third-order effect yields $\mathcal{O}(L)$ corner skin modes out of $\mathcal{O}(L^3)$ total modes (Kawabata et al., 2020). These regimes are tied to intrinsic non-Hermitian topology protected by spatial symmetry and require a modification of non-Bloch band theory in higher dimensions.

A different notion of selectivity appears in constructions of non-Hermitian Hamiltonians with real spectra of the form $H=H_0A$. For diagonal $A$, only the zero mode becomes a skin mode, while the non-zero modes remain bulk-like; this is termed the selective non-Hermitian skin effect (Ge, 2024). In that setting, the zero-mode profile is

$\psi_{0,j}=\psi_{0,j}^{(0)}s^{-(j-1)},$

and the selectivity is tied to the special relation between the zero mode and the spatially varying effective couplings.

A broader real-space classification distinguishes global skin effects, in which all or most modes accumulate at the same edge, from relative skin effects, in which different bands or channels accumulate at different boundaries depending on the sign structure of their velocities (Wei et al., 15 May 2025). This framework makes explicit that “skin pumping” can refer either to a common directional drift of the full spectrum or to mode-dependent, mutually opposed accumulations.

Regime	Defining feature	Representative source
Hybrid ST/STT	Only topological boundary modes are skin-pumped	(Zou et al., 2021)
Higher-order NHSE	Corner or hinge skin modes scale subextensively	(Kawabata et al., 2020)
Selective NHSE	Only the zero mode is a skin mode	(Ge, 2024)
Relative skin effect	Different mode sectors localize at opposite boundaries	(Wei et al., 15 May 2025)

Energy dependence adds yet another layer. With nonreciprocity beyond nearest-neighbor hopping, the direction of NHSE can reverse as the open-boundary eigenenergy crosses critical “skin effect edges,” which are determined by self-intersections of the periodic-boundary spectrum and by changes in spectral winding number (Zeng, 2022). Thus, skin pumping need not be globally fixed even within a single bandstructure.

3. Dynamical and adiabatic pumping

The time-domain counterpart of the NHSE is the dynamic skin effect. For a Gaussian wave packet in a non-Hermitian system under open boundary conditions, the time-evolved state can be written as

$\kappa(k)$0

so the exponential skin factor biases the spread-out Hermitian packet toward the skin boundary (Li et al., 2022). The packet peak moves as

$\kappa(k)$1

which implies acceleration toward the boundary even when the initial velocity is zero. This produces amplification or attenuation, non-conserved norm, and inelastic boundary reflection.

Dynamic NHSE has now been observed experimentally. In tunable one-dimensional nonreciprocal double-chain mechanical systems with glide-time symmetry, wave packets launched in the bulk were seen to migrate toward one boundary, with distinct conserved and unconserved dynamic phases (Li et al., 2023). In the conserved regime, energy localizes at an edge while the spectrum remains real; in the unconserved regime, boundary trapping is accompanied by amplification or damping depending on the imaginary parts of the dominant modes. The generalized Brillouin zone (GBZ) organizes these phases and makes the time-resolved boundary accumulation a direct dynamical realization of skin pumping.

Adiabatic pumping in non-Hermitian systems introduces a separate issue: quantization of transported charge. In generic non-Hermitian settings, quantized charge pumping is guaranteed only in a biorthogonal formalism based on left and right eigenvectors (Zhang et al., 2024). The relevant Berry curvature is

$\kappa(k)$2

and when the NHSE is present the pumped charge is controlled not by an ordinary Bloch Chern number but by a non-Bloch Chern number defined on the GBZ-time manifold. This result is important because a common misconception is that right-eigenvector transport suffices; the non-Hermitian case is generically biorthogonal.

The non-Hermitian Rice-Mele model provides a concrete setting in which adiabatic pumping, skin effect, exceptional points, and system-size dependence coexist (Kumar et al., 2021). There, the non-Hermitian parameter $\kappa(k)$3 and the system size $\kappa(k)$4 act as independent tuning knobs, and even a trivial adiabatic protocol can produce pumping with no Hermitian counterpart. The supplied analysis attributes this to the combined action of finite-size GBZ physics, point-gap topology, and skin-induced edge-state transport.

4. Quantum, continuum, and steady-state realizations

In quantum bosonic systems, skin pumping can be realized through parametric processes. Arrays of parametrically driven nonlinear resonators generate an auxiliary non-Hermitian BdG problem whose open-boundary eigenstates are skin modes (Wang et al., 2022). In a one-dimensional chain, the output mean photon number scales exponentially with chain length, and in a two-dimensional lattice the corner skin effect enables directional corner-to-corner photon amplification, with amplification by up to a factor of 30 when ports are placed at skin-mode–hosting corners. The physical couplings remain reciprocal; the directional pumping emerges in the BdG description.

A continuum realization based on Hermitian, nonlocal parametric pairing extends these ideas beyond tight-binding lattices (Bestler et al., 5 May 2025). The dynamical matrix

$\kappa(k)$5

develops exceptional points and a tilted diabolical line, while uniform local damping stabilizes the system without removing its non-Hermitian spectral character. Opening the boundaries yields right- or left-localized skin modes depending on whether $\kappa(k)$6 lies above or below $\kappa(k)$7, and a compactified spectral winding number restores a robust bulk-boundary correspondence in the continuum. This suggests that skin pumping is not intrinsically a lattice effect.

A distinct steady-state scenario appears in bosonic topological edge states under parametric driving. There, the non-Hermiticity is the intrinsic non-Hermiticity of the bosonic BdG Hamiltonian,

$\kappa(k)$8

rather than physical dissipation, and non-equilibrium Green’s functions reveal an open-boundary steady state with pronounced corner particle accumulation (Okuma, 2 Feb 2026). The steady state shows quadrature anisotropy and corner-localized pileup, linking skin pumping to genuinely quantum observables rather than only to classical mode profiles.

5. Analytical frameworks and classification

The modern theory of skin pumping is organized by several complementary frameworks. Non-Bloch band theory and the GBZ remain central for translation-invariant systems with OBC sensitivity (Zhang et al., 2024). The real-space unified theory of the generalized NHSE, however, emphasizes that many skin phenomena are more naturally characterized in terms of velocities and spatially varying non-Hermitian terms:

$\kappa(k)$9

where the sign structure of $L\times L$0 separates global from relative skin effects (Wei et al., 15 May 2025).

Systems without translational invariance require further extensions. The imaginary-Stark skin effect arises in a one-dimensional lattice with spatially increasing loss and is explicitly beyond the framework of non-Bloch band theory (Lin et al., 2024). Its spectrum is T-shaped, approximately half of the eigenstates localize at the left boundary, and each skin mode is dominated in the bulk by a single stable exponentially decaying wave rather than the two-exponential interference familiar from conventional NHSE. The transfer-matrix method becomes the appropriate tool in this regime.

For spatially inhomogeneous NHSE hoppings, phase-space GBZ constructions extend the notion of the GBZ to position-dependent systems and reveal GBZ bifurcation, scaling-dependent GBZs, and phase-space topological protection (Li, 17 Feb 2025). That framework also supports scaling-induced non-Hermitian exceptional criticality and unusual entanglement scaling. A plausible implication is that “skin pumping” in inhomogeneous systems is governed not by a single complex momentum contour but by a coupled position-momentum structure.

Another classification, developed from algebraic geometry, associates non-Hermitian pumping with graph topology of spectral branching singularities (Lee et al., 2019). In that view, conformally invariant branching structures classify open-boundary spectral phases, and gap-preserving transitions between them generate emergent band geometry and Berry-curvature discontinuities. This perspective is especially useful when multiple non-reciprocal length scales are present.

6. Experimental signatures, pumping principles, and applications

Experimental work has established that skin pumping is measurable through spatially resolved voltages, mechanical wave profiles, lasing thresholds, and center-of-mass trajectories. In topolectrical circuits, 2D and 3D lattices built from capacitors, inductors, and negative impedance converters with current inversion (INICs) realize non-reciprocal pumping, and direct voltage mapping reveals corner localization on specific sublattices in the hybrid ST and STT regimes (Zou et al., 2021). The ability to switch between hybrid skin-topological and pure skin-skin configurations is controlled by the orientation of the INICs.

In laser arrays, localized excitation exposes a nontrivial pumping principle. For non-Hermitian gauged arrays, pumping the “head” or “tail” of a skin mode can give the same lasing threshold, contrary to the conventional intuition based only on modal intensity overlap (Ge et al., 2023). The decisive overlap is instead the tripartite product of the pump, the lasing mode, and its biorthogonal partner,

$L\times L$1

and the threshold is further shaped by energy exchange at non-Hermitian coupling junctions with the photonic environment. This revises the principle of selective pumping in non-Hermitian photonics.

Cold-atom proposals reveal a different observable: abrupt kinks in semiclassical wave-packet trajectories caused by discontinuities in non-Hermitian band geometry (Qin et al., 2023). In a two-dimensional optical lattice with laser-induced loss, the center-of-mass response develops prominent kinks when the momentum trajectory crosses Berry-curvature discontinuities associated with complex-momentum deformation. These are dynamic bulk signatures of non-Hermitian pumping beyond static boundary localization.

Across platforms, the supplied literature identifies several application directions: topological switching and sensing in topolectrical circuits, directional photon amplification in driven bosonic lattices, unidirectional wave guiding and boundary trapping in mechanical systems, and threshold engineering in laser arrays (Zou et al., 2021, Wang et al., 2022, Li et al., 2023, Ge et al., 2023). A recurring theme is that skin pumping enhances boundary selectivity and response without being tied to a single microscopic mechanism. The field’s present technical emphasis is therefore not merely on observing boundary accumulation, but on controlling which modes are pumped, where they accumulate, whether the effect is static or dynamical, and which generalized topological invariant governs the phenomenon.