Topologically Protected Surface Plasma Waves

• Topologically Protected Surface Plasma Waves (TSPWs) are chiral edge modes in magnetized plasmas defined by nontrivial bulk topology rather than interface geometry.
• Multiple formulations—including electromagnetic, fluid, and operator approaches—demonstrate TSPWs through regularized Hamiltonians and phase-space topological invariants.
• Applications span plasma propulsion, nonreciprocal waveguides, and localized plasma heating, with experimental setups validated in gaseous plasmas and plasma crystals.

Searching arXiv for recent and foundational papers on topologically protected surface plasma waves in magnetized plasmas. Topologically Protected Surface Plasma Waves (TSPWs) are chiral edge modes of plasma media whose existence is enforced by bulk topological structure rather than by the geometric details of an interface. In magnetized plasmas, the static bias breaks time-reversal symmetry, opens nontrivial bulk gaps, and—through bulk–edge correspondence—produces a unidirectional surface branch without a counterpropagating partner in the same gap. The term encompasses several distinct but related realizations: the gaseous plasmon polariton at a plasma–vacuum boundary, the Topological Langmuir–Cyclotron Wave (TLCW) at a density-transition layer satisfying a local Langmuir–cyclotron resonance, chiral edge states of magnetized plasma crystals, and low-frequency ion-dominated edge waves that appear when ion motion is retained (Parker et al., 2019, Qin et al., 2022, Qian et al., 2023, Rajawat et al., 2022).

1. Emergence of the concept

The modern formulation of TSPWs in plasmas began with the observation that a homogeneous magnetized plasma can support one-way helical electromagnetic surface states at a plasma interface without recourse to photonic-crystal microstructuring. In that setting, the essential ingredient was the magnetic-field-induced separation of equi-frequency contours, which created a nontrivial gap in $k$-space and a single surface branch for fixed conserved $k_z$ (Yang et al., 2014).

The subsequent development of the field replaced the initial equi-frequency-contour argument with explicit bulk topological invariants and more realistic plasma profiles. The gaseous plasmon polariton (GPP) was identified as a topological interfacial state of a cold magnetized plasma adjacent to vacuum, with a graded density falloff rather than an ideal discontinuity. That work also introduced a regularized Hermitian fluid–Maxwell Hamiltonian and showed that the lowest gap has a nonzero gap Chern number, so the surface mode is required by bulk–edge correspondence even for experimentally realistic density ramps (Parker et al., 2019).

A second line of development established that not all plasma TSPWs are captured by ordinary momentum-space Chern theory. The TLCW was derived as a surface excitation generated by the phase-space topology of the Langmuir–cyclotron resonance in an inhomogeneous plasma, and its propagation along complex interfaces was demonstrated numerically in 2D and 3D. In parallel, periodic plasma-cylinder crystals were shown to host Chern-insulator edge states in dispersive and lossy gaseous plasmas, and a two-fluid treatment revealed a low-frequency ion-dominated family of topological edge waves absent in electron-only models (Fu et al., 2022, Qian et al., 2023, Rajawat et al., 2022).

A recurrent misconception is that any one-way plasma surface wave is therefore topological. The literature distinguishes ordinary nonreciprocal surface plasmon polaritons from edge modes whose dispersion traverses a bulk topological gap and whose existence is tied to a bulk invariant. This distinction becomes especially important in continuous media, where the topology may reside either in a regularized momentum-space formulation or, for the TLCW, in phase space rather than in momentum space alone (Parker et al., 2019, Qin et al., 2022).

2. Electromagnetic, fluid, and operator formulations

The canonical electromagnetic description uses a cold magnetized plasma with a static bias $B_0=B_0\hat z$. In the single-electron, immobile-ion model, the dielectric tensor is gyrotropic and may be written in Stix form as

$$\hat{\epsilon}(\omega)= \begin{pmatrix} S & -iD & 0\ iD & S & 0\ 0 & 0 & P \end{pmatrix},$$

with

$$S=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad D=\frac{\omega_{ce}}{\omega}\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad P=1-\frac{\omega_{pe}^2}{\omega^2}.$$

Equivalent circular-basis expressions are often used in dispersive and lossy calculations: $$\epsilon_\pm(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma\mp\omega_c)}\right],\qquad \epsilon_z(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma)}\right].$$ These forms underlie both continuous-interface studies and periodic plasma-crystal calculations (Rajawat et al., 1 Aug 2025, Qian et al., 2023).

When ion motion is included, the relative permittivity retains the same gyrotropic tensor structure but the Stix scalars become sums over species,

$$S(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad D(\omega)=\sum_s\frac{\Omega_s}{\omega}\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad P(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2},$$

and the Maxwell problem becomes a $12\times 12$ Hermitian eigenproblem for $\psi=[v_e,v_i,E,B]^T$. This two-fluid extension is what generates the low-frequency topological gap between the first two positive-frequency bands (Rajawat et al., 2022).

For inhomogeneous plasmas, especially in the TLCW literature, the wave problem is formulated as a global pseudo-differential-operator eigenproblem

$i\partial_t\psi=\hat H\psi,\qquad \psi=(v,E,B)^T,$

with

$k_z$0

Its Weyl symbol $k_z$1 defines the local bulk spectrum at fixed $k_z$2, while the semiclassical parameter is $k_z$3, the ratio of electron gyro-radius to inhomogeneity scale length. This operator framework is what makes the phase-space topological analysis of the TLCW precise (Qin et al., 2022).

3. Topological structure and bulk–edge correspondence

Two distinct topological mechanisms occur in the plasma literature. In periodic plasma crystals and in regularized continuous magnetoplasmas, one works with bulk bands over a two-dimensional $k_z$4-space and assigns Chern numbers to gaps or bands. For the plasma crystal, the first gap is inferred from a symmetry-indicator relation,

$k_z$5

and the reported eigenvalues imply that $k_z$6 is odd, specifically $k_z$7 for the first gap. In that setting, one chiral edge channel per termination crosses the gap, as expected for a Chern insulator (Qian et al., 2023).

For continuous magnetized plasmas, straightforward compactification of momentum space is subtle. One route is explicit regularization of the high-$k_z$8 response, such as replacing the plasma response by a decaying factor $k_z$9 or introducing a spatial cutoff. With that regularization, the GPP analysis found integer band Chern numbers and a lowest-gap Chern number $B_0=B_0\hat z$0, thereby predicting a single chiral interface mode at the plasma–vacuum boundary (Parker et al., 2019).

The TLCW established a qualitatively different result: in classical continuous media, momentum space is contractible, so topology there is generally trivial, and nontrivial topology appears only over non-contractible manifolds in phase space. For a one-dimensional density inhomogeneity with good quantum numbers $B_0=B_0\hat z$1 and $B_0=B_0\hat z$2, the relevant manifold is

$B_0=B_0\hat z$3

A boundary isomorphism theorem shows that the first Chern number computed on any sphere enclosing an isolated phase-space Weyl point is invariant. For the Langmuir–cyclotron resonance, the lower of the two touching positive-frequency bands carries $B_0=B_0\hat z$4 on such an $B_0=B_0\hat z$5, and Faure’s index theorem implies

$B_0=B_0\hat z$6

so the edge problem possesses exactly one net upward spectral-flow branch in the gap (Qin et al., 2022).

This phase-space result resolves a central point of interpretation. The TLCW is topological not because of a Brillouin-zone invariant, but because a resonance-induced Weyl point in $B_0=B_0\hat z$7 has nontrivial topological charge. The surface mode is therefore a spectral flow of the inhomogeneous wave operator, not merely a local surface resonance (Qin et al., 2022).

4. Principal realizations

Realization	Interface or medium	Defining result
Gaseous plasmon polariton	Magnetized plasma–vacuum boundary with graded density falloff	Lowest gap has $B_0=B_0\hat z$8; a single chiral edge state traverses the gap
Topological Langmuir–Cyclotron Wave	Density-transition layer where $B_0=B_0\hat z$9	Phase-space Weyl point has $$\hat{\epsilon}(\omega)= \begin{pmatrix} S & -iD & 0\ iD & S & 0\ 0 & 0 & P \end{pmatrix},$$0; one spectral-flow branch crosses the gap
Plasma-crystal edge state	2D square lattice of plasma cylinders in air	Full bandgap between $$\hat{\epsilon}(\omega)= \begin{pmatrix} S & -iD & 0\ iD & S & 0\ 0 & 0 & P \end{pmatrix},$$1 and $$\hat{\epsilon}(\omega)= \begin{pmatrix} S & -iD & 0\ iD & S & 0\ 0 & 0 & P \end{pmatrix},$$2 GHz with one chiral edge channel per termination
Ion-dominated low-frequency TSPW	Two-fluid plasma–vacuum interface	Low-frequency topological gap between bands 1 and 2 for $$\hat{\epsilon}(\omega)= \begin{pmatrix} S & -iD & 0\ iD & S & 0\ 0 & 0 & P \end{pmatrix},$$3
Cylindrical TSPW	Cylindrical plasma–vacuum boundary	Undamped below $$\hat{\epsilon}(\omega)= \begin{pmatrix} S & -iD & 0\ iD & S & 0\ 0 & 0 & P \end{pmatrix},$$4, continuum-damped above $$\hat{\epsilon}(\omega)= \begin{pmatrix} S & -iD & 0\ iD & S & 0\ 0 & 0 & P \end{pmatrix},$$5 by upper-hybrid modes

The GPP occupies the interface between vacuum and a realistic magnetized gaseous plasma whose density decays smoothly to zero. For $$\hat{\epsilon}(\omega)= \begin{pmatrix} S & -iD & 0\ iD & S & 0\ 0 & 0 & P \end{pmatrix},$$6 and $$\hat{\epsilon}(\omega)= \begin{pmatrix} S & -iD & 0\ iD & S & 0\ 0 & 0 & P \end{pmatrix},$$7, a gap $$\hat{\epsilon}(\omega)= \begin{pmatrix} S & -iD & 0\ iD & S & 0\ 0 & 0 & P \end{pmatrix},$$8 was reported, and the chiral interface branch crosses that gap. The analysis emphasized that the mode persists for smooth density falloff with scale length comparable to or longer than the wavelength, rather than requiring a discontinuous surface (Parker et al., 2019).

The TLCW is localized near the density-transition point where the local electron plasma frequency matches the electron cyclotron frequency. In the effective two-band reduction near the Langmuir–cyclotron Weyl point, the full $$\hat{\epsilon}(\omega)= \begin{pmatrix} S & -iD & 0\ iD & S & 0\ 0 & 0 & P \end{pmatrix},$$9 symbol reduces to a tilted $$S=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad D=\frac{\omega_{ce}}{\omega}\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad P=1-\frac{\omega_{pe}^2}{\omega^2}.$$0 Dirac cone in phase space, and the extra $$S=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad D=\frac{\omega_{ce}}{\omega}\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad P=1-\frac{\omega_{pe}^2}{\omega^2}.$$1 branch provides the spectral flow. Its dispersion is linear in $$S=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad D=\frac{\omega_{ce}}{\omega}\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad P=1-\frac{\omega_{pe}^2}{\omega^2}.$$2,

$$S=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad D=\frac{\omega_{ce}}{\omega}\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad P=1-\frac{\omega_{pe}^2}{\omega^2}.$$3

and its mode structure is a shifted Gaussian. This gives a closed-form realization of a chiral surface branch generated by a resonance rather than by periodicity (Qin et al., 2022).

In plasma crystals, the edge modes are genuinely photonic-crystalline: a square lattice of plasma cylinders in air with $$S=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad D=\frac{\omega_{ce}}{\omega}\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad P=1-\frac{\omega_{pe}^2}{\omega^2}.$$4 cm, $$S=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad D=\frac{\omega_{ce}}{\omega}\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad P=1-\frac{\omega_{pe}^2}{\omega^2}.$$5 cm, $$S=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad D=\frac{\omega_{ce}}{\omega}\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad P=1-\frac{\omega_{pe}^2}{\omega^2}.$$6 T, $$S=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad D=\frac{\omega_{ce}}{\omega}\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad P=1-\frac{\omega_{pe}^2}{\omega^2}.$$7, and $$S=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad D=\frac{\omega_{ce}}{\omega}\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad P=1-\frac{\omega_{pe}^2}{\omega^2}.$$8 GHz supports a full bandgap between $$S=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad D=\frac{\omega_{ce}}{\omega}\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2},\qquad P=1-\frac{\omega_{pe}^2}{\omega^2}.$$9 and $$\epsilon_\pm(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma\mp\omega_c)}\right],\qquad \epsilon_z(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma)}\right].$$0 GHz, approximately an $$\epsilon_\pm(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma\mp\omega_c)}\right],\qquad \epsilon_z(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma)}\right].$$1 gap. Chiral edge branches appear in a supercell calculation with PMC terminations, and their direction reverses under $$\epsilon_\pm(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma\mp\omega_c)}\right],\qquad \epsilon_z(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma)}\right].$$2 (Qian et al., 2023).

The two-fluid theory adds an ion-dominated branch at low frequency. For fixed nonzero $$\epsilon_\pm(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma\mp\omega_c)}\right],\qquad \epsilon_z(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma)}\right].$$3, a new nontrivial gap opens between bands 1 and 2, and the corresponding TSPW propagates below the ion cyclotron resonance. Its group velocity is opposite in sign to that of the high-frequency electron topological surface plasma wave for the same magnetic-field direction. Above $$\epsilon_\pm(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma\mp\omega_c)}\right],\qquad \epsilon_z(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma)}\right].$$4, the mode continues as a quasi-edge branch that can couple to a continuum of lower-hybrid resonances inside a smooth interface (Rajawat et al., 2022).

5. Localization, damping, and the meaning of protection

Topological protection in these systems means absence of backscattering within a common bulk gap and in the absence of a counterpropagating partner, not absence of attenuation. This distinction is explicit in the smooth-interface literature. For the electron-only plasma–vacuum problem, the TSPW is undamped for $$\epsilon_\pm(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma\mp\omega_c)}\right],\qquad \epsilon_z(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma)}\right].$$5, but above the cyclotron frequency a smooth density ramp produces a continuum of ramp-localized upper-hybrid resonances at $$\epsilon_\pm(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma\mp\omega_c)}\right],\qquad \epsilon_z(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma)}\right].$$6 where $$\epsilon_\pm(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma\mp\omega_c)}\right],\qquad \epsilon_z(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma)}\right].$$7. The edge state then becomes a collisionlessly damped quasi-mode, found by analytic continuation of the dispersion relation into the lower-half complex-$$\epsilon_\pm(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma\mp\omega_c)}\right],\qquad \epsilon_z(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma)}\right].$$8 plane. In the sharp-ramp limit the spatial damping rate scales approximately as $$\epsilon_\pm(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma\mp\omega_c)}\right],\qquad \epsilon_z(\omega)=\epsilon_0\!\left[1-\frac{\omega_p^2}{\omega(\omega+i\gamma)}\right].$$9, and for $$S(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad D(\omega)=\sum_s\frac{\Omega_s}{\omega}\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad P(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2},$$0 the reported quasi-mode had $$S(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad D(\omega)=\sum_s\frac{\Omega_s}{\omega}\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad P(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2},$$1 and $$S(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad D(\omega)=\sum_s\frac{\Omega_s}{\omega}\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad P(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2},$$2 (Rajawat et al., 2022).

The cylindrical extension preserved the same mechanism. For a smooth plasma–vacuum transition in a cylinder, the TSPW is undamped below $$S(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad D(\omega)=\sum_s\frac{\Omega_s}{\omega}\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad P(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2},$$3 and becomes damped above $$S(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad D(\omega)=\sum_s\frac{\Omega_s}{\omega}\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad P(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2},$$4 by resonant coupling to a continuum of upper-hybrid modes localized in the transition layer. The semi-analytical theory and 3D PIC simulations agreed for both temporal and spatial damping: $$S(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad D(\omega)=\sum_s\frac{\Omega_s}{\omega}\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad P(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2},$$5 for $$S(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad D(\omega)=\sum_s\frac{\Omega_s}{\omega}\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad P(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2},$$6, $$S(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad D(\omega)=\sum_s\frac{\Omega_s}{\omega}\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad P(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2},$$7, $$S(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad D(\omega)=\sum_s\frac{\Omega_s}{\omega}\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad P(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2},$$8, $$S(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad D(\omega)=\sum_s\frac{\Omega_s}{\omega}\frac{\omega_{ps}^2}{\omega^2-\Omega_s^2},\qquad P(\omega)=1-\sum_s\frac{\omega_{ps}^2}{\omega^2},$$9, and $12\times 12$0; and $12\times 12$1, $12\times 12$2 for $12\times 12$3, $12\times 12$4, and $12\times 12$5 (Rajawat et al., 1 Aug 2025).

The low-frequency two-fluid branch shows the analogous phenomenon at the lower-hybrid resonance rather than the upper-hybrid resonance. Below $12\times 12$6 the ion-dominated TSPW is an undamped topological edge mode; for $12\times 12$7 it becomes a quasi-edge state damped by coupling to a continuum of lower-hybrid resonant modes inside a smooth interface (Rajawat et al., 2022).

Numerical evidence for robustness is extensive. TLCWs were shown to propagate around zigzag interfaces, square and circular closed interfaces, and sharply turning phase-transition boundaries without reflection. Plasma-vacuum TSPWs in PIC simulations propagated around a rectangular discontinuity without backscattering, and in magnetically sheared configurations undamped and damped branches refracted into one another without reflection (Fu et al., 2022, Rajawat et al., 2022). The main caveat, emphasized in the 3D defect study of continuous magneto-plasma–metal interfaces, is that topology suppresses backward reflection but not side scattering. In finite-width 3D structures, energy can leak sideways or into cavity-like modes unless sidewalls, full metal cladding, or ridge geometries are used (Gangaraj et al., 2016).

6. Experimental regimes, applications, and broader context

The parameter ranges reported in the literature are laboratory-accessible. For realistic gaseous plasmas in devices such as LAPD, $12\times 12$8 T and $12\times 12$9 were identified as sufficient to produce a robust GPP gap at GHz frequencies (Parker et al., 2019). In periodic plasma crystals, the relevant band is $\psi=[v_e,v_i,E,B]^T$0–$\psi=[v_e,v_i,E,B]^T$1 GHz, with the explicit example gap at $\psi=[v_e,v_i,E,B]^T$2–$\psi=[v_e,v_i,E,B]^T$3 GHz and helium-plasma damping $\psi=[v_e,v_i,E,B]^T$4 GHz at $\psi=[v_e,v_i,E,B]^T$5 Torr (Qian et al., 2023). At higher frequencies, the continuous magneto-plasma literature also discussed THz-scale operation, including semiconductor-plasma implementations where the magnetic bias needed for large $\psi=[v_e,v_i,E,B]^T$6 becomes practical (Yang et al., 2014, Gangaraj et al., 2016).

Diagnostics proposed across the literature include localized RF launching near the boundary, B-dot probes, Langmuir probes, near-field mapping of edge-field localization, and extraction of $\psi=[v_e,v_i,E,B]^T$7 from supercell or interface measurements. The cylindrical damping work proposed monitoring $\psi=[v_e,v_i,E,B]^T$8 at the interface to extract $\psi=[v_e,v_i,E,B]^T$9 and $i\partial_t\psi=\hat H\psi,\qquad \psi=(v,E,B)^T,$0, while the plasma-crystal work emphasized direct measurement of chiral edge dispersions and their reversal under $i\partial_t\psi=\hat H\psi,\qquad \psi=(v,E,B)^T,$1 (Rajawat et al., 1 Aug 2025, Qian et al., 2023).

The applications discussed so far are correspondingly diverse. TLCWs were proposed as a mechanism to drive current and flow in magnetized plasmas because they transport directed momentum and angular momentum along complex interfaces. Spatially tailored magnetic profiles in cylindrical systems were proposed as directional energy sinks, using upper-hybrid damping for localized plasma heating and angular-momentum deposition. Plasma-crystal edge transport was linked to plasma-based lighting and plasma propulsion engines, while magneto-plasma waveguides suggest nonreciprocal components such as circulators and isolators (Fu et al., 2022, Rajawat et al., 1 Aug 2025, Qian et al., 2023).

In broader band-theoretic terms, TSPWs in plasmas connect plasma physics, topological photonics, and even geophysical or astrophysical MHD. An instructive parallel is the identification of magneto-Kelvin and magneto-Yanai waves in the solar tachocline as topologically protected equatorial edge modes, with a Chern number difference $i\partial_t\psi=\hat H\psi,\qquad \psi=(v,E,B)^T,$2 across the equator. That result does not describe a plasma–vacuum surface wave, but it reinforces the more general point that topological edge transport in plasma systems need not be confined to conventional interfaces or to photonic-crystal settings (Lier et al., 2024).