Topological Zero Modes Overview

• Topological zero modes (TZMs) are zero-energy states defined by bulk topological invariants and protecting symmetries, appearing at defects, edges, and vortices.
• They are typically analyzed via flux-insertion methods and differential equations that determine their localization, oscillation, and profile across various systems.
• TZMs manifest in multiple settings, such as unpaired Majorana modes or robust Dirac pairs in superconductors and photonic arrays, highlighting their potential in quantum applications.

Searching arXiv for the core and related papers on topological zero modes. Searching arXiv for supporting papers on smooth-domain-wall and photonic realizations of topological zero modes. Topological zero modes (TZMs) are zero-energy, zero-admittance, or symmetry-pinned mid-gap states bound to vortices, domain walls, edges, corners, dislocations, or special Landau levels, whose existence is controlled by bulk topology together with protecting symmetries. In Bogoliubov–de Gennes superconductors they appear as unpaired Majorana modes or as symmetry-enforced Dirac/Kramers pairs; in photonic, magnonic, and topolectrical settings the same label is used for defect or boundary modes with identical single-particle topology, while some works use it more broadly for in-gap defect states that approach zero in a symmetry-restoring limit or for geometry-induced zero-energy states on curved manifolds (Roy, 2010, Noh et al., 2019, Saji et al., 2024, Fonseca et al., 2015).

1. Organizing principles

A recurring structure across the literature is that topology fixes existence and counting, whereas differential equations fix localization, oscillation, and profile. Rahul Roy’s flux-insertion construction maps a BdG Hamiltonian to an associated insulator with the same single-particle spectrum and an antiunitary spectral symmetry

$S\mathcal{H}(\phi)S^{-1}=-\mathcal{H}(-\phi),$

so that a bulk Hall conductance or Chern number becomes a parity statement about zero modes bound to a $\pi$-flux or, equivalently, to a superconducting vortex. In the 1D smooth-domain-wall setting, the zero-mode envelope obeys

$\varphi''(x)+2s\,v(x)\varphi'(x)-m(x)\varphi(x)=0,$

and the counting rule is $N_{\rm ZM}=|W_R-W_L|$. This establishes a common index-theorem-like viewpoint: bulk topological data determine whether a defect must host a zero mode, while the interface profile determines how that mode looks in real space (Roy, 2010, Marra et al., 2024).

The same label does not always refer to identical physical situations. In twisted bilayer graphene, the “topological zero modes” are zero-energy Landau-level states rather than defect-bound states: two same-chirality Dirac cones within a valley produce an orbital twofold-degenerate zero-energy Landau level, and including spin and valley yields an eightfold-degenerate zero-energy Landau level that cannot be lifted by strong magnetic fields within the continuum theory (Gail et al., 2011). This broadens the notion of a TZM from a purely defect-bound state to a more general zero-energy mode protected by topological charge and symmetry.

2. Symmetry classes and vortex zero modes in superconductors

In two-dimensional superconductors, the decisive question is when a vortex binds an exactly zero-energy state and whether that state is topologically protected. Roy’s classification uses the associated-insulator flux-insertion argument to give necessary and sufficient conditions for protected vortex-core zero modes in the Altland–Zirnbauer classes relevant to superconductors. For class D, odd Chern-number parity implies an odd number of vortex zero modes and therefore one unpaired Majorana zero mode; for classes with enforced pairing or Kramers pairing, the protected object is a Dirac zero mode rather than an isolated Majorana (Roy, 2010).

Class	Condition on bulk/topological data	Protected vortex mode
D	$\sigma_{xy}2\pi\hbar/e^{2}=2k-1$	odd number of zero modes, hence one unpaired Majorana mode
C	$\sigma_{xy}2\pi\hbar/e^{2}=2(2k-1)$	odd number of pairs, i.e. protected Dirac mode
DIII	associated insulator in the nontrivial $\mathbb{Z}_2$ class	odd number of Kramers pairs, i.e. Dirac mode
CI	no protected mode	none

This framework reinterprets the spinless $p_x+ip_y$ vortex Majorana as a generic class-D consequence of odd Hall conductance/Chern-number parity rather than as a peculiarity of a soluble continuum model. The crucial point is robustness: disorder cannot change the zero-mode parity without closing the bulk gap or changing the invariant (Roy, 2010).

A complementary Dirac-side formulation is provided by the generalized Jackiw–Rossi Hamiltonian with an antilinear anticommuting symmetry. In the topological-insulator surface realization of an $s$-wave superconducting vortex, finite chemical potential, Zeeman coupling, and orbital field can all be included while retaining the spectral reflection symmetry. The normalizable vortex zero mode survives only if

$m(\infty)^2+\mu^2\ge h^2,$

so the Zeeman field can delocalize and destroy the Majorana mode once it exceeds a critical value. This sharpens the Fu–Kane vortex-zero-mode story by showing that finite $\pi$0 is not fatal, whereas sufficiently strong $\pi$1 is (Herbut et al., 2010).

3. Edge, domain-wall, and Landau-level zero modes

In one-dimensional topological superconductors and insulators, the invariant often counts edge zero modes, but the spatial form of those modes is finer data. For the extended quantum Ising/Kitaev chain with nearest- and next-nearest-neighbor couplings, the winding number $\pi$2 still gives the number of zero modes per boundary, but the characteristic roots $\pi$3 of

$\pi$4

determine whether the mode is purely exponential, oscillatory-decaying, perfectly localized on one or two sites, or even peaked away from the first site. The paper therefore distinguishes exact Majorana zero modes in semi-infinite or special perfectly localized cases from finite-chain near-zero edge modes, where left and right Majoranas hybridize and split (Pathak et al., 25 Jun 2025).

For smooth interfaces, exact wavefunctions can be obtained without assuming a sharp wall. In the Hermitian modified Jackiw–Rebbi/BdG problem, exponentially confined domain walls lead to hypergeometric zero-mode solutions classified as “no hair,” “short hair,” and “long hair,” depending on the relation between wall width $\pi$5, decay length $\pi$6, and oscillation wavelength $\pi$7. The asymptotic structure is universal: $\pi$8 and, in the oscillatory regime,

$\pi$9

Topology still fixes $\varphi''(x)+2s\,v(x)\varphi'(x)-m(x)\varphi(x)=0,$0, but broad walls can shift the localization center away from the geometric interface toward points where a local invariant changes sign (Marra et al., 2024).

The non-Hermitian extension preserves this structure for line-gapped systems. There, $\varphi''(x)+2s\,v(x)\varphi'(x)-m(x)\varphi(x)=0,$1 and $\varphi''(x)+2s\,v(x)\varphi'(x)-m(x)\varphi(x)=0,$2 are complex, the zero-mode wavefunctions are generically complex rather than Majorana-like real, and the decay/oscillation exponents are again governed by $\varphi''(x)+2s\,v(x)\varphi'(x)-m(x)\varphi(x)=0,$3. The counting rule remains a bulk-boundary statement in terms of asymptotic line-gap invariants, but the modes acquire asymmetric localization and damped oscillations characteristic of non-Hermitian spectra (Marra et al., 9 Apr 2025).

4. Realizations beyond conventional electronic superconductors

Photonic systems provide the cleanest single-particle analogs. In a distorted honeycomb waveguide lattice with a Kekulé order parameter

$\varphi''(x)+2s\,v(x)\varphi'(x)-m(x)\varphi(x)=0,$4

a unit vortex binds a single mid-gap photonic mode localized near the core. Chiral symmetry pins this mode precisely to the middle of the band gap, and adiabatically varying the global phase offset $\varphi''(x)+2s\,v(x)\varphi'(x)-m(x)\varphi(x)=0,$5 implements a braid in parameter space with

$\varphi''(x)+2s\,v(x)\varphi'(x)-m(x)\varphi(x)=0,$6

The measured $\varphi''(x)+2s\,v(x)\varphi'(x)-m(x)\varphi(x)=0,$7 geometric phase reproduces the single-particle sign structure of Majorana braiding, while the paper is explicit that this is a classical-wave simulation rather than many-body anyonic statistics (Noh et al., 2019).

In a mirror-symmetric SSH photonic array with a central defect, the defect TZM exists for both dimerizations but switches mirror parity at

$\varphi''(x)+2s\,v(x)\varphi'(x)-m(x)\varphi(x)=0,$8

For $\varphi''(x)+2s\,v(x)\varphi'(x)-m(x)\varphi(x)=0,$9 the defect mode is symmetric and has finite central amplitude; for $N_{\rm ZM}=|W_R-W_L|$0 it is antisymmetric and obeys $N_{\rm ZM}=|W_R-W_L|$1. Because a central-waveguide input is mirror-even, only the symmetric TZM has nonzero overlap and can be selectively excited. The transition is therefore not between “zero mode” and “no zero mode,” but between two topologically distinct defect zero modes in different mirror sectors (Tang et al., 2022).

Topolectrical circuits realize several further variants. The breathing kagome circuit hosts second-order corner-localized zero-admittance modes at the resonance

$N_{\rm ZM}=|W_R-W_L|$2

with bulk topology diagnosed by a quantized $N_{\rm ZM}=|W_R-W_L|$3 Berry phase and protection supplied by a generalized chiral symmetry of the tripartite lattice (Yang et al., 2020). Coupled non-Hermitian topolectrical SSH chains exhibit size-dependent TZMs that can emerge or disappear as system size changes, with a rank-deficiency criterion on zero-admittance eigenvectors distinguishing topological and trivial phases (Rafi-Ul-Islam et al., 2022). In a finite non-Hermitian SSH topolectrical chain, gain/loss and asymmetric coupling can even restore exactly zero-admittance edge modes at a finite critical size $N_{\rm ZM}=|W_R-W_L|$4, visible as pronounced impedance peaks (Rafi-Ul-Islam et al., 23 Jul 2025).

Bosonic spin systems also support defect TZMs. In a 2D honeycomb ferromagnet with sublattice anisotropy $N_{\rm ZM}=|W_R-W_L|$5, lattice dislocations bind non-propagating in-gap magnon states. Their existence is controlled by the weak $N_{\rm ZM}=|W_R-W_L|$6 defect index

$N_{\rm ZM}=|W_R-W_L|$7

and the paper reports that these dislocation-bound magnon zero-modes survive magnetic disorder and can remain even when the bulk Bott index becomes trivial (Saji et al., 2024).

5. Non-Hermitian, driven, dissipative, and imperfect regimes

Imperfections need not merely destroy topological zero modes; they can reorganize the phase structure. In proximitized double helical liquids, pairing asymmetry, interaction asymmetry, and random spin-flip backscattering change the effective symmetry class from DIII to BDI, allowing a phase with a single Majorana zero mode per corner in addition to the clean time-reversal-invariant pair. The zero-mode counting formula depends on the renormalized local pairing, crossed pairing, and spin-flip amplitudes, and the paper finds cascades such as $N_{\rm ZM}=|W_R-W_L|$8, together with a disorder- or asymmetry-enabled revival of Majorana zero modes under screening control (Ohorodnyk et al., 11 Dec 2025).

Driven-dissipative systems add another distinction: edge localization at quasienergy $N_{\rm ZM}=|W_R-W_L|$9 or $\sigma_{xy}2\pi\hbar/e^{2}=2k-1$0 is not by itself enough to establish topology. In a periodically driven dissipative Rashba nanowire, the Floquet damping matrix supports topological Majorana $\sigma_{xy}2\pi\hbar/e^{2}=2k-1$1-modes and $\sigma_{xy}2\pi\hbar/e^{2}=2k-1$2-modes, but also topologically trivial $\sigma_{xy}2\pi\hbar/e^{2}=2k-1$3- and $\sigma_{xy}2\pi\hbar/e^{2}=2k-1$4-edge modes. The MZMs and MPMs are tied to bulk topology and Floquet invariants, whereas the trivial TZMs and TPMs are associated primarily with exceptional points and carry no bulk topological invariant. Dissipation can therefore both alter and induce topological phases while simultaneously generating edge-localized but topologically trivial modes (Gogoi et al., 14 Jul 2025).

Nonlinearity can even delocalize a chiral-protected zero mode without removing its topological origin. In a hybrid Hermitian/non-Hermitian nonlinear SSH interface lattice, the zero mode remains at $\sigma_{xy}2\pi\hbar/e^{2}=2k-1$5 and on one sublattice, but its spatial support can evolve from interface-localized to partially extended, fully extended across the whole lattice, or skin-localized. The spectral localizer continues to detect its topological character, so the paper explicitly separates spectral/topological pinning from exponential localization (Cai et al., 2024).

6. Diagnostics, terminology, and recurrent ambiguities

A persistent ambiguity in the field is that “zero mode” does not always mean the same thing. The spectral-localizer literature shows this sharply: the localizer

$\sigma_{xy}2\pi\hbar/e^{2}=2k-1$6

has zero-modes not only in topological metals but generically in trivial metals as well, because the effective Dirac mass $\sigma_{xy}2\pi\hbar/e^{2}=2k-1$7 changes sign across a Fermi surface. What distinguishes topological from trivial metals is therefore not the mere presence of localizer zero-modes, but the surrounding low-lying localizer spectrum: Weyl semimetals exhibit isolated mid-gap localizer zero-modes, whereas trivial metals display the boundary-spectrum multiplets of an auxiliary Dirac problem (Franca et al., 2023).

Statistical-mechanical diagnostics raise a related issue. Real-time Fisher-zero pairing in SSH and Kitaev chains is proposed as a signature of boundary zero modes, and in the Kitaev case the deformation of Fisher-zero loops is interpreted as a braiding picture. The correspondence is presented as a physically motivated and numerically validated mechanism rather than as a fully general proof, so it should be read as a diagnostic framework rather than a replacement for bulk-boundary correspondence (Meng et al., 2 Apr 2025).

The strongest terminological caution concerns exactness. In finite long-range Kitaev chains, many low-energy edge states are only approximate Majorana zero modes because left and right modes hybridize and split (Pathak et al., 25 Jun 2025). On toroidal topological-insulator surfaces, the reported zero-energy states are geometry-induced by the spin connection and the two $\sigma_{xy}2\pi\hbar/e^{2}=2k-1$8 Berry phases of the torus, but the work does not prove a conventional index-protected zero-mode theorem (Fonseca et al., 2015). In dislocated magnon systems, the paper itself calls the defect states “zero-modes” while also stating that they are gapped by $\sigma_{xy}2\pi\hbar/e^{2}=2k-1$9 and become gapless only as parity symmetry is restored (Saji et al., 2024).

Across these usages, TZMs are best understood not as a single spectral template but as a family of symmetry- and topology-constrained zero-frequency phenomena. The unifying content is the same: bulk topology, defect topology, or spectral symmetry force a state to sit at, or to be pinned near, a distinguished spectral point that cannot be removed without changing the protecting structure.