Jackiw-Rebbi Topological Index

• The Jackiw-Rebbi topological index is defined by the sign change of a Dirac mass field across a domain wall, guaranteeing a localized zero-energy state.
• It uses explicit index formulas and winding numbers to count zero modes, linking analytical edge indices with bulk topological invariants across dimensions.
• This invariant underpins physical phenomena such as charge fractionalization, parity transitions, and topologically protected transport in photonic, acoustic, and condensed-matter systems.

Searching arXiv for recent and foundational papers on the Jackiw-Rebbi topological index and related index-theoretic formulations. The Jackiw–Rebbi topological index is the topological quantity that encodes when a Dirac operator in a spatially varying mass or scalar background must support a localized zero-energy state. Across the literature, it appears in several closely related forms: as the sign change of a one-dimensional Dirac mass across a domain wall, as an index $n_+-n_-$ counting zero modes of definite chirality, as a winding-number/edge-index correspondence in chiral one-dimensional lattices, and as a defect-classification invariant in higher-dimensional Clifford- or K-theoretic settings. In the standard one-dimensional setting, the central mechanism is that a kink or domain wall interpolates between asymptotic vacua of opposite sign, forcing a protected bound state at zero energy (Angelakis et al., 2013). More recent work also writes the Jackiw–Rebbi index explicitly as $$N_{\mathrm{JR}}=\frac{1}{2}\left[\operatorname{sgn}\phi(+\infty)-\operatorname{sgn}\phi(-\infty)\right]$$, using it as the topological count of protected fermionic zero modes and as the organizing invariant for universal defect-scaling phenomena (Pinheiro et al., 23 May 2026).

1. Canonical one-dimensional formulation

The canonical Jackiw–Rebbi setting is a one-dimensional Dirac fermion coupled to a spatially dependent scalar field or Lorentz scalar potential. In the slow-light realization, the model is written as

$$i\partial_t {\bf \Psi} = \left( \alpha c p_z + \frac{\beta m c^2}{\kappa}\phi(z) \right){\bf \Psi},$$

with $\alpha=-\sigma_z$ and $\beta=\sigma_y$ (Angelakis et al., 2013). The scalar field has degenerate vacua at $\phi=\pm\kappa$, and a kink profile

$\phi_s(z)=\kappa \tanh(\lambda z)$

interpolates between them (Angelakis et al., 2013). The effective mass satisfies $m_{\rm eff}(z)\propto \phi_s(z)$, so the mass changes sign across the kink (Angelakis et al., 2013).

In this framework, the zero mode is given explicitly by

$${\bf\Psi}_0(z) = \exp\left(-\frac{mc}{\kappa}\int_0^z dx\,\phi_s(x)\right){\bf\chi} = \exp\left[-\frac{mc}{\lambda}\ln(\cosh \lambda z)\right]{\bf\chi},$$

with the spinor constraint

$\alpha\beta\,{\bf\chi}=-i{\bf\chi}$

(Angelakis et al., 2013). Because the kink interpolates between opposite vacua, the wavefunction decays exponentially away from the kink center and is localized at the soliton (Angelakis et al., 2013).

The topological content is that the background at $$N_{\mathrm{JR}}=\frac{1}{2}\left[\operatorname{sgn}\phi(+\infty)-\operatorname{sgn}\phi(-\infty)\right]$$0 and $$N_{\mathrm{JR}}=\frac{1}{2}\left[\operatorname{sgn}\phi(+\infty)-\operatorname{sgn}\phi(-\infty)\right]$$1 lies in two different topological sectors. The index is described abstractly as

$$N_{\mathrm{JR}}=\frac{1}{2}\left[\operatorname{sgn}\phi(+\infty)-\operatorname{sgn}\phi(-\infty)\right]$$2

where $$N_{\mathrm{JR}}=\frac{1}{2}\left[\operatorname{sgn}\phi(+\infty)-\operatorname{sgn}\phi(-\infty)\right]$$3 count zero modes of definite chirality or related grading (Angelakis et al., 2013). The supplied sources emphasize that the zero mode is guaranteed by topology rather than by microscopic details of the kink profile (Angelakis et al., 2013). A concise summary stated in the materials is: trivial background implies no protected zero mode, while kink background implies one protected zero mode (Angelakis et al., 2013).

A closely aligned formulation appears in generalized one-dimensional discussions of mass domain walls, where the physically relevant invariant is the sign reversal itself. In photonic binary waveguide arrays, for example, the topological content is expressed through the sign of the Dirac mass on either side of the interface, with two distinct topological sectors satisfying $$N_{\mathrm{JR}}=\frac{1}{2}\left[\operatorname{sgn}\phi(+\infty)-\operatorname{sgn}\phi(-\infty)\right]$$4; the interface mode exists only when the mass changes sign (Tran et al., 2017). This suggests that, in practical realizations, the Jackiw–Rebbi index often functions as a domain-wall criterion rather than solely as a formal operator index.

2. Explicit index formulas and zero-mode counting

An explicit formula for the Jackiw–Rebbi topological index is given in the braneworld-oriented study of domain-wall merging: $$N_{\mathrm{JR}}=\frac{1}{2}\left[\operatorname{sgn}\phi(+\infty)-\operatorname{sgn}\phi(-\infty)\right]$$5 (Pinheiro et al., 23 May 2026). This quantity counts whether the scalar background $$N_{\mathrm{JR}}=\frac{1}{2}\left[\operatorname{sgn}\phi(+\infty)-\operatorname{sgn}\phi(-\infty)\right]$$6 interpolates between opposite vacua at $$N_{\mathrm{JR}}=\frac{1}{2}\left[\operatorname{sgn}\phi(+\infty)-\operatorname{sgn}\phi(-\infty)\right]$$7 and $$N_{\mathrm{JR}}=\frac{1}{2}\left[\operatorname{sgn}\phi(+\infty)-\operatorname{sgn}\phi(-\infty)\right]$$8 (Pinheiro et al., 23 May 2026). When the asymptotic signs differ, $$N_{\mathrm{JR}}=\frac{1}{2}\left[\operatorname{sgn}\phi(+\infty)-\operatorname{sgn}\phi(-\infty)\right]$$9, and the Jackiw–Rebbi mechanism guarantees a single normalizable fermionic zero mode; when the asymptotic signs are the same, the index vanishes and there is no protected zero mode (Pinheiro et al., 23 May 2026).

In that same formulation, the Dirac fermion is Yukawa-coupled to a kink background through

$$i\partial_t {\bf \Psi} = \left( \alpha c p_z + \frac{\beta m c^2}{\kappa}\phi(z) \right){\bf \Psi},$$0

and the static problem reduces to a supersymmetric pair of Schrödinger operators

$$i\partial_t {\bf \Psi} = \left( \alpha c p_z + \frac{\beta m c^2}{\kappa}\phi(z) \right){\bf \Psi},$$1

(Pinheiro et al., 23 May 2026). The zero mode belongs to the $$i\partial_t {\bf \Psi} = \left( \alpha c p_z + \frac{\beta m c^2}{\kappa}\phi(z) \right){\bf \Psi},$$2 sector and is

$$i\partial_t {\bf \Psi} = \left( \alpha c p_z + \frac{\beta m c^2}{\kappa}\phi(z) \right){\bf \Psi},$$3

which is normalizable precisely when the kink interpolates between opposite vacua (Pinheiro et al., 23 May 2026).

The same source emphasizes that the number of normalizable zero modes is topologically protected and indexed by $$i\partial_t {\bf \Psi} = \left( \alpha c p_z + \frac{\beta m c^2}{\kappa}\phi(z) \right){\bf \Psi},$$4 (Pinheiro et al., 23 May 2026). In the models studied there, $$i\partial_t {\bf \Psi} = \left( \alpha c p_z + \frac{\beta m c^2}{\kappa}\phi(z) \right){\bf \Psi},$$5 is fixed to $$i\partial_t {\bf \Psi} = \left( \alpha c p_z + \frac{\beta m c^2}{\kappa}\phi(z) \right){\bf \Psi},$$6, so there is one protected chiral zero mode in the single-kink case (Pinheiro et al., 23 May 2026). This is a particularly direct realization of the customary Jackiw–Rebbi statement that the asymptotic topology of the mass field determines the zero-mode count.

A higher-dimensional analogue appears in the three-dimensional Jackiw–Rebbi hedgehog problem, where the relativistic index is written as

$$i\partial_t {\bf \Psi} = \left( \alpha c p_z + \frac{\beta m c^2}{\kappa}\phi(z) \right){\bf \Psi},$$7

with $$i\partial_t {\bf \Psi} = \left( \alpha c p_z + \frac{\beta m c^2}{\kappa}\phi(z) \right){\bf \Psi},$$8 (Nishida et al., 2010). There the index counts zero modes bound to a point defect in a three-component scalar background and reduces, in the defect language, to a winding number (Nishida et al., 2010). This suggests that the one-dimensional sign-change index and higher-dimensional winding-number indices are different realizations of the same broader defect-index logic.

3. Bulk–boundary correspondence, winding numbers, and analytic index

In chiral one-dimensional lattice systems, the Jackiw–Rebbi mechanism is recast as a rigorous bulk–boundary correspondence. For a chiral-symmetric Bloch Hamiltonian

$$i\partial_t {\bf \Psi} = \left( \alpha c p_z + \frac{\beta m c^2}{\kappa}\phi(z) \right){\bf \Psi},$$9

the bulk invariant is the winding number

$\alpha=-\sigma_z$0

(Thiang, 2023). For the corresponding half-space Hamiltonian $\alpha=-\sigma_z$1, the edge index is

$\alpha=-\sigma_z$2

(Thiang, 2023). The central result is

$\alpha=-\sigma_z$3

or, equivalently,

$\alpha=-\sigma_z$4

(Thiang, 2023).

The supplied material explicitly frames this as an index theorem in the spirit of Jackiw–Rebbi: the bulk winding number is the topological index, and the edge zero modes are the analytic manifestation (Thiang, 2023). In this picture, the half-space truncation acts as a boundary defect, and the localized edge states are the analogues of Jackiw–Rebbi bound states (Thiang, 2023).

An abstract operator-theoretic formulation identifies the half-space off-diagonal block $\alpha=-\sigma_z$5 as a Toeplitz operator with symbol $\alpha=-\sigma_z$6, and the Toeplitz index theorem gives

$\alpha=-\sigma_z$7

(Thiang, 2023). The source stresses that this directly realizes the slogan

$\alpha=-\sigma_z$8

(Thiang, 2023).

This lattice formulation differs in language from the continuum Dirac kink, but the underlying structure is the same. A nontrivial bulk invariant forces localized zero-energy boundary states, and only one chirality sector survives (Thiang, 2023). A plausible implication is that the lattice winding number can be viewed as the regularized counterpart of the continuum mass-domain-wall index.

4. Generalizations to arbitrary dimension and K-theoretic classification

A broader reformulation treats generalized Jackiw–Rebbi models as a universal framework for topological free-fermion phases. In this setting, the Dirac mass satisfies

$\alpha=-\sigma_z$9

and the two half-spaces are related by a parity transformation (Meetei et al., 2014). For the complex Dirac case, the Hamiltonian is

$\beta=\sigma_y$0

with gamma matrices obeying the usual Clifford relations (Meetei et al., 2014). The parity operator can be chosen as

$\beta=\sigma_y$1

where $\beta=\sigma_y$2 acts as $\beta=\sigma_y$3 (Meetei et al., 2014).

The key statement is that determining the allowed $\beta=\sigma_y$4 is equivalent to determining the admissible mass matrix $\beta=\sigma_y$5, and this becomes a Clifford algebra extension problem (Meetei et al., 2014). In the complex case, the classifying space yields

$\beta=\sigma_y$6

which is $\beta=\sigma_y$7 for even $\beta=\sigma_y$8 and $\beta=\sigma_y$9 for odd $\phi=\pm\kappa$0 (Meetei et al., 2014). For real symmetry classes with discrete symmetry operators $\phi=\pm\kappa$1, the classifying space becomes

$\phi=\pm\kappa$2

with classification

$\phi=\pm\kappa$3

(Meetei et al., 2014).

In this generalized setting, interface zero modes satisfy

$\phi=\pm\kappa$4

(Meetei et al., 2014). The topological condition is that only one parity sector survives at the interface, whereas a naive non-topological model would allow both even and odd states (Meetei et al., 2014). The paper uses integer quantum Hall and quantum spin Hall examples to show how parity selection reproduces $\phi=\pm\kappa$5 and $\phi=\pm\kappa$6 free-fermion classifications (Meetei et al., 2014).

A more defect-oriented K-theory treatment appears in the multi-flavored $\phi=\pm\kappa$7-dimensional Jackiw–Rebbi model with an SU(2) hedgehog background. For the doublet case with $\phi=\pm\kappa$8, the relevant point-defect classification is

$\phi=\pm\kappa$9

while for the triplet case it becomes

$\phi_s(z)=\kappa \tanh(\lambda z)$0

(Ho et al., 2012). The paper interprets this as an integer topological index classifying Majorana zero modes bound to hedgehog or ’t Hooft–Polyakov monopole defects (Ho et al., 2012). In the two-flavor doublet case, the explicit analysis shows either no normalizable zero mode or one normalizable Majorana zero mode depending on mass parameters; in the four-flavor case, there can be zero, one, or two (Ho et al., 2012).

Taken together, these works show that the Jackiw–Rebbi index extends naturally from one-dimensional domain walls to general defect classifications in Clifford/K-theoretic language. The common content is that admissible mass textures define topological classes, and localized zero modes furnish the analytic realization of those classes (Meetei et al., 2014, Ho et al., 2012).

5. Physical consequences: charge fractionalization, parity, and transport

A classic physical consequence of the Jackiw–Rebbi zero mode is charge fractionalization. In the one-dimensional Dirac kink problem, occupation of the zero mode yields two degenerate vacua whose charges differ by one unit, leading to fractional charges

$\phi_s(z)=\kappa \tanh(\lambda z)$1

(Angelakis et al., 2013). The supplied source emphasizes that these are eigenvalues of the charge operator in the second-quantized theory rather than merely smeared expectation values (Angelakis et al., 2013).

In superconducting settings, the analogous topological phenomenon can become fractional fermion parity rather than fractional charge. In a one-dimensional topological superconductor with a Jackiw–Rebbi-type bound state at a domain wall, the relevant bulk invariant is written as

$\phi_s(z)=\kappa \tanh(\lambda z)$2

and this Zak-Berry-phase invariant is stated to be equivalent to Kitaev’s Pfaffian invariant (Xiong et al., 2014). The paper argues that the domain wall carries a fractional parity in the sense that two configurations differ by

$\phi_s(z)=\kappa \tanh(\lambda z)$3

(Xiong et al., 2014). The claimed consequence is a topologically protected zero-energy crossing of the Jackiw–Rebbi-type bound-state energy (Xiong et al., 2014).

In quantum spin Hall and SSH-related platforms, Jackiw–Rebbi zero modes are also tied to transport anomalies and non-Abelian exchange physics. In the quantum spin Hall constriction study, the zero mode arises from a domain wall between phases controlled by the relative strengths of Zeeman coupling and inter-edge tunneling, with an effective transition Hamiltonian

$\phi_s(z)=\kappa \tanh(\lambda z)$4

(Wu et al., 2019). There the paper does not write an explicit index formula, but treats the interface state as the standard mass-domain-wall realization of the Jackiw–Rebbi principle (Wu et al., 2019). The zero-energy nature manifests in a $\phi_s(z)=\kappa \tanh(\lambda z)$5-periodic Aharonov–Bohm oscillation at resonance (Wu et al., 2019).

A later transport-focused work on SSH/JR zero modes formulates the topological protection operationally in the $\phi_s(z)=\kappa \tanh(\lambda z)$6 limit, where a unitary symmetry $\phi_s(z)=\kappa \tanh(\lambda z)$7 forbids mixing between two Majorana flavors (Ge et al., 19 Dec 2025). The braiding fidelity is

$\phi_s(z)=\kappa \tanh(\lambda z)$8

and at resonance the Fano factor satisfies

$\phi_s(z)=\kappa \tanh(\lambda z)$9

(Ge et al., 19 Dec 2025). The same source states that the central topological mechanism remains the standard Jackiw–Rebbi one: a sign-changing mass term in a one-dimensional Dirac problem yields a protected zero-energy bound state (Ge et al., 19 Dec 2025).

These examples show that the Jackiw–Rebbi index is not only a static zero-mode count. It also governs fractional quantum numbers, parity switching, resonant transport, and, in symmetry-protected settings, the conditions under which zero modes remain suitable for braiding protocols (Angelakis et al., 2013, Xiong et al., 2014, Wu et al., 2019, Ge et al., 19 Dec 2025).

6. Realizations in photonic, acoustic, and condensed-matter platforms

The supplied materials document numerous experimental and synthetic realizations in which the Jackiw–Rebbi index is implemented as a sign-changing effective mass.

In photonic binary waveguide arrays, the interface is formed by two regions described by Dirac models of mass $m_{\rm eff}(z)\propto \phi_s(z)$0 and $m_{\rm eff}(z)\propto \phi_s(z)$1, obtained from the coupled-mode system $\alpha\beta\,{\bf\chi}=-i{\bf\chi}$7 which maps to the one-dimensional nonlinear Dirac equation $\alpha\beta\,{\bf\chi}=-i{\bf\chi}$8 The source states that the Jackiw–Rebbi state exists only when the mass changes sign and is topologically robust in both linear and nonlinear regimes, including focusing and defocusing nonlinearity (Tran et al., 2017).

In all-dielectric photonic chains controlled by bianisotropy, the effective Hamiltonian near the Dirac-like points is

$m_{\rm eff}(z)\propto \phi_s(z)$2

where $m_{\rm eff}(z)\propto \phi_s(z)$3 is the effective mass term proportional to the bianisotropy (Gorlach et al., 2018). Flipping half of the meta-atoms reverses the sign of $m_{\rm eff}(z)\propto \phi_s(z)$4, creating a domain wall and interface states inside the gap (Gorlach et al., 2018). The paper characterizes the relevant topological invariant as the sign change of the mass across the interface rather than as a Chern number (Gorlach et al., 2018).

In photonic van der Waals heterostructures based on stacked WS$m_{\rm eff}(z)\propto \phi_s(z)$5 gratings, the effective Dirac model is

$m_{\rm eff}(z)\propto \phi_s(z)$6

and the Jackiw–Rebbi state appears when the band gap is fully closed and reopened so that the two domains have opposite band ordering (Randerson et al., 4 Jun 2025). The paper reports a linewidth of $m_{\rm eff}(z)\propto \phi_s(z)$7 meV, angular bandwidth of $m_{\rm eff}(z)\propto \phi_s(z)$8, and directional enhancement of excitonic emission of up to $m_{\rm eff}(z)\propto \phi_s(z)$9 times that of uncoupled monolayer WSe$${\bf\Psi}_0(z) = \exp\left(-\frac{mc}{\kappa}\int_0^z dx\,\phi_s(x)\right){\bf\chi} = \exp\left[-\frac{mc}{\lambda}\ln(\cosh \lambda z)\right]{\bf\chi},$$0 (Randerson et al., 4 Jun 2025).

Acoustic metagratings realize a real-space analogue with effective Dirac mass

$${\bf\Psi}_0(z) = \exp\left(-\frac{mc}{\kappa}\int_0^z dx\,\phi_s(x)\right){\bf\chi} = \exp\left[-\frac{mc}{\lambda}\ln(\cosh \lambda z)\right]{\bf\chi},$$1

and the paper uses

$${\bf\Psi}_0(z) = \exp\left(-\frac{mc}{\kappa}\int_0^z dx\,\phi_s(x)\right){\bf\chi} = \exp\left[-\frac{mc}{\lambda}\ln(\cosh \lambda z)\right]{\bf\chi},$$2

as a direct real-space topological invariant (Xia et al., 25 Apr 2025). The interface state is associated with a constant $${\bf\Psi}_0(z) = \exp\left(-\frac{mc}{\kappa}\int_0^z dx\,\phi_s(x)\right){\bf\chi} = \exp\left[-\frac{mc}{\lambda}\ln(\cosh \lambda z)\right]{\bf\chi},$$3 phase jump in reflection, a transmitted phase $${\bf\Psi}_0(z) = \exp\left(-\frac{mc}{\kappa}\int_0^z dx\,\phi_s(x)\right){\bf\chi} = \exp\left[-\frac{mc}{\lambda}\ln(\cosh \lambda z)\right]{\bf\chi},$$4, and experimentally observed localization with peak transmittance $${\bf\Psi}_0(z) = \exp\left(-\frac{mc}{\kappa}\int_0^z dx\,\phi_s(x)\right){\bf\chi} = \exp\left[-\frac{mc}{\lambda}\ln(\cosh \lambda z)\right]{\bf\chi},$$5 near $${\bf\Psi}_0(z) = \exp\left(-\frac{mc}{\kappa}\int_0^z dx\,\phi_s(x)\right){\bf\chi} = \exp\left[-\frac{mc}{\lambda}\ln(\cosh \lambda z)\right]{\bf\chi},$$6 kHz (Xia et al., 25 Apr 2025).

In topological-insulator nanowires, the effective one-dimensional mass in angular-momentum channel $${\bf\Psi}_0(z) = \exp\left(-\frac{mc}{\kappa}\int_0^z dx\,\phi_s(x)\right){\bf\chi} = \exp\left[-\frac{mc}{\lambda}\ln(\cosh \lambda z)\right]{\bf\chi},$$7 is

$${\bf\Psi}_0(z) = \exp\left(-\frac{mc}{\kappa}\int_0^z dx\,\phi_s(x)\right){\bf\chi} = \exp\left[-\frac{mc}{\lambda}\ln(\cosh \lambda z)\right]{\bf\chi},$$8

and magnetic flux tunes the mass through zero (Jana et al., 2019). The paper states that a Jackiw–Rebbi zero mode appears when the mass changes sign across a junction of different radii, and the zero-bias conductance peak can reach $${\bf\Psi}_0(z) = \exp\left(-\frac{mc}{\kappa}\int_0^z dx\,\phi_s(x)\right){\bf\chi} = \exp\left[-\frac{mc}{\lambda}\ln(\cosh \lambda z)\right]{\bf\chi},$$9 under optimal contact placement (Jana et al., 2019).

In topological Josephson junctions with magnetic islands, two opposite magnetic domains realize a sign-changing Dirac mass along a helical edge, producing a Jackiw–Rebbi resonance (Gresta et al., 2020). The source states that the thermal conductance can show a negative slope just above the superconducting critical temperature as a signature of the Jackiw–Rebbi peak (Gresta et al., 2020).

Across these platforms, the recurring invariant is the same: the interface or defect separates regions with opposite effective mass sign or opposite topological ordering. The physical degrees of freedom vary—photons, phonons, helical electrons, slow light, or proximitized nanowire modes—but the index logic is unchanged (Tran et al., 2017, Gorlach et al., 2018, Randerson et al., 4 Jun 2025, Xia et al., 25 Apr 2025, Jana et al., 2019, Gresta et al., 2020).

7. Current extensions, interpretations, and limitations

Recent work extends the Jackiw–Rebbi index beyond existence proofs for isolated zero modes. In the braneworld-inspired study of merging domain walls, the index $\alpha\beta\,{\bf\chi}=-i{\bf\chi}$0 controls the universality class of chiral-mode hybridization: the spatial separation of left- and right-handed zero modes obeys

$\alpha\beta\,{\bf\chi}=-i{\bf\chi}$1

with $\alpha\beta\,{\bf\chi}=-i{\bf\chi}$2 argued to be determined solely by the Jackiw–Rebbi topological sector (Pinheiro et al., 23 May 2026). For the sine-Gordon model, the overlap integral is derived exactly as

$\alpha\beta\,{\bf\chi}=-i{\bf\chi}$3

and comparison across sine-Gordon and double sine-Gordon models with $\alpha\beta\,{\bf\chi}=-i{\bf\chi}$4 gives $\alpha\beta\,{\bf\chi}=-i{\bf\chi}$5 (Pinheiro et al., 23 May 2026). The paper interprets this as a universal consequence of the index-fixed zero-mode structure (Pinheiro et al., 23 May 2026).

Other recent work emphasizes that not all uses of “Jackiw–Rebbi index” refer to a single explicit formula. In several condensed-matter realizations, the topological content is presented operationally through domain-wall zero modes and symmetry-protected degeneracies rather than through a standalone invariant expression (Ge et al., 19 Dec 2025, Wu et al., 2019). The provided materials repeatedly note when a paper “does not explicitly write a winding number or $\alpha\beta\,{\bf\chi}=-i{\bf\chi}$6 invariant formula” but still relies on “the standard Jackiw-Rebbi domain-wall mechanism” or “the standard Jackiw-Rebbi counting” (Ge et al., 19 Dec 2025, Jana et al., 2019).

A further limitation concerns unavailable source material. For arXiv record (Rubiano, 2019), the supplied content states that no PDF and no source are available, so no technical claims about its proposed magnetostatic analogy can be verified from the provided material (Rubiano, 2019). This means that, within the present evidence base, that work cannot be used to establish any explicit relation between the Poisson equation, one-dimensional Dirac zero modes, or a topological invariant (Rubiano, 2019).

The accumulated record nevertheless supports a coherent definition. The Jackiw–Rebbi topological index is the topological count of protected zero-energy states enforced by a sign-changing mass or order-parameter texture. In the simplest one-dimensional form, it is encoded by the asymptotic sign difference of the mass field (Angelakis et al., 2013, Pinheiro et al., 23 May 2026). In lattice and operator-theoretic language, it becomes a winding number equal to an edge Fredholm index (Thiang, 2023). In higher-dimensional free-fermion and defect problems, it is promoted to a Clifford/K-theoretic classification of admissible mass textures and defect-bound zero modes (Meetei et al., 2014, Ho et al., 2012). Its physical manifestations include charge fractionalization, fractional fermion parity, protected transport resonances, and robust interface states across photonic, acoustic, and electronic platforms (Angelakis et al., 2013, Xiong et al., 2014, Gorlach et al., 2018, Xia et al., 25 Apr 2025).