Jackiw-Rebbi Modes in Topological Systems

• Jackiw-Rebbi modes are localized fermionic states that emerge at defects where the Dirac mass changes sign, creating robust zero-energy solutions.
• They exhibit characteristic spectral features such as symmetric bound states, energy gaps, and charge fractionalization, which are essential for understanding topological transitions.
• Their generalization to higher dimensions underpins the classification of topological phases and guides experimental realizations in platforms like topological insulators, metamaterials, and superconductors.

Jackiw-Rebbi modes are localized fermionic states bound to defects or interfaces where a Dirac mass term changes sign. In the original $1+1$-dimensional setting this structure produces an unpaired zero-energy solution of a Dirac Hamiltonian with a Lorentz-scalar background, while in higher dimensions the same mass-domain-wall construction yields gapless boundary states extended along the interface. The unifying content is that the relevant mode is fixed by the asymptotic sign structure of the mass background rather than by the microscopic profile that interpolates between the two sides (Gonzalez et al., 2017, Meetei et al., 2014).

1. Defining mechanism in one dimension

The standard one-dimensional Jackiw-Rebbi problem is built from the Dirac Hamiltonian

$\hat H_D=\sigma_y \hat p+\sigma_x \varphi(x),$

with $\{\hat H_D,\sigma_z\}=0$. Because of this anticommutation relation, nonzero eigenvalues occur in $\pm \mathcal E$ pairs, so a state at $\mathcal E=0$ can be unpaired. For zero energy the Dirac equation reduces to

$$(-\partial_x+\varphi)\psi_2=0,\qquad (\partial_x+\varphi)\psi_1=0,$$

with solutions

$$\psi_1(x)=C_- \exp\!\left[-\int^x \varphi(x')\,dx'\right],\qquad \psi_2(x)=C_+ \exp\!\left[+\int^x \varphi(x')\,dx'\right].$$

For the domain-wall profile $\varphi(x)=m\,\mathrm{sgn}(x)$, one obtains

$\psi_1=C_-e^{-m|x|},\qquad \psi_2=C_+e^{+m|x|},$

and normalizability forces $C_+=0$, leaving the familiar localized zero mode

$\hat H_D=\sigma_y \hat p+\sigma_x \varphi(x),$0

The existence of this state is robust against the detailed shape of $\hat H_D=\sigma_y \hat p+\sigma_x \varphi(x),$1 so long as the scalar background asymptotically approaches opposite signs as $\hat H_D=\sigma_y \hat p+\sigma_x \varphi(x),$2 (Gonzalez et al., 2017).

That robustness is not unconditional once an explicit fermion mass is added. In the massive Jackiw-Rebbi model, where a bare mass $\hat H_D=\sigma_y \hat p+\sigma_x \varphi(x),$3 is combined with the kink background, the zero mode exists only when

$\hat H_D=\sigma_y \hat p+\sigma_x \varphi(x),$4

Equivalently, the asymptotic effective masses $\hat H_D=\sigma_y \hat p+\sigma_x \varphi(x),$5 must have opposite signs. The massive model therefore makes explicit a point that is implicit in the standard massless case: the relevant condition is asymptotic mass inversion, not merely the presence of a localized defect profile (Charmchi et al., 2014).

2. Spectral structure, exact solvability, and fractionalization

The one-dimensional model admits a complete spectral analysis. In the $\hat H_D=\sigma_y \hat p+\sigma_x \varphi(x),$6-kink background

$\hat H_D=\sigma_y \hat p+\sigma_x \varphi(x),$7

the exact bound-state energies are

$\hat H_D=\sigma_y \hat p+\sigma_x \varphi(x),$8

The full spectrum is symmetric about $\hat H_D=\sigma_y \hat p+\sigma_x \varphi(x),$9 because the model has charge and particle conjugation symmetries, and the isolated zero mode is self charge-conjugate. Weak and strong forms of the Levinson theorem were used to confirm that the number of zero-energy fermionic modes is exactly one in the standard model (Charmchi et al., 2014).

A particularly instructive result of the massive generalization is the appearance of an energy gap in the form of a triangle where no bound states exist. In that formulation the zero mode is not ever-present: it is formed from the union of two threshold bound states at $\{\hat H_D,\sigma_z\}=0$0. This sharpens the distinction between the standard massless Jackiw-Rebbi problem, where the zero mode exists for any nontrivial kink, and the massive deformation, where the kink must first overcome the bare mass scale (Charmchi et al., 2014).

The isolated zero mode is the standard origin of fractional fermion number. In the fermionic field-theory setting, the two degenerate soliton-sector ground states differ by one fermion, and charge-conjugation symmetry then fixes their charges to

$\{\hat H_D,\sigma_z\}=0$1

This is the classic Jackiw-Rebbi fractionalization mechanism (Angelakis et al., 2013). A common misconception is that every condensed-matter realization necessarily manifests this half-charge directly. That is not generally so. In a quantum spin Hall insulator constriction, for example, the zero mode is present and symmetry-protected at zero energy, but charge fractionalization is explicitly stated not to be manifested in that setup; the experimentally emphasized consequences there are interference and braiding rather than half-charge (Wu et al., 2019).

3. Higher-dimensional generalization and topological classification

The Jackiw-Rebbi construction extends naturally to arbitrary spatial dimension $\{\hat H_D,\sigma_z\}=0$2 by taking a Dirac Hamiltonian

$\{\hat H_D,\sigma_z\}=0$3

with one distinguished coordinate $\{\hat H_D,\sigma_z\}=0$4 and an odd mass profile

$\{\hat H_D,\sigma_z\}=0$5

In this form, the interface at $\{\hat H_D,\sigma_z\}=0$6 separates two gapped bulks with opposite mass sign. The localized state is no longer necessarily a pointlike zero mode: in $\{\hat H_D,\sigma_z\}=0$7 it becomes a gapless branch dispersing along the $\{\hat H_D,\sigma_z\}=0$8 directions parallel to the interface (Meetei et al., 2014).

A central organizing object in this generalized formulation is the parity operator

$\{\hat H_D,\sigma_z\}=0$9

for complex fermions, or

$\pm \mathcal E$0

in the real Majorana case, with $\pm \mathcal E$1. Determining admissible $\pm \mathcal E$2 is equivalent to determining admissible mass matrices $\pm \mathcal E$3 or $\pm \mathcal E$4, and that problem is exactly a Clifford-algebra problem. In the complex case this reproduces the free-fermion classification

$\pm \mathcal E$5

which is $\pm \mathcal E$6 for even $\pm \mathcal E$7 and $\pm \mathcal E$8 for odd $\pm \mathcal E$9. In the real case the classification is

$\mathcal E=0$0

Within this viewpoint, Jackiw-Rebbi modes are the universal boundary states of topologically distinct free-fermion insulators (Meetei et al., 2014).

The interface-state selection rule is equally important. For the zero-energy mode with $\mathcal E=0$1, one obtains

$\mathcal E=0$2

so the physical interface mode occupies one definite parity sector of the naive massless edge theory. This is the sense in which topology removes half of the naive boundary Hilbert space. The generalized construction reproduces, for example, the chiral edge mode of the integer quantum Hall effect and the helical Kramers pair of the quantum spin Hall effect within the same domain-wall framework (Meetei et al., 2014).

4. Defects, Majorana descendants, and parity fractionalization

Beyond codimension-one interfaces, Jackiw-Rebbi physics also appears for point defects in three dimensions. In the $\mathcal E=0$3D relativistic Bogoliubov-de Gennes version,

$\mathcal E=0$4

a hedgehog defect in the three-component order parameter $\mathcal E=0$5 carries a winding number

$\mathcal E=0$6

For $\mathcal E=0$7, the index theorem gives the number of zero modes exactly; when $\mathcal E=0$8, chiral symmetry is broken and generally only one zero mode survives for odd $\mathcal E=0$9 (Nishida et al., 2010).

That same paper shows that the nonrelativistic limit of the $$(-\partial_x+\varphi)\psi_2=0,\qquad (\partial_x+\varphi)\psi_1=0,$$0D Jackiw-Rebbi model produces a $$(-\partial_x+\varphi)\psi_2=0,\qquad (\partial_x+\varphi)\psi_1=0,$$1 superconductor with Hamiltonian

$$(-\partial_x+\varphi)\psi_2=0,\qquad (\partial_x+\varphi)\psi_1=0,$$2

with

$$(-\partial_x+\varphi)\psi_2=0,\qquad (\partial_x+\varphi)\psi_1=0,$$3

The defect-bound zero mode survives as a Majorana mode when $$(-\partial_x+\varphi)\psi_2=0,\qquad (\partial_x+\varphi)\psi_1=0,$$4 and $$(-\partial_x+\varphi)\psi_2=0,\qquad (\partial_x+\varphi)\psi_1=0,$$5 form a hedgehoglike structure and

$$(-\partial_x+\varphi)\psi_2=0,\qquad (\partial_x+\varphi)\psi_1=0,$$6

This establishes a direct continuum link between Jackiw-Rebbi zero modes and nonrelativistic topological superconductors (Nishida et al., 2010).

In a different superconducting extension, a Jackiw-Rebbi-type bound state localized at a domain wall can coexist with Majorana zero-energy states at the geometric ends of a one-dimensional topological superconductor. The distinctive claim there is that the domain-wall state carries not universal fractional charge but half of the parity difference between a uniform chain and a chain containing two well-separated domain walls. This fractional fermion parity enforces a topologically protected zero-energy crossing of the Jackiw-Rebbi-type bound state as a control parameter is varied (Xiong et al., 2014).

A related high-energy perspective is provided by the multi-flavored $$(-\partial_x+\varphi)\psi_2=0,\qquad (\partial_x+\varphi)\psi_1=0,$$7D Jackiw-Rebbi model in hedgehog or ’t Hooft-Polyakov monopole backgrounds. Under charge-conjugation and time-reversal constraints, K-theory predicts $$(-\partial_x+\varphi)\psi_2=0,\qquad (\partial_x+\varphi)\psi_1=0,$$8 Majorana zero modes for the SU(2)-doublet case with $$(-\partial_x+\varphi)\psi_2=0,\qquad (\partial_x+\varphi)\psi_1=0,$$9, and the explicit mode analysis realizes $$\psi_1(x)=C_- \exp\!\left[-\int^x \varphi(x')\,dx'\right],\qquad \psi_2(x)=C_+ \exp\!\left[+\int^x \varphi(x')\,dx'\right].$$0, $$\psi_1(x)=C_- \exp\!\left[-\int^x \varphi(x')\,dx'\right],\qquad \psi_2(x)=C_+ \exp\!\left[+\int^x \varphi(x')\,dx'\right].$$1, or $$\psi_1(x)=C_- \exp\!\left[-\int^x \varphi(x')\,dx'\right],\qquad \psi_2(x)=C_+ \exp\!\left[+\int^x \varphi(x')\,dx'\right].$$2 normalizable Majorana zero modes for suitable two- and four-flavor mass matrices (Ho et al., 2012).

5. Realizations and probes across physical platforms

Jackiw-Rebbi modes have been implemented or emulated in a wide range of settings. The common operational step is always the same: engineer an effective Dirac mass that reverses sign across space, then measure a midgap or interface-localized state.

Platform	Domain-wall mechanism	Probe or hallmark
One-dimensional electrostatics	$$\psi_1(x)=C_- \exp\!\left[-\int^x \varphi(x')\,dx'\right],\qquad \psi_2(x)=C_+ \exp\!\left[+\int^x \varphi(x')\,dx'\right].$$3 from a charged sheet between dielectrics	Zero-mode profile read directly from $$\psi_1(x)=C_- \exp\!\left[-\int^x \varphi(x')\,dx'\right],\qquad \psi_2(x)=C_+ \exp\!\left[+\int^x \varphi(x')\,dx'\right].$$4 (Gonzalez et al., 2017)
Slow-light simulator	Two-photon detuning $$\psi_1(x)=C_- \exp\!\left[-\int^x \varphi(x')\,dx'\right],\qquad \psi_2(x)=C_+ \exp\!\left[+\int^x \varphi(x')\,dx'\right].$$5 as optical mass kink	Midgap transmission peak in an otherwise reflecting gap (Angelakis et al., 2013)
QSHI heterostructure and ring	$$\psi_1(x)=C_- \exp\!\left[-\int^x \varphi(x')\,dx'\right],\qquad \psi_2(x)=C_+ \exp\!\left[+\int^x \varphi(x')\,dx'\right].$$6	$$\psi_1(x)=C_- \exp\!\left[-\int^x \varphi(x')\,dx'\right],\qquad \psi_2(x)=C_+ \exp\!\left[+\int^x \varphi(x')\,dx'\right].$$7 Aharonov-Bohm period at resonance; non-Abelian braiding under chiral-symmetry protection (Wu et al., 2019)
TI nanowire junction	Flux-tuned sign reversal of $$\psi_1(x)=C_- \exp\!\left[-\int^x \varphi(x')\,dx'\right],\qquad \psi_2(x)=C_+ \exp\!\left[+\int^x \varphi(x')\,dx'\right].$$8 across different radii	Zero-bias conductance peak up to $$\psi_1(x)=C_- \exp\!\left[-\int^x \varphi(x')\,dx'\right],\qquad \psi_2(x)=C_+ \exp\!\left[+\int^x \varphi(x')\,dx'\right].$$9 under optimal coupling (Jana et al., 2019)
Topological Josephson junction with magnetic domains	Magnetic mass inversion for relative angle $\varphi(x)=m\,\mathrm{sgn}(x)$0	Negative slope of thermal conductance just above $\varphi(x)=m\,\mathrm{sgn}(x)$1 for JR resonances (Gresta et al., 2020)
All-dielectric and acoustic metamaterials	Sign flip of bianisotropy, or sign flip of $\varphi(x)=m\,\mathrm{sgn}(x)$2 at an interface	Localized photonic interface state; acoustic interface state with constant phase jump (Gorlach et al., 2018, Xia et al., 25 Apr 2025)

These realizations do not all reproduce the same aspect of the original fermionic model. The electrostatic construction is explicitly a correspondence for the zero-energy sector rather than a full realization of the entire Jackiw-Rebbi spectrum (Gonzalez et al., 2017). The slow-light simulator reproduces the mass-kink Hamiltonian and uses transmission spectroscopy as a direct midgap diagnostic (Angelakis et al., 2013). The QSHI realization uses chiral symmetry rather than superconducting particle-hole symmetry, which is why its zero mode can show non-Abelian braiding while remaining distinct from a Majorana mode (Wu et al., 2019). In the TI nanowire proposal, the effective one-dimensional masses come from angular-momentum quantization and axial flux, so the domain wall is engineered geometrically rather than through a scalar soliton field (Jana et al., 2019).

The metamaterial realizations make the same mechanism visible in classical waves. In the all-dielectric chain, flipping half of the mirror-asymmetric meta-atoms reverses the sign of the effective bianisotropic mass term and creates a photonic Jackiw-Rebbi interface state (Gorlach et al., 2018). In the acoustic metagrating, the effective mass is

$\varphi(x)=m\,\mathrm{sgn}(x)$3

and the interface state is tied to a real-space topological invariant $\varphi(x)=m\,\mathrm{sgn}(x)$4, robust imaginary band degeneracy, and a $\varphi(x)=m\,\mathrm{sgn}(x)$5 reflected-phase jump (Xia et al., 25 Apr 2025).

6. Hybridization, coherent dynamics, and current extensions

When more than one Jackiw-Rebbi mode is present, overlap lifts exact zero-energy degeneracy and produces coupled-mode physics. In a modified SSH chain with a four-site unit cell, the low-energy Dirac theory around $\varphi(x)=m\,\mathrm{sgn}(x)$6 gives an effective mass

$\varphi(x)=m\,\mathrm{sgn}(x)$7

which changes sign twice when $\varphi(x)=m\,\mathrm{sgn}(x)$8. The two resulting Jackiw-Rebbi modes hybridize into symmetric and antisymmetric combinations with energies $\varphi(x)=m\,\mathrm{sgn}(x)$9, and the occupation oscillates coherently between the two interfaces with period

$\psi_1=C_-e^{-m|x|},\qquad \psi_2=C_+e^{+m|x|},$0

The same work makes an important negative point: the quadratic gap closing at $\psi_1=C_-e^{-m|x|},\qquad \psi_2=C_+e^{+m|x|},$1 does not support a topological domain wall because the induced mass there is $\psi_1=C_-e^{-m|x|},\qquad \psi_2=C_+e^{+m|x|},$2, which never changes sign (Mandal, 1 Jul 2026).

Hybridization also governs the loss of chirality localization when two domain walls merge. In a $\psi_1=C_-e^{-m|x|},\qquad \psi_2=C_+e^{+m|x|},$3-dimensional Jackiw-Rebbi setting used as a braneworld analogue, the spatial separation between the left- and right-handed localized combinations scales as

$\psi_1=C_-e^{-m|x|},\qquad \psi_2=C_+e^{+m|x|},$4

For the five studied $\psi_1=C_-e^{-m|x|},\qquad \psi_2=C_+e^{+m|x|},$5 models, the reported values lie in

$\psi_1=C_-e^{-m|x|},\qquad \psi_2=C_+e^{+m|x|},$6

and for sine-Gordon the exact overlap integral is

$\psi_1=C_-e^{-m|x|},\qquad \psi_2=C_+e^{+m|x|},$7

The paper interprets $\psi_1=C_-e^{-m|x|},\qquad \psi_2=C_+e^{+m|x|},$8 as the crossover plateau of a local effective exponent and argues that the rate of chirality loss is controlled by the Jackiw-Rebbi index rather than by integrability, mass gap, or detailed wall profile (Pinheiro et al., 23 May 2026).

A fully dynamical extension appears in a trapped-ion simulator where the kink itself is not frozen but realized by the soft zigzag mode of an ion crystal. There the Jackiw-Rebbi zero mode still carries fractional fermion number $\psi_1=C_-e^{-m|x|},\qquad \psi_2=C_+e^{+m|x|},$9, but fermion back-reaction modifies the Peierls-Nabarro potential, can pin the kink, suppress its quantum spreading, and qualitatively affect kink-antikink collisions. In that setting the fractionalized charge can remain attached to a moving kink or, under nonadiabatic collisions and large fermionic bandwidth, partially decouple and propagate away as fast fermionic wavefronts (Kahan et al., 8 Dec 2025).

Current extensions also broaden the underlying mathematics and wave physics. Non-Hermitian dielectric gratings support coupled radiating Jackiw-Rebbi-like resonances whose hybridization splits not only the real part of the spectrum but also the radiative loss, producing a sub-radiant bonding mode and a super-radiant antibonding mode with distinct far-field emission (Wang et al., 24 Feb 2025). A bosonic analog replaces the fermionic mass profile by a field-dependent kinetic function $C_+=0$0; the sign change of

$C_+=0$1

plays the role of the Jackiw-Rebbi mass inversion and selects which bosonic $C_+=0$2D component remains massless on the wall (Arai et al., 2018). At a more formal level, a $C_+=0$3-adic version replaces the real line by $C_+=0$4 and the Dirac Hamiltonian by a non-local operator on $C_+=0$5, yielding localized wavefunctions and long-range interactions while aiming to reproduce the same predictions as the standard model (Zúñiga-Galindo, 17 Mar 2026).

Taken together, these developments show that “Jackiw-Rebbi modes” now denotes more than the original one-dimensional zero mode. The term covers a family of defect-bound states generated by sign-changing Dirac masses or their precise analogs, ranging from solitonic zero modes and higher-dimensional boundary branches to Majorana descendants, radiating non-Hermitian hybrids, bosonic wall modes, and mathematically nonlocal generalizations. What remains invariant across these forms is the domain-wall logic: asymptotically distinct gapped sectors force a localized state whose existence is fixed by the topology of the defect configuration rather than by microscopic interpolation details.