Majorana Edge Reconstruction

• Majorana edge reconstruction is the reorganization of boundary Majorana modes under perturbations like magnetic fields, dissipation, or finite-size effects.
• It leads to composite edge structures such as spin–charge-separated states where Majorana zero modes couple with fractional fermions.
• This phenomena reshapes observable properties including edge dispersion, velocity, and transport signatures, with implications for engineered non-Abelian operations.

Searching arXiv for papers on Majorana edge reconstruction and closely related edge Majorana phenomena. Majorana edge reconstruction denotes a reorganization of boundary Majorana degrees of freedom caused by additional structure beyond the minimal topological-superconductor picture. In different parts of the literature, the term is used in several distinct but related senses: a Majorana end mode can be reconstructed into a composite spin–charge-separated boundary object; a chiral Majorana edge channel can have its dispersion, velocity, and transport spectrum reshaped by magnetic field or chemical potential; a nominal Majorana edge state can be reconfigured by finite size, dissipation, or periodic driving; and a non-Abelian quantum Hall edge can undergo a reconstruction confined to the neutral Majorana sector while leaving the charge sector unchanged (Zhang et al., 2016, Tiwari et al., 2014, Ezawa, 2023, Thakurathi et al., 2013, Lotrič et al., 9 Jul 2025).

1. Conceptual scope

In the most general usage, Majorana edge reconstruction refers to a change in the structure of edge-localized Majorana modes under perturbations that do not simply remove the boundary degrees of freedom, but reorganize them. The reorganization may involve their quantum numbers, their spatial support, the number of effective edge channels, or the way they couple to other boundary excitations. The literature contains at least four technically distinct realizations.

First, in interacting one-dimensional fermion systems, a Majorana edge mode can be reconstructed into a composite spin–charge-separated boundary structure, with the spin sector carrying Majorana zero modes and the charge sector carrying inversion-protected fractional fermions (Zhang et al., 2016). Second, in superconducting hybrid structures, the boundary Majorana channel can be reconstructed at the single-particle level by tuning the magnetic field, chemical potential, confinement, or magnetic texture, which changes its dispersion, localization length, and edge transport signatures (Tiwari et al., 2014, Akzyanov et al., 2015, Chen et al., 2015). Third, in finite or driven systems, the reconstruction appears as a reorganization of edge spectra under finite-size hybridization, dissipation, or Floquet driving, including the appearance and disappearance of edge zero modes or the conversion of exact zero modes into finite-energy resonances (Ezawa, 2023, Thakurathi et al., 2013). Fourth, in reconstructed quantum Hall edges, the Majorana sector itself can reconstruct while the charge sector remains fixed; in the recent $\nu=5/2$ context, this possibility was proposed as a resolution of the non-Abelian thermal Hall puzzle (Iyer et al., 2023, Lotrič et al., 9 Jul 2025).

A recurring theme is that edge reconstruction is not synonymous with the destruction of Majorana physics. In several settings, the reconstructed edge retains topological degeneracy, but the observable boundary theory differs from the minimal Kitaev-chain or chiral-$p$-wave expectation. This suggests that the boundary theory can be more structurally elaborate than the bulk topological label alone would indicate.

2. Spin–charge-separated reconstruction in one dimension

A particularly explicit many-body realization appears in a one-dimensional spin-$1/2$ fermion chain with pair-hopping interactions at half filling, described by the lattice Hamiltonian

$\begin{aligned} H_{\text{lat} &= \sum_{j,\sigma} \big[-t\big(c_{j\sigma}^\dagger c_{j+1\sigma} + \text{h.c.}\big) - \mu\, c_{j\sigma}^\dagger c_{j\sigma}\big] \ &\quad + W_1 \sum_j \big(c_{j\uparrow}^\dagger c_{j\downarrow}^\dagger c_{j+1\downarrow} c_{j+1\uparrow} + \text{h.c.}\big) \ &\quad + W_2 \sum_j \big( c_j^\dagger c_{j+1}^\dagger c_{j+1} c_j + \text{h.c.}\big), \end{aligned}$

with $c_j=(c_{j\uparrow},c_{j\downarrow})^T$. Functional RG and mean-field theory identify a bond-centered spin-density-wave order, yielding an effective Hamiltonian that decomposes into two SSH-like chains in the eigenbasis of $\sigma_x$. One sector is topological and the other trivial, and at the special point $t=|M|$ the topological sector contains two dangling edge sites hosting exactly localized zero modes (Zhang et al., 2016).

The low-energy field theory separates into a charge sector and a spin sector. At the Luther–Emery point, the charge sector refermionizes to an SSH model,

$$H_c = \int dx\, \chi_c^\dagger(-iv_c\partial_x\tau_3 - g_c \tau_2)\chi_c,$$

while the spin sector refermionizes to a Kitaev-chain form,

$$H_s = \int dx \left\{ \chi_s^\dagger(-iv_s\partial_x\tau_3)\chi_s - \frac{h_s}{2}\big[\chi_s^\dagger \tau_2 (\chi_s^\dagger)^T + \text{h.c.}\big]\right\}.$$

The boundary then carries two distinct zero-mode structures: SSH-like fractional-fermion operators $F_{L,R}$ in the charge sector and Majorana operators $p$0 in the spin sector (Zhang et al., 2016).

In this formulation, each edge supports a local fractional charge $p$1 and participates in a nonlocal pair of spin-sector Majorana zero modes. The fourfold ground-state degeneracy is the product of a twofold fractional-fermion occupancy and a twofold spin-parity degeneracy. The edge degrees of freedom can therefore be reorganized as

$p$2

The paper characterizes this as a reconstruction of a “simple” Majorana edge mode into a composite object: a spin Majorana mode “decorated” by a fractional charge localized at one edge (Zhang et al., 2016).

Symmetry protection is correspondingly factorized. The charge-sector fractional fermions are protected by inversion $p$3, while the spin-sector Majoranas are protected by spin parity $p$4. Breaking inversion destabilizes the fractional fermions; breaking spin parity allows the Majoranas to hybridize. In this sense the fourfold edge structure is a product SPT, $p$5 from inversion and $p$6 from spin parity (Zhang et al., 2016).

The same work also maps the Haldane phase of a spin-1 XXZ chain to a constrained spin-$p$7 fermion model and argues that the familiar spin-$p$8 edge spinons may be reinterpreted as spin–charge-separated edge zero modes, with Majorana modes in one sector and fractional fermions in the other after exchanging $p$9. This provides a fermionic reinterpretation of Haldane-edge physics as another form of Majorana edge reconstruction (Zhang et al., 2016).

3. Boundary-spectrum reconstruction by confinement, fields, and textures

In superconducting hybrid structures, the phrase often refers to a reconfiguration of the edge spectrum rather than to symmetry-fractionalized many-body edge content. A representative example is a superconductor–topological-insulator interface in a perpendicular magnetic field. There the boundary between a proximitized superconducting region and a Landau-quantized normal region supports a chiral Majorana mode governed by a Dirac–Bogoliubov–de Gennes Hamiltonian. The edge velocity

$1/2$0

is not fixed: it depends sensitively on $1/2$1, $1/2$2, and $1/2$3. At $1/2$4, the zeroth Landau level is flat and $1/2$5; increasing $1/2$6 makes the dispersion near $1/2$7 increasingly linear and steep. When $1/2$8 is large enough, higher Landau levels enter the low-energy window and multiple edge modes appear. In that setting, edge reconstruction means changing the dispersion $1/2$9, the edge velocity $\begin{aligned} H_{\text{lat} &= \sum_{j,\sigma} \big[-t\big(c_{j\sigma}^\dagger c_{j+1\sigma} + \text{h.c.}\big) - \mu\, c_{j\sigma}^\dagger c_{j\sigma}\big] \ &\quad + W_1 \sum_j \big(c_{j\uparrow}^\dagger c_{j\downarrow}^\dagger c_{j+1\downarrow} c_{j+1\uparrow} + \text{h.c.}\big) \ &\quad + W_2 \sum_j \big( c_j^\dagger c_{j+1}^\dagger c_{j+1} c_j + \text{h.c.}\big), \end{aligned}$0, the effective level spacing around the superconducting perimeter, and therefore the pattern of resonant Andreev-reflection peaks in transport (Tiwari et al., 2014).

A related geometric realization occurs for a superconducting island on a topological-insulator surface in a magnetic field. For odd vorticity $\begin{aligned} H_{\text{lat} &= \sum_{j,\sigma} \big[-t\big(c_{j\sigma}^\dagger c_{j+1\sigma} + \text{h.c.}\big) - \mu\, c_{j\sigma}^\dagger c_{j\sigma}\big] \ &\quad + W_1 \sum_j \big(c_{j\uparrow}^\dagger c_{j\downarrow}^\dagger c_{j+1\downarrow} c_{j+1\uparrow} + \text{h.c.}\big) \ &\quad + W_2 \sum_j \big( c_j^\dagger c_{j+1}^\dagger c_{j+1} c_j + \text{h.c.}\big), \end{aligned}$1, one Majorana mode is localized near the vortex core and another near the island edge. Their overlap produces a splitting

$\begin{aligned} H_{\text{lat} &= \sum_{j,\sigma} \big[-t\big(c_{j\sigma}^\dagger c_{j+1\sigma} + \text{h.c.}\big) - \mu\, c_{j\sigma}^\dagger c_{j\sigma}\big] \ &\quad + W_1 \sum_j \big(c_{j\uparrow}^\dagger c_{j\downarrow}^\dagger c_{j+1\downarrow} c_{j+1\uparrow} + \text{h.c.}\big) \ &\quad + W_2 \sum_j \big( c_j^\dagger c_{j+1}^\dagger c_{j+1} c_j + \text{h.c.}\big), \end{aligned}$2

As the island radius $\begin{aligned} H_{\text{lat} &= \sum_{j,\sigma} \big[-t\big(c_{j\sigma}^\dagger c_{j+1\sigma} + \text{h.c.}\big) - \mu\, c_{j\sigma}^\dagger c_{j\sigma}\big] \ &\quad + W_1 \sum_j \big(c_{j\uparrow}^\dagger c_{j\downarrow}^\dagger c_{j+1\downarrow} c_{j+1\uparrow} + \text{h.c.}\big) \ &\quad + W_2 \sum_j \big( c_j^\dagger c_{j+1}^\dagger c_{j+1} c_j + \text{h.c.}\big), \end{aligned}$3 changes, the system crosses between a regime of strong core–edge hybridization, a regime of an isolated edge Majorana with a sizable edge minigap, and a regime with many edge subgap states whose spacing scales as $\begin{aligned} H_{\text{lat} &= \sum_{j,\sigma} \big[-t\big(c_{j\sigma}^\dagger c_{j+1\sigma} + \text{h.c.}\big) - \mu\, c_{j\sigma}^\dagger c_{j\sigma}\big] \ &\quad + W_1 \sum_j \big(c_{j\uparrow}^\dagger c_{j\downarrow}^\dagger c_{j+1\downarrow} c_{j+1\uparrow} + \text{h.c.}\big) \ &\quad + W_2 \sum_j \big( c_j^\dagger c_{j+1}^\dagger c_{j+1} c_j + \text{h.c.}\big), \end{aligned}$4. In that sense the low-energy edge theory is reconstructed by geometry and magnetic field, even though the parity condition for Majorana existence remains fixed by odd vorticity (Akzyanov et al., 2015).

Magnetic-texture engineering gives another meaning to the term. At interfaces between an $\begin{aligned} H_{\text{lat} &= \sum_{j,\sigma} \big[-t\big(c_{j\sigma}^\dagger c_{j+1\sigma} + \text{h.c.}\big) - \mu\, c_{j\sigma}^\dagger c_{j\sigma}\big] \ &\quad + W_1 \sum_j \big(c_{j\uparrow}^\dagger c_{j\downarrow}^\dagger c_{j+1\downarrow} c_{j+1\uparrow} + \text{h.c.}\big) \ &\quad + W_2 \sum_j \big( c_j^\dagger c_{j+1}^\dagger c_{j+1} c_j + \text{h.c.}\big), \end{aligned}$5-wave superconducting thin film and a noncollinear magnetic insulator, a local spin rotation converts the exchange texture into spin-dependent hopping. In the large-$\begin{aligned} H_{\text{lat} &= \sum_{j,\sigma} \big[-t\big(c_{j\sigma}^\dagger c_{j+1\sigma} + \text{h.c.}\big) - \mu\, c_{j\sigma}^\dagger c_{j\sigma}\big] \ &\quad + W_1 \sum_j \big(c_{j\uparrow}^\dagger c_{j\downarrow}^\dagger c_{j+1\downarrow} c_{j+1\uparrow} + \text{h.c.}\big) \ &\quad + W_2 \sum_j \big( c_j^\dagger c_{j+1}^\dagger c_{j+1} c_j + \text{h.c.}\big), \end{aligned}$6 regime, the low-energy theory reduces to a spinless $\begin{aligned} H_{\text{lat} &= \sum_{j,\sigma} \big[-t\big(c_{j\sigma}^\dagger c_{j+1\sigma} + \text{h.c.}\big) - \mu\, c_{j\sigma}^\dagger c_{j\sigma}\big] \ &\quad + W_1 \sum_j \big(c_{j\uparrow}^\dagger c_{j\downarrow}^\dagger c_{j+1\downarrow} c_{j+1\uparrow} + \text{h.c.}\big) \ &\quad + W_2 \sum_j \big( c_j^\dagger c_{j+1}^\dagger c_{j+1} c_j + \text{h.c.}\big), \end{aligned}$7-wave superconductor. Cycloidal, helical, conical, tilted-conical, Bloch-wall, and Néel-wall textures induce $\begin{aligned} H_{\text{lat} &= \sum_{j,\sigma} \big[-t\big(c_{j\sigma}^\dagger c_{j+1\sigma} + \text{h.c.}\big) - \mu\, c_{j\sigma}^\dagger c_{j\sigma}\big] \ &\quad + W_1 \sum_j \big(c_{j\uparrow}^\dagger c_{j\downarrow}^\dagger c_{j+1\downarrow} c_{j+1\uparrow} + \text{h.c.}\big) \ &\quad + W_2 \sum_j \big( c_j^\dagger c_{j+1}^\dagger c_{j+1} c_j + \text{h.c.}\big), \end{aligned}$8-wave pairing and support Majorana edge modes without adjusting the chemical potential, while a skyrmion texture induces a $\begin{aligned} H_{\text{lat} &= \sum_{j,\sigma} \big[-t\big(c_{j\sigma}^\dagger c_{j+1\sigma} + \text{h.c.}\big) - \mu\, c_{j\sigma}^\dagger c_{j\sigma}\big] \ &\quad + W_1 \sum_j \big(c_{j\uparrow}^\dagger c_{j\downarrow}^\dagger c_{j+1\downarrow} c_{j+1\uparrow} + \text{h.c.}\big) \ &\quad + W_2 \sum_j \big( c_j^\dagger c_{j+1}^\dagger c_{j+1} c_j + \text{h.c.}\big), \end{aligned}$9-wave-like state with a bulk persistent current and a topological edge current. Changing the magnetic texture changes the momentum range of the edge modes, their localization profile, and the accompanying current pattern, which amounts to a reconstruction of the edge Majorana sector by the interfacial magnetic order (Chen et al., 2015).

A further boundary-engineering realization is provided by zigzag edges of monolayer transition-metal dichalcogenides. The bulk is topologically trivial, but the chalcogen-terminated zigzag edge hosts a single isolated metallic band with strong spin–orbit coupling. Adding proximity-induced $c_j=(c_{j\uparrow},c_{j\downarrow})^T$0-wave pairing and an in-plane Zeeman field reconstructs this electronic edge into an effective one-dimensional topological superconductor, with Majorana bound states at the ends of the edge segment when $c_j=(c_{j\uparrow},c_{j\downarrow})^T$1. In this usage, the reconstruction is an engineered conversion of a trivial edge channel into a Majorana-supporting one-dimensional topological boundary (Chu et al., 2013).

4. Finite-size, dissipative, and Floquet reconstructions

Finite systems provide a controlled setting in which edge reconstruction becomes a spectral problem. For a finite Kitaev chain, an isospectral matrix reduction yields an effective two-band Majorana-edge Hamiltonian

$c_j=(c_{j\uparrow},c_{j\downarrow})^T$2

whose spectrum is $c_j=(c_{j\uparrow},c_{j\downarrow})^T$3. This quantity $c_j=(c_{j\uparrow},c_{j\downarrow})^T$4 encodes the overlap of the two edge Majoranas through bulk virtual processes. At $c_j=(c_{j\uparrow},c_{j\downarrow})^T$5,

$c_j=(c_{j\uparrow},c_{j\downarrow})^T$6

while at $c_j=(c_{j\uparrow},c_{j\downarrow})^T$7, $c_j=(c_{j\uparrow},c_{j\downarrow})^T$8 for odd $c_j=(c_{j\uparrow},c_{j\downarrow})^T$9 and $\sigma_x$0 for even $\sigma_x$1. Moving away from the topological sweet spot therefore reconstructs the edge spectrum from exact or exponentially sharp zero modes into finite-energy Andreev-like edge states (Ezawa, 2023).

Open-system coupling yields a non-Hermitian version of the same phenomenon. For local dissipation, the edge-mode energies acquire

$\sigma_x$2

so the shift is parametrically suppressed with chain length. For global dissipation,

$\sigma_x$3

independent of $\sigma_x$4, which makes the Majorana edges fragile. The paper therefore distinguishes weak reconstruction of the edge resonance under local dissipation from strong reconstruction under global dissipation (Ezawa, 2023).

Periodic driving can reconstruct edge modes even when the corresponding equilibrium edge is trivial. In the Kitaev honeycomb model with a periodic $\sigma_x$5-function kick of the $\sigma_x$6 bond, the Floquet operator can generate edge Majorana modes on zigzag edges in parameter regimes where the equilibrium Hamiltonian supports only armchair-edge modes. The critical frequencies satisfy

$\sigma_x$7

together with

$\sigma_x$8

Crossing these lines changes the number and character of Floquet edge modes, including the appearance of both $\sigma_x$9- and $t=|M|$0-Majoranas, which is a distinctly dynamical form of edge reconstruction (Thakurathi et al., 2013).

An exactly solvable one-dimensional spin model derived from a spin-orbit-coupled three-band Hubbard system exhibits a different finite-size mechanism. Its edge Majoranas are protected by a $t=|M|$1 invariant tied to lattice parity: $t=|M|$2 For odd $t=|M|$3, exact Majorana end modes survive at zero energy; for even $t=|M|$4, the end states reconstruct into finite-energy bound states. In the same model, adiabatically moving a domain wall reconstructs the spatial support of the edge Majoranas in a way that realizes braiding in a strictly geometric one-dimensional chain (Dong et al., 2014).

5. Reconstructed quantum Hall edges and the neutral Majorana sector

In quantum Hall systems, edge reconstruction has long referred to smooth-edge charge rearrangement, but recent work isolates a version that acts solely in the neutral Majorana sector. A useful precursor is the reconstructed $t=|M|$5 edge with an adjacent $t=|M|$6 side strip. When alternating superconducting and ferromagnetic gaps are imposed, a naive parafermion expectation is defeated by bulk constraints requiring integer charge and spin in each proximitized segment. The would-be parafermionic sector is reduced to an additional Majorana sector, so each SC–FM domain wall hosts two decoupled Majoranas rather than a single parafermion. The resulting topological degeneracy is $t=|M|$7, and the Josephson spectrum remains $t=|M|$8-periodic, with extra structure visible when the edge velocities $t=|M|$9 and $$H_c = \int dx\, \chi_c^\dagger(-iv_c\partial_x\tau_3 - g_c \tau_2)\chi_c,$$0 differ (Iyer et al., 2023).

A more recent and more explicit use of the phrase appears in the $$H_c = \int dx\, \chi_c^\dagger(-iv_c\partial_x\tau_3 - g_c \tau_2)\chi_c,$$1 thermal Hall context. There the experimentally observed $$H_c = \int dx\, \chi_c^\dagger(-iv_c\partial_x\tau_3 - g_c \tau_2)\chi_c,$$2 matches the PH-Pfaffian edge theory, even though bulk numerics favor Pfaffian or AntiPfaffian order. The proposal is that the charge sector remains fixed as a single downstream chiral boson with

$$H_c = \int dx\, \chi_c^\dagger(-iv_c\partial_x\tau_3 - g_c \tau_2)\chi_c,$$3

while only the neutral Majorana sector reconstructs (Lotrič et al., 9 Jul 2025).

The proposed mechanism involves an edge region of width $$H_c = \int dx\, \chi_c^\dagger(-iv_c\partial_x\tau_3 - g_c \tau_2)\chi_c,$$4 containing alternating Pf and aPf stripes, with density oscillations of period $$H_c = \int dx\, \chi_c^\dagger(-iv_c\partial_x\tau_3 - g_c \tau_2)\chi_c,$$5. DMRG on an infinite cylinder with screened Coulomb interactions and a smooth gate-defined confining potential finds both those density oscillations and an orbital entanglement spectrum whose chirality alternates with the density, consistent with successive Pf and aPf regions. Each Pf/aPf interface carries four co-propagating Majorana modes, and the coupled interface network is described by a quadratic Majorana Hamiltonian

$$H_c = \int dx\, \chi_c^\dagger(-iv_c\partial_x\tau_3 - g_c \tau_2)\chi_c,$$6

In a broad region of coupling space, this interface network has a class-D Chern number $$H_c = \int dx\, \chi_c^\dagger(-iv_c\partial_x\tau_3 - g_c \tau_2)\chi_c,$$7, so two chiral Majorana modes remain near the bulk side of the reconstructed region while the physical edge itself retains only a single upstream Majorana, producing an effective PH-Pfaffian-like edge theory (Lotrič et al., 9 Jul 2025).

The important point is that bulk–boundary correspondence is satisfied by the composite structure: the physical edge plus the “deep” neutral modes together reproduce the Pf or aPf bulk order, yet transport probes that couple only to the outer boundary see the PH-Pfaffian signature. This proposal therefore treats Majorana edge reconstruction not as a correction to equilibration, but as an actual reorganization of neutral edge degrees of freedom over a mesoscopic length scale (Lotrič et al., 9 Jul 2025).

6. Detection, manipulation, and implications

Several observables recur across these realizations. In superconducting topological-insulator hybrids, the phase accumulated by a chiral Majorana loop,

$$H_c = \int dx\, \chi_c^\dagger(-iv_c\partial_x\tau_3 - g_c \tau_2)\chi_c,$$8

produces Majorana-induced resonant Andreev reflection, so reconstructing the edge velocity $$H_c = \int dx\, \chi_c^\dagger(-iv_c\partial_x\tau_3 - g_c \tau_2)\chi_c,$$9 with $$H_s = \int dx \left\{ \chi_s^\dagger(-iv_s\partial_x\tau_3)\chi_s - \frac{h_s}{2}\big[\chi_s^\dagger \tau_2 (\chi_s^\dagger)^T + \text{h.c.}\big]\right\}.$$0 or $$H_s = \int dx \left\{ \chi_s^\dagger(-iv_s\partial_x\tau_3)\chi_s - \frac{h_s}{2}\big[\chi_s^\dagger \tau_2 (\chi_s^\dagger)^T + \text{h.c.}\big]\right\}.$$1 shifts the differential-conductance peaks (Tiwari et al., 2014). For superconducting islands on topological insulators, the edge Majorana is distinguished from ordinary Dirac edge states by stability against gate-voltage variations and by sensitivity to odd versus even vorticity (Akzyanov et al., 2015). In TMD zigzag-edge platforms, the reconstruction into a one-dimensional topological superconductor implies standard end-Majorana probes such as zero-bias anomalies and length-dependent splitting (Chu et al., 2013).

Edge reconstruction can also be operational rather than merely spectral. In a finite $$H_s = \int dx \left\{ \chi_s^\dagger(-iv_s\partial_x\tau_3)\chi_s - \frac{h_s}{2}\big[\chi_s^\dagger \tau_2 (\chi_s^\dagger)^T + \text{h.c.}\big]\right\}.$$2-type superconducting heterostructure with a vortex, tuning point-like constriction junctions changes which bricks share a common boundary. The edge Majorana then spreads over, splits among, or localizes on different connected edge segments. This controlled reconfiguration enables generation, fusion, transport, and braiding of edge Majorana fermions by gate voltages alone, with the time-dependent Bogoliubov–de Gennes dynamics reproducing the non-Abelian exchange rule

$$H_s = \int dx \left\{ \chi_s^\dagger(-iv_s\partial_x\tau_3)\chi_s - \frac{h_s}{2}\big[\chi_s^\dagger \tau_2 (\chi_s^\dagger)^T + \text{h.c.}\big]\right\}.$$3

In that setting, Majorana edge reconstruction is literally the programmable reconfiguration of the edge graph supporting the zero modes (Liang et al., 2011).

In Kitaev magnets, Majorana edge modes can be detected indirectly through “Majorana edge magnetization.” For zigzag edges in the $$H_s = \int dx \left\{ \chi_s^\dagger(-iv_s\partial_x\tau_3)\chi_s - \frac{h_s}{2}\big[\chi_s^\dagger \tau_2 (\chi_s^\dagger)^T + \text{h.c.}\big]\right\}.$$4 phase, the flat band of edge zero modes can be localized into Wannier Majoranas $$H_s = \int dx \left\{ \chi_s^\dagger(-iv_s\partial_x\tau_3)\chi_s - \frac{h_s}{2}\big[\chi_s^\dagger \tau_2 (\chi_s^\dagger)^T + \text{h.c.}\big]\right\}.$$5, and a weak edge field selects a ground-state superposition with a nonzero magnetization in only one spin direction. The resulting edge spin behaves as a peculiar free spin with unidirectional response, and changes in edge geometry or symmetry class reconstruct that response by reconstructing the underlying flat band (Mizoguchi et al., 2018).

Taken together, these works imply that the edge theory of a Majorana-supporting phase is often more malleable than the minimal bulk label suggests. Reconstruction can be driven by interactions, symmetry, electrostatics, magnetic texture, dissipation, or geometry; it can enrich, screen, relocate, or gap the nominal edge Majorana sector; and in several cases it supplies the experimentally relevant boundary theory even when the bulk topological order remains unchanged (Zhang et al., 2016, Lotrič et al., 9 Jul 2025).