Extended Kitaev Chain Models

• Extended Kitaev chain is a one-dimensional model that generalizes spinless p-wave superconductivity by incorporating longer-range hopping, pairing, and inhomogeneities.
• The model exhibits rich topological features including multiple Majorana zero modes, massive edge states, and a complex bulk-edge correspondence sensitive to symmetry and interaction range.
• Analytical and numerical methods, such as winding numbers and finite-size scaling, provide practical diagnostics for phase boundaries and edge localization in these extended systems.

The extended Kitaev chain is a family of one-dimensional Kitaev-chain models in which the standard spinless $p$-wave superconducting chain is generalized beyond the nearest-neighbor, uniform, Hermitian setting. In the literature cited here, the extension appears in several distinct but related forms: longer-range hopping and pairing, next-nearest-neighbor terms, dimerized or trimerized unit cells, spatially varying bond phases, quantum-dot or Andreev-bound-state inhomogeneities, and non-Hermitian deformations. The common consequence is a richer bulk-edge structure than in the basic nearest-neighbor chain, including multiple topological sectors, multicriticality, massive edge modes, non-endpoint Majorana bound states, and, in broader spin-chain usages, Jordan–Wigner-related Majorana descriptions of bond-dependent spin models (Alecce et al., 2017, Rahul et al., 2023, He et al., 2023).

1. Canonical extended-range formulation

A canonical fermionic definition is the spinless $p$-wave superconducting chain with couplings beyond nearest neighbors,

$$H= -\sum_{j=1}^{L}\mu\left(a_j^\dagger a_j-\frac12\right) +\sum_{\ell=1}^{r}\sum_{j=1}^{L-\ell} \left( -w_\ell e^{i\varphi_\ell} a_j^\dagger a_{j+\ell} +\Delta_\ell a_j a_{j+\ell} +\text{h.c.} \right),$$

where $r$ is the maximum number of coupled neighbors, $w_\ell$ and $\Delta_\ell$ are hopping and pairing amplitudes, and $\varphi_\ell$ can break time-reversal symmetry (Alecce et al., 2017). A closely related truncated-range formulation writes

$H = - \mu \sum_{j = 1}^N c_j^\dagger c_j^{\,} - \sum_{j = 1}^{N - 1} \sum_{\ell =1}^{q} \big( t_\ell^{} c_j^\dagger c_{j+\ell}^{\,} - \Delta_\ell^{} c_{j}^\dagger c_{j+\ell}^\dagger + \mbox{H.c.} \big),$

with algebraic couplings

$$\Delta_\ell = \Delta \left(\frac{1}{\ell}\right)^\alpha, \qquad t_\ell = t \left(\frac{1}{\ell}\right)^\beta,$$

so that $\alpha$ and $p$0 control the decay of pairing and hopping (Widniczck et al., 8 Jun 2026).

Within this range-extended setting, the truncated-range scenario has as many distinct topological phases as the number of coupled neighboring sites (Widniczck et al., 8 Jun 2026). A particularly important special case is the extended Kitaev chain with next-nearest-neighbor hopping and pairing, where the real-space Hamiltonian includes nearest-neighbor amplitudes $p$1, next-nearest-neighbor amplitudes $p$2, and, in the non-Hermitian deformation, a complex chemical potential $p$3 (Rahul et al., 2023). In that model the next-nearest-neighbor term enlarges the topological structure from the usual two-phase picture of the basic Kitaev chain to three phases with winding numbers

$p$4

(Rahul et al., 2023).

This family already shows that “extended” is not merely quantitative. Adding further-neighbor couplings changes the allowed winding sectors, the number of phase boundaries, and the multiplicity of edge modes. In the short-range limit $p$5, these formulations reduce to the standard nearest-neighbor Kitaev chain (Alecce et al., 2017, Widniczck et al., 8 Jun 2026).

2. Bulk topology and phase structure

For the finite-range time-reversal-symmetric chain, the Bloch/BdG Hamiltonian can be written as

$p$6

with

$p$7

and the corresponding BDI invariant is the winding number

$p$8

(Alecce et al., 2017). When time-reversal symmetry is broken by $p$9, the symmetry class reduces to class D and the relevant invariant becomes

$$H= -\sum_{j=1}^{L}\mu\left(a_j^\dagger a_j-\frac12\right) +\sum_{\ell=1}^{r}\sum_{j=1}^{L-\ell} \left( -w_\ell e^{i\varphi_\ell} a_j^\dagger a_{j+\ell} +\Delta_\ell a_j a_{j+\ell} +\text{h.c.} \right),$$0

(Alecce et al., 2017).

A central finite-range result is that, with time-reversal symmetry preserved and with both hopping and pairing extended to $$H= -\sum_{j=1}^{L}\mu\left(a_j^\dagger a_j-\frac12\right) +\sum_{\ell=1}^{r}\sum_{j=1}^{L-\ell} \left( -w_\ell e^{i\varphi_\ell} a_j^\dagger a_{j+\ell} +\Delta_\ell a_j a_{j+\ell} +\text{h.c.} \right),$$1 neighbors, one can have as many Majorana zero modes per edge as the number of coupled neighbors $$H= -\sum_{j=1}^{L}\mu\left(a_j^\dagger a_j-\frac12\right) +\sum_{\ell=1}^{r}\sum_{j=1}^{L-\ell} \left( -w_\ell e^{i\varphi_\ell} a_j^\dagger a_{j+\ell} +\Delta_\ell a_j a_{j+\ell} +\text{h.c.} \right),$$2 (Alecce et al., 2017). By contrast, if only hopping or only pairing is extended, one still gets at most one Majorana zero mode per edge (Alecce et al., 2017). In the specific uniform-coupling case analyzed there, the winding number satisfies

$$H= -\sum_{j=1}^{L}\mu\left(a_j^\dagger a_j-\frac12\right) +\sum_{\ell=1}^{r}\sum_{j=1}^{L-\ell} \left( -w_\ell e^{i\varphi_\ell} a_j^\dagger a_{j+\ell} +\Delta_\ell a_j a_{j+\ell} +\text{h.c.} \right),$$3

(Alecce et al., 2017).

Dimerized extensions reorganize the same topological content in a different way. For the SSH-Kitaev chain at $$H= -\sum_{j=1}^{L}\mu\left(a_j^\dagger a_j-\frac12\right) +\sum_{\ell=1}^{r}\sum_{j=1}^{L-\ell} \left( -w_\ell e^{i\varphi_\ell} a_j^\dagger a_{j+\ell} +\Delta_\ell a_j a_{j+\ell} +\text{h.c.} \right),$$4, an exact real-space extended Bogoliubov transformation decomposes a $$H= -\sum_{j=1}^{L}\mu\left(a_j^\dagger a_j-\frac12\right) +\sum_{\ell=1}^{r}\sum_{j=1}^{L-\ell} \left( -w_\ell e^{i\varphi_\ell} a_j^\dagger a_{j+\ell} +\Delta_\ell a_j a_{j+\ell} +\text{h.c.} \right),$$5-site chain into two independent $$H= -\sum_{j=1}^{L}\mu\left(a_j^\dagger a_j-\frac12\right) +\sum_{\ell=1}^{r}\sum_{j=1}^{L-\ell} \left( -w_\ell e^{i\varphi_\ell} a_j^\dagger a_{j+\ell} +\Delta_\ell a_j a_{j+\ell} +\text{h.c.} \right),$$6-site Kitaev chains, with the full winding number given by

$$H= -\sum_{j=1}^{L}\mu\left(a_j^\dagger a_j-\frac12\right) +\sum_{\ell=1}^{r}\sum_{j=1}^{L-\ell} \left( -w_\ell e^{i\varphi_\ell} a_j^\dagger a_{j+\ell} +\Delta_\ell a_j a_{j+\ell} +\text{h.c.} \right),$$7

so that the full chain supports phases with $$H= -\sum_{j=1}^{L}\mu\left(a_j^\dagger a_j-\frac12\right) +\sum_{\ell=1}^{r}\sum_{j=1}^{L-\ell} \left( -w_\ell e^{i\varphi_\ell} a_j^\dagger a_{j+\ell} +\Delta_\ell a_j a_{j+\ell} +\text{h.c.} \right),$$8 (He et al., 2023). The phase boundaries follow from the standard subchain condition $$H= -\sum_{j=1}^{L}\mu\left(a_j^\dagger a_j-\frac12\right) +\sum_{\ell=1}^{r}\sum_{j=1}^{L-\ell} \left( -w_\ell e^{i\varphi_\ell} a_j^\dagger a_{j+\ell} +\Delta_\ell a_j a_{j+\ell} +\text{h.c.} \right),$$9, which in SSH-Kitaev parameters gives four critical curves: $r$0 (He et al., 2023).

Long-range models add further distinctions. In the infinite-range cases, standard bulk invariants do not always fully distinguish between massless Majorana zero modes and massive edge modes, so exact diagonalization and finite-size scaling are additionally required (Alecce et al., 2017). In the time-reversal-symmetric long-range hopping-and-pairing case, the critical lines are

$r$1

while for sufficiently long-ranged couplings the formal winding number can become $r$2 because of nonanalyticity at $r$3 (Alecce et al., 2017).

3. Edge modes, finite-size diagnostics, and spatial distribution

The extended Kitaev chain is defined as much by its edge phenomenology as by its bulk winding. In the generalized transfer-matrix treatment for arbitrary neighbor range $r$4, zero-energy wavefunctions take the form

$r$5

and the number of independent zero-mode solutions per edge is

$r$6

where $r$7 and $r$8 count roots inside and outside the unit circle (Alecce et al., 2017). This real-space result matches the bulk statement that the maximum number of Majorana zero modes per edge is controlled by the interaction range (Alecce et al., 2017).

Extended chains can also support massive edge modes. In the infinite-range models studied in the same work, these are edge-localized states with finite energy that remain separated from the bulk by a finite gap, and they can coexist with Majorana zero modes in crossover regions (Alecce et al., 2017). This sharpens the distinction between “edge-localized” and “zero-energy Majorana” in long-range superconducting chains.

A more recent finite-chain analysis introduces two additional diagnostics: the Majorana average position and the occupation of edge-to-edge non-local fermion states, defined as an effective parity (Widniczck et al., 8 Jun 2026). That work reports a direct correlation between the ground-state fermion parity and the edge occupation numbers, which are translated into localization and delocalization of the Majorana average position (Widniczck et al., 8 Jun 2026). In that formulation, the distribution of Majorana modes is not reducible to the bulk winding number alone, but is tied to how non-local fermion occupation reorganizes edge localization in finite systems (Widniczck et al., 8 Jun 2026).

Operational diagnostics can also be formulated directly in terms of end operators. For the nearest-neighbor open Kitaev chain, the end-to-end observable

$r$9

defines the thermodynamic indicator

$w_\ell$0

with the empirical expression

$w_\ell$1

(Reslen, 2018). That work explicitly notes that this formula is not derived for extended chains and should not be transferred unchanged to longer-range hopping or pairing without separate derivation (Reslen, 2018).

4. Dimerized, trimerized, phase-textured, and inhomogeneous variants

One major branch of the subject replaces the uniform lattice by an enlarged unit cell. In the SSH-Kitaev chain, the exact decomposition into two subchains gives a direct explanation of why the full system can host $w_\ell$2, $w_\ell$3, or $w_\ell$4 Majorana zero modes per edge (He et al., 2023). The same work also introduces two BCS-pair order parameters, $w_\ell$5 and $w_\ell$6, associated with the two decomposed sectors, so that the full ground state is a product of two BCS condensates (He et al., 2023).

A trimerized extension goes further. The three-sublattice model obtained by hybridizing a modified SSH chain with trimerized unit cells and the standard Kitaev chain has a hexamer structure in the Majorana basis and supports not only Majorana zero modes but also nonzero-energy edge-localized modes that are still Majorana in nature (Ghuneim et al., 4 Sep 2025). Its BdG Hamiltonian has a $w_\ell$7 structure, remains in class BDI, and uses a winding number

$w_\ell$8

for zero-energy topology, while open-boundary spectra reveal the additional $w_\ell$9 edge-mode pairs (Ghuneim et al., 4 Sep 2025).

A different extension is generated not by unit-cell enlargement but by a spatial phase texture. In the linearly phase-modulated chain

$\Delta_\ell$0

a hopping phase difference $\Delta_\ell$1 generates zero-energy bound states localized at non-endpoint sites (Zhang et al., 2024). In the finite-size calculations of that work, twofold-degenerate zero-energy states first emerge at $\Delta_\ell$2; for $\Delta_\ell$3 there are two non-endpoint Majorana bound states, for $\Delta_\ell$4 fourfold-degenerate Majorana bound states appear, and for $\Delta_\ell$5 sixfold-degenerate Majorana bound states appear (Zhang et al., 2024). In ring geometries, the same phase control can move interior Majoranas and implement braiding without T/Y junctions (Zhang et al., 2024).

An inhomogeneous end segment provides yet another extension. For a finite Kitaev chain coupled to a quantum dot through a step-like potential $\Delta_\ell$6 and position-dependent pairing $\Delta_\ell$7,

$\Delta_\ell$8

a robust Andreev bound state localized in the quantum-dot region appears as the generic lowest-energy solution in the topologically trivial phase (Zeng et al., 2018). That state does not exist in the bare uniform chain and is attributed to a partial decoupling of the component Majorana bound states over the length of the dot potential (Zeng et al., 2018). The same work stresses that the signatures of this trivial ABS in local tunneling are identical to the signatures of topologically protected Majorana zero modes (Zeng et al., 2018).

5. Non-Hermitian and fractional generalizations

Non-Hermitian extensions have produced a distinct body of “extended Kitaev chain” results. One route is asymmetric pair creation and annihilation,

$\Delta_\ell$9

with $\varphi_\ell$0 (Li et al., 2017). In the unbroken time-reversal-symmetric region the spectrum is real, the topology is characterized by a biorthogonal extended Zak phase,

$\varphi_\ell$1

and Majorana edge zero modes survive under open boundary conditions (Li et al., 2017). The topological phase is destroyed in the broken region, where eigenstates coalesce at exceptional points (Li et al., 2017).

A more explicitly range-extended non-Hermitian chain adds next-nearest-neighbor hopping and pairing together with a complex chemical potential $\varphi_\ell$2 (Rahul et al., 2023). There the next-nearest-neighbor term produces the three phases $\varphi_\ell$3, while the non-Hermitian factor qualitatively reorganizes the critical structure. The gap-closing momentum moves continuously along the critical line, multiple gapless points can appear, one Hermitian critical line collapses into an isolated critical point, and the number of multicritical points is reduced (Rahul et al., 2023). The paper also reports unconventional critical exponents,

$\varphi_\ell$4

in sharp contrast to the Hermitian extended Kitaev chain, where different multicritical points can have $\varphi_\ell$5 or $\varphi_\ell$6 (Rahul et al., 2023).

A more algebraic extension replaces ordinary Pauli matrices in the BdG blocks by rational powers, producing a centrally extended Clifford algebra and a pseudo-metallic regime with rational-valued winding number (Basa et al., 2022). In that construction the critical twist is

$\varphi_\ell$7

the bulk gap closes at

$\varphi_\ell$8

and the topological Majorana boundary mode of the untwisted chain evolves into an extended midgap state in the pseudo-metallic phase (Basa et al., 2022). The paper interprets this as a projective or fractional extension of Kitaev-chain topology rather than a longer-range coupling extension (Basa et al., 2022).

6. Artificial realizations, transport, and engineered chain links

Artificial implementations have made the extended Kitaev chain a device-level concept. In an alternating array of normal quantum dots and proximitized hybrid segments hosting Andreev bound states, the low-energy spin-polarized dot orbital and the spin-polarized ABS quasiparticle become the two sites of an effective two-site Kitaev chain,

$\varphi_\ell$9

with

$H = - \mu \sum_{j = 1}^N c_j^\dagger c_j^{\,} - \sum_{j = 1}^{N - 1} \sum_{\ell =1}^{q} \big( t_\ell^{} c_j^\dagger c_{j+\ell}^{\,} - \Delta_\ell^{} c_{j}^\dagger c_{j+\ell}^\dagger + \mbox{H.c.} \big),$0

(Miles et al., 2023). When the system is scaled to three sites, second-order virtual tunneling generates next-nearest-neighbor couplings,

$H = - \mu \sum_{j = 1}^N c_j^\dagger c_j^{\,} - \sum_{j = 1}^{N - 1} \sum_{\ell =1}^{q} \big( t_\ell^{} c_j^\dagger c_{j+\ell}^{\,} - \Delta_\ell^{} c_{j}^\dagger c_{j+\ell}^\dagger + \mbox{H.c.} \big),$1

so the longer device is explicitly an extended Kitaev chain rather than an ideal nearest-neighbor one (Miles et al., 2023). The same work reports that the effective coupling between sites at distance $H = - \mu \sum_{j = 1}^N c_j^\dagger c_j^{\,} - \sum_{j = 1}^{N - 1} \sum_{\ell =1}^{q} \big( t_\ell^{} c_j^\dagger c_{j+\ell}^{\,} - \Delta_\ell^{} c_{j}^\dagger c_{j+\ell}^\dagger + \mbox{H.c.} \big),$2 scales as

$H = - \mu \sum_{j = 1}^N c_j^\dagger c_j^{\,} - \sum_{j = 1}^{N - 1} \sum_{\ell =1}^{q} \big( t_\ell^{} c_j^\dagger c_{j+\ell}^{\,} - \Delta_\ell^{} c_{j}^\dagger c_{j+\ell}^\dagger + \mbox{H.c.} \big),$3

so the longer-range terms are exponentially short-ranged (Miles et al., 2023).

A complementary implementation is the flux-controlled two-site Kitaev chain, where two spin-polarized quantum dots are coupled through extended ABSs in a flux-tunable Josephson junction (Kulesh et al., 27 Jan 2025). The phase difference

$H = - \mu \sum_{j = 1}^N c_j^\dagger c_j^{\,} - \sum_{j = 1}^{N - 1} \sum_{\ell =1}^{q} \big( t_\ell^{} c_j^\dagger c_{j+\ell}^{\,} - \Delta_\ell^{} c_{j}^\dagger c_{j+\ell}^\dagger + \mbox{H.c.} \big),$4

controls the ABS energy and coherence factors, thereby tuning the ECT-to-CAR ratio (Kulesh et al., 27 Jan 2025). The poor man’s Majorana sweet spot is identified by

$H = - \mu \sum_{j = 1}^N c_j^\dagger c_j^{\,} - \sum_{j = 1}^{N - 1} \sum_{\ell =1}^{q} \big( t_\ell^{} c_j^\dagger c_{j+\ell}^{\,} - \Delta_\ell^{} c_{j}^\dagger c_{j+\ell}^\dagger + \mbox{H.c.} \big),$5

and the paper reports a continuous sweet-spot line in $H = - \mu \sum_{j = 1}^N c_j^\dagger c_j^{\,} - \sum_{j = 1}^{N - 1} \sum_{\ell =1}^{q} \big( t_\ell^{} c_j^\dagger c_{j+\ell}^{\,} - \Delta_\ell^{} c_{j}^\dagger c_{j+\ell}^\dagger + \mbox{H.c.} \big),$6 space rather than isolated points (Kulesh et al., 27 Jan 2025). An additional middle probe detects a zero-bias conductance peak from the hybrid region, indicating that the two poor man’s Majorana wavefunctions $H = - \mu \sum_{j = 1}^N c_j^\dagger c_j^{\,} - \sum_{j = 1}^{N - 1} \sum_{\ell =1}^{q} \big( t_\ell^{} c_j^\dagger c_{j+\ell}^{\,} - \Delta_\ell^{} c_{j}^\dagger c_{j+\ell}^\dagger + \mbox{H.c.} \big),$7 and $H = - \mu \sum_{j = 1}^N c_j^\dagger c_j^{\,} - \sum_{j = 1}^{N - 1} \sum_{\ell =1}^{q} \big( t_\ell^{} c_j^\dagger c_{j+\ell}^{\,} - \Delta_\ell^{} c_{j}^\dagger c_{j+\ell}^\dagger + \mbox{H.c.} \big),$8 both reside partially in the ABS region (Kulesh et al., 27 Jan 2025).

Transport calculations through longer-range chains connected to metallic leads show that range extension and time-reversal-symmetry breaking are not interchangeable perturbations. In the N-TS-N junction built from a chain with

$H = - \mu \sum_{j = 1}^N c_j^\dagger c_j^{\,} - \sum_{j = 1}^{N - 1} \sum_{\ell =1}^{q} \big( t_\ell^{} c_j^\dagger c_{j+\ell}^{\,} - \Delta_\ell^{} c_{j}^\dagger c_{j+\ell}^\dagger + \mbox{H.c.} \big),$9

the short-range proxy $$\Delta_\ell = \Delta \left(\frac{1}{\ell}\right)^\alpha, \qquad t_\ell = t \left(\frac{1}{\ell}\right)^\beta,$$0 and the long-range case $$\Delta_\ell = \Delta \left(\frac{1}{\ell}\right)^\alpha, \qquad t_\ell = t \left(\frac{1}{\ell}\right)^\beta,$$1 respond differently as $$\Delta_\ell = \Delta \left(\frac{1}{\ell}\right)^\alpha, \qquad t_\ell = t \left(\frac{1}{\ell}\right)^\beta,$$2 varies (Banerjee et al., 29 Jun 2026). In the short-range chain a quantized zero-bias peak of height $$\Delta_\ell = \Delta \left(\frac{1}{\ell}\right)^\alpha, \qquad t_\ell = t \left(\frac{1}{\ell}\right)^\beta,$$3 survives for $$\Delta_\ell = \Delta \left(\frac{1}{\ell}\right)^\alpha, \qquad t_\ell = t \left(\frac{1}{\ell}\right)^\beta,$$4 and $$\Delta_\ell = \Delta \left(\frac{1}{\ell}\right)^\alpha, \qquad t_\ell = t \left(\frac{1}{\ell}\right)^\beta,$$5, weakens at $$\Delta_\ell = \Delta \left(\frac{1}{\ell}\right)^\alpha, \qquad t_\ell = t \left(\frac{1}{\ell}\right)^\beta,$$6, and disappears when the bulk gap closes at $$\Delta_\ell = \Delta \left(\frac{1}{\ell}\right)^\alpha, \qquad t_\ell = t \left(\frac{1}{\ell}\right)^\beta,$$7 (Banerjee et al., 29 Jun 2026). In the long-range chain there is no quantized zero-bias peak even at $$\Delta_\ell = \Delta \left(\frac{1}{\ell}\right)^\alpha, \qquad t_\ell = t \left(\frac{1}{\ell}\right)^\beta,$$8, because long-range interactions convert zero-energy Majoranas into massive subgap Dirac modes, and for nonzero $$\Delta_\ell = \Delta \left(\frac{1}{\ell}\right)^\alpha, \qquad t_\ell = t \left(\frac{1}{\ell}\right)^\beta,$$9 the subgap states disappear from the gap region (Banerjee et al., 29 Jun 2026).

7. Related spin-chain usages and broader generalizations

In a broader but well-established usage, “extended Kitaev chain” also refers to bond-dependent spin chains whose Jordan–Wigner images are Majorana or BdG chains. The spin-$\alpha$0 Kitaev-XX-$\alpha$1 chain is one such example. After Jordan–Wigner transformation it becomes a quadratic spinless-fermion Hamiltonian with a two-site unit cell, and the exact solution yields six phases: four gapped ordered phases and two gapless phases (Zhuang et al., 26 Sep 2025). The gapless phases contain two branches of helical Majorana fermions, and the transition lines include deconfined quantum critical lines with $\alpha$2 and quadratic critical lines with $\alpha$3 (Zhuang et al., 26 Sep 2025).

A more materials-oriented example is the $\alpha$4 chain proposed for CoNb$\alpha$5O$\alpha$6, where the dominant microscopic exchange is ferromagnetic Kitaev and a sizable antiferromagnetic $\alpha$7 term fixes the easy axis (Churchill et al., 2024). In that work the chain is described as a one-dimensional Kitaev chain in disguise as an Ising chain, with local-frame Hamiltonian

$\alpha$8

and fitted parameters

$\alpha$9

(Churchill et al., 2024).

Coupled-chain extensions of the spin Kitaev-$p$00 chain show a different kind of enlargement. In the anisotropic honeycomb model with $p$01 corresponding to decoupled Kitaev-$p$02 chains, the emergent $p$03 Tomonaga–Luttinger liquid persists for finite interchain coupling and develops into an extended quantum spin liquid with spinon-like excitations (Gohlke et al., 2022). That phase differs from the conventional Kitaev spin liquid and is explicitly compared with sliding Luttinger liquids (Gohlke et al., 2022).

Other spin-chain descendants probe transport and entanglement rather than bulk topology. The Heisenberg–Kitaev chain interpolates between a one-dimensional spin Kitaev chain and an XXZ Heisenberg chain, and its high-temperature energy transport shows ballistic response at the integrable points, a topological gap near the Kitaev point, and quantum-chaotic regions away from integrability (Steinigeweg et al., 2013). The spin-$p$04 twisted Kitaev chain and generalized bond-dependent XY chain can be rewritten in terms of Majorana degrees of freedom and driven by local pulse sequences into rainbow-like states of long-distance entangled Majorana pairs, with maximal bipartite entanglement entropy and mirror-symmetric Majorana pairing (Xu et al., 2023).

These spin-chain usages are not identical to the standard fermionic extended Kitaev chain with longer-range $p$05-wave hopping and pairing. They nevertheless belong to the same conceptual field because they preserve the defining combination of one-dimensionality, bond-dependent or BdG-like structure, and Majorana-based descriptions of edge, bulk, or entanglement phenomena (Zhuang et al., 26 Sep 2025, Churchill et al., 2024, Gohlke et al., 2022, Steinigeweg et al., 2013, Xu et al., 2023).