Self-Consistent Long-Range Kitaev Chain

• The paper demonstrates that the self-consistent long-range Kitaev chain yields a sparse gap matrix that organizes correlations into a short-range bulk band and nonlocal edge corner blocks.
• It reveals non-local edge mode hybridization where direct end-to-end coupling produces algebraic Majorana splitting despite exponentially localized edge states.
• The model highlights practical implications for topological qubits, emphasizing finite-size corrections that scale as a power-law with the interaction exponent.

The self-consistent long-range Kitaev chain, or seco-LRKC, is a one-dimensional spinless $p$-wave superconducting model in which the pairing field is generated by a long-range attractive density-density interaction and determined self-consistently rather than imposed as a prescribed bilinear kernel. In the formulation introduced in "Non-local edge mode hybridization in the long-range interacting Kitaev chain" (Haink et al., 30 Sep 2025), the defining feature is that self-consistency filters the algebraic interaction into a sparse Bogoliubov–de Gennes gap matrix composed of a short-range bulk band and exponentially localized edge-to-edge corner blocks. This structure produces a finite Majorana splitting in finite systems through direct nonlocal coupling between the two ends of the chain, even when the Majorana wavefunctions themselves remain exponentially localized and have negligible spatial overlap.

1. Microscopic formulation and self-consistent definition

The starting point is a spinless 1D BCS Hamiltonian with nearest-neighbor hopping and an attractive long-range density-density interaction,

$H = H_0 + H_{\text{int}},$

with

$$H_0=-\frac{1}{2}\sum_{x=1}^{n-1}\tau\, c_x^\dagger c_{x+1}+\text{h.c.} -\frac{1}{2}\sum_{x=1}^{n}\mu\left(c_x^\dagger c_x-\frac{1}{2}\right),$$

and

$$H_{\text{int}}=\frac{1}{2}\sum_{x\neq x'} c_x^\dagger c_{x'}^\dagger V_{x,x'} c_{x'} c_x, \qquad V_{x,x'}=-U_0 |x-x'|^{-\nu},$$

where $U_0>0$ and $\nu>0$ is the interaction exponent (Haink et al., 30 Sep 2025).

A standard mean-field decoupling yields a pairing term with real-space gap matrix

$$\Delta_{x,x'} = V_{x,x'} \langle c_{x'} c_x\rangle.$$

Using the Nambu spinor

$$\bm{\Psi}=(c_1,\dots,c_n,c_1^\dagger,\dots,c_n^\dagger)^T,$$

the BdG Hamiltonian is

$$\mathcal H= \begin{pmatrix} h/2 & \Delta \ \Delta^\dagger & -h/2 \end{pmatrix},$$

where $h_{x,x}=-\mu$, $H = H_0 + H_{\text{int}},$0, and all other entries vanish.

What distinguishes the seco-LRKC from fixed long-range pairing models is the determination of the anomalous correlation matrix

$H = H_0 + H_{\text{int}},$1

through minimization of the nonlinear functional

$H = H_0 + H_{\text{int}},$2

The seco-LRKC is therefore the self-consistent mean-field theory defined by this variational problem, not a quadratic chain with a power-law pairing kernel inserted by hand.

2. Structure of the self-consistent gap matrix

For $H = H_0 + H_{\text{int}},$3 and $H = H_0 + H_{\text{int}},$4, the self-consistent correlation matrix $H = H_0 + H_{\text{int}},$5, and hence the gap matrix $H = H_0 + H_{\text{int}},$6, splits into two qualitatively distinct components (Haink et al., 30 Sep 2025). The first is a short-range bulk band near the main diagonal,

$H = H_0 + H_{\text{int}},$7

which corresponds to the bulk-like translationally invariant solution. The second is a set of long-range anti-diagonal corner blocks near $H = H_0 + H_{\text{int}},$8 and $H = H_0 + H_{\text{int}},$9,

$$H_0=-\frac{1}{2}\sum_{x=1}^{n-1}\tau\, c_x^\dagger c_{x+1}+\text{h.c.} -\frac{1}{2}\sum_{x=1}^{n}\mu\left(c_x^\dagger c_x-\frac{1}{2}\right),$$0

These corner blocks are exponentially localized in matrix space, but their amplitude inherits the interaction tail: $$H_0=-\frac{1}{2}\sum_{x=1}^{n-1}\tau\, c_x^\dagger c_{x+1}+\text{h.c.} -\frac{1}{2}\sum_{x=1}^{n}\mu\left(c_x^\dagger c_x-\frac{1}{2}\right),$$1 The resulting gap matrix is therefore sparse rather than dense: a short-range diagonal band coexists with exponentially localized end-to-end corner blocks. This is the central structural result of the self-consistent construction.

A common simplification in long-range Kitaev-chain discussions is to identify long-range interactions with a pairing matrix that decays algebraically throughout real space. The seco-LRKC does not realize that pattern. Instead,

$$H_0=-\frac{1}{2}\sum_{x=1}^{n-1}\tau\, c_x^\dagger c_{x+1}+\text{h.c.} -\frac{1}{2}\sum_{x=1}^{n}\mu\left(c_x^\dagger c_x-\frac{1}{2}\right),$$2

because self-consistency reorganizes correlations into a short-range bulk component and a highly structured nonlocal edge component.

3. Non-local edge-mode hybridization

In the standard short-range Kitaev chain, the splitting of the two Majorana zero modes is controlled by the exponentially small overlap of their edge wavefunctions, giving

$$H_0=-\frac{1}{2}\sum_{x=1}^{n-1}\tau\, c_x^\dagger c_{x+1}+\text{h.c.} -\frac{1}{2}\sum_{x=1}^{n}\mu\left(c_x^\dagger c_x-\frac{1}{2}\right),$$3

The seco-LRKC departs from this mechanism while preserving exponential edge localization (Haink et al., 30 Sep 2025).

The edge eigenstates remain exponentially localized at the two ends of the chain, with a decay length set by $$H_0=-\frac{1}{2}\sum_{x=1}^{n-1}\tau\, c_x^\dagger c_{x+1}+\text{h.c.} -\frac{1}{2}\sum_{x=1}^{n}\mu\left(c_x^\dagger c_x-\frac{1}{2}\right),$$4, so their direct wavefunction overlap can be negligibly small. Nevertheless, the self-consistent gap matrix contains direct end-to-end pairing elements,

$$H_0=-\frac{1}{2}\sum_{x=1}^{n-1}\tau\, c_x^\dagger c_{x+1}+\text{h.c.} -\frac{1}{2}\sum_{x=1}^{n}\mu\left(c_x^\dagger c_x-\frac{1}{2}\right),$$5

These anti-diagonal corner terms provide a matrix element that couples the two edge modes directly within the BdG Hamiltonian.

This mechanism is termed non-local edge mode hybridization. The hybridization is nonlocal because it is mediated by the long-range corner blocks of the self-consistent gap matrix rather than by the spatial overlap of exponentially localized Majorana wavefunctions. The finite Majorana mass in a finite chain is therefore a Hamiltonian-level edge-edge coupling effect, not an overlap effect.

An important consequence is conceptual as well as spectral: exponential localization of the edge wavefunctions does not by itself imply exponentially small splitting. In the seco-LRKC, localization and decoupling are distinct properties.

4. Edge-mode mass scaling, phases, and thermodynamic behavior

The main asymptotic result is that the finite-size Majorana mass scales algebraically with system size,

$$H_0=-\frac{1}{2}\sum_{x=1}^{n-1}\tau\, c_x^\dagger c_{x+1}+\text{h.c.} -\frac{1}{2}\sum_{x=1}^{n}\mu\left(c_x^\dagger c_x-\frac{1}{2}\right),$$6

for any $$H_0=-\frac{1}{2}\sum_{x=1}^{n-1}\tau\, c_x^\dagger c_{x+1}+\text{h.c.} -\frac{1}{2}\sum_{x=1}^{n}\mu\left(c_x^\dagger c_x-\frac{1}{2}\right),$$7 (Haink et al., 30 Sep 2025). This follows from the scaling of the long-range corner blocks,

$$H_0=-\frac{1}{2}\sum_{x=1}^{n-1}\tau\, c_x^\dagger c_{x+1}+\text{h.c.} -\frac{1}{2}\sum_{x=1}^{n}\mu\left(c_x^\dagger c_x-\frac{1}{2}\right),$$8

together with first-order perturbation theory, which implies that the edge-mode splitting inherits the same exponent.

For finite $$H_0=-\frac{1}{2}\sum_{x=1}^{n-1}\tau\, c_x^\dagger c_{x+1}+\text{h.c.} -\frac{1}{2}\sum_{x=1}^{n}\mu\left(c_x^\dagger c_x-\frac{1}{2}\right),$$9, the edge mode is therefore not exactly at zero energy. In the thermodynamic limit,

$$H_{\text{int}}=\frac{1}{2}\sum_{x\neq x'} c_x^\dagger c_{x'}^\dagger V_{x,x'} c_{x'} c_x, \qquad V_{x,x'}=-U_0 |x-x'|^{-\nu},$$0

so the mode remains massless asymptotically, but the approach to zero is power-law rather than exponential. The paper analytically motivates and numerically demonstrates that fitting the finite-size data to

$$H_{\text{int}}=\frac{1}{2}\sum_{x\neq x'} c_x^\dagger c_{x'}^\dagger V_{x,x'} c_{x'} c_x, \qquad V_{x,x'}=-U_0 |x-x'|^{-\nu},$$1

gives

$$H_{\text{int}}=\frac{1}{2}\sum_{x\neq x'} c_x^\dagger c_{x'}^\dagger V_{x,x'} c_{x'} c_x, \qquad V_{x,x'}=-U_0 |x-x'|^{-\nu},$$2

in the self-consistent model. The numerical finite-size scaling plots place the red data points on the line $$H_{\text{int}}=\frac{1}{2}\sum_{x\neq x'} c_x^\dagger c_{x'}^\dagger V_{x,x'} c_{x'} c_x, \qquad V_{x,x'}=-U_0 |x-x'|^{-\nu},$$3.

This behavior persists for all $$H_{\text{int}}=\frac{1}{2}\sum_{x\neq x'} c_x^\dagger c_{x'}^\dagger V_{x,x'} c_{x'} c_x, \qquad V_{x,x'}=-U_0 |x-x'|^{-\nu},$$4, despite the fact that the edge wavefunctions remain exponentially localized. The coexistence of exponential wavefunction localization with algebraic energy splitting is one of the defining features of the seco-LRKC.

The topological structure remains comparatively close to the standard Kitaev chain. The model has only the trivial and topological phases, and the winding number does not depend on $$H_{\text{int}}=\frac{1}{2}\sum_{x\neq x'} c_x^\dagger c_{x'}^\dagger V_{x,x'} c_{x'} c_x, \qquad V_{x,x'}=-U_0 |x-x'|^{-\nu},$$5. This places the long-range effect primarily in the finite-size edge-sector energetics rather than in a radical restructuring of the bulk topological classification.

5. Relation to non-self-consistent long-range Kitaev chains

The seco-LRKC should be distinguished from long-range Kitaev chains in which pairing is imposed directly as a bilinear power law. In one such formulation,

$$H_{\text{int}}=\frac{1}{2}\sum_{x\neq x'} c_x^\dagger c_{x'}^\dagger V_{x,x'} c_{x'} c_x, \qquad V_{x,x'}=-U_0 |x-x'|^{-\nu},$$6

and the study explicitly states that there is no self-consistent pairing calculation, no iteration for $$H_{\text{int}}=\frac{1}{2}\sum_{x\neq x'} c_x^\dagger c_{x'}^\dagger V_{x,x'} c_{x'} c_x, \qquad V_{x,x'}=-U_0 |x-x'|^{-\nu},$$7, and no order-parameter equation (Jha et al., 11 Feb 2026). Earlier long-range Kitaev-chain analyses likewise considered quadratic models with nearest-neighbor hopping and power-law pairing, or related fixed long-range kernels, without a self-consistent gap equation (Regemortel et al., 2015, Dutta et al., 2017).

Feature	Non-seco LRKC	seco-LRKC
Pairing construction	Imposed by hand as power law	Determined self-consistently
Gap structure	Dense algebraic long-range pairing	Sparse band plus corner blocks
Edge hybridization	Often tied to imposed long-range kernel	Direct edge-edge coupling through self-consistent corners
$$H_{\text{int}}=\frac{1}{2}\sum_{x\neq x'} c_x^\dagger c_{x'}^\dagger V_{x,x'} c_{x'} c_x, \qquad V_{x,x'}=-U_0 |x-x'|^{-\nu},$$8 regime	Massive Dirac modes can emerge	Algebraic Majorana mass decay persists for all $$H_{\text{int}}=\frac{1}{2}\sum_{x\neq x'} c_x^\dagger c_{x'}^\dagger V_{x,x'} c_{x'} c_x, \qquad V_{x,x'}=-U_0 |x-x'|^{-\nu},$$9

This distinction has direct consequences for edge physics. In non-self-consistent models, for $U_0>0$0 with $U_0>0$1, the long-range pairing can be strong enough to generate massive Dirac fermions and alter the spectrum and topology more dramatically. For $U_0>0$2, Majorana zero modes persist, but the finite-size scaling can differ from the bare interaction exponent. In the self-consistent model, by contrast, algebraic edge-mode decay with system size persists for all $U_0>0$3, the bulk/topological structure stays much closer to the standard Kitaev chain, and the winding number is independent of $U_0>0$4.

The reason given is structural rather than purely spectral. Self-consistency strongly filters the interaction-induced pairing, preventing the dense long-range pairing matrix that underlies the $U_0>0$5 massive-Dirac regime in non-seco models. The seco-LRKC is therefore not a minor variant of the prescribed power-law model; it realizes a different organization of correlations.

6. Physical implications, misconceptions, and experimental relevance

A central implication is that finite-energy edge modes can remain topological. The seco-LRKC shows that a topological edge state need not be exactly at zero energy in a finite sample, and that its finite-size mass can be controlled by the interaction tail rather than by the localization length alone (Haink et al., 30 Sep 2025).

This directly addresses a common misconception: exponentially localized Majorana wavefunctions do not guarantee exponentially small splitting. In the seco-LRKC, the edge modes remain exponentially localized, yet the splitting is algebraic because the BdG Hamiltonian contains direct long-range edge-edge couplings. A plausible implication is that edge localization diagnostics alone are insufficient to assess finite-size protection in long-range interacting topological superconductors.

The finite-size correction

$U_0>0$6

can remain substantial in mesoscopic systems, especially for small $U_0>0$7. The paper notes that such finite mass can induce unwanted dynamical phases,

$U_0>0$8

during braiding or parity manipulation, which can lead to dephasing and reduced qubit fidelity. Long-range interaction tails may therefore impose a practical limitation for topological qubits in mesoscopic devices.

The experimental settings mentioned as plausible contexts for such long-range tails are quantum-dot Kitaev chains, incomplete electrostatic screening in low-dimensional devices, interactions near quantum criticality, intrinsic quasi-1D triplet superconductors such as $U_0>0$9, and dipolar polar-molecule platforms with effective $\nu>0$0 interactions (Haink et al., 30 Sep 2025). Related long-range Kitaev-chain physics can also arise effectively in planar Josephson junctions proximitized to a 2DEG with Rashba spin-orbit coupling and Zeeman field, where integrating out transverse structure produces effective long-range hopping and pairing terms (Liu et al., 2018). This suggests that self-consistent long-range edge-coupling effects are most relevant in settings where effective couplings extend significantly beyond nearest neighbors and the system size remains mesoscopic rather than asymptotically large.