Longer-Range Kitaev Chain

• Longer-range Kitaev chain is a one-dimensional topological superconducting model that extends pairing and hopping beyond nearest neighbors.
• It enriches the phase space by supporting winding numbers beyond Z₂ and enabling multiple Majorana zero modes, including finite-energy Dirac end modes in slow-decay regimes.
• The model utilizes finite-range and algebraically decaying interactions analyzed via BdG formalism and transfer matrices to understand disorder effects and bulk-boundary correspondence.

The longer-range Kitaev chain is a one-dimensional spinless-fermion $p$-wave superconducting lattice model in which hopping and pairing extend beyond nearest neighbors. In the narrow sense, “longer-range” denotes couplings to a finite number $r>1$ of neighbors; in a broader usage it includes the truly long-range limit in which hopping and pairing decay algebraically over all separations. Relative to the nearest-neighbor Kitaev chain, these extensions enlarge the space of topological phases, allow winding numbers beyond the standard $\mathbb Z_2$ structure, generate multiple Majorana zero modes per edge, and, in slow-decay regimes, produce finite-energy massive Dirac end modes, singular momentum-space structure, and weakened bulk-boundary correspondence (Kartik et al., 2020, Cinnirella et al., 27 Aug 2025).

1. Model class and Hamiltonian structure

A general finite-range formulation allows hopping and pairing up to range $r$: $$H \;=\; -\;\mu\sum_{n=1}^{L}\chi_{n}^{\dagger}\chi_{n} \;-\;\sum_{j=1}^{r}\; \sum_{n=1}^{L-j}\Bigl[ \,t_{j}\,\chi_{n}^{\dagger}\chi_{n+j} \;+\;\Delta_{j}\,\chi_{n}\,\chi_{n+j} \;+\;{\rm h.c.}\Bigr].$$ Here $\mu$ is the chemical potential, $t_j$ the hopping amplitude to the $j$-th neighbor, and $\Delta_j$ the corresponding $p$-wave pairing amplitude. In the frequently studied two-range Kitaev chain (2LRK), one sets $r>1$0 and often assumes $r>1$1, $r>1$2, with $r>1$3 (Cinnirella et al., 27 Aug 2025).

A closely related formulation writes

$r>1$4

with $r>1$5 and $r>1$6 as a finite-range truncation of algebraically decaying couplings. In the truly long-range limit $r>1$7, these sums generate polylogarithmic structures in momentum space; $r>1$8 recovers the nearest-neighbor chain (Kartik et al., 2020).

The infinite-range, power-law formulation commonly used in the literature keeps nearest-neighbor hopping and promotes pairing to an algebraic tail, for example

$r>1$9

Other variants include algebraic hopping and algebraic pairing simultaneously, dimerized chains with intracell and intercell couplings, and self-consistent long-range models derived from power-law density-density attraction rather than imposed pairing kernels (Vodola et al., 2014, Jäger et al., 2020, Li et al., 2020, Haink et al., 30 Sep 2025).

The self-consistent formulation is conceptually distinct. Starting from

$\mathbb Z_2$0

one introduces

$\mathbb Z_2$1

and solves the nonlinear gap equation together with the BdG problem. This produces a gap matrix whose structure is not imposed a priori and, as a result, alters edge-mode hybridization in a way not captured by dense power-law pairing matrices (Haink et al., 30 Sep 2025).

2. Bulk topology and phase structure

For translationally invariant longer-range chains, Fourier transformation leads to a BdG Hamiltonian with a two-component pseudospin. In one standard notation,

$\mathbb Z_2$2

and the spectrum is

$\mathbb Z_2$3

The winding number

$\mathbb Z_2$4

classifies the corresponding BDI phases. For an $\mathbb Z_2$5-neighbor model, phases with $\mathbb Z_2$6 can occur; each additional pair of hopping and pairing terms can allow the pseudospin trajectory to wind one extra time around the origin (Kartik et al., 2020).

In the clean 2LRK at fixed $\mathbb Z_2$7, the phase diagram contains three sectors: $\mathbb Z_2$8 trivial, $\mathbb Z_2$9 with a single Majorana per edge, and $r$0 with two Majoranas per edge. The clean critical lines follow from gap closings at high-symmetry points,

$r$1

This already shows the chief distinction from the nearest-neighbor chain, which supports only $r$2 (Cinnirella et al., 27 Aug 2025).

Dimerization further enriches the phase diagram. For a chain with two sites per unit cell, intracell hopping $r$3, intercell hopping $r$4, intracell pairing $r$5, intercell pairing $r$6, and chemical potential $r$7, the gap closes only at $r$8, with phase boundaries

$r$9

The BDI winding number $$H \;=\; -\;\mu\sum_{n=1}^{L}\chi_{n}^{\dagger}\chi_{n} \;-\;\sum_{j=1}^{r}\; \sum_{n=1}^{L-j}\Bigl[ \,t_{j}\,\chi_{n}^{\dagger}\chi_{n+j} \;+\;\Delta_{j}\,\chi_{n}\,\chi_{n+j} \;+\;{\rm h.c.}\Bigr].$$0 takes values $$H \;=\; -\;\mu\sum_{n=1}^{L}\chi_{n}^{\dagger}\chi_{n} \;-\;\sum_{j=1}^{r}\; \sum_{n=1}^{L-j}\Bigl[ \,t_{j}\,\chi_{n}^{\dagger}\chi_{n+j} \;+\;\Delta_{j}\,\chi_{n}\,\chi_{n+j} \;+\;{\rm h.c.}\Bigr].$$1. At $$H \;=\; -\;\mu\sum_{n=1}^{L}\chi_{n}^{\dagger}\chi_{n} \;-\;\sum_{j=1}^{r}\; \sum_{n=1}^{L-j}\Bigl[ \,t_{j}\,\chi_{n}^{\dagger}\chi_{n+j} \;+\;\Delta_{j}\,\chi_{n}\,\chi_{n+j} \;+\;{\rm h.c.}\Bigr].$$2, it factorizes into two scalar windings $$H \;=\; -\;\mu\sum_{n=1}^{L}\chi_{n}^{\dagger}\chi_{n} \;-\;\sum_{j=1}^{r}\; \sum_{n=1}^{L-j}\Bigl[ \,t_{j}\,\chi_{n}^{\dagger}\chi_{n+j} \;+\;\Delta_{j}\,\chi_{n}\,\chi_{n+j} \;+\;{\rm h.c.}\Bigr].$$3 and $$H \;=\; -\;\mu\sum_{n=1}^{L}\chi_{n}^{\dagger}\chi_{n} \;-\;\sum_{j=1}^{r}\; \sum_{n=1}^{L-j}\Bigl[ \,t_{j}\,\chi_{n}^{\dagger}\chi_{n+j} \;+\;\Delta_{j}\,\chi_{n}\,\chi_{n+j} \;+\;{\rm h.c.}\Bigr].$$4, making the $$H \;=\; -\;\mu\sum_{n=1}^{L}\chi_{n}^{\dagger}\chi_{n} \;-\;\sum_{j=1}^{r}\; \sum_{n=1}^{L-j}\Bigl[ \,t_{j}\,\chi_{n}^{\dagger}\chi_{n+j} \;+\;\Delta_{j}\,\chi_{n}\,\chi_{n+j} \;+\;{\rm h.c.}\Bigr].$$5 phase transparent as the simultaneous activation of two independent winding channels (Li et al., 2020).

In the power-law limit, the phase structure departs from the short-range template. For $$H \;=\; -\;\mu\sum_{n=1}^{L}\chi_{n}^{\dagger}\chi_{n} \;-\;\sum_{j=1}^{r}\; \sum_{n=1}^{L-j}\Bigl[ \,t_{j}\,\chi_{n}^{\dagger}\chi_{n+j} \;+\;\Delta_{j}\,\chi_{n}\,\chi_{n+j} \;+\;{\rm h.c.}\Bigr].$$6, one recovers the standard topological and trivial sectors, while for sufficiently slow decay the momentum-space singularity at $$H \;=\; -\;\mu\sum_{n=1}^{L}\chi_{n}^{\dagger}\chi_{n} \;-\;\sum_{j=1}^{r}\; \sum_{n=1}^{L-j}\Bigl[ \,t_{j}\,\chi_{n}^{\dagger}\chi_{n+j} \;+\;\Delta_{j}\,\chi_{n}\,\chi_{n+j} \;+\;{\rm h.c.}\Bigr].$$7 changes both the topology and the edge phenomenology. Several studies identify half-integer or noninteger winding in the long-range regime, for example $$H \;=\; -\;\mu\sum_{n=1}^{L}\chi_{n}^{\dagger}\chi_{n} \;-\;\sum_{j=1}^{r}\; \sum_{n=1}^{L-j}\Bigl[ \,t_{j}\,\chi_{n}^{\dagger}\chi_{n+j} \;+\;\Delta_{j}\,\chi_{n}\,\chi_{n+j} \;+\;{\rm h.c.}\Bigr].$$8 or $$H \;=\; -\;\mu\sum_{n=1}^{L}\chi_{n}^{\dagger}\chi_{n} \;-\;\sum_{j=1}^{r}\; \sum_{n=1}^{L-j}\Bigl[ \,t_{j}\,\chi_{n}^{\dagger}\chi_{n+j} \;+\;\Delta_{j}\,\chi_{n}\,\chi_{n+j} \;+\;{\rm h.c.}\Bigr].$$9, and emphasize that phase transitions may occur even when the bulk gap does not close in the standard short-range manner (Cinnirella et al., 2024, Francica et al., 2022, Bhattacharya et al., 2018).

A recurrent point of interpretation is the status of bulk-boundary correspondence. In the Aubry–André–Harper-modulated long-range chain, the bulk winding or Chern invariant can remain quantized while edge crossings are replaced by avoided crossings or while the number of edge-state crossings changes. The literature treats this as a weakening, rather than a simple failure, of the conventional local bulk-boundary correspondence in the presence of nonlocal pairing kernels (Fraxanet et al., 2020).

3. Boundary modes, multiplicity, and hybridization

The defining boundary signature of the longer-range Kitaev chain is the proliferation and deformation of end modes. In finite-range models, the clean phase can host $\mu$0, $\mu$1, or $\mu$2 Majorana zero modes per edge. In the minimal longer-range model with nearest- and next-nearest-neighbor couplings, a transfer-matrix treatment yields a zero-mode recursion

$\mu$3

with eigenvalues

$\mu$4

If both satisfy $\mu$5, the edge supports two normalizable Majorana zero modes; if only one does, there is a single Majorana; if none do, the edge is topologically trivial (Lieu et al., 2018).

The dimerized longer-range chain provides an explicit realization of a $\mu$6 phase with two independent Majorana zero modes at each end. In the $\mu$7 sector there is precisely one Majorana zero mode per boundary, while in the $\mu$8 sector the intercell hopping and pairing terms create a second edge-localization channel. This is the sense in which longer-range couplings can magnify a Kitaev-like phase into a twofold-degenerated Majorana phase (Li et al., 2020).

Truly long-range pairing introduces a qualitatively different edge regime. For $\mu$9, open chains retain massless Majorana end modes in the thermodynamic limit. For $t_j$0, several works report that the boundary excitations are no longer exact zero modes but massive Dirac end modes: the two Majoranas hybridize into a finite-energy subgap fermion whose energy remains nonzero as $t_j$1. This finite mass also controls transport signatures and distinguishes the long-range topological phase from the short-range topological phase (Giuliano et al., 2017, Bhattacharya et al., 2018, Cinnirella et al., 2024).

The spatial profile of these boundary states is model-dependent. In the analytically solvable chain with algebraic tunneling and pairing, edge modes are purely exponential in the isotropic case $t_j$2 and $t_j$3, whereas anisotropy produces an exponential core together with an algebraic tail governed by $t_j$4. The zero-mode splitting scales as $t_j$5 in the isotropic case and as $t_j$6 in the anisotropic case (Jäger et al., 2020).

The self-consistent long-range Kitaev chain sharpens this distinction. There, the gap matrix decomposes into an exponentially localized short-range band and a long-range corner cluster localized near $t_j$7 and $t_j$8, with

$t_j$9

The individual edge wavefunctions remain exponentially localized,

$j$0

yet the edge-mode mass obeys

$j$1

Hence exponential localization of the wavefunction and algebraic decay of the energy splitting coexist. This directly contrasts with non-self-consistent dense power-law pairing models, where the edge wavefunctions themselves acquire algebraic tails and the massive Dirac regime appears for $j$2 (Haink et al., 30 Sep 2025).

4. Disorder, open-system diagnostics, and transport

Disorder reorganizes the longer-range phase diagram rather than merely destroying it. In the disordered 2LRK, three disorder classes have been analyzed: Anderson on-site disorder, Aubry–André chemical-potential disorder, and correlated bond disorder acting simultaneously on $j$3, $j$4, and $j$5. The central diagnostic is the dc conductance of a lead-chain-reservoir setup in the non-equilibrium steady state; in the weak-coupling limit, the zero-bias conductance is quantized in units of a reference conductance $j$6, with

$j$7

This makes the winding number directly accessible without translational invariance (Cinnirella et al., 27 Aug 2025).

The resulting disorder phase diagrams are strongly nonmonotonic. For Anderson disorder, a reentrant $j$8 island intrudes into the $j$9 region, while $\Delta_j$0 is suppressed for moderate disorder and all topology is destroyed at large disorder. For Aubry–André disorder, weak disorder leaves topology essentially unchanged, intermediate disorder generates reentrant $\Delta_j$1, and stronger modulation retreats the $\Delta_j$2 sector. Correlated bond disorder behaves differently: even weak disorder suppresses $\Delta_j$3 and $\Delta_j$4 in favor of $\Delta_j$5, while strong disorder can generate a new reentrant $\Delta_j$6 region with reduced fluctuations (Cinnirella et al., 27 Aug 2025).

A complementary semi-analytic approach uses Lyapunov exponents of random transfer matrices. In the nearest-plus-next-nearest chain, disorder can induce local zero modes absent in the clean system, generating $\Delta_j$7 or $\Delta_j$8 transitions. The clean direct $\Delta_j$9 boundary is unstable: any finite disorder splits it into $p$0, creating a tricritical point at the clean $p$1-$p$2 boundary. Entanglement-spectrum degeneracy and dynamical qubit correlators reproduce the same phase boundaries, with nonlocal correlators diagnosing the first Majorana and local correlators diagnosing the second (Lieu et al., 2018).

Open-system analyses of power-law chains exhibit analogous reentrance. In the disordered long-range pairing Kitaev model attached to metallic leads and Lindblad baths, the massive topological phase initially localizes under increasing disorder, driving the finite-energy edge level downward toward zero; for the parameters studied, a critical disorder $p$3 marks a transition into a short-range-topological-like regime with $p$4, followed by eventual destruction of subgap states at larger $p$5 (Cinnirella et al., 2024).

Transport observables are especially sensitive to the distinction between massless Majorana and massive Dirac edges. In an NSN setup for the long-range pairing chain, the low-bias Fano factor satisfies $p$6 in the short-range topological phase, $p$7 in the long-range topological phase with finite edge mass, and $p$8 on the critical line $p$9. The same finite edge mass reorganizes electrical, thermal, and thermoelectric responses in open long-range chains: at parameter values where the short-range model is gapless, the long-range model can still show a finite threshold set by the edge Dirac mass before charge or heat current turns on (Giuliano et al., 2017, Banerjee et al., 20 May 2025).

5. Correlations, entanglement, and non-equilibrium criticality

Longer-range pairing reshapes equal-time correlations even in gapped phases. In the algebraic pairing chain, two qualitatively distinct regimes occur. For $r>1$00, correlation functions display a hybrid structure: exponential decay at short distance and algebraic decay at long distance. For $r>1$01, the decay is purely algebraic throughout the gapped phase. This long-distance structure is accompanied by conformal-symmetry breaking along critical lines for sufficiently small $r>1$02, with standard $r>1$03 Ising behavior recovered only for sufficiently fast decay (Vodola et al., 2014).

Entanglement mirrors these changes. In gapped short-range-like regimes the half-chain entropy obeys an area law, but for $r>1$04 a logarithmic violation of the area law persists even though the system remains gapped. Exactly at $r>1$05, the coefficient of the logarithmic growth becomes non-universal: it depends nontrivially on the Rényi index and on the field $r>1$06, and cannot be written in the usual conformal form $r>1$07. The underlying mathematical reason is that the block-Toeplitz symbol develops a non-commuting jump, requiring an extended Fisher–Hartwig analysis (Ares et al., 2018).

The long-range chain also supports genuine bulk long-range entanglement. For two macroscopically separated blocks, the mutual information scales as

$r>1$08

where $r>1$09 is the decay exponent of the pairing correlator. When $r>1$10, which occurs for $r>1$11 in the pairing sector analyzed, the mutual information saturates to a nonzero constant in the thermodynamic limit. This provides a sharp criterion for a long-range entangled phase in one dimension (Francica et al., 2022).

Out-of-equilibrium dynamics are likewise nonlocal. After a global quench, the long-range chain violates the Lieb–Robinson bound in the strict sense, but most information still propagates on a ballistic light cone determined by dominant peaks of the quasiparticle group-velocity distribution. For $r>1$12, ultrafast modes create immediate but small long-distance mutual information, while slow modes produce persistent correlations at time-like separations and very slow equilibration toward the generalized Gibbs ensemble (Regemortel et al., 2015).

Dynamical critical phenomena also deviate from short-range expectations. Dynamical quantum phase transitions can occur even when a quench does not cross a static phase boundary but does move between short-range and genuinely long-range regimes. Under slow ramps of the chemical potential, the defect density obeys the short-range scaling $r>1$13 only beyond the long-range crossover; for $r>1$14, the exponent becomes $r>1$15, and the adiabatic dynamics are governed by a topological healing length rather than the ordinary correlation length. Sudden quenches in the same regime can generate multi-cusp structures in the Loschmidt rate function, including regions with three cusp singularities per Fisher-zero sector, which shrink and disappear as $r>1$16 (Dutta et al., 2017, Francica et al., 2022).

6. Realizations, mappings, and broader uses

Several physical platforms realize, or effectively map onto, longer-range Kitaev chains. A planar Josephson junction proximitized to a Rashba 2DEG in an in-plane Zeeman field can be reinterpreted as an effective one-dimensional Kitaev chain with long-range hopping and pairing. The effective couplings are obtained from the Fourier components of the chiral angle,

$r>1$17

and numerically decay with oscillatory power-law tails,

$r>1$18

A real-space Clifford pseudospectrum index reproduces the clean winding plateaux and remains robust under moderate disorder (Guo et al., 2018).

Other realizations emphasized in the literature include trapped-ion simulators with tunable power-law interactions, Rydberg arrays, photonic lattices, helical Shiba chains of magnetic adatoms on superconductors, semiconductor nanowires proximitized by $r>1$19-wave superconductors, and ultracold-atom implementations with Aubry–André–Harper modulation and engineered long-range attraction. These platforms are used in different papers to motivate measurements of correlation decay, conductance quantization, subgap transport thresholds, or edge-mode spectroscopy (Vodola et al., 2014, Cinnirella et al., 2024, Fraxanet et al., 2020).

The longer-range chain has also become a theoretical laboratory for questions beyond static topology. It supports disorder-protected and disorder-induced zero modes, nonlocal hybridization mechanisms, higher winding sectors, and anomalous Floquet edge states at quasienergy $r>1$20. In metrological settings, it provides an exactly solvable many-body model in which the quantum Fisher information for pairing-strength estimation remains Heisenberg-limited for power-law profiles with any $r>1$21, while slower logarithmic decay profiles can yield

$r>1$22

or $r>1$23 at $r>1$24 (Bhattacharya et al., 2018, Yang et al., 2021).

Taken together, these results establish the longer-range Kitaev chain as a unifying framework for extended-range topological superconductivity. In finite-range form it enlarges the integer winding hierarchy and stabilizes multiple local Majorana channels. In the truly long-range limit it generates half-integer topology, massive Dirac end modes, nonlocal hybridization without wavefunction overlap, and altered entanglement and dynamical scaling. The central open theme running across this literature is that extending the range of pairing and hopping does not merely perturb the nearest-neighbor Kitaev chain; it changes the meaning of locality itself within the topological superconducting problem (Kartik et al., 2020, Haink et al., 30 Sep 2025).