Non-Hermitian Kitaev Chain

Updated 12 April 2026

Non-Hermitian Kitaev Chain is a 1D topological superconductor defined by complex potentials, asymmetric hopping, and pairing imbalance.
It exhibits unique spectral transitions, including real-to-complex eigenvalue changes and exceptional points that alter bulk-boundary correspondence.
Advanced invariants like the biorthogonal Zak phase, Pfaffian, and winding numbers predict topological phase transitions and Majorana mode robustness.

The non-Hermitian Kitaev chain generalizes the paradigmatic one-dimensional topological superconductor by incorporating non-Hermitian terms—complex potentials, pairing imbalance, asymmetric hopping, and gain/loss—thus enabling a fundamentally new class of quantum phases, topological invariants, and edge-state phenomena beyond traditional Hermitian classification. Through a rigorous analytic framework, the non-Hermitian Kitaev chain encompasses both spinless and spinful models, captures effects of Rashba spin-orbit coupling, and admits extensions involving spatially modulated or interacting (many-body) terms. Distinctive features include real-to-complex spectral transitions, exceptional points, altered bulk–boundary correspondence, and regimes in which Majorana zero modes acquire novel robustness or fragility.

1. Model Classes and Non-Hermitian Terms

The minimal non-Hermitian Kitaev chain takes the form of a 1D lattice of spinless or spinful fermions with non-Hermitian modifications to hopping, pairing, and on-site potentials. The general second-quantized Hamiltonian for the spinless case may be written as: $H = -\sum_{j} t_R\,c_{j+1}^\dagger c_j + t_L\,c_j^\dagger c_{j+1} + \Delta_c\,c_{j+1}^\dagger c_j^\dagger + \Delta_a\,c_j\,c_{j+1} -\sum_{j} (\mu + i\gamma)\,c_j^\dagger c_j$ where:

$t_R$ , $t_L$ : right/left hopping amplitudes (possibly complex/asymmetric)
$\Delta_c$ , $\Delta_a$ : $p$ -wave pair creation/annihilation strengths ( $\Delta_c \neq \Delta_a$ introduces pairing imbalance)
$\mu$ : chemical potential
$\gamma$ : uniform gain/loss rate (imaginary potential)

Spinful extensions incorporate Rashba spin–orbit coupling (RSOC), resulting in Hamiltonians of the type: $H = H_0 + H_{\mathrm{RSOC}} + H_{\mathrm{NH}}$ with $t_R$ 0 containing spin-flip hopping (strength $t_R$ 1).

On-site non-Hermiticity can also be spatially modulated (quasiperiodic potentials), and pairing/gain terms can alternate in space or be complex-valued, allowing for staggered and dimerized models (Shahab et al., 10 Dec 2025, Li et al., 2017, Shi et al., 2023, Zhang et al., 2 Feb 2026).

2. Bulk Spectra, Exceptional Points, and Gap Structures

Non-Hermiticity enriches spectral topology beyond Hermitian analogues in several critical ways:

The energy spectrum $t_R$ 2 is generically complex. Regions of parameter space with purely real spectrum are termed $t_R$ 3-unbroken or “time-reversal symmetric” (TRS unbroken); complex spectra indicate $t_R$ 4-broken phases with exceptional points (EPs).
For models with pairing imbalance (e.g., $t_R$ 5), the single-particle spectrum takes the form

$t_R$ 6

with reality of the spectrum controlled by $t_R$ 7 and $t_R$ 8 (Li et al., 2017).

Exceptional lines or surfaces in parameter space—such as the hyperbola $t_R$ 9 in the imaginary pairing Kitaev chain—mark where eigenvalues and eigenvectors coalesce at true non-diagonalizable (Jordan block) degeneracies, driving real-to-complex transitions and unorthodox quantum critical points (Yang et al., 2019).
With next-nearest neighbor couplings or similar extensions, critical lines collapse into isolated multicritical points upon introduction of non-Hermiticity, and gap-closing momenta $t_L$ 0 become tunable, not fixed at high-symmetry points as in Hermitian cases. The dynamical exponent at criticality universally flows to $t_L$ 1 (subdiffusive) (Rahul et al., 2023).

3. Topological Invariants: Biorthogonal Zak Phase, Pfaffian, and Winding Numbers

Topological classification in non-Hermitian Kitaev chains relies on biorthogonal (left/right eigenbasis) generalizations of Hermitian invariants:

Extended Zak (Berry) phase: For diagonalizable, real-spectrum regimes, a biorthogonal Zak phase

$t_L$ 2

remains quantized; e.g., $t_L$ 3 for nontrivial, $t_L$ 4 for trivial phases (Li et al., 2017).

Winding number for point/line gaps: For complex spectra, invariants are constructed from the winding of $t_L$ 5 (point gap) or projectors (line gap) around $t_L$ 6. For example, the Skew-symmetric (Pfaffian) invariant at $t_L$ 7 remains well-defined, with phase boundaries set by vanishing Pfaffian (Sakaguchi et al., 2022, Zhang et al., 2 Feb 2026).
Physical interpretation: These invariants correctly predict the presence or absence of zero-energy boundary modes, generalizing the usual $t_L$ 8 invariant of class D to non-Hermitian settings.

In many cases, biorthogonality of the eigenvectors is essential, especially near EPs where right/right overlaps vanish, and only the biorthogonal topology survives (Li et al., 2017).

4. Majorana Zero Modes, Skin Effect, and Bulk–Boundary Correspondence

Non-Hermitian Kitaev chains exhibit new forms of boundary phenomena, including:

Majorana edge modes: Topological (nontrivial) phases host zero-energy Majorana modes exponentially localized at the chain ends. For asymmetric or imbalanced models, such modes often appear as reciprocal localization pairs (localized at opposite ends), resulting in a symmetric combined density—showing exact cancellation of the non-Hermitian skin effect for zero modes (Raj et al., 2 Jan 2026). Excited states, in contrast, generically exhibit the full non-Hermitian skin effect, with particle and hole components at opposite ends.
Defective/coalescing Majorana modes: At EPs, Majorana zero modes coalesce—leading to “defective” modes of algebraic multiplicity and breakdown of standard bulk–boundary correspondence, such that the number of boundary zero modes can interpolate continuously between two and one as non-Hermitian parameters are tuned (Zhao et al., 2020, Li et al., 2017).
Stability and robustness: In staggered or spatially modulated non-Hermitian systems, fixed points (“fixed lines”) arise where the topological index (winding number) is constant over a finite parameter region, insensitive to the strength or rate of non-Hermitian perturbations (Shi et al., 2023).
Interaction with RSOC and many-body effects: Spinful chains with RSOC show that RSOC renormalizes the effective pairing gap $t_L$ 9, shrinking the topological region and shifting phase boundaries in both Hermitian and non-Hermitian models (Shahab et al., 10 Dec 2025).

5. Phase Diagrams, Analytical Phase Boundaries, and Critical Loci

A unifying feature across non-Hermitian Kitaev chains is a rich and precisely calculable phase structure, with phase boundaries that differ sharply from the Hermitian limit.

Uniform gain/loss models: The topological phase boundary in $\Delta_c$ 0 space is generically an ellipse, e.g.,

$\Delta_c$ 1

for the spinful RSOC chain (Shahab et al., 10 Dec 2025).

Staggered/alternating pairing models: Topological regimes as a function of chemical potential and pairing imbalance ( $\Delta_c$ 2) can extend to arbitrarily large $\Delta_c$ 3 for $\Delta_c$ 4, with boundary lines at $\Delta_c$ 5 and $\Delta_c$ 6 (Zhang et al., 2 Feb 2026).
Dimerized/interacting chains: Interaction-induced topological phases shrink and may be annihilated at critical non-Hermitian interaction strength, as determined by exact gap-closing conditions (Sayyad et al., 2023).
Multicriticality and exceptional loci: Non-Hermitian chains generically reduce multicritical points, localize them in parameter space, and universally produce linear gap closing with dynamical exponent $\Delta_c$ 7 at nontrivial transitions (Rahul et al., 2023).

6. Many-Body, Floquet, and Bosonic Extensions

Finally, the non-Hermitian Kitaev paradigm encompasses interacting and bosonic systems, as well as periodically driven variants.

Many-body Hubbard interactions: Exact mappings and Pfaffian invariants classify many-body degeneracy and persistence of topological phase under complex Hubbard interactions, up to a point-gap closure (Sayyad et al., 2023).
Bosonic Kitaev chains: Despite Hermiticity at the Hamiltonian level, non-Hermitian excitation (dynamical) matrices generate robust nontrivial topology (e.g., skin effect, topological amplification), with quantized winding numbers extracted from the one-body spectrum (Fortin et al., 2024, Bomantara et al., 21 May 2025).
Floquet non-Hermitian phases: Periodic driving in bosonic chains admits Floquet analogues of the non-Hermitian skin effect, supports zero and $\Delta_c$ 8 edge modes, and realizes coexistence of skin and topological phenomena in excitation spectra (Bomantara, 19 Nov 2025).

Topological edge states and the non-Hermitian winding invariant remain robust against disorder and, in some regimes, finite onsite frequency, though the skin effect is generically suppressed by nonzero bosonic frequency and can be partially revived by disorder (Fortin et al., 2024, Bomantara et al., 21 May 2025, Bomantara, 19 Nov 2025).

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