Bosonic Kitaev Chain: Non-Hermitian Topology

Updated 4 December 2025

Bosonic Kitaev chain is a one-dimensional quadratic bosonic lattice model that features p-wave–like pairing and non-Hermitian dynamical effects.
It exhibits phenomena such as the non-Hermitian skin effect, topologically protected edge states, and spectral winding, which are crucial for understanding quantum criticality.
Experimental implementations in superconducting, nano-optomechanical, and cold atom systems demonstrate its potential for robust sensing and quantum information applications.

The bosonic Kitaev chain is a class of one-dimensional quadratic bosonic lattice models that generalize the fermionic Kitaev chain to systems with bosonic statistics. These models feature nearest-neighbor hopping and $p$ -wave–like pairing terms for bosons, giving rise to unconventional phases of matter that display emergent non-Hermitian phenomena, exotic topological invariants, and distinct many-body zero modes. Despite being governed by Hermitian Hamiltonians, the bosonic commutation relations, combined with pairing interactions, yield dynamical matrices with complex spectra responsible for features such as the non-Hermitian skin effect, topological amplification, and exceptional points. Bosonic Kitaev chains thus serve as prototypical platforms for exploring non-Hermitian topology, quantum criticality, and robust sensing in synthetic bosonic systems ranging from cold atom chains to nanomechanical and superconducting circuits.

1. Model Hamiltonians and Quadrature Representation

The prototypical bosonic Kitaev chain (BKC) is defined by the quadratic Hamiltonian

$\hat H = \sum_{j=1}^{N-1}\left[t\,\hat a_{j+1}^\dagger \hat a_j + \Delta\,\hat a_{j+1}^\dagger \hat a_j^\dagger + \mathrm{h.c.}\right] + \omega\sum_{j=1}^N \hat a_j^\dagger \hat a_j.$

Here, $\hat a_j$ is the annihilation operator for a bosonic mode at site $j$ , $t$ is the hopping amplitude, $\Delta$ is the nearest-neighbor pairing (two-mode squeezing) amplitude, and $\omega$ the on-site bosonic frequency (which is often set to zero in idealized treatments) (Fortin et al., 12 Dec 2024, Busnaina et al., 2023, McDonald et al., 2018).

Expressing $\hat a_j$ in terms of canonical quadratures,

$\hat x_j = \frac{1}{2}(\hat a_j + \hat a_j^\dagger), \quad \hat p_j = \frac{1}{2i} (\hat a_j - \hat a_j^\dagger),$

the Hamiltonian maps to a model where $x$ and $p$ quadratures are coupled with asymmetric spatial structure: $\hat H = \sum_j \left[(t+\Delta)\hat x_j \hat p_{j+1} - (t-\Delta)\hat x_{j+1} \hat p_j\right].$ The pairing terms induce effective non-Hermitian dynamics in the Heisenberg equations of motion for the quadratures, a key departure from fermionic analogues (Fortin et al., 12 Dec 2024, McDonald et al., 2018).

2. Non-Hermitian Excitation Spectrum and Skin Effect

A central property of BKC models is that, despite Hermitian Hamiltonians, the operator dynamics—formulated as first-order Heisenberg equations—are governed by non-Hermitian dynamical matrices. For the quadratic case,

$\dot{\psi} = M \psi,$

with

$M = \begin{pmatrix} 0 & T \ -V & 0 \end{pmatrix},$

where $T$ and $V$ encode the hopping and pairing structure in the quadrature basis (Wang et al., 23 Sep 2025, Bomantara, 19 Nov 2025).

This leads to key phenomena:

Non-Hermitian Skin Effect: Under open boundary conditions (OBC), all bulk eigenmodes exponentially localize at one end of the chain. In contrast, under periodic boundary conditions (PBC), the spectrum forms loops in the complex plane, capturing nontrivial spectral winding (Fortin et al., 12 Dec 2024, Slim et al., 2023, McDonald et al., 2018, Bomantara et al., 21 May 2025).
Topological Spectral Winding Number: For point-gapped non-Hermitian bands, the winding number

$\nu(E) = \frac{1}{2\pi i} \int_{-\pi}^\pi dk\, \partial_k \log[E(k)-E_b]$

classifies the phase. For appropriate parameters, $\nu = \pm 1$ for the two quadrature sectors ( $M_x$ , $M_p$ ), predicting topologically protected amplification channels (Fortin et al., 12 Dec 2024, Slim et al., 2023, Busnaina et al., 2023).

3. Topological Zero Modes and Edge States

Unlike the fermionic Kitaev chain, whose topological phase hosts Majorana zero modes protected by a $\mathbb Z_2$ invariant, the BKC supports a range of edge phenomena:

Bosonic Zero Modes in Interacting Fermionic Models: In the symmetric interacting Kitaev chain, non-local bosonic zero-mode operators $\hat O_B$ arise in the topologically trivial ( $|U|>4t$ ) phase. These operators commute with the total fermion parity and are exponentially localized at boundaries. Explicit infinite series constructions in the Majorana basis have been derived, and their coexistence with fermionic zero modes is established in dimerized (Kitaev–SSH) models (Wang et al., 2021).
Topologically Protected Edge Modes in Modified BKCs: Extensions, such as SSH–type models, support zero-energy (midgap) edge modes whenever a non-Hermitian winding number is nonzero. Two distinct classes of edge modes exist ("intercell"- and "intracell"-dominated), with differential stability to onsite potential and disorder (Bomantara et al., 21 May 2025, Bomantara, 19 Nov 2025).

A table summarizing types of zero modes in the BKC context:

Model Context	Type of Mode	Protection/Signature
Original BKC (free)	Skin modes, no MZMs	Winding number, amplification
Interacting Fermi-KC	Bosonic Zero Mode	Commuting with parity, edge-local
Modified BKC–SSH	Topological Zero	Non-Hermitian winding, midgap edge

4. Phase Diagrams, Quantum Criticality, and Exceptional Points

The BKC exhibits rich phase diagrams distinguished by dynamical stability, spectral reality, and topological properties:

Boundary Sensitivity: The spectrum with OBC is purely real for $t > \Delta$ , while with PBC, Im $[E_k]\neq0$ for all $k$ at any $\Delta\neq0$ . This boundary-condition dependence is absent in the fermionic chain (McDonald et al., 2018, Fortin et al., 12 Dec 2024).
Quantum Phase Transitions: For interacting bosonic Kitaev–Hubbard chains, inclusion of on-site two- and three-body repulsions yields direct Ising transitions between bond-pairing insulator and trivial insulator, with critical exponents $\nu \approx 1$ (Wang et al., 2022).
Exceptional Points (EPs): The non-Hermitian Bloch core matrices of BKCs exhibit EPs where eigenvalues and eigenvectors coalesce. These EPs precisely coincide with many-body localization–delocalization transitions in Fock space, and their analytic locations provide phase boundaries across the $(t,\Delta,g)$ parameter space (Wang et al., 11 May 2025, Wang et al., 23 Sep 2025).

5. Floquet Engineering and Robustness of Topological Phenomena

Periodically driven (Floquet) generalizations of the BKC yield an enriched spectrum of non-Hermitian topological phenomena:

Floquet Topological Modes: Piecewise-constant or binary drives realize time-periodic models in which zero and $\pi$ quasienergy topological boundary modes coexist with the non-Hermitian skin effect. Floquet winding numbers ( $\nu_0$ , $\nu_\pi$ ) are computed in proper chiral symmetry frames (Bomantara, 19 Nov 2025).
Stability to Perturbations: Topological edge modes are generically robust against finite onsite frequency and spatial disorder in certain Floquet BKC models, while the skin effect can be quenched (onsite frequency) or revived (disorder), depending on drive protocol and coupling symmetry (Bomantara, 19 Nov 2025, Bomantara et al., 21 May 2025).
Phase Diagram Complexity: The number of edge modes may be tuned and proliferate with repeated crossings of drive parameters, revealing Floquet-specific topological transitions lacking static analogs (Bomantara, 19 Nov 2025).

6. Experimental Realizations and Observables

BKCs have been realized across optical, nano-optomechanical, and superconducting platforms:

Superconducting Parametric Cavities: Frequency modes are mapped to synthetic lattice sites, with in situ tunable hopping and pairing via parametric pumping, enabling direct measurements of chiral transport, skin localization, and boundary sensitivity (Busnaina et al., 2023).
Nano-Optomechanics: Modulated laser fields induce effective hopping and pairing between nanomechanical modes, revealing topological amplification, spectral winding, exponential response scaling, and robust end-to-end gain (Slim et al., 2023).
Cold Atom Arrays: Bose–Hubbard chains with engineered pair-tunneling realize $\mathbb Z_2$ symmetry–broken phases and zero-energy edge excitations, observable by spectroscopic probes and single-site imaging (Vishveshwara et al., 2020).

Observable quantities include:

Local quadrature susceptibilities (chiral gain)
Spatial profiles of quadrature wavefunctions (edge localization)
Singular value spectra of response matrices (bulk–boundary correspondence)
End-to-end correlation functions and midgap mode splittings

7. Comparison to Fermionic Kitaev Chains and Further Generalizations

The bosonic Kitaev chain fundamentally differs from its fermionic counterpart:

Statistical Constraints: While the fermionic chain supports true Majorana zero modes and $\mathbb Z_2$ topology, the bosonic version cannot realize bona fide Majorana bosons; instead, it supports non-Hermitian dynamical topology and skin modes (Slim et al., 2023, Wang et al., 2022).
Hard-Core Limit: For infinite on-site repulsion, the BKC maps to an anisotropic XY spin chain and, via Jordan–Wigner, to a fermionic Kitaev chain. The non-Hermitian skin effect disappears in this hard-core limit (Wang et al., 2022).
Criticality and Edge Phenomena: Both allow robust boundary zero modes under specific symmetry and parameter regimes, but their physical content (superselection sector, localization, amplification) is distinct (Wang et al., 2021, Vishveshwara et al., 2020).

Extensions include staggered/dimerized BKCs, compass-model engineering, and hybrid models with coexisting bosonic and fermionic zero modes (Wang et al., 2021, Wang et al., 11 May 2025).

The bosonic Kitaev chain thus constitutes a paradigmatic quantum platform for exploring the interplay of statistics, non-Hermitian topology, dynamical amplification, and quantum many-body phase transitions, with broad applicability to quantum sensing, information processing, and synthetic matter engineering (Fortin et al., 12 Dec 2024, Wang et al., 23 Sep 2025, Bomantara et al., 21 May 2025, Bomantara, 19 Nov 2025).