Hermitian Bosonic Kitaev Chains

• Hermitian bosonic Kitaev chains are one-dimensional pairing systems whose Hermitian many-body Hamiltonians yield non-Hermitian dynamical matrices exhibiting exceptional points, skin effects, and chiral transport.
• They are realized through quadratic bosonic models and mapped spin or fermion chains, revealing rich topological phases and stability transitions via analytical and experimental methods.
• Experimental platforms, including superconducting circuits and optomechanical systems, demonstrate that non-Hermitian phenomena can emerge without violating the underlying Hermitian dynamics.

Hermitian bosonic Kitaev chains are one-dimensional pairing lattices whose second-quantized Hamiltonians are self-adjoint, yet whose bosonic Bogoliubov or dynamical matrices are generically non-Hermitian. In this sense, they are bosonic analogues of the fermionic Kitaev chain, but with a decisive structural difference: the relevant linear generator for bosonic Heisenberg dynamics can exhibit complex spectra, exceptional points, skin effects, and boundary-sensitive stability even when the many-body Hamiltonian satisfies $\hat H^\dagger=\hat H$ (Lee et al., 9 Jul 2026). A second, distinct usage of the term appears in interacting spin and fermion models that map exactly onto Hermitian Kitaev-chain forms while supporting emergent bosonic edge excitations, such as bosonic zero modes that are even under total fermion parity (Francica, 2021).

1. Canonical definitions and model classes

A canonical bosonic Kitaev chain is the nearest-neighbor quadratic pairing Hamiltonian

$$\hat{H}_{\rm BKC} = \frac{1}{2}\sum_{j} \left( i w\,\hat{a}_{j+1}^{\dag}\hat{a}_{j} + i\Delta\,\hat{a}_{j+1}^{\dag}\hat{a}_{j}^{\dag} + {\rm H.c.} \right),$$

with bosonic modes $\hat a_j$, real $w$, and real $\Delta$ (Lee et al., 9 Jul 2026). A more general translationally invariant family allows normal and pairing couplings at arbitrary range,

$$\hat{H} = \frac{1}{2} \sum_{d\ge 0} \sum_{j} \Big( (g_d + i w_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j + (i\Delta_d + \eta_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j^{\dag} + {\rm H.c.} \Big),$$

with real $g_d,w_d,\Delta_d,\eta_d$; the bare BKC corresponds to $w_d = w\,\delta_{d,1}$, $\Delta_d = \Delta\,\delta_{d,1}$, and $g_d=\eta_d=0$ (Lee et al., 9 Jul 2026).

The same label also covers several closely related Hermitian constructions. A minimal dimer version is

$$\hat{H}_{\rm BKC} = \frac{1}{2}\sum_{j} \left( i w\,\hat{a}_{j+1}^{\dag}\hat{a}_{j} + i\Delta\,\hat{a}_{j+1}^{\dag}\hat{a}_{j}^{\dag} + {\rm H.c.} \right),$$0

which already displays exceptional points in its effective bosonic core matrices (He et al., 26 Feb 2025). A staggered chain introduces two sublattices and alternating hopping, pairing, and on-site potentials, leading to a $$\hat{H}_{\rm BKC} = \frac{1}{2}\sum_{j} \left( i w\,\hat{a}_{j+1}^{\dag}\hat{a}_{j} + i\Delta\,\hat{a}_{j+1}^{\dag}\hat{a}_{j}^{\dag} + {\rm H.c.} \right),$$1 Bloch core matrix in Nambu space (Wang et al., 11 May 2025). A modified two-sublattice bosonic Kitaev chain with intracell and intercell couplings yields an excitation Hamiltonian exactly mappable to a non-Hermitian SSH model (Bomantara et al., 21 May 2025). In a different interacting direction, a spin-$$\hat{H}_{\rm BKC} = \frac{1}{2}\sum_{j} \left( i w\,\hat{a}_{j+1}^{\dag}\hat{a}_{j} + i\Delta\,\hat{a}_{j+1}^{\dag}\hat{a}_{j}^{\dag} + {\rm H.c.} \right),$$2 fermion chain with $$\hat{H}_{\rm BKC} = \frac{1}{2}\sum_{j} \left( i w\,\hat{a}_{j+1}^{\dag}\hat{a}_{j} + i\Delta\,\hat{a}_{j+1}^{\dag}\hat{a}_{j}^{\dag} + {\rm H.c.} \right),$$3 antiferromagnetic interaction maps onto a Hermitian Kitaev chain at half filling and supports a bosonic zero mode protected by spin-up parity (Francica, 2021).

This multiplicity of constructions suggests that “Hermitian bosonic Kitaev chain” is best understood as a family resemblance rather than a single model class. The common thread is a Hermitian many-body Hamiltonian with hopping-plus-pairing structure whose effective low-energy or dynamical description reproduces hallmark Kitaev phenomena in a bosonic setting.

2. Hermiticity of the Hamiltonian and non-Hermiticity of the dynamics

In this literature, “Hermitian” refers to the many-body Hamiltonian, not to the bosonic dynamical matrix. For quadratic bosonic systems written in Nambu form,

$$\hat{H}_{\rm BKC} = \frac{1}{2}\sum_{j} \left( i w\,\hat{a}_{j+1}^{\dag}\hat{a}_{j} + i\Delta\,\hat{a}_{j+1}^{\dag}\hat{a}_{j}^{\dag} + {\rm H.c.} \right),$$4

the dynamical matrix is

$$\hat{H}_{\rm BKC} = \frac{1}{2}\sum_{j} \left( i w\,\hat{a}_{j+1}^{\dag}\hat{a}_{j} + i\Delta\,\hat{a}_{j+1}^{\dag}\hat{a}_{j}^{\dag} + {\rm H.c.} \right),$$5

and the Heisenberg equations read $$\hat{H}_{\rm BKC} = \frac{1}{2}\sum_{j} \left( i w\,\hat{a}_{j+1}^{\dag}\hat{a}_{j} + i\Delta\,\hat{a}_{j+1}^{\dag}\hat{a}_{j}^{\dag} + {\rm H.c.} \right),$$6 (Lee et al., 9 Jul 2026). Even if $$\hat{H}_{\rm BKC} = \frac{1}{2}\sum_{j} \left( i w\,\hat{a}_{j+1}^{\dag}\hat{a}_{j} + i\Delta\,\hat{a}_{j+1}^{\dag}\hat{a}_{j}^{\dag} + {\rm H.c.} \right),$$7 is Hermitian, $$\hat{H}_{\rm BKC} = \frac{1}{2}\sum_{j} \left( i w\,\hat{a}_{j+1}^{\dag}\hat{a}_{j} + i\Delta\,\hat{a}_{j+1}^{\dag}\hat{a}_{j}^{\dag} + {\rm H.c.} \right),$$8 is generically non-Hermitian because of the bosonic symplectic structure. A recurring misconception is that Hermiticity of $$\hat{H}_{\rm BKC} = \frac{1}{2}\sum_{j} \left( i w\,\hat{a}_{j+1}^{\dag}\hat{a}_{j} + i\Delta\,\hat{a}_{j+1}^{\dag}\hat{a}_{j}^{\dag} + {\rm H.c.} \right),$$9 should force Hermiticity of the BdG generator; the bosonic case is precisely the counterexample (Lee et al., 9 Jul 2026, Flynn et al., 2020).

For the bare BKC, the momentum-space dynamical matrix is

$\hat a_j$0

with eigenvalues

$\hat a_j$1

so periodic-boundary spectra are generically complex when $\hat a_j$2 (Lee et al., 9 Jul 2026). In the quadrature basis,

$\hat a_j$3

the BKC becomes block diagonal: $\hat a_j$4 so the $\hat a_j$5 and $\hat a_j$6 quadratures evolve independently (Lee et al., 9 Jul 2026). This decoupling is equivalent to an effective particle-hole symmetry of the dynamical matrix,

$\hat a_j$7

and each quadrature sector is a single-band non-Hermitian Hatano–Nelson problem (Lee et al., 9 Jul 2026).

The same structure underlies the phase-dependent chiral transport emphasized in the bosonic Kitaev-Majorana chain. In the quadrature representation, the equations of motion decouple and acquire asymmetric nearest-neighbor couplings, so one quadrature propagates preferentially in one direction and the other in the opposite direction; as $\hat a_j$8, the transport becomes strictly one-way in the idealized limit (McDonald et al., 2018). The superconducting-circuit realization later observed this behavior directly as chiral transport and quadrature-dependent localization (Busnaina et al., 2023).

3. Symmetry structure, topology, and skin effects

Two effective symmetries organize much of the Hermitian bosonic Kitaev-chain phenomenology. The first is sublattice symmetry, defined at the second-quantized level by

$\hat a_j$9

For translationally invariant bosonic pairing Hamiltonians, this symmetry implies an effective transpose-type time-reversal symmetry of the dynamical matrix. In the BKC, that relation can be written as

$w$0

or, in the folded-zone formulation, $w$1 with $w$2 (Lee et al., 9 Jul 2026). This effective time-reversal symmetry is not physical time-reversal symmetry, but it supports a $w$3 invariant $w$4 and a symmetry-protected skin effect (Lee et al., 9 Jul 2026).

The second organizing symmetry is quadrature particle-hole symmetry, which is equivalent to decoupling of $w$5 and $w$6 dynamics. When it holds, the problem reduces to two single-band non-Hermitian sectors with winding numbers

$w$7

and the BKC non-Hermitian skin effect can be understood as two decoupled Hatano–Nelson chains, one for each quadrature (Lee et al., 9 Jul 2026).

These symmetry principles clarify why Hermitian bosonic chains display non-Hermitian bulk-boundary phenomena. In the bare BKC, periodic-boundary eigenmodes are extended plane waves, whereas open-boundary eigenmodes become exponentially localized at one edge; at the same time, periodic boundaries can be dynamically unstable while open boundaries are completely stable over a broad parameter regime (Lee et al., 9 Jul 2026). This boundary-sensitive stability was already identified in the bosonic Kitaev-Majorana chain, where the boundary-less system has delocalized unstable modes while a finite open chain can be dynamically stable and described by localized modes (McDonald et al., 2018).

The modified bosonic Kitaev chain sharpens the topological content. For $w$8, its excitation Hamiltonian is exactly similar to a non-Hermitian SSH model with effective couplings

$w$9

and the topologically nontrivial regime is

$\Delta$0

In that regime, zero-energy edge modes coexist with a skin effect in the excitation Hamiltonian; finite onsite potential $\Delta$1 rapidly destroys the skin effect and some, but not all, edge modes (Bomantara et al., 21 May 2025). Disorder does not simply wash out this structure: at $\Delta$2 the topological zero modes remain robust, and disorder in $\Delta$3 can partially recover skin-effect-like localization at nonzero onsite potential (Bomantara et al., 21 May 2025).

4. Exceptional points, stability transitions, and dynamical braiding

Exceptional-point physics is central to Hermitian bosonic Kitaev chains because the non-Hermitian object is the dynamical matrix, not the many-body Hamiltonian. The minimal dimer provides the simplest analytic example. Its Nambu core matrices

$\Delta$4

have eigenvalues

$\Delta$5

with exceptional points at $\Delta$6 (He et al., 26 Feb 2025). These EPs partition parameter space into four regions: two regions with two harmonic-oscillator sectors, and two mixed regions with one harmonic oscillator and one inverted harmonic oscillator. The nonequilibrium quantum phase transition is then detected through the second-order intensity correlation $\Delta$7; the derivative of its long-time average with respect to $\Delta$8 develops pronounced valleys near $\Delta$9, which deepen with averaging time (He et al., 26 Feb 2025).

The extended bosonic Kitaev chain shows a related hidden-EP mechanism. For the periodic model

$$\hat{H} = \frac{1}{2} \sum_{d\ge 0} \sum_{j} \Big( (g_d + i w_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j + (i\Delta_d + \eta_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j^{\dag} + {\rm H.c.} \Big),$$0

the Nambu core matrix

$$\hat{H} = \frac{1}{2} \sum_{d\ge 0} \sum_{j} \Big( (g_d + i w_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j + (i\Delta_d + \eta_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j^{\dag} + {\rm H.c.} \Big),$$1

becomes non-diagonalizable at

$$\hat{H} = \frac{1}{2} \sum_{d\ge 0} \sum_{j} \Big( (g_d + i w_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j + (i\Delta_d + \eta_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j^{\dag} + {\rm H.c.} \Big),$$2

That exceptional point coincides with a localization-delocalization transition in an equivalent single-particle problem formulated in Fock space, obtained by projecting onto a BCS-like pairing basis (He et al., 2024). In the localized regime $$\hat{H} = \frac{1}{2} \sum_{d\ge 0} \sum_{j} \Big( (g_d + i w_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j + (i\Delta_d + \eta_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j^{\dag} + {\rm H.c.} \Big),$$3, a Bogoliubov vacuum exists and the inverse participation ratio stays finite; for $$\hat{H} = \frac{1}{2} \sum_{d\ge 0} \sum_{j} \Big( (g_d + i w_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j + (i\Delta_d + \eta_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j^{\dag} + {\rm H.c.} \Big),$$4, the standard Bogoliubov vacuum ceases to exist and the equivalent Fock-space eigenstates delocalize (He et al., 2024). The same hidden-EP scenario appears in the Dicke-model reduction to a two-site bosonic Kitaev model, where the average photon number after a quench from the empty state detects the transition (He et al., 2024).

A staggered bosonic Kitaev chain extends this logic to a $$\hat{H} = \frac{1}{2} \sum_{d\ge 0} \sum_{j} \Big( (g_d + i w_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j + (i\Delta_d + \eta_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j^{\dag} + {\rm H.c.} \Big),$$5 non-Hermitian Bloch core matrix,

$$\hat{H} = \frac{1}{2} \sum_{d\ge 0} \sum_{j} \Big( (g_d + i w_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j + (i\Delta_d + \eta_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j^{\dag} + {\rm H.c.} \Big),$$6

with $$\hat{H} = \frac{1}{2} \sum_{d\ge 0} \sum_{j} \Big( (g_d + i w_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j + (i\Delta_d + \eta_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j^{\dag} + {\rm H.c.} \Big),$$7 and $$\hat{H} = \frac{1}{2} \sum_{d\ge 0} \sum_{j} \Big( (g_d + i w_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j + (i\Delta_d + \eta_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j^{\dag} + {\rm H.c.} \Big),$$8 (Wang et al., 11 May 2025). In analytically tractable regimes, the EP conditions reduce to simple algebraic relations such as

$$\hat{H} = \frac{1}{2} \sum_{d\ge 0} \sum_{j} \Big( (g_d + i w_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j + (i\Delta_d + \eta_d)\,\hat{a}_{j+d}^{\dag}\hat{a}_j^{\dag} + {\rm H.c.} \Big),$$9

or

$g_d,w_d,\Delta_d,\eta_d$0

and these EP loci coincide with sharp localization-delocalization transitions of collective eigenstates in effective Fock-space networks, diagnosed by layer-resolved block inverse participation ratios (Wang et al., 11 May 2025).

A broader stability theory is supplied by pseudo-Hermiticity, generalized $g_d,w_d,\Delta_d,\eta_d$1 symmetry, and Krein stability theory. For any quadratic bosonic Hamiltonian,

$g_d,w_d,\Delta_d,\eta_d$2

so the effective BdG generator is pseudo-Hermitian and therefore generalized-$g_d,w_d,\Delta_d,\eta_d$3-symmetric (Flynn et al., 2020). Dynamical stability corresponds to the unbroken generalized-$g_d,w_d,\Delta_d,\eta_d$4 phase, where $g_d,w_d,\Delta_d,\eta_d$5 is diagonalizable and all eigenvalues are real. Instability appears either through exceptional points or through Krein collisions, where degenerate real eigenvalues split into complex-conjugate pairs while the matrix remains diagonalizable (Flynn et al., 2020). The Krein phase rigidity introduced there vanishes at both kinds of transition and extends standard non-Hermitian phase rigidity to bosonic indefinite-metric problems (Flynn et al., 2020).

These non-Hermitian spectral features can be organized topologically. In multiband Hermitian bosonic Kitaev chains, complex dynamical eigenvalues can braid in the complex-frequency plane when loops in parameter space encircle exceptional points. Explicit two-strand and three-strand braids arise from square-root and cubic-root branch structures near second- and third-order EPs, and symmetry constrains the braids by pairing $g_d,w_d,\Delta_d,\eta_d$6 with $g_d,w_d,\Delta_d,\eta_d$7 at fixed momentum and relating $g_d,w_d,\Delta_d,\eta_d$8 to $g_d,w_d,\Delta_d,\eta_d$9 by complex conjugation (Wang et al., 23 Sep 2025).

5. Interacting, mapped, and spin-chain realizations

One major interacting realization begins from a spin-$w_d = w\,\delta_{d,1}$0 fermion chain with nearest-neighbor hopping and $w_d = w\,\delta_{d,1}$1 antiferromagnetic interaction. After a Majorana representation, a Jordan–Wigner transformation on a doubled chain, a Mattis–Nam transformation, and a second Jordan–Wigner transformation, the $w_d = w\,\delta_{d,1}$2 sector reduces exactly to

$w_d = w\,\delta_{d,1}$3

which is a Hermitian Kitaev chain at half filling and $w_d = w\,\delta_{d,1}$4 (Francica, 2021). The corresponding zero mode

$w_d = w\,\delta_{d,1}$5

is “bosonic” in the sense that it is built from an even number of microscopic fermions and is even under total fermion parity $w_d = w\,\delta_{d,1}$6, while remaining odd under spin-up parity $w_d = w\,\delta_{d,1}$7, the protecting $w_d = w\,\delta_{d,1}$8 symmetry of the topological phase (Francica, 2021). In that model, the bosonic zero mode is accompanied by spontaneous breaking of $w_d = w\,\delta_{d,1}$9, long-range end-to-end order in $\Delta_d = \Delta\,\delta_{d,1}$0, and at least twofold entanglement-spectrum degeneracy in the ordered phase (Francica, 2021). When the hopping term is turned on, the system undergoes a continuous transition to a trivial phase; at $\Delta_d = \Delta\,\delta_{d,1}$1, finite-size scaling gives $\Delta_d = \Delta\,\delta_{d,1}$2 and $\Delta_d = \Delta\,\delta_{d,1}$3, consistent with the 2D classical Ising universality class (Francica, 2021).

A related but explicitly bosonic interacting construction is the paired Bose–Hubbard chain

$\Delta_d = \Delta\,\delta_{d,1}$4

which supports a gapped $\Delta_d = \Delta\,\delta_{d,1}$5 Ising phase with number fluctuation but no off-diagonal long-range order (Vishveshwara et al., 2020). In the strongly interacting limit, the low-energy Hilbert space truncates to two occupations per site, yielding an anisotropic XY spin chain and, by Jordan–Wigner, the fermionic Kitaev chain. The bosonic and fermionic systems then share identical energy spectra, including the ground-state doublet, while their wavefunctions remain markedly different (Vishveshwara et al., 2020). The bosonic doubly degenerate phase is described there as a gap-protected macroscopic qubit (Vishveshwara et al., 2020).

The bosonic Kitaev-Hubbard chain adds explicit on-site interactions and, in some cases, a three-body constraint. In the hard-core limit it maps to a spin-$\Delta_d = \Delta\,\delta_{d,1}$6 XY chain with Dzyaloshinskii–Moriya interaction,

$\Delta_d = \Delta\,\delta_{d,1}$7

and the non-Hermitian skin effect of the free-boson BdG description disappears exactly in this hard-core regime (Wang et al., 2022). For chains with a three-body constraint and for unconstrained soft-core bosons, DMRG finds direct transitions between bond-pairing insulators and trivial insulators with correlation-length exponent $\Delta_d = \Delta\,\delta_{d,1}$8, again consistent with Ising criticality (Wang et al., 2022).

Spin-chain generalizations broaden the class of Hermitian bosonic Kitaev analogues. The spin-1 Kitaev-AKLT chain

$\Delta_d = \Delta\,\delta_{d,1}$9

with $g_d=\eta_d=0$0 and $g_d=\eta_d=0$1, is Hermitian and exactly solvable at $g_d=\eta_d=0$2, where it becomes a sum of projectors onto maximal spin components along bond directions (Raja et al., 14 Oct 2025). At that point it ունի an exponential ground-state degeneracy $g_d=\eta_d=0$3, admits a fractionalized spin-$g_d=\eta_d=0$4 description, and each ground state can be written as a matrix product state (Raja et al., 14 Oct 2025). This suggests that Hermitian bosonic Kitaev-chain physics extends naturally from quadratic bosons to constrained spin systems with projector structures, fractionalization, and exact tensor-network ground states.

6. Experimental implementations and broader significance

The most direct experimental realization to date is a multimode superconducting parametric cavity implementing the bosonic Kitaev chain in synthetic dimensions. There, lattice sites are encoded in cavity frequency modes, while complex hopping and pairing are generated in situ by parametric pumping at mode-difference and mode-sum frequencies, respectively (Busnaina et al., 2023). The experiment demonstrated chiral transport, quadrature wavefunction localization, and sensitivity to boundary conditions, all identified as precursors of nontrivial topology and the non-Hermitian skin effect in a Hermitian bosonic chain (Busnaina et al., 2023).

Several other platforms are explicitly discussed in the literature surveyed here. The Hermitian bosonic dimer can be realized by two coupled cavities or mechanical resonators with both beam-splitter and two-mode-squeezing interactions, with $g_d=\eta_d=0$5 controlled by detunings or onsite frequencies; in that setting, second-order coherence $g_d=\eta_d=0$6 serves as an EP witness (He et al., 26 Feb 2025). The hidden-EP chain analysis maps naturally onto the Dicke model, where the time evolution of the average photon number from the empty state detects the localization-delocalization transition associated with the exceptional point (He et al., 2024). The modified bosonic Kitaev chain admits an optical-cavity implementation using alternating cavities embedded in a $g_d=\eta_d=0$7 nonlinear medium with parametrically modulated frequencies, giving tunable intracell and intercell hopping and pairing amplitudes (Bomantara et al., 21 May 2025). More generally, optomechanical systems, circuit QED architectures, and cold-atom settings recur as natural platforms for realizing bosonic Kitaev chains and their generalizations (He et al., 26 Feb 2025, Wang et al., 11 May 2025).

Two broad conclusions emerge from this body of work. First, Hermitian bosonic Kitaev chains provide controlled access to non-Hermitian topology without abandoning Hermitian many-body dynamics: skin effects, exceptional points, spectral winding, symmetry-protected skin effects, and dynamical braiding all arise from the bosonic dynamical matrix rather than from explicit gain-loss terms in the Hamiltonian (Lee et al., 9 Jul 2026, Wang et al., 23 Sep 2025). Second, interacting and mapped realizations show that “bosonic Kitaev physics” is not confined to quadratic bosons: it also appears in spinful fermion chains, paired Bose–Hubbard models, hard-core bosonic Kitaev-Hubbard systems, and spin-1 projector chains, where the protected low-energy object may be a bosonic zero mode, a bond-pairing Ising phase, or a fractionalized spin-sector edge degree of freedom rather than a literal bosonic Majorana excitation (Francica, 2021, Vishveshwara et al., 2020, Wang et al., 2022, Raja et al., 14 Oct 2025).

In that broader sense, Hermitian bosonic Kitaev chains define a research program rather than a single model: they are Hermitian one-dimensional pairing systems whose bosonic structure turns the usual distinction between Hermitian and non-Hermitian physics into a problem of representation, symmetry, and stability.