Analytical diagonalization of the open-boundary bosonic Kitaev chain: An asymmetric plane-wave ansatz approach

AI summary

Overview

Research area: Quantum physics — analytical diagonalization of quadratic bosonic Hamiltonians, specifically the open-boundary bosonic Kitaev chain (OBKC). The work sits at the intersection of non-Hermitian physics, bosonic Bogoliubov–de Gennes theory, and exactly solvable lattice models.

Technical level: Advanced. The paper assumes working familiarity with Bogoliubov transformations, bosonic commutation relations, non-Hermitian matrices, and plane-wave ansatze for tridiagonal lattice problems.

Scope in one sentence: The paper gives a closed-form, purely algebraic derivation of the full quasiparticle spectrum and eigenvectors of the finite open-boundary bosonic Kitaev chain, replacing the geometric "local squeezing" solution with an asymmetric plane-wave ansatz for a 2N × 2N non-Hermitian associated matrix.

What This Paper Is About

The bosonic Kitaev chain is a quadratic bosonic model with purely imaginary nearest-neighbor hopping i t and pairing i Δ that has attracted attention for its non-Hermitian boundary physics, non-Hermitian skin effects, and chiral transport, despite being governed by a Hermitian Hamiltonian. The known solution diagonalizes it by a local squeezing transformation in the position–momentum representation, which works but is geometrically motivated and leaves the non-uniqueness of the transformation (an arbitrary real constant λ) largely undiscussed.

The authors ask whether the same model can be diagonalized completely and explicitly using only standard bosonic Bogoliubov machinery inside Fock space, without the (x, p) representation or the generalized Brillouin zone formalism. They answer yes for any finite chain length N, and in doing so they produce explicit closed-form eigenvalues and eigenvectors plus a systematic treatment of a twofold degeneracy that had not been fully elaborated.

Key Contributions

1. A purely algebraic solution of the OBKC. The Hamiltonian is cast into standard bosonic quadratic form and a conventional Bogoliubov transformation is applied, producing a 2N × 2N non-Hermitian "associated matrix" M. No (x, p) representation, squeezing argument, or generalized Brillouin zone is used.

2. An asymmetric plane-wave ansatz. Because the two plane-wave branches give different expressions for the eigenvalue in a non-Hermitian matrix, the authors generalize the usual symmetric ansatz to V_j = X e^{i p j} + Y e^{-i p' j} with unequal right- and left-moving complex momenta p and p′. This yields explicit quantization conditions and closed-form eigenvalues for arbitrary finite N.

3. A rigorous degeneracy-resolution procedure. Each eigenvalue is shown to be twofold degenerate in the left eigenspace, with two linearly independent eigenvectors. The authors prove algebraically that the bosonic commutation relations admit only a single physical Bogoliubov mode built from the two degenerate eigenvectors, and they construct the N physical quasiparticle operators.

4. Characterization of non-uniqueness. The resulting Bogoliubov transformations carry N/2 real free parameters for even N and (N−1)/2 for odd N, which is shown to reduce, in a special case, to the single arbitrary constant λ of the squeezing transformation — reflecting the absence of a well-defined ground state.

Main Findings

• Model and stability condition. The Hamiltonian uses δ_± ≡ (t ± Δ)/2 with t > 0 and Δ > 0, and the paper explicitly excludes the special case t = Δ. Diagonalizability via Bogoliubov transformation requires δ_- > 0; for δ_- < 0 the model is "dynamically unstable" and cannot be diagonalized, consistent with Colpa's earlier discussion of bosonic quadratic forms and with the squeezing-transformation analysis.

• Illustrative N = 2 interpretation. For N = 2, H_B|_{δ_->0} is proportional to the spatial rotation generator x₁p₂ − x₂p₁ (an angular momentum), while H_B|_{δ_-<0} is proportional to the Lorentz boost generator x₁p₂ + x₂p₁, which has continuous spectrum — explaining non-diagonalizability.

• Two decoupled blocks. A real unitary transformation Q block-diagonalizes the associated matrix as M̃ = Q M Q⁻¹ = diag(F₁, F₂) with F₁ = A + B and F₂ = A − B. Each of F₁, F₂ describes a single-particle tight-binding Hatano–Nelson model with purely imaginary asymmetric hopping strengths. For δ_- > 0, both are similar to the same Hermitian tridiagonal matrix via S = diag(√(δ₊/δ₋), (√(δ₊/δ₋))², …, (√(δ₊/δ₋))^N) — a similarity transformation that exactly reflects the local squeezing transformation, though the authors deliberately do not exploit it.

• Quantization conditions from the boundary equations. The ansatz with p ≡ α + iβ and p′ ≡ α′ + iβ′ gives α + α′ = 2πm/(N+1) with integer m, β = −β′, and α = α′ ± π. Reality of the eigenvalue forces the key relation δ₊/δ₋ = e^{2β}, which uniquely fixes β and simultaneously shows why δ_- < 0 fails.

• Closed-form spectrum. The eigenvalues are Λ(α) = −2 sin α √(δ₊δ₋), with allowed momenta α_m = −π/2 + mπ/(N+1) for m = 1, 2, …, N (for both even and odd N). Eigenvectors are V_j(α_m) = 2(−i)^{j−1} sin(πmj/(N+1)) e^{−βj} and W_j(α_m) = 2(−i)^{j−1} sin(πmj/(N+1)) e^{+βj}.

• Edge localization. V is localized near the left edge of the chain and W near the right edge (shown in the insets of Fig. 1 for N = 10 and N = 11). For even N all α_m are nonzero; for odd N there is a zero mode at α_{(N+1)/2} = 0.

• Zero-mode handling. For even N the eigenvalue Λ cannot be zero, since the boundary equations would force a null vector. For odd N a unique normalized left eigenvector at Λ = 0 is given explicitly, and it is shown to be recoverable from the general eigenvector formula by setting α = 0.

• Pairing of eigenvalues. For even N the eigenvalues split into N/2 pairs {Λ(α_n), Λ(α_{N+1−n}) = −Λ(α_n)} for n = 1, …, N/2. For odd N the N−1 nonzero eigenvalues form (N−1)/2 such pairs, alongside the zero mode.

• Degenerate eigenvectors and orthogonality identities. The two degenerate left eigenvectors of M at each Λ(α_m) are χ(α_m) ≡ (V(α_m), V(α_m)) and ξ(α_m) ≡ (W(α_m), −W(α_m)), with χ(α_m)ξ†(α_m) = 0. The key identities used are V(α_m)W†(α_n) = W(α_m)V†(α_n) = 2(N+1)δ_{mn} and V(α_m)W^T(α_n) = W(α_m)V^T(α_n) = 2(N+1)δ_{m+n,N+1}.

• Connection to non-Bloch band theory. In the thermodynamic limit N → ∞, α becomes continuous on (−π/2, π/2), reproducing the continuous open-boundary band of Yokomizo and Murakami. Their generalized Brillouin zone for the OBKC consists of two circles with radii |z₊| = √(δ₊/δ₋) = e^β and |z₋| = √(δ₋/δ₊) = e^{−β} in the two bosonic Bogoliubov–de Gennes sectors — exactly the skin factors in V and W.

• Relation to the earlier dispersion. The explicit spectrum reproduces the structure of the squeezing-transformation result E_n = 2√(δ₊δ₋) cos(nπ/(N+1)), with even-N pairs {E_n, E_{N+1−n} = −E_n} and an odd-N zero mode.

Methodology in Plain English

A quadratic bosonic Hamiltonian can be diagonalized by mixing annihilation and creation operators (a Bogoliubov transformation), and the mixing coefficients are found by solving an eigenvalue problem for a matrix built from the hopping and pairing amplitudes. For this model that matrix is twice the size of the number of sites and is non-Hermitian, which is the technical obstacle.

The authors guess that the eigenvectors look like waves running along the chain, but because the matrix is non-Hermitian, a wave running left and a wave running right do not behave symmetrically. So they let the left-moving and right-moving waves have different (complex) momenta. Plugging this guess into the bulk equations gives two expressions for the energy; demanding they agree and are real fixes the imaginary part of the momenta in terms of the ratio δ₊/δ₋. Plugging the guess into the two end-of-chain boundary equations then quantizes the real part of the momentum, giving an explicit list of N allowed values.

The catch is that the doubled matrix produces two eigenvectors for every energy, whereas a chain of N sites has only N physical modes. The authors show algebraically that the bosonic commutation relations eliminate one of the two combinations, leaving exactly one legitimate quasiparticle per energy, and they construct these N operators. The leftover freedom in how one pairs up the degenerate solutions is not a bug but a genuine feature: it produces N/2 (or (N−1)/2) free real parameters, generalizing the single free constant λ of the squeezing approach.

Why This Matters

Impact on research. The paper supplies what earlier treatments left implicit: exact closed-form eigenvalues and eigenvectors at finite N, an explicit construction of the Bogoliubov transformation that non-Bloch band theory guaranteed to exist but did not write down, and a systematic resolution of the degenerate eigenspaces. It also makes the non-uniqueness of the transformation — tied to the absence of a well-defined ground state — an explicit, counted feature rather than a side remark. The asymmetric plane-wave ansatz is presented as a general tool, not a one-off trick.

Real-world applications (as framed by the paper):

• Driven-dissipative quantum systems, for which the OBKC serves as an effective Hamiltonian and where the absence of a lower energy bound is acceptable rather than pathological.
• Non-Hermitian skin-effect physics, since the two wave functions V and W are exponentially localized at opposite ends with skin factors e^{±β}.
• Chiral (directional) transport in bosonic lattices, a hallmark of the model despite its Hermitian Hamiltonian.
• Platforms where the model's signatures have been experimentally observed, as cited in the paper (references [3] and [4]); the specific experimental systems are identified only by those citations in the provided content.

Industry relevance. The paper does not itself propose a device or application. Its relevance to quantum hardware is indirect: quadratic bosonic models of this type underpin the modeling and calibration of driven-dissipative photonic and circuit-QED lattices, and exact analytical spectra are useful for benchmarking numerical simulations of such arrays. Any stronger industry claim would go beyond what the paper states.

Future Directions

• Inhomogeneous chains. The authors state that the asymmetric plane-wave ansatz should extend to certain inhomogeneous open-boundary bosonic pairing models with nonuniform on-site energies, or with hopping/pairing strengths varying near the chain ends, where inhomogeneous non-Hermitian associated matrices arise. They demonstrate this on a minimal inhomogeneous OBKC with nonuniform hopping and pairing on the first bond; the details of that demonstration fall outside the portion of the paper available here.

• Inhomogeneous pairing or hopping more generally. The abstract explicitly states that the ansatz and degeneracy-resolution technique are not limited to the present model and can be generalized to other bosonic pairing systems, including those with inhomogeneous pairing or hopping terms — an open program beyond the minimal example.

• Full counting of the free parameters. The paper establishes that there are N/2 (even N) or (N−1)/2 (odd N) real free parameters, versus a single λ in the squeezing method. How these parameters should be fixed physically — for instance by a choice of vacuum or by stability requirements — is left as a consequence of the missing ground state rather than resolved.

• Comparison with the generalized Brillouin zone at finite size. The paper matches its finite-N result to the N → ∞ continuous band and to the two-circle generalized Brillouin zone. A systematic finite-N correspondence, including how the degenerate eigenvectors map onto the two Bogoliubov–de Gennes sectors, is a natural continuation.

Target Audience

This paper is aimed at researchers and graduate students in condensed-matter and quantum-optics theory who work with quadratic bosonic Hamiltonians, non-Hermitian lattice models, or bosonic Bogoliubov–de Gennes problems and who want explicit, finite-size, closed-form solutions rather than thermodynamic-limit or numerically obtained ones. It will also interest readers comparing algebraic and geometric (squeezing-based) routes to diagonalization, and those who need the degeneracy structure of the associated matrix spelled out for their own models. Readers without a background in Bogoliubov transformations and non-Hermitian eigenvalue problems will find the technical sections demanding.

Authors’ abstract

The bosonic Kitaev chain under open boundary conditions has attracted recent attention due to its realization in driven-dissipative systems and its intriguing non-Hermitian boundary physics. The model is known to be solvable via local squeezing transformations in the position-momentum representation. In this paper, we present an alternative, purely algebraic solution that relies entirely on the standard bosonic Bogoliubov transformation. For an $N$-site chain, we propose an asymmetric plane-wave ansatz with unequal left- and right-moving momenta to analytically solve the associated $2N\times 2N$ non-Hermitian ``associated matrix". The left eigenvalue problem yields $N$ distinct eigenvalues, each of which is twofold degenerate. By carefully resolving these degeneracies using the bosonic commutation relations, we construct the $N$ physical Bogoliubov quasiparticle operators. The construction reveals non-uniquenesses that in special cases exactly mirror the freedom in the local squeezing transformations of the original approach. The diagonal form of the Hamiltonian is obtained explicitly and is shown to be equivalent to the original Hamiltonian. The proposed asymmetric plane-wave ansatz and degeneracy-resolution technique are not limited to the present model and can be generalized to other bosonic pairing systems, including those with inhomogeneous pairing or hopping terms.