Symmetry as a route to generalized bosonic Kitaev chains

Abstract: The bosonic Kitaev chain (BKC) model is a deceptively simple looking quadratic pairing Hamiltonian. Despite being purely Hermitian, it exhibits a number of striking non-Hermitian topological phenomena, including skin effects. We show here how symmetries play a key role in this model, and how identifying these allows one to develop generalized BKC-like models. We emphasize the surprising fact that any quadratic bosonic pairing Hamiltonian with a sublattice (chiral) symmetry necessarily has a dynamical matrix with an effective time reversal symmetry. This symmetry is unrelated to physical time-reversal, but enables non-trivial topological invariants. We also discuss how this symmetry is unrelated to another key property of the BKC, the decoupling of quadrature dynamics. This feature can instead be connected to a distinct symmetry, namely an effective particle-hole symmetry of the dynamical matrix. We discuss non-trivial generalized BKC models that only keep one of these two effective symmetries intact. We also provide a classification of all translationally-invariant 1D pairing Hamiltonians, and show connections between the BKC and a well-studied non-Hermitian fermionic system, the symplectic Hatano-Nelson model.

• The paper identifies two distinct symmetry mechanisms—quadrature decoupling (qPHS) and sublattice symmetry (SLS)—that underpin topology and dynamical stability.
• The paper demonstrates that symmetry-protected sublattice operations ensure perturbative stability under open boundary conditions while inducing non-Hermitian skin effects.
• The paper maps generalized bosonic models to non-Hermitian tenfold symmetry classes, facilitating experimental access to fermionic-like topological phenomena.

Symmetry-Protected Topological Structure and Stability in Generalized Bosonic Kitaev Chains

Introduction and Motivation

The study presented in "Symmetry as a route to generalized bosonic Kitaev chains" (2607.08638) addresses the remarkable interplay between symmetry, non-Hermitian topology, and dynamical stability in quadratic bosonic lattice systems, with a particular focus on the bosonic Kitaev chain (BKC) and its generalizations. While the BKC is described by a Hermitian Hamiltonian, it nonetheless manifests non-Hermitian phenomena, such as the non-Hermitian skin effect (NHSE), typically absent in closed, lossless quantum systems. The central aim is to clarify the specific symmetries underpinning these behaviors, demonstrate how these symmetries permit the construction of a broader class of BKC-like models, and elucidate the theoretical foundation for dynamical stability in the presence of open versus periodic boundary conditions.

Key Symmetries and Topological Structure

Quadrature Decoupling and Particle-Hole Symmetry

A central insight of the work is the identification of two fundamentally distinct symmetry mechanisms that independently produce topological and dynamical features associated with the BKC:

• Quadrature decoupling corresponds to a non-standard particle-hole symmetry (qPHS) in the dynamical matrix. This symmetry reflects the fact that, for the BKC and related models, the linearized equations of motion in terms of canonical quadratures $(\hat{q},\hat{p})$ are block-diagonal, resulting in two dynamically decoupled subsystems. Each subsystem simulates an effective Hatano-Nelson chain—a paradigmatic non-Hermitian model exhibiting the NHSE. The existence of qPHS allows for the definition of independent winding invariants for each quadrature sector, diagnosing the presence of skin effects in a manner analogous to non-Hermitian single-band models.

Sublattice (Chiral) Symmetry and Effective Time-Reversal

• Sublattice symmetry (SLS) (site-dependent sign flip), familiar from the classification of Hermitian band structures (e.g., the SSH model), imposes a different set of constraints. The authors demonstrate that any quadratic bosonic pairing Hamiltonian with SLS automatically inherits an effective time-reversal symmetry (TRS) at the level of its dynamical matrix—a transpose-type symmetry distinguished from the standard physical anti-unitary time-reversal of many-body Hamiltonians. This emergent effective TRS enables the existence of $\mathbb{Z}_2$-valued topological invariants for spectral point gaps, resulting in a symmetry-protected skin effect (SPSE). Such skin effects are not reducible to the physics of independent single-band systems and remain robust even when the quadrature decoupling is broken.

Distinction From Physical Symmetries

Notably, the effective TRS induced by SLS is not, in general, equivalent to a physical time-reversal operation (local antiunitary symmetry) at the level of the second-quantized Hamiltonian. This distinction becomes manifest in generalized models: it is possible to break physical TRS while preserving the effective TRS (and vice versa), underscoring the primacy of dynamical matrix symmetries in establishing non-Hermitian topology in bosonic chains.

Classification and Construction of BKC-like Models

The symmetry analysis naturally suggests a classification of general quadratic bosonic pairing Hamiltonians in 1D according to the presence or absence of (i) quadrature decoupling (qPHS) and (ii) sublattice symmetry. The combination (or selective breaking) of these symmetries allows for the systematic design of models with tailored topological and dynamical properties:

• Models with both SLS and qPHS (e.g., the standard BKC and its extensions with odd-range imaginary hopping or pairing) exhibit simultaneous SPSE and decoupled skin effects, and are perturbatively protected against dynamical instability under OBC.
• Models with SLS but broken qPHS (e.g., addition of real odd-range hopping) break quadrature decoupling but retain SPSE; these can show localization transitions and phase-sensitive topological transitions not seen in the standard BKC.
• Models with qPHS and broken SLS (e.g., even-range imaginary pairing) display phase-selective skin effects without symmetry protection, are highly sensitive to disorder, and are generically unstable to SLS-breaking perturbations.
• Models breaking both symmetries (e.g., uniform chemical potential) are generically topologically trivial and lack both the skin effect and OBC stability.

Beyond this classification, the authors provide a comprehensive mapping to the non-Hermitian tenfold (Kawabata 38-fold) symmetry classes [Kawabata et al., Phys. Rev. X 9, 041015 (2019)], elucidating the precise symmetry content of each BKC-like model within the systematic non-Hermitian topological paradigm.

Dynamical Stability and Boundary Sensitivity

A critical practical and theoretical issue is the dynamical stability of the BKC and related models, especially under OBC versus PBC. The authors show that the sublattice symmetry plays a dual role:

• It ensures eigenvalues appear in symmetric pairs about zero, making the system perturbatively marginal against instability (as pairing terms couple zero-energy pairs).
• Simultaneously, for OBC, SLS leads to the vanishing of the relevant matrix elements responsible for dynamical instabilities to all orders in perturbation theory, robustly protecting the system.

In contrast, under PBC, degeneracies enable hybridization and instability is generic—a result borne out both analytically and in explicit numerical diagonalization of finite systems. This symmetry-protected boundary sensitivity provides a rigorous foundation for the practical stability of BKC phases and their fragility to symmetry-breaking perturbations.

Relation to the Symplectic Hatano-Nelson Model

A further significant contribution is the demonstration of a concrete mapping between a generalized BKC with real hopping (the $g_1$-BKC) and the symplectic Hatano-Nelson (SHN) model for non-Hermitian fermions. This mapping shows that the effective TRS of the BKC dynamical matrix, while non-physical in the bosonic system, corresponds to a bona fide physical TRS protecting topological phases in the SHN. This provides conceptual continuity between non-Hermitian topological physics in bosonic and fermionic chains and establishes the utility of BKC-like models as experimentally accessible simulators of distinctive non-Hermitian fermionic phenomena.

Strong Numerical Results and Contradictory Claims

The authors present explicit numerical evidence for stability protection in OBC systems with SLS, the existence and topological characterization of generalized skin effects with both phase- and frequency-selectivity, and the breakdown of both stability and the skin effect upon breaking the requisite symmetries. They report that for certain generalizations (e.g., $g_3$-BKC), finite-range dynamical stability persists well beyond the perturbative regime, suggesting the presence of additional non-perturbative protection mechanisms. Remarkably, they identify and clarify scenarios where OBC instability persists even when the corresponding PBC system remains stable—contrary to the standard NHSE phenomenology.

Implications and Future Directions

Practical Implications:

• The symmetry-based framework developed here enables the engineering of bosonic lattices with controlled amplification, phase-sensitive and frequency-selective gain, and directional transport, holding promise for quantum-limited amplifiers, non-reciprocal devices, and parametric sensors.
• The stability protection mechanism elucidated can be exploited for robust implementation of non-Hermitian topological phases in photonic, circuit-QED, or optomechanical platforms.

Theoretical Implications and Open Questions:

• The construction clarifies the precise link between non-Hermitian spectral topology and band topology in bosonic systems, with potential extensions to dynamically unstable and interacting regimes.
• The mapping to fermionic SHN physics raises the prospect of quantum simulation of entanglement phase transitions and other non-Hermitian phenomena beyond the constraints of parity or post-selection.
• Open problems include the extension of this symmetry classification to the genuinely interacting regime and the relationship between spectral winding-based and wavefunction-based topological invariants, as well as non-perturbative mechanisms for OBC stability.

Conclusion

This work rigorously identifies and exploits the symmetries responsible for non-Hermitian topology and boundary-sensitive dynamical stability in the BKC, providing an extensible blueprint for the design, classification, and theoretical interpretation of generalized bosonic pairing chains. The decoupling of sublattice and quadrature symmetries uncovers a much richer phenomenology—including novel localization transitions, phase- and frequency-selective amplification, and counterintuitive stability diagrams—highlighting both the fundamental and applied utility of symmetry-guided approaches in non-Hermitian many-body quantum systems.

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References

See (2607.08638) and the extensive reference list in the source document for foundational and contemporary literature on non-Hermitian topology, bosonic pairing systems, and the NHSE.