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arXiv:2607.08638

Symmetry as a route to generalized bosonic Kitaev chains

Gideon Lee, Tony Jin, Aashish A. Clerk

quant-phcond-mat.mes-hallSubmitted Jul 9, 2026• Announced Jul 10, 2026
arXivHTMLPDF

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.

Comments
19 pages, 6 figures

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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 topological classifications of quadratic bosonic systems according to the symmetries of the Hamiltonian, the dynamic matrix from the equation of motion, and the effective Hamiltonian from the Lindblad equation are summarized. The Hermitian Hamiltonian leads to a three-fold way table while the non-Hermitian dynamic matrix and effective Hamiltonian lead to a ten-fold way table. The difference comes from to the bosonic commutation relation on the particle-hole and chiral symmetries. Moreover, the system-reservoir coupling may cause a system with or without coupling to a reservoir to fall into different classes. We present a 1D bosonic pairing model inspired by the fermionic Kitaev chain to contrast the topological behavior from the Hamiltonian, dynamic matrix, and effective Hamiltonian. In contrast, a 2D Chern insulator is insensitive to the different classifications.
We investigate the non-Hermitian Kitaev chain with non-reciprocal hopping amplitudes and asymmetric superconducting pairing. We work out the symmetry structure of the model and show that particle-hole symmetry (PHS) is preserved throughout the entire parameter regime. As a consequence of PHS, the topological phase transition point of a finite open chain coincides with that of the periodic (infinite) system. By explicitly constructing the zero-energy wave functions (Majorana modes), we show that Majorana modes necessarily occur as reciprocal localization pairs accumulating on opposite boundaries, whose combined probability density exhibits an exact cancellation of the non-Hermitian skin effect for the zero energy modes. Excited states, by contrast, generically display skin-effect localization, with particle and hole components accumulating at opposite ends of the system. At the level of bulk topology, we further construct a Z2\mathbb{Z}_2Z2​ topological invariant in restricted parameter regimes that correctly distinguishes the topological and trivial phases. Finally, we present the topological phase diagram of the non-Hermitian Kitaev chain across a broad range of complex parameters and delineate the associated phase boundaries.
Superconducting quantum circuits are a natural platform for quantum simulations of a wide variety of important lattice models describing topological phenomena, spanning condensed matter and high-energy physics. One such model is the bosonic analog of the well-known fermionic Kitaev chain, a 1D tight-binding model with both nearest-neighbor hopping and pairing terms. Despite being fully Hermitian, the bosonic Kitaev chain exhibits a number of striking features associated with non-Hermitian systems, including chiral transport and a dramatic sensitivity to boundary conditions known as the non-Hermitian skin effect. Here, using a multimode superconducting parametric cavity, we implement the bosonic Kitaev chain in synthetic dimensions. The lattice sites are mapped to frequency modes of the cavity, and the in situ tunable complex hopping and pairing terms are created by parametric pumping at the mode-difference and mode-sum frequencies, respectively. We experimentally demonstrate important precursors of nontrivial topology and the non-Hermitian skin effect in the bosonic Kitaev chain, including chiral transport, quadrature wavefunction localization, and sensitivity to boundary conditions. Our experiment is an important first step towards exploring genuine many-body non-Hermitian quantum dynamics.
We propose a Hermitian quadratic bosonic model (QBH) whose dynamical matrix exhibits distinct topological and dynamical phenomena depending on whether the hopping and pairing amplitudes are real or purely imaginary. In the real-parameter regime, the dynamical matrix is unitarily equivalent to four decoupled copies of the sublattice-symmetric non-Hermitian Su-Schrieffer-Heeger (nSSH2) model, thereby inheriting its topological phases and energy spectrum, including the M\"obius phase, a gapless topological phase with fractional winding number, having no Hermitian counterpart. We show that the dynamics generated by the QBH Hamiltonian naturally reproduces non-Hermitian time evolution, without invoking nonlinear Schr\"odinger dynamics or ad hoc normalization. It is demonstrated by analytically calculating the Loschmidt amplitude and computing the dynamical topological order parameter under periodic boundary conditions, which displays a distinct chiral response in the M\"obius phase. In contrast, when the hopping and pairing terms are taken to be purely imaginary, the dynamical matrix becomes unitarily equivalent to a different version of the sublattice-symmetric non-Hermitian Su-Schrieffer-Heeger (nSSH1) model that supports only two topological phases, trivial and nontrivial, and the M\"obius phase disappears. The latter system exhibits sublattice-dependent chiral amplification under open boundary conditions. We show that this amplification arises from the nontrivial topology of the dynamical matrix, establishing a clear link between topological phase and amplification behavior in the imaginary-parameter regime.
Abstract Unlike their fermionic counterparts, the dynamics of Hermitian quadratic bosonic Hamiltonians are governed by a generally non-Hermitian Bogoliubov-de Gennes effective Hamiltonian. This underlying non-Hermiticity gives rise to a dynamically stable regime, whereby all observables undergo bounded evolution in time, and a dynamically unstable one, whereby evolution is unbounded for at least some observables. We show that stability-to-instability transitions may be classified in terms of a suitably generalized P T symmetry , which can be broken when diagonalizability is lost at exceptional points in parameter space, but also when degenerate real eigenvalues split off the real axis while the system remains diagonalizable. By leveraging tools from Krein stability theory in indefinite inner-product spaces, we introduce an indicator of stability phase transitions, which naturally extends the notion of phase rigidity from non-Hermitian quantum mechanics to the bosonic setting. As a paradigmatic example, we fully characterize the stability phase diagram of a bosonic analogue to the Kitaev–Majorana chain under a wide class of boundary conditions. In particular, we establish a connection between phase-dependent transport properties and the onset of instability, and argue that stable regions in parameter space become of measure zero in the thermodynamic limit. Our analysis also reveals that boundary conditions that support Majorana zero modes in the fermionic Kitaev chain are precisely the same that support stability in the bosonic chain.
We propose a class of periodically driven (Hermitian) modified bosonic Kitaev chains that effectively hosts rich nonHermitian Floquet topological phenomena. Two particular models are investigated in details as case studies. The first of these represents a minimal topologically nontrivial model in which nonHermitian skin effect, topological zero modes, and topological πππ modes coexist. The other displays a more sophisticated model that supports multiple topological zero modes and topological πππ modes in a tunable manner. By subjecting both models to perturbations such as a finite onsite bosonic frequency and spatial disorder, these features exhibit distinct responses. In particular, while generally all topological edge modes are robust against such perturbations, the nonHermitian skin effect is easily suppressed and revived by, respectively, the onsite bosonic frequency and spatial disorder in the first model, but it could be insensitive to both perturbations in the second model. Our studies thus demonstrate the prospect of a periodically driven bosonic Kitaev chain as a starting point in exploring various nonHermitian Floquet topological phases through the lens of a Hermitian system.