Many-body symmetry-protected zero boundary modes of synthetic photo-magnonic crystals

The topological classification of insulators and superconductors, the “ten-fold way”, is grounded on fermionic many-body symmetries and has had a dramatic impact on many fields of physics. Therefore, it seems equally important to investigate a similar approach for bosons as tightly analogous to the fermionic prototype as possible. There are, however, several obstacles coming from the fundamental physical differences between fermions and bosons. Here, we propose a theory of free boson topology (topological classification and bulk-boundary correspondence) protected by bosonic many-body symmetry operations, namely, squeezing transformations, particle number, and bosonic time reversal. We identify two symmetry classes that are topologically non-trivial in one dimension. They include key models like the bosonic Kitaev chain, protected by a squeezing symmetry within our framework, and the celebrated bosonic SSH model, protected by a squeezing symmetry and particle number. To provide a robust experimental platform for testing our theory, we introduce a new quantum meta-material: photo-magnonic crystals. They are remarkable for their experimental flexibility and natural affinity for displaying band topological physics at microwave frequencies. We engineer a many-body symmetry-protected topological photo-magnonic chain with boundary modes mandated by a Pfaffian invariant. Using an electromagnetic finite-element modelling, we simulate its reflection and transmission and identify experimental signatures of its boundary modes. The experimental tuning of the crystal to its symmetry-protected topological phase is also addressed. Our modelling of the photo-magnonic chain provides a thorough blueprint for its experimental realisation and the unambiguous observation of its exotic physics.

💡 Research Summary

This paper develops a comprehensive theory of free‑boson topology that is rooted in physical many‑body symmetries rather than the abstract single‑particle symmetries used in the fermionic ten‑fold way. The authors identify three fundamental bosonic symmetry operations: (i) a squeezing transformation (S) that mixes creation and annihilation operators, (ii) particle‑number conservation (N), and (iii) bosonic time‑reversal (T). By examining all possible combinations of these operations they construct eight symmetry classes and, crucially, derive the associated topological invariants in one dimension. Two classes turn out to be non‑trivial: class {S} is characterized by an integer winding number, while class {N,S} is protected by a Z₂‑valued Pfaffian invariant. The paper shows how these invariants emerge from index theorems applied to the dynamical (Bogoliubov‑de Gennes) matrices of bosonic systems, deliberately dropping thermodynamic‑stability constraints so that the index theorems remain applicable even when the effective Hamiltonian is non‑Hermitian.

The theoretical framework is illustrated with two paradigmatic models that have already been realized experimentally: the bosonic Kitaev chain and the bosonic Su‑Schrieffer‑Heeger (SSH) chain. In the Kitaev chain only the squeezing symmetry is present, leading to a winding‑number protected zero‑energy edge mode. In the SSH chain both squeezing and particle‑number symmetries are present, and the presence or absence of a zero‑energy edge mode is dictated by the sign of the Pfaffian invariant. Numerical diagonalization confirms the bulk‑boundary correspondence in both cases.

To provide a realistic experimental platform, the authors introduce “photo‑magnonic crystals”, a new class of quantum meta‑materials formed by alternating microwave cavities and magnetic (magnon) resonators. These structures naturally combine heavy, low‑mobility magnons with highly mobile photons, and allow the engineering of synthetic gauge fields and strong squeezing interactions within each unit cell. Using electromagnetic finite‑element modelling, the authors compute reflection and transmission spectra for a one‑dimensional chain of such cavities. They demonstrate that, when the system parameters are tuned into the topological regime identified by the Pfaffian invariant, a sharply localized zero‑frequency edge mode appears, manifesting as a pronounced dip in transmission and a peak in reflection at the chain’s ends. The simulations show excellent agreement with the effective bosonic Hamiltonian derived analytically, confirming that the classical electromagnetic model faithfully reproduces the quantum topological physics.

The paper also details practical tuning protocols: external magnetic fields control the magnon frequency and thus the squeezing strength; cavity spacing and coupling capacitors adjust the hopping amplitude; and low‑loss superconducting materials minimize dissipation to preserve the sharp spectral features. By sweeping these parameters, experimentalists can drive the system across the topological phase transition, directly observing the Pfaffian sign change via the emergence or disappearance of the edge mode.

In conclusion, the work (i) establishes a bosonic analogue of the ten‑fold way based on concrete many‑body symmetries, (ii) provides explicit bulk‑boundary correspondences that survive non‑Hermitian effects, (iii) validates the theory on known bosonic models, and (iv) delivers a detailed blueprint for realizing and detecting symmetry‑protected topological phases in a microwave‑frequency photo‑magnonic crystal. The authors suggest extensions to higher dimensions, inclusion of interactions beyond the quadratic level, and exploration of quantum information applications such as protected microwave quantum channels. This integrated theory‑simulation‑experiment approach positions photo‑magnonic crystals as a versatile platform for bosonic topological physics.