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

TL;DR

This work develops a native bosonic topological framework based on many-body symmetries—particle number ${\cal N}$, squeezing ${\cal S}$, and bosonic time reversal ${\cal T}$—to classify 1D quadratic bosonic Hamiltonians. It identifies two nontrivial 1D symmetry classes, ${\cal S}$ (with a winding invariant) and ${\cal N},{\cal S}$ (with a Pfaffian invariant), and demonstrates a bulk-boundary correspondence that mandates symmetry-protected boundary modes. Using photo-magnonic crystals—arrays of coupled microwave cavities with magnons—the authors blend finite-element electromagnetic simulations with an effective Hamiltonian and input-output formalism to predict and observe edge states via $S$-parameters, providing a concrete experimental blueprint. They also analyze two bosonic topological models (the bosonic Kitaev chain and bosonic SSH chain) within this framework, highlighting fundamental differences from fermionic counterparts due to the distinct protecting symmetries. The results establish a scalable platform for exploring robust bosonic edge modes, with implications for topological photonics, non-Hermitian dynamics, and microwave circuit design, and point toward higher-dimensional and Floquet-engineered extensions.

Abstract

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 potentially optimal way forward: 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 consist of arrays of interconnected photo-magnonic cavities. 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.

Figures (25)

• Figure 1: (a) Setup of a cavity magnonics experiment, where a microwave cavity (metallic grey box) is mounted inside an electromagnet applying a static magnetic field, and connected to a vector network analyser to monitor reflection and transmission coefficients. (b) Interior of the re-entrant two-post cavity 2014Goryachev, loaded with a YIG sphere placed between the two posts, where the cavity magnetic field $\vb{b}$ is strongest (indicated by the colour gradient). Two loop antennas on the left and right sides of the cavity couple to the cavity mode. The lid of the cavity is not shown. (c) Zoom on the YIG sphere, acting as a macrospin and coupling to the static applied magnetic field, and the cavity mode's RF magnetic field.
• Figure 2: Simulation of transmission amplitude ($\abs{S_{21}}$) of the unit cell of the crystal using COMSOL. The level repulsion signals the strong magnon-photon coupling. The dashed red lines correspond to the spectrum of the Hamiltonian of \ref{['eq:magnon-photon-hamiltonian:bosonic']} with parameters $\omega_a/2\pi=9.999$ GHz and $g/2\pi=122.5$ MHz. The dashed white lines are the frequencies of the uncoupled photon and magnon modes, $\omega_a/2\pi$ and $\omega_m/2\pi$ respectively. Inset: frequency cut at resonance $\omega_m=\omega_a$ corresponding to the intersection of the two dashed white lines. The frequency spacing of 225 MHz between the two resonances is twice the coupling strength, hence $g/2\pi=122.5$ MHz.
• Figure 3: (a) COMSOL model of two cavities (without YIG spheres) coupled with a rectangular iris. (b) Simulated reflection and transmission parameters of the cavity unit cell without a YIG sphere. The grey vertical line indicates the location of the estimated resonance frequency located at $\omega_a/2\pi=9.998$ GHz. (c) Simulated reflection and transmission parameters of two reentrant cavities coupled through an iris as per (a). The grey vertical line indicates the location of the estimated resonances located at $9.9656$ GHz and $9.991$ GHz.
• Figure 4: Schematic of the proposed 1D photo-magnonic crystal with $N$ unit cells. (a) Closed system described by \ref{['eq:H0:bosonic']} illustrating the coupling between the modes. (b) Illustration of the dissipation channels in the crystal considered in the electromagnetic simulations and the input-output formalism.
• Figure 5: Simulated transmission amplitude through the photo-magnonic crystal using COMSOL for (a) $N=2$, (b) $N=4$, and (c) $N=8$ lattice sites. The red dashed lines correspond to the numerical diagonalisation of \ref{['eq:H0:bosonic']} with the parameters $\omega_a/2\pi=9.9783$ GHz, $\abs{g}/2\pi=112.5$ MHz, $\abs{t}/2\pi=12.7$ MHz and a magnon linewidth $\kappa_m/2\pi=10$ MHz. For (a), the magnon frequency is shifted by 30 MHz, see \ref{['app:fem']} for a discussion. The diagonal dashed white line corresponds to $\omega=\omega_m$.