Topology and energy dependence of Majorana bound states in a photonic cavity

Light-matter interaction plays a crucial role in modifying the properties of quantum materials. In this work, we investigate the effect of cavity induced photon fields on a topological superconductor hosting Majorana bound states (MBS). We model the system using a Peierls substitution of the photonic operator in the kinetic and spin-orbit terms, and utilize an exact diagonalization of Hamiltonian for a finite number of photons to investigate the coupled system. We find that the MBS persist even in the presence of a cavity field and notably appear at finite and tunable energy, in contrast to a usual 1D topological superconductor. The MBS energy is shifted by two processes: the cavity photon energy adds a constant energy shift, while the light-matter interaction induces additional parameter dependencies, such that the MBS experience a pseudo-dispersion as a function of both light-matter interaction and magnetic field. Additionally, we find that the MBS energy oscillations are suppressed with increasing light-matter interaction and that disorder stability is not impacted by the light-matter interaction. Combined, these offer additional tunability and stability of the MBS. As a second result, we establish a modified spectral localizer formalism as an essential tool for topological characterization of quantum matter in a cavity. The spectral localizer allows characterization at arbitrary energies, which is needed for probing different photon sectors. However, hybridization between different photon sectors in the low-frequency regime limits a straightforward application of a standard spectral localizer. We fully resolve this issue by judiciously applying an energy shift to the spectral localizer. Our work thus introduces a new avenue for controlling MBS via light-matter coupling and provides a framework for exploring cavity-modified topologies.

💡 Research Summary

In this work the authors investigate how a single‑mode photonic cavity modifies the physics of a one‑dimensional topological superconductor (1DTSC) that hosts Majorana bound states (MBS). The electronic system is the standard Rashba nanowire proximitized by an s‑wave superconductor, described by a tight‑binding Hamiltonian with chemical potential μ, Zeeman field B, hopping t, spin‑orbit coupling α and induced pairing Δ. To incorporate the cavity they perform a fully quantum Peierls substitution, replacing the hopping and spin‑orbit amplitudes by (t,e^{i\gamma(b^\dagger+b)}) and (\alpha,e^{i\gamma(b^\dagger+b)}), where (b^\dagger, b) are photon creation and annihilation operators and (\gamma) quantifies the light‑matter coupling strength. The total Hamiltonian thus contains the electronic part, the photon mode of frequency ω (including the zero‑point term (\hbar\omega/2)), and the interaction term that couples different photon‑number sectors.

Because the Peierls factor is an operator, the Hilbert space expands into a block‑matrix structure (H_\infty) where each block corresponds to a fixed photon number N. Diagonal blocks (N = M) describe the electronic Hamiltonian dressed by N photons, shifted in energy by (\hbar\omega(N+½)). Off‑diagonal blocks (N ≠ M) encode processes that change the photon number and generate a Floquet‑like ladder of bands, but here the ladder is fully quantized rather than periodic in time. For numerical work the authors truncate the photon space to a maximum photon number (N_{\rm ph}^{\max}) and perform exact diagonalization of the resulting finite matrix (H_{N_{\rm ph}}).

A central question is whether the topological edge modes survive and how their energies are altered. The authors find two distinct contributions to the MBS energy shift: (i) the photon energy (\hbar\omega) adds a constant offset, moving the whole spectrum upward; (ii) the light‑matter coupling (\gamma) introduces a pseudo‑dispersion, i.e. the MBS energy becomes a smooth function of both (\gamma) and the Zeeman field B. Consequently, the Majorana modes appear at finite, tunable energies (E_{MBS}\approx (N+½)\hbar\omega + f(\gamma,B)) rather than at zero energy. Importantly, increasing (\gamma) suppresses the characteristic oscillations of the MBS energy with wire length (or with B), which are usually caused by overlap of the two end states. This suppression is interpreted as a photon‑induced reduction of the wave‑function overlap, effectively stabilizing the Majoranas.

The authors also test robustness against on‑site disorder. Even with random variations of μ, the Majorana modes persist; the topological gap shrinks slightly but the disorder‑induced stability is not compromised by the cavity. Thus the cavity provides an extra knob to tune the Majorana energy without degrading protection.

To characterize topology in a system where edge states appear at non‑zero energies, the paper adopts the spectral localizer (SL) formalism. The SL operator (L_{x,E}= \kappa (X-xI)\tau_x + (H_\infty -EI)\tau_y) combines position and Hamiltonian, with (\kappa) a scaling factor. Its minimal singular value (\sigma(x,E)) acts as a local gap: it vanishes where a state at energy E is localized at position x. The signature of the Hermitian matrix (\tilde X + i(H_\infty -EI)) (with (\tilde X = \kappa (X-xI))) yields an integer topological invariant (\nu(x,E)) that is both spatially and energetically resolved. In the high‑frequency regime ((\hbar\omega \gg) electronic bandwidth) the photon sectors are well separated, and the SL correctly identifies the non‑trivial sector (typically the 0‑photon sector) while the others remain trivial. In the low‑frequency regime, however, photon sectors overlap and the standard SL becomes non‑Hermitian, producing spurious phase diagrams. The authors resolve this by restricting the energy argument to the expected photon‑sector energies (E = (N+½)\hbar\omega) and applying an explicit energy shift to the SL, restoring Hermiticity and yielding clean, physically meaningful topological maps.

Overall, the paper demonstrates that embedding a 1DTSC in a quantum cavity offers a novel method to control Majorana bound states: the cavity photon energy provides a uniform energy offset, while the light‑matter coupling allows continuous tuning of the Majorana energy and suppression of finite‑size oscillations, all without sacrificing disorder resilience. The modified spectral localizer emerges as a powerful, computationally efficient tool for diagnosing topology in hybrid light‑matter systems where edge modes reside at arbitrary energies. This work opens a pathway toward “cavity‑engineered topological quantum devices” and suggests that photonic environments could become integral components of future Majorana‑based quantum information platforms.