Topological protection of Majorana polaritons in a cavity

Cavity embedding is an emerging paradigm for the control of quantum matter, offering avenues to manipulate electronic states and potentially drive topological phase transitions. In this work, we address the stability of a one-dimensional topological superconducting phase to the vacuum quantum fluctuations brought by a global cavity mode. By employing a quasiadiabatic analytical approach completed by density matrix renormalization group calculations, we show that the Majorana end modes evolve into composite polaritonic modes while maintaining the topological order intact and robust to disorder. These Majorana polaritons keep their non-Abelian exchange properties and protect a twofold exponentially degenerate ground state for an open chain. They become, however, weak edge modes in the sense that they no longer commute with the full Hamiltonian and protect the exponential degeneracy only in the ground-state manifold.

Figure 1

Sketch of the two different cavity embeddings. The ladder couples either (a) to a quantized electric field, or (b) to a quantized magnetic field. (c) Band structure of the cavity-free Hamiltonian ${\widehat{H}}_0$ with (full green line) and without (dashed black line) superconducting pairing.

Figure 2

The top (bottom) row shows DMRG results for the magnetic (electric) coupling. (a), (d) Ground-state energy splitting $\mathrm{\Delta }{E}_{gs}=|E_0^{gs}-E_1^{gs}|$ for different coupling strength. The mean-field (MF) result is also shown for comparison. (b), (e) Entanglement spectra ${\xi }_s=-\text{log}{\lambda }_s$ for a half-chain bipartition of the ground state in the even parity sector. The twofold degeneracies come from (even, even) and (odd, odd) parity resolved partitions. (c), (f) Correlation function for the top leg of the ladder geometry. $L=48$ except for (a), (e).

Figure 3

Magnetic cavity: DMRG results ($g_B>0$) and exact results ($g_B=0$) for topological markers as a function of disorder strength $W$. (a) Edge-edge correlator $Q$ from Eq. (5) and (c) ground-state degeneracy $\mathrm{\Delta }{E}_{gs}$ for a single disorder realization at each strength $W$. The corresponding disorder averaged quantities are shown in (b), (d) with $N_{\text{dis}}=20\left(1000\right)$ realizations for $g_B=0.1(0.)$. Error bars indicate two standard deviations and $L=48$.

Figure 4

Energy gap to the first excited state in the two parity sectors $p=0$ (full lines) and $p=1$ (dashed lines) calculated via DMRG. The red lines represent the mean-field predicted values of gaps. (a) and (b) show respectively the result for the magnetic coupling and electric coupling. $L=48$.

Figure 5

(a), (c) Connected matrix elements revealing the hybrid nature of Majorana polaritons for (a) magnetic coupling $g_B=0.1$ and (c) electric coupling $g_E=0.4$, with zero ($\mathcal{O}={\widehat{c}}_{+,j}$) and one photon ($\mathcal{O}={\widehat{c}}_{+,j}\widehat{a}$). (b), (d) Evolution of the total zero- and one-photon weights $N_0$ and $N_1$, Eq. (12), with light-matter coupling for the magnetic (b) and electric (d) cavities. Here $L=48$.