Weakened Topological Protection of the Quantum Hall Effect in a Cavity

We study the quantum Hall effect in a two-dimensional homogeneous electron gas coupled to a quantum cavity field. As initially pointed out by Kohn, Galilean invariance for a homogeneous quantum Hall system implies that the electronic center of mass (c.m.) decouples from the electron-electron interaction, and the energy of the c.m. mode, also known as Kohn mode, is equal to the single particle cyclotron transition. In this work, we point out that strong light-matter hybridization between the Kohn mode and the cavity photons gives rise to collective hybrid modes between the Landau levels and the photons. We provide the exact solution for the collective Landau polaritons and we demonstrate the weakening of topological protection at zero temperature due to the existence of the lower polariton mode which is softer than the Kohn mode. This provides an intrinsic mechanism for the recently observed topological breakdown of the quantum Hall effect in a cavity [F. Appugliese et al., Breakdown of topological protection by cavity vacuum fields in the integer quantum Hall effect, Science 375, 1030 (2022).]. Importantly, our theory predicts the cavity suppression of the thermal activation gap in the quantum Hall transport. Our work paves the way for future developments in cavity control of quantum materials.

Figure 1

(a) Two-dimensional material confined in a cavity. The distance between the cavity mirrors is $L_z$. The system is placed perpendicular to a homogeneous magnetic field $\mathbf{B}$. (b) Imaginary part of the response function ${\chi }_{yy}(w)$ for IQH system in the cavity. The radial cavity frequency $\omega =2\pi \times 0.14\mathrm{THz}$, the 2D electron density $n_{2\mathrm{D}}=2\times {10}^{11}{\mathrm{cm}}^{-2}$, the effective electron mass $m=0.07m_e$ and the temperature $T=50\mathrm{mK}$ are chosen according to the experiment in Ref. [23]. We observe the upper ${\mathrm{\Omega }}_{+}$ and lower ${\mathrm{\Omega }}_{-}$ polariton, with normalized Rabi splitting ${\mathrm{\Omega }}_R/\omega =0.33$. The lower polariton is softer than the cyclotron mode ${\omega }_c=eB/m$. This signals the weakened topological protection of the hybrid system.

Figure 2

Quantum Hall transport in a cavity at $T=0$ with a finite broadening $\delta =2\pi \times 5\times {10}^{-3}\mathrm{THz}$ for two different values of magnetic field strength $B=1T$ and $B=2T$ which correspond to the filling factors $\nu =8$ and $\nu =4$, respectively. The 2D electron density is $n_{2\mathrm{D}}=2\times {10}^{11}{\mathrm{cm}}^{-2}$ as in Ref. [23]. The Hall and the longitudinal conductivities deviate from the topologically expected values. For the smaller value of the magnetic field (higher filling factor) the deviations of the IQH transport due to the cavity are enhanced.

Figure 3

Low temperature transport at magnetic field strength $B=1T$. In (a) and (b) $n_{2\mathrm{D}}=2\times {10}^{11}{\mathrm{cm}}^{-2}$ while in (c) and (d) $n_{2\mathrm{D}}=4\times {10}^{11}{\mathrm{cm}}^{-2}$. The light-matter coupling strongly affects the quantum Hall transport. For the smaller cavity frequency $\omega =2\pi \times 0.1\mathrm{THz}$ and the larger electron density the deviation from the topologically protected values maximizes. This relates to the behavior of ${\mathrm{\Omega }}_{-}$ which controls the thermal activation in the system. The broadening parameter is chosen very small $\delta =2\pi \times {10}^{-4}\mathrm{THz}$ to avoid influencing transport, but guarantee numerical convergence.