Electrostatics-induced breakdown of the integer quantum Hall effect in cavity QED

TL;DR

This work identifies an electrostatic boundary mechanism—image-charge potentials from a nearby metallic split-ring boundary—that can destabilize the integer quantum Hall plateaus in a cavity-QED setting. A minimal single-electron model shows that image-induced edge-pocket potentials generate counter-propagating edge channels and backscattering, with a characteristic backscattering energy Γ_ℓ that scales with edge distance as ~d_edge^−2 and competes with LL spacings and Zeeman energies. The authors corroborate the mechanism with Kwant-based transport simulations, and show that vacuum-field (purely photonic) corrections are several orders of magnitude smaller under realistic conditions, though nano-cavities could enhance them. The results align with experimental observations and offer clear experimental tests, such as varying ω_LC while keeping geometry fixed to distinguish electrostatic from vacuum contributions, highlighting the pivotal role of electrostatic boundary effects in cavity-modified quantum materials.

Abstract

We analyze the recently observed breakdown of the integer quantum Hall effect in a two-dimensional electron gas embedded in a metallic split-ring resonator. By accounting for both the quantized vacuum field and electrostatic boundary modifications, we identify a mechanism that could potentially explain this breakdown in terms of non-chiral edge channels arising from electrostatic boundary effects. For experimentally relevant geometries, a minimal single-electron model of this mechanism predicts characteristic signatures and energy scales consistent with those observed in experiments. These predictions can be directly tested against alternative, purely vacuum-induced explanations to shed further light on the origin of this puzzling phenomenon.

Figures (11)

• Figure 1: (a) Sketch of a cavity quantum Hall system, where a 2DEG in an external magnetic field $\textbf{B}$ is strongly coupled to the quantized field $\mathcal{E}_{\rm cav}$ of a split-ring resonator. (b) The resonator is modeled as a lumped-element $LC$ circuit with the Hall bar inside the two capacitor plates separated by a distance $d$. (c) Example of the disorder potential $U_{\rm dis}({\bf r})$ with a sketch of the combined potentials $U_{\rm conf}({\bf r})+U_{\rm im}({\bf r})$ (upper panel, blue solid line).
• Figure 2: (a) Spectrum of the LLL against eigenstate position $\langle{n|x|n\rangle}$ close to the edge and distorted by an external potential $U(x) = Ax^6 - e\mathcal{E}_{\rm ext} x$ mimicking the action of the confinement and image charge potentials. Here $e\mathcal{E}_{\rm ext} = \Gamma / l_B$. (b) Wavefunction $|\psi_n(\mathbf{r})|^2$ as a function of position for the marked eigenstates. With PBC on the left, with open boundary condition (OBC) on the right. Parameters in the Technical Information (TI).
• Figure 3: (a) Conductance $G(E_F)$ as a function of the Fermi energy, for a reference system (black dashed) and a system with a metallic plate at distance $d_{\rm{edge}}/l_B=3.5-30$ (red-green), that is including the image charge potential in Eq. \ref{['eq:single_plate_image_pot']}. (b) Deviation from quantization of the conductance at $\ell=0$ (left) and $\ell=1$ (right) plateaus. The dashed line is an exponential fit of the curve at $d_{\rm edge}/l_B = 30$, using the fitting function $f_{\rm fit}(E_F|\Delta , A)=\exp[ - E_F/(\hbar \Delta_{\ell}) + A ]$ with $\Delta_{\ell}$, $A$ as free parameters. (c) Extracted exponential slope $\Delta_{\ell}$ as a function of the plate distance $d_{\rm edge}$. The black solid line is the scaling of $2\Gamma$, the blue solid line is a numerical fit over the curve at $\ell=0$, indicating a scaling $\sim d_{\rm edge}^{-1.7}$. All the conductance curves are shifted such that the minimum energy eigenvalue is at 0. Parameters can be found in the TI, they realize $\Gamma_{\ell=0}/\hbar \omega_B \sim 0.4$ at $d_\mathrm{edge}/l_B=10$.
• Figure 4: $1/G$ at finite $T$ as a function of the magnetic field parameter $\alpha$ fixing either (a) the chemical potential $\mu$ or (b) the number of particles $N_e$. The red solid and black dashed lines correspond to the same setup with and without image charges, respectively. The vertical purple dashed-dot lines highlight the integer filling factors $\nu=\alpha N_e/((N_x-3)(N_y-3))$, where $\nu=3$ is the rightmost line (here we correct for the finite size raciunas_creating_2018). (c) Sketch of the mechanism for destruction of Zeeman plateaus and elongation of cyclotron plateaus. The red shaded areas represent the states within their respective backscattering energy, the blue dashed lines represent examples of Fermi energy level, highlighting the states involved. Insets in (a) and (b) show the odd Zeeman plateaus with an estimated $\Gamma_{\ell=2}/(\hbar \omega_B) \sim 0.87$, $\Gamma_{\ell=2}/E_Z \sim 3$; for clear comparison, the red curve is shifted artificially on the black one. Other parameters in the TI.
• Figure S1: (a) Conductance $G(E_F)$ as a function of the Fermi energy, in units of conductance quantum $e^2/h$. The black-dashed line is the system without the image charge potential term, while the red solid line is obtained under the same parameters but also including the image charge potential in Eq. \ref{['eq:single_plate_image_pot']}. The curves are shifted such that the minimum energy eigenvalue is at 0. The yellow shaded area is the energy range where $E_F<U_*$. Inset: zoom over the first $\ell=0$ plateau. The blue solid line is the analytic estimate using Eq. \ref{['eq:rams_T_approx']} with $U_* = \Gamma$. (b) $\ell=0$ plateau with disorder, averaged over $N_{\rm dis}=50$ realizations. The green dot-dashed line is a phenomenological fit using Eq. \ref{['eq:rams_T_approx']} with $U_* = \Gamma, f\approx 8, \gamma\approx 2$. (c) Comparison between the $\ell=0$ plateau for a clean system with images (red solid line), without images (black dashed line), and with images and disorder (green solid line), plotted in log-scale as $|1-G|$ (in units of $e^2/h$). Parameters: (clean system)$N_x=200$,$N_y=50$, (disordered system)$N_x=100$, $N_y=50$, $\alpha=1/40$, $\omega_B/J \approx 0.31$, $l_B/l_0\approx 2.5$, $\Gamma/(\hbar \omega_B)=0.2$, $d_{\rm edge}/l_B=20$, $E_{\rm dis}/(\hbar \omega_B) = 0.15$ (when present), $\xi_c/l_B = 1$.