Robustness of quantized Hall resistivity under cavity coupling at zero temperature

Technical Summary: Robustness of Quantized Hall Resistivity under Cavity Coupling at Zero Temperature

Problem Statement
Recent experiments involving quantum Hall (QH) systems embedded in terahertz cavities have demonstrated that strong light–matter coupling can modify transport properties through the formation of Landau polaritons. While theoretical models have shown that the Hall conductivity ($\sigma_{xy}$) can deviate from its quantized value due to the combination of ultrastrong light–matter interaction, finite polariton lifetime (broadening), and finite temperature, experimental probes primarily measure the Hall resistivity ($\rho_{xy}$). A critical question remains regarding the robustness of topological protection in these hybrid systems: does the modification observed in conductivity translate to the resistivity tensor, which is the standard observable in transport experiments? Previous findings suggested a breakdown of topological protection in integer QH states, yet high-precision measurements of the von Klitzing constant at low temperatures have shown no renormalization. This paper addresses the apparent discrepancy between theoretical predictions for conductivity and experimental observations of resistivity.

Methodology
The authors employ a phenomenological model based on the Pauli-Fierz Hamiltonian for a two-dimensional electron gas (2DEG) coupled to a single-mode homogeneous cavity field and a strong magnetic field. Key methodological steps include:

1. Center-of-Mass (CM) Decoupling: Utilizing Galilean invariance, the system is reduced to a Hamiltonian describing the coupling of the center-of-mass coordinate to the cavity field, effectively decoupling it from relative electron motion.
2. Hopfield Transformation: The resulting CM Hamiltonian, representing two coupled harmonic oscillators (Landau level transition and cavity photons), is diagonalized using the Hopfield transformation. This yields two hybrid quasiparticle modes: upper ($\Omega_+$) and lower ($\Omega_-$) Landau polaritons.
3. Kubo Formalism: At zero temperature ($T=0$), the system is treated in the polariton vacuum state. The authors compute the current-current correlation functions using the standard Kubo formalism to derive the optical conductivities ($\sigma_{ab}$) in the frequency domain, incorporating a phenomenological broadening parameter $\delta$ to model finite polariton lifetime.
4. Tensor Inversion: The resistivity tensor ($\rho = \sigma^{-1}$) is analytically derived from the calculated conductivity tensor components, explicitly accounting for the tensor inversion rather than simple scalar reciprocity.

Key Contributions and Results
The paper establishes a fundamental asymmetry between the response of Hall conductivity and Hall resistivity under strong light–matter interaction at zero temperature:

• Immunity of Hall Resistivity: The primary result is that the Hall resistivity ($\rho_{yx}$) remains strictly quantized at $h/e^2\nu$ (where $\nu$ is the filling factor) regardless of the light–matter coupling strength or the magnitude of the polariton broadening $\delta$. Unlike the Hall conductivity, which exhibits deviations for any non-zero $\delta$, the Hall resistivity is completely immune to cavity-induced modifications arising from polariton broadening.
• Asymmetry in Transport Properties: The study reveals that modifications to the transport properties in the cavity-coupled QH system are confined entirely to the longitudinal resistivity component ($\rho_{xx}$) along the cavity field polarization. The Hall resistivity remains unaffected, while the longitudinal resistivity acquires cavity-induced corrections dependent on the cavity frequency and broadening.
• Non-Commutativity of Limits: The authors analyze the zero-frequency limit ($\omega \to 0$) where the lower polariton gap closes. They demonstrate that the limits of zero dissipation ($\delta \to 0$) and zero cavity frequency are non-commutative for the Hall conductivity. However, for the Hall resistivity, the quantized value is independent of the order of these limits, remaining robust even when the lower polariton gap closes.
• Anisotropy: The model identifies an anisotropic behavior in the resistivity tensor. While the Hall component is isotropic and robust, the longitudinal resistivity is affected by the preferred polarization direction of the cavity photon field, indicating that cavity photons can selectively break rotational symmetry in electronic transport without destroying topological invariants.

Significance and Claims
The paper claims that these results provide a deeper explanation for the absence of renormalization in the von Klitzing constant observed in recent experiments probing even integer QH plateaus via Hall resistivity measurements at low temperatures. The findings clarify that:

1. Topological Robustness: In cavity QH systems, topological protection manifests strictly at the level of the resistivity tensor, independent of light–matter interaction strength and polariton broadening, provided the temperature is zero.
2. Observable Selection: The Hall resistivity emerges as the correct observable for assessing topological protection in driven or hybrid light–matter platforms, distinguishing it from the Hall conductivity which is sensitive to dissipation and coupling strength.
3. Physical Distinction: The work underscores that conductivity and resistivity encode fundamentally different physical information in hybrid light–matter systems, necessitating a careful reassessment of which observables faithfully reflect topological robustness.

The authors note that while their zero-temperature model explains the stability of the resistivity, combined effects of temperature, finite polariton lifetime, and ultrastrong coupling remain relevant for explaining activated transport observed in experiments. They also mention that a recent cavity QH hydrodynamical approach reaches similar conclusions.