Perfect Nonreciprocal Axion-polaritons

Technical Summary: Perfect Nonreciprocal Axion-polaritons

Problem Statement
Axion electrodynamics describes a topological electromagnetic response characterized by a $\theta_0 \mathbf{E} \cdot \mathbf{B}$ term. While the static axion field $\theta_0$ is quantized to $0$ or $\pi$ by time-reversal symmetry (TRS) or inversion symmetry, the breaking of both symmetries renders the axion field dynamical. This dynamical axion, arising from fluctuations of the magnetic order parameter, couples to electromagnetic fields to form hybrid collective modes known as axion-polaritons.

Despite prior observations of axion signatures in Weyl/Dirac semimetals and magnetic systems (e.g., MnBi$_2$Te$_4$), nonreciprocal axion-polaritons—propagating modes where the dispersion relation is asymmetric, $\omega(k) \neq \omega(-k)$—have not been identified. Previous theoretical and experimental work on axion-polaritons relied solely on static magnetic fields. While magnetic fields break TRS, they preserve inversion symmetry, which is insufficient to generate directional asymmetry in the dispersion. Similarly, high-energy physics searches (e.g., PVLAS, CAST) utilize magnetic fields alone, failing to produce the necessary directional asymmetry. A regime capable of simultaneously breaking both inversion and TRS to generate nonreciprocal axion-polaritons has remained unexplored.

Methodology
The authors propose a theoretical framework for a dynamical axion insulator subjected to mutually perpendicular static external electric ($\mathbf{E}_0$) and magnetic ($\mathbf{B}_0$) fields. The system is described by a Lagrangian comprising the electromagnetic field energy, the static axion term, and the dynamical axion field energy:
$$\mathcal{L} = \mathcal{L}_{EM} + \mathcal{L}_{\theta_0} + \mathcal{L}_{\theta}$$
where the coupling term is proportional to $(\theta_0 + \theta)\mathbf{E} \cdot \mathbf{B}$.

The study derives the coupled equations of motion for the vector potential $\mathbf{A}$ and the dynamical axion field $\theta$ in the Coulomb gauge. By assuming a plane-wave ansatz $(\mathbf{A}, \theta) \propto e^{i(kx-\omega t)}$ with $\mathbf{E}_0 \parallel \hat{y}$ and $\mathbf{B}_0 \parallel \hat{z}$, the authors reduce the system to a $2 \times 2$ matrix equation for the $z$-component of the vector potential and the axion field. Solving the resulting characteristic equation yields the dispersion relation $\omega(k)$.

To analyze the physical implications, the authors:

1. Calculate the group velocities $|v_g| = |\partial_k \omega|$ for the upper and lower polariton branches.
2. Define a direction-dependent coupling strength $g_a(k)$ to quantify the hybridization between photons and axions.
3. Introduce a dissipative term ($\sim \Gamma \partial_t \theta$) to model the finite lifetime of magnons and compute the complex transmission coefficients ($t_{LR}$ and $t_{RL}$) for a slab geometry, effectively modeling the system as an optical isolator.

Key Contributions and Results

1. Intrinsic Nonreciprocal Dispersion:
   The authors demonstrate that the simultaneous application of static $\mathbf{E}$ and $\mathbf{B}$ fields breaks both inversion and TRS, leading to a nonreciprocal dispersion relation $\omega(k) \neq \omega(-k)$. This asymmetry arises from a term in the dispersion equation linear in both frequency and momentum ($2EBk\omega$).

   • Mechanism: The nonreciprocity is driven by direction-dependent axion-photon coupling strengths ($g_+ \neq g_-$). Counter-propagating photons hybridize with the axion quasiparticle with unequal amplitudes.
   • Observables: The group velocity $|v_g|$ becomes asymmetric with respect to momentum. For instance, the group velocity for the lower branch vanishes at different rates for $+k$ and $-k$, and the upper branch exhibits a shift in the momentum where the group velocity vanishes.

2. Perfect Nonreciprocity Regime:
   A critical regime is identified where the external fields satisfy the condition $|\mathbf{E}_0| = c' |\mathbf{B}_0|$ (where $c'$ is the photon velocity in the medium). In this limit:

   • One-Way Decoupling: The right-moving photon ($+k$) becomes completely decoupled from the axion mode ($g_+ = 0$), propagating as a free photon with an unrenormalized dispersion.
   • Strong Hybridization: Conversely, the left-moving photon ($-k$) hybridizes strongly with the axion mode.
   • Asymmetric Coupling: This state represents a "perfect nonreciprocity" where the photon acts as a source for the axion in one direction but not vice versa, a feature distinct from trivial decoupling.

3. Optical Isolator Realization:
   The nonreciprocal dispersion manifests as an optical isolator. When a small dissipation is introduced:

   • Transmission Asymmetry: The right-moving photon ($LR$) propagates freely through the slab with high transmissivity ($|t_{LR}| \approx 1$, limited only by Fresnel reflection), as it does not couple to the lossy axion mode.
   • Attenuation: The left-moving photon ($RL$) hybridizes with the dissipative axion mode, acquiring a finite imaginary part to its wavevector. This leads to exponential attenuation and strongly suppressed transmissivity ($|t_{RL}| \ll 1$) near the avoided crossing.
   • Requirement: This isolation effect strictly requires the simultaneous breaking of inversion and TRS; it vanishes when $\mathbf{E}_0 = 0$.

Significance and Claims
The paper establishes nonreciprocal axion-polaritons as a fundamentally new route to optical nonreciprocity. Unlike conventional isolators that rely on external biasing, engineered dissipation, active modulation, or nanoscale confinement, this mechanism arises intrinsically from the direction-dependent axion-photon coupling in a bulk material tunable by static fields.

The authors claim that these results:

• Provide a powerful, independent probe for establishing the existence of axion quasiparticles, complementing existing signatures like Kerr rotation.
• Suggest experimentally accessible routes for detection using standard spectroscopic measurements and vector network analyzers (VNAs) to measure amplitude and phase of transmission.
• Identify specific material platforms, such as van der Waals compounds (MnBi$_2$Te$_4$) and magnetoelectric antiferromagnets (Cr$_2$O$_3$), as viable candidates. The authors estimate that the required static fields (e.g., $B \sim 8$ mT and $E \sim 10^5$ V/m) are achievable in laboratory settings.
• Open new directions for exploring nonreciprocal slow light and potentially enhancing sensitivity for dark matter axion detection.