Axion Electrodynamics: Theory & Implications

• Axion electrodynamics is a framework that integrates a dynamic pseudoscalar (axion) field with Maxwell's equations, leading to modifications in electromagnetic behavior.
• The theoretical model introduces changes in dispersion relations, polarization rotation, and mode splitting, with clear predictions for optical phenomena in constrained geometries.
• This approach underpins experimental searches by predicting suppressed yet measurable effects, including induced currents and Casimir energy corrections in dark matter and topological phase studies.

Axion electrodynamics is a generalized framework for electromagnetism in which a dynamical pseudoscalar field—the axion—couples to the electromagnetic sector through a topological density proportional to $a(x)\, \mathbf{E}\cdot\mathbf{B}$ or, covariantly, $a(x) F_{\mu\nu}\tilde F^{\mu\nu}$, where $F_{\mu\nu}$ is the field-strength tensor and $\tilde F^{\mu\nu}$ its dual. This extension modifies classical Maxwell theory, leads to nontrivial optical effects, forms the basis for the electromagnetic response of axionic topological phases in condensed matter, and underlies many laboratory and astrophysical searches for axionic dark matter. The axion–photon effective coupling, denoted $g_{a\gamma\gamma}$, sets the strength of this interaction and is model-dependent.

1. Lagrangian Structure and Modified Maxwell Equations

The general Lagrangian for axion electrodynamics in a homogeneous medium of constant permittivity $\varepsilon$ and permeability $\mu$ is

$$\mathcal{L} = -\frac{1}{4} F_{\mu\nu}H^{\mu\nu} + A_\mu J^\mu - \frac{1}{2} (\partial_\mu a)(\partial^\mu a) - \frac{1}{2} m_a^2 a^2 - \frac{1}{4} g_{a\gamma\gamma} a(x) F_{\mu\nu} \tilde F^{\mu\nu}$$

Here, $H^{\mu\nu}$ encodes the medium response ($$\mathbf{D} = \varepsilon \mathbf{E}, \; \mathbf{B} = \mu \mathbf{H}$$), and $a(x)$ is the axion field. The axion–photon coupling is

$$g_{a\gamma\gamma} = g_\gamma \frac{\alpha}{\pi} \frac{1}{f_a}$$

with $\alpha$ the fine-structure constant, $f_a$ the axion decay constant, and $g_\gamma$ a model-dependent number ($\sim 0.36$ in the DFSZ model) (Brevik, 2022).

Varying this action gives the modified Maxwell's equations: $$\begin{aligned} \nabla \cdot \mathbf{D} &= \rho - g_{a\gamma\gamma} \mathbf{B} \cdot \nabla a \ \nabla \times \mathbf{H} &= \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} + g_{a\gamma\gamma} \left( \frac{\partial a}{\partial t} \mathbf{B} + \nabla a \times \mathbf{E} \right) \ \nabla \cdot \mathbf{B} &= 0 \ \nabla \times \mathbf{E} &= -\frac{\partial\mathbf{B}}{\partial t} \end{aligned}$$ Similar forms arise in curved spacetime and in magnetohydrodynamics with general $f(\phi)F\tilde F$ couplings (Hwang et al., 2022).

In the most common application—where $a$ is nearly homogeneous and time-dependent as predicted for ambient axion dark matter—the dominant new source is an effective current $J_{\text{axion}} \propto \dot{a} \mathbf{B}_0$ in the presence of a static magnetic field (Brevik, 2022).

2. Optical Effects: Birefringence, Polarization Rotation, and Mode Splitting

The axion coupling leads to a range of optical phenomena. For a time-dependent axion background ($a=a(t)$), a linearly polarized electromagnetic wave propagating through the medium experiences circular birefringence, resulting in a rotation of the polarization plane. Explicitly, the dispersion relation for right- and left-circular polarizations becomes

$$\omega_{\pm} \approx |\mathbf{k}| \pm \frac{1}{2} g_{a\gamma\gamma} \dot a$$

leading to a rotation rate $$\frac{d\varphi}{dz} = \frac{g_{a\gamma\gamma}}{2} \dot a$$ (Favitta et al., 2023).

In a finite slab of axion dielectric with thickness $L$, for constant $\dot\theta = g_{a\gamma\gamma}\dot a$, the net rotation is

$$\Delta\theta = \frac{1}{2} \sqrt{\frac{\mu}{\varepsilon}} \dot\theta L,$$

directly measurable as optical activity (Brevik et al., 2023).

In systems with spatially varying axion amplitudes—e.g., between parallel metallic plates with an axion field $a(z)$—the electromagnetic mode structure is split: one branch remains unperturbed, while the other splits into "superluminal" and "subluminal" modes with a frequency splitting $$\Delta\omega \sim g_{a\gamma\gamma} a_0 / (L\varepsilon)$$ (Brevik, 2022, Brevik et al., 17 Feb 2024). The axion-induced splitting is directly visible in the Casimir effect.

3. Casimir Effect and Zero-Point Energy Corrections

Axion electrodynamics introduces corrections to the zero-point energy and associated Casimir pressure between plates. For a simple one-dimensional configuration with a static spatial gradient ($\beta = g_{a\gamma\gamma} \partial_z a$), the Casimir energy per unit area in scalar electrodynamics is

$$E/L^2 = -\frac{\pi^2}{1440 \sqrt{\varepsilon \mu}\, L^3} \mp \frac{\beta}{4 \sqrt{\varepsilon \mu} L^2} + \cdots,$$

with the first term recovering the standard ($L^{-3}$) Casimir effect and the second representing the leading axion-gradient correction ($L^{-2}$ scaling). The sign depends on the polarization; the net axionic contribution cancels if both split modes are equally populated (Brevik et al., 17 Feb 2024, Brevik, 2022).

In time-dependent axion backgrounds, the correction to the Casimir energy at zero temperature is

$$\Delta E_C(L, \dot a) \sim -\frac{7}{320} (g_{a\gamma\gamma} \dot a)^2 / L,$$

also remarkably suppressed compared to the standard term (Favitta et al., 2023, Brevik et al., 2023).

4. Phenomenology: Laboratory and Astrophysical Consequences

In strong static backgrounds (e.g., haloscope experiments), the axion-induced current density is

$$J_{\text{axion}}(t) = -\frac{g_{a\gamma\gamma} \omega a_0}{\mu c} B_0 \sin(\omega t)$$

with $\omega \sim m_a c^2/\hbar$. In realistic laboratory scenarios (e.g., $B_0=10$ T, $a_0/f_a \lesssim 10^{-19}$ from nEDM bounds), the resulting current density is at most $$|J_{\text{axion}}(t)| \sim 3.4\times 10^{-19} \sin(\omega t)$$ A/m$^2$, far below direct detection thresholds (Brevik, 2022).

In parallel, axion electrodynamics underlies proposals for laboratory measurements targeting the induced polarization rotation and Casimir-force corrections under controlled axion backgrounds. All these effects are suppressed by the smallness of $g_{a\gamma\gamma}$ and realistic axion field amplitudes, leading to negligible corrections under laboratory-accessible conditions (Brevik, 2022, Brevik et al., 17 Feb 2024, Brevik et al., 2023).

5. Boundary Conditions, Hybrid Fields, and Generalizations

Boundary-value problems in axion electrodynamics are most economically treated using the hybrid (Hehl–Obukhov) field definitions: $$\mathbf{D}_\gamma = \varepsilon\mathbf{E} - \theta \mathbf{B}, \quad \mathbf{H}_\gamma = \frac{1}{\mu}\mathbf{B} + \theta \mathbf{E}$$ where $\theta(x) = g_{a\gamma\gamma} a(x)$ (Brevik et al., 17 Feb 2024, Brevik et al., 2023). Perfect conductor boundaries enforce continuity of tangential $\mathbf{E}$ and $\mathbf{H}_\gamma$, and normal $\mathbf{D}_\gamma$ and $\mathbf{B}$. This off-diagonal mixing produces matching conditions that underlie the existence of axion-induced image charges and monopoles near interfaces (Brevik et al., 17 Feb 2024, Brevik et al., 2023).

In systems with constant axion gradients, hybrid fields diagonalize the Maxwell equations. For time- and space-dependent $\theta(x)$, the full Maxwell equations acquire $\propto \dot\theta$ and $\nabla\theta$ sources in Gauss's and Ampère's laws (Brevik et al., 2023).

6. Relation to Topological Phases and Higher-Form Symmetry

The $\theta F\tilde F$ term in axion electrodynamics encodes the electromagnetic response of three-dimensional topological insulators ($\theta = \pi$) and axion insulators with dynamical $\theta(\mathbf{r}, t)$. Interfaces with surface $\Delta\theta$ support quantized surface Hall conductance and half-integer quantum Hall modes, and axion-induced nonreciprocal photonics (Sekine et al., 2020, Planelles, 2021).

Extension to higher-form symmetry reveals the fusion rules and defect algebra that classify domain walls, monopoles, strings, and related Witten and anomalous Hall effects in a unified higher-group theoretical setting (Hidaka et al., 2020, Yokokura, 2022). These aspects underlie both the bulk-boundary correspondence in topological phases and the behavior of extended objects in QFT.

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Axion electrodynamics thus provides a rigorous and unified theoretical framework that predicts a rich structure of electromagnetic phenomena in the presence of pseudoscalar backgrounds, relevant to both high-energy and condensed matter domains. Its most distinctive physical signatures—mode splitting, polarization rotation, and axion-induced forces—emerge through the interplay of boundary conditions, dispersion relations, and quantization in confined geometries. The feeble magnitude of the axion coupling, however, places all such effects far below current experimental reach, but the general formalism underpins ongoing searches in both dark matter detection and topological material science (Brevik, 2022, Brevik et al., 2023, Brevik et al., 17 Feb 2024).