Nonreciprocal Gyrotropic Forces

Updated 24 August 2025

Nonreciprocal gyrotropic forces are physical effects arising from broken time-reversal and parity symmetry, leading to direction-dependent wave transmission and interaction forces.
They enable giant nonreciprocal contrasts—exceeding 90 dB in some microwave experiments—with mode coalescence near exceptional points that enhance sensitivity.
Applications span metamaterial design, superconducting devices, and nanoscale actuation, leveraging tailored gyrotropy for efficient, unidirectional energy management.

Nonreciprocal gyrotropic forces are physical effects in which the presence of gyrotropy—typically breaking time-reversal and/or parity symmetry, via material biasing or designed metamaterial structure—leads to direction-dependent transport, interaction forces, or modal properties in classical, quantum, and statistical electromagnetic and wave systems. These forces underlie a broad suite of phenomena including dramatic nonreciprocal transmission, fluctuation-induced lateral Casimir forces, spontaneous torques, asymmetric optical phase responses, and nonreciprocal energy conversion at both macroscopic and nanoscale, platform-agnostic levels. Their theoretical and practical manifestation relies heavily on the interplay between modal structure, symmetry, non-Hermitian dynamics, and material engineering (including gain, loss, and magnetic or mechanical bias).

1. Symmetry Foundations and Gyrotropic Constitutive Responses

Gyrotropy refers to the property of a material wherein its electromagnetic (or wave) response tensors—typically permittivity $\hat\varepsilon$ , permeability $\hat\mu$ , or the magnetoelectric tensor—acquire off-diagonal, antisymmetric (or more generally, nonreciprocal) components. These emerge under static magnetic bias (as in ferrites), mechanical rotation (as in rotating plasmas), or time-spatially modulated metamaterial designs. Such symmetry breaking yields constitutive relations of the form:

$\mathbf{D} = \epsilon_0\,\epsilon_r\,\mathbf{E} - (\boldsymbol{\Gamma}\times\mathbf{H}), \qquad \mathbf{B} = \mu_0\,\mu_r\,\mathbf{H} + (\boldsymbol{\Gamma}\times\mathbf{E})\,,$

where $\boldsymbol{\Gamma}$ is the magnetoelectric-gyrotopy vector; alternatively, gyrotropic plasmonic or ferrite materials are described by permeability or permittivity tensors like:

$\hat \mu = \begin{bmatrix} \mu_r & 0 & i\kappa_r \ 0 & 1 & 0 \ - i\kappa_r & 0 & \mu_r \end{bmatrix} \,,\qquad \hat \varepsilon = \begin{bmatrix} \epsilon & i\gamma & 0 \ - i\gamma & \epsilon & 0 \ 0 & 0 & \epsilon_z \end{bmatrix}\,.$

The presence of off-diagonal imaginary entries encodes the fundamental nonreciprocal character.

Such nonreciprocity manifests not only in the breaking of Lorentz reciprocity ( $S_{21}\ne S_{12}$ for a scattering matrix $S$ ), but also fundamentally influences fluctuation-dissipation relations and modal orthogonality. Theoretical treatments must generalize classical and quantum response theory, as standard symmetric Green function expansions are replaced with spectral representations respecting antisymmetric tensor structure (Lakhtakia, 2018, Thomas et al., 2017, Milton et al., 4 Dec 2024).

2. Giant Nonreciprocity Near Exceptional Points

When gyrotropy is combined with balanced gain and loss—especially under temporal or spatial symmetry (e.g., combined mirror-time or parity-time, ${\cal PT}$ , symmetry)—non-Hermitian degeneracies called exceptional points (EPs) become accessible. In such systems, the spectrum of coupled modes (e.g., in photonic resonators or lumped-circuit analogues) is governed by non-Hermitian coupling matrices, and the eigenvalues coalesce at the EP:

$\omega_{s,a}^{(0)} = \omega_0 \mp \sqrt{\Omega_0^2 - (\rho \gamma)^2},\qquad\text{with}\quad \gamma_{\rm EP} = \Omega_0/\rho\,.$

At $\gamma=\gamma_{\rm EP}$ , not only do eigenvalues coalesce, but modal orthogonality collapses; consequently, response functions and transmission parameters become extremely sensitive to nonreciprocal (gyrotropic) perturbations.

When coupled to a waveguide, such EP-adjacent degeneracies enable destructive interference in one direction (e.g., backward transmission) and constructive in the other, yielding giant nonreciprocal contrast—which can exceed $90$~dB in microwave resonator experiments (Thomas et al., 2017). The combination of mode coalescence and asymmetric coupling induced by gyrotropy universally enables high-performance, reconfigurable nonreciprocal devices across not only electromagnetics, but also in acoustics and lumped electronic circuits.

3. Fluctuation-Induced Forces and Quantum Nonreciprocity

Fluctuation-induced gyrotropic forces arise in Casimir–Polder and related quantum electromagnetic phenomena, when a small object (e.g., an atom or nanoparticle) is placed near or within a gyrotropic, nonreciprocal medium. The total optical force comprises resonant (recoil/emission) and nonresonant (vacuum fluctuation) contributions, both given in terms of the system Green function $G(\mathbf{r},\mathbf{r}';\omega)$ :

$\mathcal{F}^R_i = 2\,\mathrm{Re}\left\{\vec\gamma^* \cdot \big[ -i\omega \partial_i G(\mathbf{r},\mathbf{r}_0;\omega)\big]_{\omega=\omega_0 + i0^+; \mathbf{r}=\mathbf{r}_0}\, \vec\gamma\right\},$

$\mathcal{E}_C = -\frac{1}{4\pi}\int_{-\infty}^{\infty} d\xi\, \mathrm{tr}\{\tilde\alpha(i\xi) [-i\omega G]_{\omega=i\xi}\},$

with the nonreciprocal response of $G$ being crucial when the environment possesses gyrotropy. In gyrotropic plasmas or photonic topological materials, the surface plasmon-polariton (SPP) modes become nonreciprocal, yielding dispersions:

$\omega_\theta = \frac{\omega_c}{2}\cos\theta + \left[\frac{\omega_p^2}{2} + \frac{\omega_c^2}{4}(1+\sin^2\theta)\right]^{1/2},$

so the density and directionality of SPPs lead directly to finite lateral quantum recoil forces, even on laterally invariant interfaces (Silveirinha et al., 2017). The sign, magnitude, and spatial decay ( $\propto 1/d^4$ ) of these forces can be tuned by external magnetic fields, with quasi-static surface mode resonances dominating at short distances.

4. Nonreciprocal Wave Propagation, Topological Modes, and Metamaterials

Classical nonreciprocal gyrotropic forces are central to the unidirectional propagation of surface or edge states in topological photonic systems. In waveguide structures composed of gyrotropic media (e.g., biased InSb or ferrites), interfaces exhibit nontrivial topological invariants (e.g., nonzero Chern numbers). This enforces the existence of one-way surface modes, immune to backscattering, as a direct result of the broken time-reversal and parity symmetries.

When multiple topological interfaces are placed in proximity, their surface modes couple and the modal spectrum rearranges. As interfaces merge, a characteristic splitting occurs: only one mode remains accessible (finite $k_x$ ), while the other diverges (infinite $k_x$ and impedance), resulting in robust channeling of energy in a single direction—a haLLMark of nonreciprocal gyrotropic "force" in the context of modal energy flow (Gangaraj et al., 2018).

Energy self-reliant gyric metamaterials further leverage local angular momentum (mechanically modulated, as in spinning-rotor lattices) to break reciprocal wave symmetry via gyroscopic (Coriolis-type) coupling. Space-time periodic modulation of rotor angular momentum $h(t)$ enables nonreciprocal band structure tunability, bandgap formation, and robust energy conservation over modulation cycles—even in the nonlinear regime (Attarzadeh et al., 2019).

5. Thermodynamic Constraints, Fluctuation-Induced Propulsion, and Quantum Friction

Utilizing nonreciprocal media for nanoscale propulsion, heat conversion, or refrigeration relies on the breaking of detailed balance in photon momentum exchange. In setups where nonreciprocal (gyrotropic) dielectric plates are held at different temperatures, thermal photon exchange generates a lateral Casimir force:

$F_y = \frac{2\hbar}{\pi}\int_0^\infty d\omega \frac{1}{e^{\hbar\omega/k_BT_1} - 1} \int \frac{dk_y}{2\pi} k_y S(k_y),$

where $S(k_y)$ , the spectral heat flux, is antisymmetric in nonreciprocal systems ( $S(k_y)\ne S(-k_y)$ ). This enables rectified thermal forces and heat engines that approach Carnot-limited efficiency, with characteristic velocities (for optimal operation) $v_{\rm opt} \sim \eta_c\, c$ in the far field and $v_{\rm opt} \sim \eta_c\bar\omega d$ in the near field, for Carnot efficiency $\eta_c$ and separation $d$ (Gelbwaser-Klimovsky et al., 2020).

In spontaneous nonequilibrium settings, nonreciprocal bodies experience vacuum torques proportional to the antisymmetric part of the susceptibility under temperature gradients:

$\tau_i = \int \frac{d\omega}{2\pi} \frac{\omega^3}{6\pi}\varepsilon_{ijk} \mathrm{Re}\,\alpha_{jk}(\omega) [\coth(\beta\omega/2) - \coth(\beta'\omega/2)]\,,$

but no net force to first order in susceptibility unless translational symmetry is broken by a nearby surface, at which point lateral propulsion emerges (Milton et al., 2023, Milton et al., 4 Dec 2024).

Modal analysis in nonreciprocal systems reveals unique signatures: in reciprocal (symmetric $S$ ) systems, characteristic mode eigenvectors are real or equiphase and far fields obey inversion symmetry, while in gyrotropic or moving media (with $S\ne S^T$ ), characteristic eigenvectors develop progressive (non-real) phase and far-field patterns lose symmetry:

$F_n(-\hat{\mathbf{r}}) = -F_n^*(\hat{\mathbf{r}}) \qquad \text{(reciprocal)},\qquad \text{but not necessarily otherwise}.$

The decomposition of the scattering matrix $S$ into symmetric and antisymmetric parts allows the definition of a nonreciprocity indicator

$\nu = P^{\text{nrec}}/P_o\,,$

with $P^{\text{nrec}}$ the outgoing power from the nonreciprocal component. Such indicators quantify, e.g., progressive phase behavior in ideal circulators or the loss of time-reversal symmetry in rotating cylinders (Wingren et al., 2023).

7. Applications: Metamaterial Engineering, Superconducting Devices, and Device Designs

Recent advances leverage intrinsic and extrinsic gyrotropy for magnet-free microwave and photonic nonreciprocal devices. Self-biased gyrotropic metamaterials employ hard magnets (e.g., NdFeB) embedded in soft ferrite matrices (e.g., YIG) to realize zero-net-magnetization media exhibiting robust Faraday rotation and high isolation, while eliminating external bias (Pyvovar et al., 11 Mar 2025). Similarly, circuit-based hybrids composed of superconducting-semiconducting voltage-tunable junctions and on-chip SQUID arrays use flux-charge coupling or spatio-temporal modulation to realize nonreciprocal couplings and gyrator behavior even in quantum-coherent platforms (Leroux et al., 2022, Tu et al., 24 Jun 2025).

Superconducting structures with tailored gyrotropy enable both nonreciprocal transport/DC diode effect and a direct link between spin Hall physics and nonreciprocal Josephson transport (Kokkeler et al., 2023). Nonreciprocal metasurfaces and nanostructures, constructed from self-biased meta-atoms or designed for maximal nonreciprocity even in the presence of loss (as in cavity-embedded plasmonic particles), further extend this principle to broad frequency ranges, including THz and optical regimes (Sadzi et al., 28 Oct 2024).

Applications include compact isolators, circulators, unidirectional waveguides, radiative heat engines, noncontact nanoactuators, and full-duplex wireless communication systems, with demonstrated isolation $>58$ ~dB and Faraday rotation up to $45^\circ$ over the entire X-band.