Frequency Domain Berry Curvature

• Frequency Domain Berry Curvature is a geometric invariant that extends Berry phase to frequency-dispersive systems, bridging spectral and topological properties.
• It enables observable effects such as frequency-dependent Hall responses, nonlinear harmonic generation, and transverse photon swings in dispersive media.
• It plays a crucial role in experimental spectroscopies and device applications by providing direct probes of topological order in photonic, semiconductor, and metamaterial systems.

Frequency domain Berry curvature generalizes the geometric phase concepts of wave mechanics and quantum materials by characterizing the geometrical properties—not only in real space or momentum space, but also as a function of frequency or under frequency-dispersive or temporally modulated conditions. Frequency domain Berry curvature emerges in diverse physical systems including dispersive photonics, topological materials with non-trivial band geometry, multiband semiconductors exposed to strong fields, and metamaterials with engineered dispersion and symmetry. In the frequency domain, Berry curvature can generate a variety of exotic observable effects, such as frequency-dependent Hall responses, nonlinear harmonic generation, “swinging” photon trajectories, and quantized pumped transport during driven adiabatic cycles, establishing a key connection between temporal, spectral, and topological properties of waves and quasiparticles.

1. Foundations of Frequency Domain Berry Curvature

Berry curvature, in its canonical form, describes a geometric property of eigenstates parameterized by certain variables, typically crystal momentum ($\mathbf{k}$) in the context of band theory: $$\Omega(\mathbf{k}) = \nabla_{\mathbf{k}} \times \langle u_{\mathbf{k}}| i \nabla_{\mathbf{k}} |u_{\mathbf{k}} \rangle$$, where $|u_{\mathbf{k}}\rangle$ is the periodic cell function of the Bloch state.

In frequency-dispersive or explicitly time-dependent systems, the parameter space naturally includes the frequency $\omega$ and its conjugate variable, time $t$. In dispersive photonic systems, the dielectric tensor $\epsilon(\omega)$ introduces a $\omega$ dependence directly into the operator structure, making the eigenvalue problem non-standard: the eigenvalue (frequency) appears inside the operator (Deng et al., 18 Aug 2025). This leads to Berry curvature components involving frequency derivatives and cross-derivatives with respect to frequency and momentum or spatial coordinates: $$\Omega_{pq} = -2\mathrm{Im}\{\partial_p (\sqrt{\theta_c} f)^\dagger \cdot \partial_q (\sqrt{\theta_c} f)\}$$ where $p,q$ span $\omega$, $\mathbf{k}$, $t$, and $\mathbf{r}$, $f$ is the normalized polarization (energy-normalized), and $\theta_c = \partial_\omega(\omega \Theta_c)$. This geometric structure inherently couples frequency and momentum variables, and can generate non-trivial gauge fields in frequency-momentum (or frequency-time) space.

The mathematical formalism must respect the non-standard spectral orthogonality and normalization that dispersive media imply, requiring energy-density weighted inner products over extended parameter spaces.

2. Effects in Dispersive Photonic and Magnetoplasmonic Systems

In dispersive media such as magnetoplasmon-polariton systems, the frequency-dependent dielectric function $\epsilon(\omega)$ gives rise to Berry curvature in the frequency domain. When the electromagnetic wavepacket is constructed from the off-shell eigenstates of the generalized Maxwell operator, adiabatic changes in either frequency (due to medium modulation) or wavevector (due to spatial inhomogeneity or time-dependent drives) induce Berry connections and curvatures that couple frequency and spatial degrees of freedom (Deng et al., 18 Aug 2025).

The dynamical equations for the wavepacket center contain terms such as

$$\begin{aligned} \dot{\mathbf{x}} &= v_g - \big[ (\Omega_{\mathbf{k}, \mathbf{r}} + v_g \Omega_{\omega, \mathbf{r}})\cdot \dot{\mathbf{r}} + (\Omega_{\mathbf{k}, t} + v_g \Omega_{\omega, t}) \big] \ \dot{\mathbf{k}} &= \mathbf{F} + \big[ (\Omega_{\mathbf{r}, \mathbf{r}} - \mathbf{F} \Omega_{\omega, \mathbf{r}})\cdot \dot{\mathbf{r}} + (\Omega_{\mathbf{r}, t} - \mathbf{F} \Omega_{\omega, t}) \big] \ \dot{\omega} &= P + \big[ (\Omega_{t, \mathbf{r}} + P\Omega_{\omega, \mathbf{r}})\cdot \dot{\mathbf{r}} + (\Omega_{t, \mathbf{k}} + P \Omega_{\omega, \mathbf{k}}) \big] \end{aligned}$$

where $v_g$ is the group velocity, $\mathbf{F}$ is the effective force, $P$ is the frequency chirp rate, and the various Berry curvatures ($\Omega_{pq}$) intermix frequency, wavevector, position, and time. These Berry curvature terms alter photon trajectories, leading to effects such as:

• Transverse ray deflection (photon swing): Even for a straight initial propagation, frequency domain Berry curvature can cause the photon trajectory or the energy centroid to curve or swing transversely, a quantum-optical analog of the anomalous velocity for electrons in crystals.
• Time refraction: When the material properties (e.g., plasma frequency) are modulated in time—while a static magnetic field is present—the interplay of frequency domain Berry curvature and medium evolution can enhance or suppress refraction-like phenomena in the time domain (Deng et al., 18 Aug 2025).

In magnetoplasmon-polariton systems, the presence of time-reversal symmetry breaking terms (e.g., magneto-optical coupling) modulates the Berry curvature landscape and directly affects these transverse displacements. The measurable consequences include photon trajectory bending and group velocity shifts upon temporal modulation of the dispersive parameters.

3. Nonlinear and Frequency-Harmonic Hall Responses from Multipole Berry Curvatures

In topological electronic systems, Berry curvature not only underlies the quantum (anomalous) Hall effect but, when distributed with spatial derivatives (multipoles), produces nonlinear anomalous Hall effects (NLAHE) that are inherently frequency domain phenomena (Zhang et al., 2020, Sankar et al., 2023).

The key Berry curvature multipoles are:

• Dipole: $$D_{\alpha} = \int \partial_{k_\alpha} \Omega_z(\mathbf{k}) f_0\, d\mathbf{k}$$, generating a second-harmonic Hall response.
• Quadrupole: $$Q_{\alpha \beta \gamma} = \int \partial_{k_\alpha}\partial_{k_\beta} \Omega_\gamma(\mathbf{k}) f_0\, d\mathbf{k}$$, giving a third harmonic Hall response.
• Hexapole and higher: Successive derivatives generate higher harmonic responses.

Electrical transport experiments in systems such as kagome antiferromagnets (e.g., FeSn) detect a third-harmonic Hall voltage ($V^{3\omega}_y$), tied to the Berry curvature quadrupole, when a longitudinal $ac$ current drive $I_x(\omega)$ is applied (Sankar et al., 2023). The voltage scales as $V_y^{(n\omega)} \propto I_x^n$, and magnetic symmetry analysis determines which multipole moment is first allowed by the symmetry group. These frequency-multiplied voltages are clear frequency-domain signatures of Berry curvature multipoles and provide direct probes of topological order and spontaneous symmetry breaking.

4. Frequency Domain Berry Curvature in Optical Lattices and Driven Systems

In ultracold atomic lattices, photonic crystals, and moiré superlattices, frequency domain signatures of Berry curvature emerge through collective dynamics or engineered external modulation.

• Bloch oscillations and Berry Curvature Spectroscopy: In superlattices with large real-space periodicity, electrons driven by a strong static field perform Bloch oscillations at a characteristic frequency $\omega_B = (eE_0L)/\hbar$. The optical Hall conductivity, when probed at various frequencies, exhibits resonances at integer multiples of $\omega_B$ whose heights are proportional to the lattice Fourier components of the Berry curvature: $$\Omega_{\mathbf{k}} = \sum_\mathbf{R} \Omega_\mathbf{R} e^{i\mathbf{k}\cdot \mathbf{R}}$$ (Beule et al., 2023). By rotating the field or probe, one can selectively couple to different Fourier components and reconstruct the underlying Berry curvature profile across the Brillouin zone.
• Pumped and Floquet Systems: Temporally modulated or Floquet systems generate frequency-dependent Berry curvatures in the quasienergy bands, leading to quantized pumped currents (e.g., Thouless pump), and geometric phases that encode the frequency (or period) of the external drive (Zhang et al., 2020, Sommer et al., 8 May 2024). Higher-form Berry curvatures, such as $(d+2)$-forms in $d$-dimensional lattice systems, govern the response to complex cyclical driving protocols, resulting in quantized higher harmonic and multi-photon frequency response.

5. Measurement Protocols and Experimental Access

Frequency domain Berry curvature effects can be experimentally accessed by a suite of nonlinear and time-resolved spectroscopies:

• Lock-in detection of higher harmonic voltages in Hall bar geometries to isolate Berry curvature multipole–induced NLAHE (Sankar et al., 2023).
• Time-resolved Faraday and Kerr rotation and THz ellipticity measurements to resolve Berry curvature–induced polarizations at resonance with Bloch oscillations (Beule et al., 2023).
• Pump-probe and time-resolved transport protocols to track photon ‘swing’ and observe sideband spectral shifts induced by frequency domain Berry curvature (Deng et al., 18 Aug 2025).
• Quantum gas microscopy and wavepacket tracking in ultracold atomic gases to directly visualize Berry curvature effects on Bloch oscillation patterns (Price et al., 2011).

A critical advantage of frequency domain measurement protocols is the ability to spectrally separate linear from nonlinear (harmonic) responses, eliminating many parasitic effects and allowing clean extraction of geometric information. Such protocols naturally facilitate the observation of quantized topological invariants, frequency multiplication, and controlled adiabatic pumping.

6. Theoretical and Mathematical Formulation

Key mathematical frameworks for frequency domain Berry curvature include:

• Generalized Berry curvature with frequency-dependent parameters:

$$\Omega_{pq} = -2\,\mathrm{Im} \left\{ \partial_p (\sqrt{\theta_c} f)^\dagger \cdot \partial_q (\sqrt{\theta_c} f) \right\}$$

• Wavepacket equations of motion with frequency domain corrections:

$$\dot{\mathbf{x}} = v_g - \ldots(\text{Berry curvature corrections in }\omega,\mathbf{k},t,\mathbf{r})$$

• Berry curvature multipole:

$$Q_{\alpha \beta \gamma} = \int d\mathbf{k}\; f_0\, \partial_{k_\alpha}\partial_{k_\beta} \Omega_\gamma(\mathbf{k})$$

• Hall response at harmonic frequencies:

$$\mathbf{V}_y^{(n\omega)} \propto I_x^n \;\;\; n \text{ determined by leading multipole}$$

• Higher-form Berry curvatures for lattice systems:

$$F^{(d+2)} = -\frac{1}{(d+1)!} \hbar d^{(d+1)} \mathrm{Im} \langle \psi | d\psi\rangle$$

yielding integrated quantized invariants when contracted with suitable cochains and integrated over parameter space (Sommer et al., 8 May 2024).

7. Physical Significance and Implications

Frequency domain Berry curvature effects unify topological, spectral, and dynamical properties:

• Topological charges and quantized pumping become encoded in higher-form frequency-domain Berry curvatures, connecting bulk band geometry to quantized transport in driven or modulated many-body systems (Sommer et al., 8 May 2024).
• Nonlinear signal generation at harmonics provides a new avenue for device design: e.g., frequency multipliers, high-efficiency rectifiers, or topologically robust photodetectors.
• Photonics and magnonics: Geometric effects in dispersive media or hybrid light-matter systems (such as magnon-polariton bands) lend themselves to field-tunable, chirality-selective, or dynamically reconfigurable topological photonic devices (Okamoto et al., 2020, Deng et al., 18 Aug 2025).
• Material characterization: Frequency–domain responses measured in experiment directly probe hidden geometric or topological aspects of the band structure and can serve as spectroscopic fingerprints for identifying and engineering novel topological materials.

Frequency domain Berry curvature thus expands the landscape of geometric response in quantum and classical wave systems, enabling high-fidelity spectroscopic access to topological invariants and providing a versatile platform for nonlinear and dynamic topological physics.