Intrinsic Gyrotropic Magnetic Current

• IGMC is a dissipationless, symmetry-governed electric current arising from time-dependent magnetic fields in materials lacking inversion symmetry.
• It is driven by the Zeeman quantum geometric tensor that couples Bloch state momentum and spin textures, enabling a probe of quantum geometry and magnetic order.
• IGMC supports applications in quantum materials by detecting hidden magnetic orders and facilitating symmetry-selective, ultrafast spintronic transport.

An intrinsic gyrotropic magnetic current (IGMC) is a dissipationless, symmetry-governed electric current generated by a time-dependent magnetic field in materials lacking inversion symmetry. This effect is fundamentally distinct from conventional chiral or anomalous magnetic responses (such as the chiral magnetic effect) and arises from the interplay between the geometric structure of Bloch states—specifically, their momentum and spin textures—and external magnetic perturbations. The IGMC is dictated by geometric quantum mechanical invariants of the band structure, most notably the Zeeman quantum geometric tensor (ZQGT), and is independent of extrinsic scattering. It serves as a powerful probe of unconventional magnetic orders, quantum geometry, and topological phases in solids, with direct experimental relevance in a wide range of quantum materials.

1. Theoretical Foundations and Quantum Geometric Structure

The IGMC is rigorously formulated by linear response theory in the presence of a time-dependent Zeeman field. The key invariant is the Zeeman quantum geometric tensor $T_{nm}^{\alpha\beta}(\mathbf{k})$, which extends the conventional quantum geometric tensor by incorporating combined momentum and spin rotations:

$$T_{nm}^{\alpha\beta}(\mathbf{k}) = \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | \partial_{k_\beta} u_{m,\mathbf{k}} \rangle - \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | u_{m,\mathbf{k}} \rangle \langle u_{m,\mathbf{k}} | \partial_{k_\beta} u_{n,\mathbf{k}} \rangle$$

This tensor decomposes into a symmetric quantum metric $Q_{nm}^{\alpha\beta}$ and an antisymmetric Zeeman Berry curvature $Z_{nm}^{\alpha\beta}$:

$$T_{nm}^{\alpha\beta} = Q_{nm}^{\alpha\beta} - \frac{i}{2} Z_{nm}^{\alpha\beta}$$

where

$$Q_{nm}^{\alpha\beta} = \frac{1}{2}[ r_{nm}^\alpha \sigma_{mn}^\beta + r_{mn}^\alpha \sigma_{nm}^\beta ],\quad Z_{nm}^{\alpha\beta} = i[ r_{nm}^\alpha \sigma_{mn}^\beta - r_{mn}^\alpha \sigma_{nm}^\beta ]$$

with $$r_{nm}^\alpha = \langle u_n | i\partial_{k_\alpha} | u_m \rangle$$ and $$\sigma_{nm}^\beta = \langle u_n | \sigma_\beta | u_m \rangle$$ (Chakraborti et al., 20 Aug 2025, Xiang et al., 2023).

The physical significance of the ZQGT lies in its description of the quantum geometric response to combined momentum-space displacement and spin rotation, fundamentally generalizing the role of the conventional charge quantum geometric tensor.

2. Linear Response Formalism and IGMC Conductivity

In the presence of an oscillating magnetic field $\mathbf{B}(t)$, the IGMC is expressed as a linear superposition of two tensors—conduction $(\sigma^C)$ and displacement $$T_{nm}^{\alpha\beta}(\mathbf{k}) = \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | \partial_{k_\beta} u_{m,\mathbf{k}} \rangle - \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | u_{m,\mathbf{k}} \rangle \langle u_{m,\mathbf{k}} | \partial_{k_\beta} u_{n,\mathbf{k}} \rangle$$0—which capture distinct physical mechanisms:

$$T_{nm}^{\alpha\beta}(\mathbf{k}) = \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | \partial_{k_\beta} u_{m,\mathbf{k}} \rangle - \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | u_{m,\mathbf{k}} \rangle \langle u_{m,\mathbf{k}} | \partial_{k_\beta} u_{n,\mathbf{k}} \rangle$$1

Explicitly, these conductivities are given by

$$T_{nm}^{\alpha\beta}(\mathbf{k}) = \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | \partial_{k_\beta} u_{m,\mathbf{k}} \rangle - \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | u_{m,\mathbf{k}} \rangle \langle u_{m,\mathbf{k}} | \partial_{k_\beta} u_{n,\mathbf{k}} \rangle$$2

where $$T_{nm}^{\alpha\beta}(\mathbf{k}) = \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | \partial_{k_\beta} u_{m,\mathbf{k}} \rangle - \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | u_{m,\mathbf{k}} \rangle \langle u_{m,\mathbf{k}} | \partial_{k_\beta} u_{n,\mathbf{k}} \rangle$$3 is the Fermi-Dirac distribution and $$T_{nm}^{\alpha\beta}(\mathbf{k}) = \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | \partial_{k_\beta} u_{m,\mathbf{k}} \rangle - \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | u_{m,\mathbf{k}} \rangle \langle u_{m,\mathbf{k}} | \partial_{k_\beta} u_{n,\mathbf{k}} \rangle$$4 is the drive frequency (Chakraborti et al., 20 Aug 2025, Xiang et al., 2023). The conduction piece is Fermi-surface dominated and synchronous with the drive; the displacement piece is Fermi-sea dominated, proportional to $$T_{nm}^{\alpha\beta}(\mathbf{k}) = \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | \partial_{k_\beta} u_{m,\mathbf{k}} \rangle - \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | u_{m,\mathbf{k}} \rangle \langle u_{m,\mathbf{k}} | \partial_{k_\beta} u_{n,\mathbf{k}} \rangle$$5, and appears as a geometric displacement current.

This structure is τ-independent (“intrinsic”) and persists even in the clean limit, in contrast to extrinsic processes (e.g., Drude/Fermi-surface oscillations) that scale with the carrier scattering time.

3. Symmetry Constraints and Distinction from Conventional Responses

Symmetry analysis is pivotal in determining which IGMC tensor elements are nonzero in a given material. The parity and time-reversal (T) properties of the ZQGT components are sharply distinct from their charge counterparts:

• The Zeeman Berry curvature $$T_{nm}^{\alpha\beta}(\mathbf{k}) = \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | \partial_{k_\beta} u_{m,\mathbf{k}} \rangle - \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | u_{m,\mathbf{k}} \rangle \langle u_{m,\mathbf{k}} | \partial_{k_\beta} u_{n,\mathbf{k}} \rangle$$6 is P-odd, T-even.
• The Zeeman quantum metric $$T_{nm}^{\alpha\beta}(\mathbf{k}) = \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | \partial_{k_\beta} u_{m,\mathbf{k}} \rangle - \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | u_{m,\mathbf{k}} \rangle \langle u_{m,\mathbf{k}} | \partial_{k_\beta} u_{n,\mathbf{k}} \rangle$$7 is P-odd, T-odd.
• Conventional Berry curvature is always antisymmetric and cannot drive longitudinal (diagonal) responses.

Consequently, the IGMC can generate both transverse (Hall-like) and longitudinal transport channels, depending on the magnetic and crystallographic symmetries. For example, in time-reversal-broken $$T_{nm}^{\alpha\beta}(\mathbf{k}) = \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | \partial_{k_\beta} u_{m,\mathbf{k}} \rangle - \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | u_{m,\mathbf{k}} \rangle \langle u_{m,\mathbf{k}} | \partial_{k_\beta} u_{n,\mathbf{k}} \rangle$$8 altermagnets, only transverse Hall-like IGMCs survive, whereas mixed $$T_{nm}^{\alpha\beta}(\mathbf{k}) = \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | \partial_{k_\beta} u_{m,\mathbf{k}} \rangle - \langle \partial_{k_\alpha} u_{n,\mathbf{k}} | s_z | u_{m,\mathbf{k}} \rangle \langle u_{m,\mathbf{k}} | \partial_{k_\beta} u_{n,\mathbf{k}} \rangle$$9-wave altermagnets (with $Q_{nm}^{\alpha\beta}$0 having both $Q_{nm}^{\alpha\beta}$1 and $Q_{nm}^{\alpha\beta}$2 components) support all four conduction and displacement channels, including those absent in conventional quantum geometry (Chakraborti et al., 20 Aug 2025, Xiang et al., 2023).

Furthermore, unlike the chiral magnetic effect (CME), which is governed by Berry curvature and requires nonequilibrium population imbalance (μ_L ≠ μ_R) or broken T, the IGMC persists in equilibrium and is present in P-broken but T-preserving systems, directly probing magnetic geometry rather than topological charge (Zhong et al., 2015).

4. Microscopic Realizations and Model Applications

The IGMC is realized in a diverse set of quantum materials:

• Unconventional 2D Magnets: Square-lattice models with momentum-dependent $Q_{nm}^{\alpha\beta}$3 and strong Rashba spin–orbit coupling showcase longitudinal and transverse IGMC channels depending on the magnetic form factor symmetry (e.g., $Q_{nm}^{\alpha\beta}$4-wave, $Q_{nm}^{\alpha\beta}$5-wave, mixed orders). The IGMC is nonzero even when the conventional Berry curvature vanishes by symmetry (Chakraborti et al., 20 Aug 2025).
• Chiral Magnets and Altermagnets: In metallic chiral magnets, the IGMC traces to Fermi-surface integrals of the Bloch orbital/spin magnetic moments and appears as a low-frequency limit of gyrotropy (natural optical activity), with the sign tracking the handedness (chirality) of the magnetic texture (Paul et al., 28 Apr 2025).
• Heterostructures: At magnetic topological insulator–unconventional magnet interfaces, the IGMC provides a symmetry fingerprint of the proximate magnetic order via the Zeeman quantum metric and Berry curvature. The transverse displacement current exhibits harmonics encoding the magnetic form factor’s parity and angular momentum (e.g., $Q_{nm}^{\alpha\beta}$6-wave: four-fold, $Q_{nm}^{\alpha\beta}$7-wave: twelve-fold) (Chakraborti et al., 21 Dec 2025).
• Superconductors: In gyrotropic superconductors, a supercurrent-induced IGMC arises from the magnetoelectric (Edelstein) tensor, with the symmetry-enforced structure of the response intimately tied to crystal point group (He et al., 2019).
• Classical Spin Systems: Emergent gauge fields associated with Dzyaloshinskii-Moriya interactions mediate IGMC in zig-zag classical dipolar lattices, where the antisymmetric coupling acts as a vector potential for the magnetic current (Mellado et al., 2022).

5. Experimental Manifestations and Detection Protocols

The intrinsic IGMC generates signatures readily accessible to modern experimental techniques:

• Transport Measurements: mV-scale Hall voltages in RuO$Q_{nm}^{\alpha\beta}$8-class altermagnets under modest oscillating B-fields ($Q_{nm}^{\alpha\beta}$920 G) at 100 Hz are predicted (Chakraborti et al., 20 Aug 2025). Displacement IGMCs require higher frequencies (THz pump–probe) but yield $Z_{nm}^{\alpha\beta}$0V-scale signals.
• Magneto-Optical Detection: Current-induced Faraday or Kerr rotations in chiral semiconductors (e.g., p-Te) provide corroborating evidence for the IGMC (Tsirkin et al., 2017).
• AC Probes and Symmetry Readout: By varying the direction, frequency, and symmetry-breaking parameter (e.g., strain, gating), the different IGMC channels can be disentangled, yielding direct signatures of the symmetry and texture of the underlying magnetic order (Chakraborti et al., 21 Dec 2025).
• Orbital vs. Spin Origin: In PT-symmetric antiferromagnets (CuMnAs), the orbital displacement IGMC dominates and strictly reverses sign with Néel vector reversal, offering a high-fidelity probe of antiferromagnetic domain orientation (Ghorai et al., 8 Jan 2026).
• Floquet Engineering: Application of bicircular light to nodal-line semimetals (e.g., compressed black phosphorus) creates Floquet-engineered Weyl nodes, substantially amplifying the IGMC and making it controllable by the light’s phase/amplitude (Zhan et al., 2024).

Estimates show that experimentally relevant devices (e.g., 100 μm RuO$Z_{nm}^{\alpha\beta}$1 or Bi$Z_{nm}^{\alpha\beta}$2Se$Z_{nm}^{\alpha\beta}$3 films) can exhibit IGMC-induced voltages of tens of μV to mV, within capabilities of lock-in and pump–probe setups.

6. Nonlinear Response and Extensions

Recent work extends the IGMC concept into the nonlinear regime, where spin-rotation quantum geometric tensors govern second-order (quadratic in B) gyrotropic currents. The nonlinear IGMC arises from a further hierarchy of quantum geometric tensors, including the spin-rotation quantum metric and Berry curvature, as well as Zeeman symplectic and metric connections. These are selectively activated by particular point group and magnetic symmetries and allow for higher-order rectification and frequency-mixed responses, enriching the design space for quantum geometric transport (Chakraborti et al., 29 Jan 2026).

7. Implications for Material Design and Spintronics

The IGMC, by virtue of its intrinsic, dissipationless, and symmetry-selective nature, constitutes a robust probe and potential control mechanism for quantum geometric and topological phases:

• The persistence of IGMC in the absence of net magnetization enables identification and readout of “hidden” magnetic orders (e.g., altermagnetism, unconventional density waves).
• The displacement (quantum-metric) channel encodes high-fidelity symmetry fingerprints, surpassing conventional probes in discriminating between competing magnetic textures (Chakraborti et al., 21 Dec 2025).
• In superconductors and antiferromagnets, the IGMC serves as a noninvasive, ultrafast, and symmetry-selective readout of magnetic domain orientation and ordering vectors (He et al., 2019, Ghorai et al., 8 Jan 2026).
• The scalability and disorder-resilience (intrinsic origin) of the effect positions IGMC-based detection and control as a foundation for next-generation spintronics and optoelectronic devices exploiting quantum geometry (Chakraborti et al., 20 Aug 2025, Chakraborti et al., 29 Jan 2026).

In summary, IGMC is a fundamentally quantum-geometric transport phenomenon, emerging at the intersection of symmetry, topology, and band structure, and enabling direct access to exotic magnetic orders and topological states inaccessible by conventional means. Its theory and experimental realization span condensed matter, magnetism, and quantum materials science.