Zeeman Quantum Geometry: Spin-Momentum Coupling

• Zeeman quantum geometry is a framework extending conventional quantum geometry by incorporating spin rotations alongside momentum translations, revealing novel transport and optical phenomena.
• It introduces the Zeeman quantum geometric tensor with symmetric and antisymmetric components that systematically classify electromagnetic responses such as the spin planar Hall effect and magnetononlinear Hall effect.
• The approach applies to unconventional magnets and can be extended to non-Hermitian and mixed-state systems, with experimental signatures including quantized IGMC responses and resonant spin phenomena.

Zeeman quantum geometry is a generalization of conventional quantum geometry that unifies the momentum-space structure of Bloch electrons with spin degrees of freedom, enabling a geometrically controlled description of transport and optical responses in quantum materials where spin rotations and momentum translations are intertwined. Central to this framework is the Zeeman quantum geometric tensor (ZQGT), which encodes the joint response of Bloch states to infinitesimal momentum translations and spin rotations, producing novel symmetry-determined contributions to phenomena such as the intrinsic gyrotropic magnetic current (IGMC) and the spin planar Hall effect. Experimental and theoretical developments have linked Zeeman quantum geometry to measurable responses in unconventional magnets, especially those with momentum-dependent spin splitting and vanishing net magnetization.

1. Foundations of Zeeman Quantum Geometry

The ZQGT extends the conventional quantum geometric tensor—which measures the infinitesimal “distance” in Hilbert space between Bloch states at neighboring momenta—by additionally incorporating the response to infinitesimal spin rotations. For a cell-periodic Bloch eigenstate $\lvert u_{m\mathbf k}^\xi\rangle$, two displacement generators are introduced:

• Momentum translation: $$U_{d\mathbf k}=e^{-i d\mathbf k \cdot \hat{\mathbf r}}$$
• Spin rotation: $$U_{d\boldsymbol\theta}=e^{-i \frac{d\boldsymbol\theta \cdot \hat{\boldsymbol\sigma}}{2}}$$

The quantum distance is given by: $$ds^2 = \big\lVert\,U_{d\theta}\,U_{d\mathbf k}\,\lvert u_{m\mathbf k}^\xi\rangle - \lvert u_{m\mathbf k}^\xi\rangle\big\rVert^2$$ which, expanded to second order, reveals cross terms: $$ds^2 = \sum_{p\neq m} g_{mp}^{ab}\,dk_a\,dk_b +\frac14\sum_{p\neq m}\Sigma_{mp}^{ab}\,d\theta_a\,d\theta_b +\sum_{p\neq m}\left(z_{mp}^{ba}+z_{pm}^{ba}\right)d\theta_a\,dk_b$$ The crucial cross term,

$z_{mp}^{ab}=r_{mp}^a\,\sigma_{pm}^b$

with $$r_{mp}^a = \langle u_{m\mathbf{k}}^\xi|i\partial_{k_a}|u_{p\mathbf{k}}^\xi\rangle$$ (position operator) and $$\sigma_{pm}^b = \langle u_{p\mathbf{k}}^\xi|\hat\sigma_b|u_{m\mathbf{k}}^\xi\rangle$$ (spin operator), defines the ZQGT. Its real part yields the Zeeman quantum metric $Q_{mp}^{ab}$, and its imaginary part the Zeeman Berry curvature $Z_{mp}^{ab}$: $$U_{d\mathbf k}=e^{-i d\mathbf k \cdot \hat{\mathbf r}}$$0 The ZQGT contains both symmetric and antisymmetric components in the composite indices $$U_{d\mathbf k}=e^{-i d\mathbf k \cdot \hat{\mathbf r}}$$1, with new structures absent in conventional QGT, such as $$U_{d\mathbf k}=e^{-i d\mathbf k \cdot \hat{\mathbf r}}$$2 and $$U_{d\mathbf k}=e^{-i d\mathbf k \cdot \hat{\mathbf r}}$$3 (Chakraborti et al., 20 Aug 2025, Ezawa, 5 Dec 2025).

2. Decomposition and Physical Meaning

By expanding the overlap of $$U_{d\mathbf k}=e^{-i d\mathbf k \cdot \hat{\mathbf r}}$$4, the mixed term $$U_{d\mathbf k}=e^{-i d\mathbf k \cdot \hat{\mathbf r}}$$5 gives rise to distinct symmetric and antisymmetric parts under $$U_{d\mathbf k}=e^{-i d\mathbf k \cdot \hat{\mathbf r}}$$6:

• $$U_{d\mathbf k}=e^{-i d\mathbf k \cdot \hat{\mathbf r}}$$7 (symmetric)
• $$U_{d\mathbf k}=e^{-i d\mathbf k \cdot \hat{\mathbf r}}$$8 (antisymmetric)

In Zeeman geometry, novel cross-symmetric and cross-antisymmetric sectors appear: $$U_{d\mathbf k}=e^{-i d\mathbf k \cdot \hat{\mathbf r}}$$9 The antisymmetric component $$U_{d\boldsymbol\theta}=e^{-i \frac{d\boldsymbol\theta \cdot \hat{\boldsymbol\sigma}}{2}}$$0 generalizes Berry curvature into spin-momentum space, while the symmetric $$U_{d\boldsymbol\theta}=e^{-i \frac{d\boldsymbol\theta \cdot \hat{\boldsymbol\sigma}}{2}}$$1 extends the quantum metric to include spin response. These structures govern linear and nonlinear electromagnetic responses beyond those predicted by conventional geometry (Chakraborti et al., 20 Aug 2025, Xiang et al., 3 Oct 2025, Ezawa, 5 Dec 2025).

3. Zeeman Quantum Geometry in Electromagnetic and Magnetotransport Response

The ZQGT directly dictates transport coefficients in external fields that couple to both spin and orbital degrees of freedom. Under an oscillating magnetic field $$U_{d\boldsymbol\theta}=e^{-i \frac{d\boldsymbol\theta \cdot \hat{\boldsymbol\sigma}}{2}}$$2, coupling as $$U_{d\boldsymbol\theta}=e^{-i \frac{d\boldsymbol\theta \cdot \hat{\boldsymbol\sigma}}{2}}$$3, the intrinsic IGMC is: $$U_{d\boldsymbol\theta}=e^{-i \frac{d\boldsymbol\theta \cdot \hat{\boldsymbol\sigma}}{2}}$$4

$$U_{d\boldsymbol\theta}=e^{-i \frac{d\boldsymbol\theta \cdot \hat{\boldsymbol\sigma}}{2}}$$5

Here, $$U_{d\boldsymbol\theta}=e^{-i \frac{d\boldsymbol\theta \cdot \hat{\boldsymbol\sigma}}{2}}$$6 (conduction) is a Fermi-surface integral of $$U_{d\boldsymbol\theta}=e^{-i \frac{d\boldsymbol\theta \cdot \hat{\boldsymbol\sigma}}{2}}$$7, and $$U_{d\boldsymbol\theta}=e^{-i \frac{d\boldsymbol\theta \cdot \hat{\boldsymbol\sigma}}{2}}$$8 (displacement) is a Fermi-sea integral of $$U_{d\boldsymbol\theta}=e^{-i \frac{d\boldsymbol\theta \cdot \hat{\boldsymbol\sigma}}{2}}$$9. Both terms are intrinsic and independent of relaxation time $$ds^2 = \big\lVert\,U_{d\theta}\,U_{d\mathbf k}\,\lvert u_{m\mathbf k}^\xi\rangle - \lvert u_{m\mathbf k}^\xi\rangle\big\rVert^2$$0.

In the context of bilinear electromagnetic responses such as the spin planar Hall effect (PHE) and magnetononlinear Hall effect (MNHE), the Zeeman Berry curvature and quantum metric organize the linear and nonlinear conductivity tensors:

• The Zeeman Berry curvature dipole controls the extrinsic spin PHE ($$ds^2 = \big\lVert\,U_{d\theta}\,U_{d\mathbf k}\,\lvert u_{m\mathbf k}^\xi\rangle - \lvert u_{m\mathbf k}^\xi\rangle\big\rVert^2$$1-scaled).
• The Zeeman quantum-metric dipole underlies the intrinsic spin MNHE ($$ds^2 = \big\lVert\,U_{d\theta}\,U_{d\mathbf k}\,\lvert u_{m\mathbf k}^\xi\rangle - \lvert u_{m\mathbf k}^\xi\rangle\big\rVert^2$$2-scaled). This formalism enables a quantum-geometric classification of all bilinear charge and spin responses under combined electric and magnetic fields (Xiang et al., 3 Oct 2025).

4. Application to Unconventional Magnets and $$ds^2 = \big\lVert\,U_{d\theta}\,U_{d\mathbf k}\,\lvert u_{m\mathbf k}^\xi\rangle - \lvert u_{m\mathbf k}^\xi\rangle\big\rVert^2$$3-wave Magnets

Zeeman quantum geometry is crucial in describing unconventional 2D magnets with zero net magnetization but momentum-dependent spin splitting, including:

• $$ds^2 = \big\lVert\,U_{d\theta}\,U_{d\mathbf k}\,\lvert u_{m\mathbf k}^\xi\rangle - \lvert u_{m\mathbf k}^\xi\rangle\big\rVert^2$$4 altermagnets (TRS breaking)
• $$ds^2 = \big\lVert\,U_{d\theta}\,U_{d\mathbf k}\,\lvert u_{m\mathbf k}^\xi\rangle - \lvert u_{m\mathbf k}^\xi\rangle\big\rVert^2$$5-wave magnets (TRS preserved, inversion symmetry broken)
• Mixed $$ds^2 = \big\lVert\,U_{d\theta}\,U_{d\mathbf k}\,\lvert u_{m\mathbf k}^\xi\rangle - \lvert u_{m\mathbf k}^\xi\rangle\big\rVert^2$$6-wave altermagnets (both TRS and inversion broken)

Key IGMC response features for these magnets are:

Model	Nonzero IGMC Channels	ZQGT Contributions
$$ds^2 = \big\lVert\,U_{d\theta}\,U_{d\mathbf k}\,\lvert u_{m\mathbf k}^\xi\rangle - \lvert u_{m\mathbf k}^\xi\rangle\big\rVert^2$$7	$$ds^2 = \big\lVert\,U_{d\theta}\,U_{d\mathbf k}\,\lvert u_{m\mathbf k}^\xi\rangle - \lvert u_{m\mathbf k}^\xi\rangle\big\rVert^2$$8 (transverse), $$ds^2 = \big\lVert\,U_{d\theta}\,U_{d\mathbf k}\,\lvert u_{m\mathbf k}^\xi\rangle - \lvert u_{m\mathbf k}^\xi\rangle\big\rVert^2$$9 (longitudinal)	$$ds^2 = \sum_{p\neq m} g_{mp}^{ab}\,dk_a\,dk_b +\frac14\sum_{p\neq m}\Sigma_{mp}^{ab}\,d\theta_a\,d\theta_b +\sum_{p\neq m}\left(z_{mp}^{ba}+z_{pm}^{ba}\right)d\theta_a\,dk_b$$0, $$ds^2 = \sum_{p\neq m} g_{mp}^{ab}\,dk_a\,dk_b +\frac14\sum_{p\neq m}\Sigma_{mp}^{ab}\,d\theta_a\,d\theta_b +\sum_{p\neq m}\left(z_{mp}^{ba}+z_{pm}^{ba}\right)d\theta_a\,dk_b$$1
$$ds^2 = \sum_{p\neq m} g_{mp}^{ab}\,dk_a\,dk_b +\frac14\sum_{p\neq m}\Sigma_{mp}^{ab}\,d\theta_a\,d\theta_b +\sum_{p\neq m}\left(z_{mp}^{ba}+z_{pm}^{ba}\right)d\theta_a\,dk_b$$2-wave	$$ds^2 = \sum_{p\neq m} g_{mp}^{ab}\,dk_a\,dk_b +\frac14\sum_{p\neq m}\Sigma_{mp}^{ab}\,d\theta_a\,d\theta_b +\sum_{p\neq m}\left(z_{mp}^{ba}+z_{pm}^{ba}\right)d\theta_a\,dk_b$$3, $$ds^2 = \sum_{p\neq m} g_{mp}^{ab}\,dk_a\,dk_b +\frac14\sum_{p\neq m}\Sigma_{mp}^{ab}\,d\theta_a\,d\theta_b +\sum_{p\neq m}\left(z_{mp}^{ba}+z_{pm}^{ba}\right)d\theta_a\,dk_b$$4 (transverse)	Off-diagonal $$ds^2 = \sum_{p\neq m} g_{mp}^{ab}\,dk_a\,dk_b +\frac14\sum_{p\neq m}\Sigma_{mp}^{ab}\,d\theta_a\,d\theta_b +\sum_{p\neq m}\left(z_{mp}^{ba}+z_{pm}^{ba}\right)d\theta_a\,dk_b$$5, vanishing $$ds^2 = \sum_{p\neq m} g_{mp}^{ab}\,dk_a\,dk_b +\frac14\sum_{p\neq m}\Sigma_{mp}^{ab}\,d\theta_a\,d\theta_b +\sum_{p\neq m}\left(z_{mp}^{ba}+z_{pm}^{ba}\right)d\theta_a\,dk_b$$6
Mixed $$ds^2 = \sum_{p\neq m} g_{mp}^{ab}\,dk_a\,dk_b +\frac14\sum_{p\neq m}\Sigma_{mp}^{ab}\,d\theta_a\,d\theta_b +\sum_{p\neq m}\left(z_{mp}^{ba}+z_{pm}^{ba}\right)d\theta_a\,dk_b$$7-wave	$$ds^2 = \sum_{p\neq m} g_{mp}^{ab}\,dk_a\,dk_b +\frac14\sum_{p\neq m}\Sigma_{mp}^{ab}\,d\theta_a\,d\theta_b +\sum_{p\neq m}\left(z_{mp}^{ba}+z_{pm}^{ba}\right)d\theta_a\,dk_b$$8, $$ds^2 = \sum_{p\neq m} g_{mp}^{ab}\,dk_a\,dk_b +\frac14\sum_{p\neq m}\Sigma_{mp}^{ab}\,d\theta_a\,d\theta_b +\sum_{p\neq m}\left(z_{mp}^{ba}+z_{pm}^{ba}\right)d\theta_a\,dk_b$$9, $z_{mp}^{ab}=r_{mp}^a\,\sigma_{pm}^b$0, $z_{mp}^{ab}=r_{mp}^a\,\sigma_{pm}^b$1	Symmetric/antisymmetric ZQGT pieces

These responses occur even when conventional Berry curvature vanishes, and are fully classified by the underlying symmetries and momentum- and spin-space structure of the ZQGT (Chakraborti et al., 20 Aug 2025, Ezawa, 5 Dec 2025).

In the broader family of $z_{mp}^{ab}=r_{mp}^a\,\sigma_{pm}^b$2-wave magnets ($z_{mp}^{ab}=r_{mp}^a\,\sigma_{pm}^b$3 = $z_{mp}^{ab}=r_{mp}^a\,\sigma_{pm}^b$4, $z_{mp}^{ab}=r_{mp}^a\,\sigma_{pm}^b$5, $z_{mp}^{ab}=r_{mp}^a\,\sigma_{pm}^b$6, $z_{mp}^{ab}=r_{mp}^a\,\sigma_{pm}^b$7, $z_{mp}^{ab}=r_{mp}^a\,\sigma_{pm}^b$8), Zeeman quantum geometry governs Hall and planar Hall conductivities, tunneling magnetoresistance (TMR), and spin Drude responses, with effects determined by the symmetry order $z_{mp}^{ab}=r_{mp}^a\,\sigma_{pm}^b$9 and structure of the off-diagonal ZQGT (Ezawa, 5 Dec 2025).

5. Classification of Zeeman Quantum Geometry in Response Functions

Zeeman quantum geometry organizes multipole moments—such as dipole and quadrupole terms—of the ZQGT, capturing the full hierarchy of intrinsic and extrinsic electromagnetic responses:

• Zeeman Berry curvature dipole $$r_{mp}^a = \langle u_{m\mathbf{k}}^\xi|i\partial_{k_a}|u_{p\mathbf{k}}^\xi\rangle$$0 extrinsic spin PHE
• Zeeman quantum metric dipole $$r_{mp}^a = \langle u_{m\mathbf{k}}^\xi|i\partial_{k_a}|u_{p\mathbf{k}}^\xi\rangle$$1 intrinsic spin MNHE
• Quantum metric quadrupole corrections arise even in the ordinary Hall effect.

Explicitly, for Bloch bands $$r_{mp}^a = \langle u_{m\mathbf{k}}^\xi|i\partial_{k_a}|u_{p\mathbf{k}}^\xi\rangle$$2,

$$r_{mp}^a = \langle u_{m\mathbf{k}}^\xi|i\partial_{k_a}|u_{p\mathbf{k}}^\xi\rangle$$3

$$r_{mp}^a = \langle u_{m\mathbf{k}}^\xi|i\partial_{k_a}|u_{p\mathbf{k}}^\xi\rangle$$4

The conductivity tensors are then assembled from Fermi-surface or Fermi-sea integrals over these ZQGT components, weighted by group velocities and energy denominators appropriate to the physical response under consideration (Xiang et al., 3 Oct 2025, Chakraborti et al., 20 Aug 2025).

6. Experimental Signatures and Observable Consequences

Zeeman quantum geometry predicts distinct experimental signatures across materials such as RuO$$r_{mp}^a = \langle u_{m\mathbf{k}}^\xi|i\partial_{k_a}|u_{p\mathbf{k}}^\xi\rangle$$5, CrSb, MnTe, and the surfaces of 3D topological insulators. For typical parameters, IGMC voltages are in the mV range for low-frequency magnetic drives and can be controlled by symmetry tuning (e.g., varying the parity or wave admixture of the altermagnetic order). In planar Hall effect geometries, the Zeeman Berry curvature dipole yields quantized plateau-like signals robust to changes in chemical potential for realistic device parameters.

Experiments sensitive to geometric oscillations and “geometric dephasing” in spin resonance probe the underlying quantum metric $$r_{mp}^a = \langle u_{m\mathbf{k}}^\xi|i\partial_{k_a}|u_{p\mathbf{k}}^\xi\rangle$$6 and curvature $$r_{mp}^a = \langle u_{m\mathbf{k}}^\xi|i\partial_{k_a}|u_{p\mathbf{k}}^\xi\rangle$$7 on the Bloch sphere. These geometric responses are directly reflected in measurable quantities such as pumped population probabilities in NMR/ESR and qubit setups, with undamped oscillations arising far beyond standard resonance regimes (Song et al., 2024, Chakraborti et al., 20 Aug 2025).

7. Extensions: Non-Hermitian, Mixed-State, and Quantum Information Geometry

Zeeman quantum geometry generalizes naturally to:

• Non-Hermitian band structures, by replacing inner products with biorthogonal pairs and defining generalized Berry connection and metric.
• Quantum information geometry, via the Uhlmann/Bures metric for density matrices, yielding the quantum Fisher metric and mean Uhlmann curvature as generalized quantum geometric tensors. In the $$r_{mp}^a = \langle u_{m\mathbf{k}}^\xi|i\partial_{k_a}|u_{p\mathbf{k}}^\xi\rangle$$8 limit, the Uhlmann construction reduces to conventional quantum metric and Berry curvature, demonstrating the unifying nature of geometric approaches in both pure and mixed-state quantum systems (Ezawa, 5 Dec 2025).