Zeeman Quantum Geometric Tensor

• Zeeman Quantum Geometric Tensor (ZQGT) is an extension of the conventional quantum geometric tensor that incorporates both momentum translations and spin (Zeeman) rotations.
• It decomposes into symmetric (quantum metric-like) and antisymmetric (Berry curvature-like) components, governing intrinsic gyrotropic magnetic responses in unconventional magnets.
• ZQGT enables the detection of hidden spin-split band structures and extends to non-Hermitian and mixed-state settings, offering a unified framework for novel magnetic phases.

The Zeeman Quantum Geometric Tensor (ZQGT) is a generalization of the conventional quantum geometric tensor, designed to encode the geometry arising from combined momentum translations and spin (Zeeman) rotations in Bloch bands. In contrast to the standard quantum metric and Berry curvature, which stem from infinitesimal momentum displacements alone, the ZQGT embodies the nontrivial quantum distances resulting from both momentum and spin-space rotations. This expanded geometric structure becomes essential in the study of “unconventional magnets”—materials with zero net magnetization but nontrivial, momentum-dependent spin splitting, where it governs intrinsic linear transport phenomena such as the intrinsic gyrotropic magnetic current (IGMC) in the presence of spin-orbit coupling, even when the conventional Berry curvature vanishes by symmetry (Chakraborti et al., 20 Aug 2025, Ezawa, 5 Dec 2025).

1. Mathematical Formulation and Decomposition

The ZQGT is defined via the quantum distance between two infinitesimally separated Bloch states under both a momentum shift $dk$ and spin (Zeeman) rotation $d\theta$: $$ds^2 = \| e^{-i (d\theta\cdot\hat{\sigma})/2} e^{-i\,dk\cdot\hat{r}} |u_{m,k}^{\xi}\rangle - |u_{m,k}^{\xi}\rangle \|^2.$$ Expanding to second order yields: $$ds^2 = \sum_{p\neq m} g^{ab}_{mp}\,dk_a\,dk_b + \frac{1}{4}\sum_{p,m}\Sigma^{ab}_{pm}\,d\theta_a\,d\theta_b + \frac{1}{2}\sum_{p\neq m} [z^{ba}_{mp} + z^{ba}_{pm}]\,d\theta_a\,dk_b,$$ where $g^{ab}_{mp}$ is the quantum metric, $\Sigma^{ab}_{pm}$ is the spin metric, and $z^{ab}_{mp}$ are matrix elements defining the ZQGT: $$z^{ab}_{mp} \equiv r^a_{mp} \sigma^b_{pm} = Q^{ab}_{mp} - \frac{i}{2}Z^{ab}_{mp}.$$ Here,

• $$Q^{ab}_{mp} = \frac{1}{2}(r^a_{mp} \sigma^b_{pm} + r^a_{pm} \sigma^b_{mp})$$ (real part, quantum-metric-like),
• $$Z^{ab}_{mp} = i [r^a_{mp} \sigma^b_{pm} - r^a_{pm} \sigma^b_{mp}]$$ (imaginary part, Berry-curvature-like). Symmetric and antisymmetric components in spatial indices are defined as $d\theta$0 and $d\theta$1.

The ZQGT is not constrained to be purely symmetric (quantum metric) or antisymmetric (Berry curvature); rather, it allows both symmetric Zeeman Berry curvature $d\theta$2 and antisymmetric Zeeman metric $d\theta$3 components, expanding the accessible geometric structure relative to conventional band geometry (Chakraborti et al., 20 Aug 2025).

2. Physical Consequences: Intrinsic Gyrotropic Magnetic Response

In two-dimensional unconventional magnets, the ZQGT governs the intrinsic gyrotropic magnetic conductivity (IGMC), which describes current responses linear in a time-dependent magnetic field $d\theta$4. The ZQGT leads to two distinct IGMC contributions: $d\theta$5 where $d\theta$6 is the “conduction-type” (Fermi surface) response, and $d\theta$7 is the “displacement-type” (Fermi sea) response. Both are intrinsic, i.e., independent of scattering time, and vanish if the ZQGT vanishes (e.g., when spin-orbit coupling is absent) (Chakraborti et al., 20 Aug 2025). The IGMC persists even when the conventional Berry curvature vanishes by symmetry, making it a unique probe of spin-split bands hidden from traditional geometrical diagnostics.

3. Prototypical Models and Symmetry Analysis

Three canonical two-dimensional models exemplify the role and structure of the ZQGT:

Case	Order Parameter $d\theta$8	Symmetry	IGMC Features
$d\theta$9 altermagnet	$$ds^2 = \| e^{-i (d\theta\cdot\hat{\sigma})/2} e^{-i\,dk\cdot\hat{r}} |u_{m,k}^{\xi}\rangle - |u_{m,k}^{\xi}\rangle \|^2.$$0	$$ds^2 = \| e^{-i (d\theta\cdot\hat{\sigma})/2} e^{-i\,dk\cdot\hat{r}} |u_{m,k}^{\xi}\rangle - |u_{m,k}^{\xi}\rangle \|^2.$$1 broken, $$ds^2 = \| e^{-i (d\theta\cdot\hat{\sigma})/2} e^{-i\,dk\cdot\hat{r}} |u_{m,k}^{\xi}\rangle - |u_{m,k}^{\xi}\rangle \|^2.$$2, $$ds^2 = \| e^{-i (d\theta\cdot\hat{\sigma})/2} e^{-i\,dk\cdot\hat{r}} |u_{m,k}^{\xi}\rangle - |u_{m,k}^{\xi}\rangle \|^2.$$3	$$ds^2 = \| e^{-i (d\theta\cdot\hat{\sigma})/2} e^{-i\,dk\cdot\hat{r}} |u_{m,k}^{\xi}\rangle - |u_{m,k}^{\xi}\rangle \|^2.$$4 (transverse), $$ds^2 = \| e^{-i (d\theta\cdot\hat{\sigma})/2} e^{-i\,dk\cdot\hat{r}} |u_{m,k}^{\xi}\rangle - |u_{m,k}^{\xi}\rangle \|^2.$$5 (longitudinal)
$$ds^2 = \| e^{-i (d\theta\cdot\hat{\sigma})/2} e^{-i\,dk\cdot\hat{r}} |u_{m,k}^{\xi}\rangle - |u_{m,k}^{\xi}\rangle \|^2.$$6-wave magnet	$$ds^2 = \| e^{-i (d\theta\cdot\hat{\sigma})/2} e^{-i\,dk\cdot\hat{r}} |u_{m,k}^{\xi}\rangle - |u_{m,k}^{\xi}\rangle \|^2.$$7	$$ds^2 = \| e^{-i (d\theta\cdot\hat{\sigma})/2} e^{-i\,dk\cdot\hat{r}} |u_{m,k}^{\xi}\rangle - |u_{m,k}^{\xi}\rangle \|^2.$$8 preserved, $$ds^2 = \| e^{-i (d\theta\cdot\hat{\sigma})/2} e^{-i\,dk\cdot\hat{r}} |u_{m,k}^{\xi}\rangle - |u_{m,k}^{\xi}\rangle \|^2.$$9 broken	$$ds^2 = \sum_{p\neq m} g^{ab}_{mp}\,dk_a\,dk_b + \frac{1}{4}\sum_{p,m}\Sigma^{ab}_{pm}\,d\theta_a\,d\theta_b + \frac{1}{2}\sum_{p\neq m} [z^{ba}_{mp} + z^{ba}_{pm}]\,d\theta_a\,dk_b,$$0 (both), $$ds^2 = \sum_{p\neq m} g^{ab}_{mp}\,dk_a\,dk_b + \frac{1}{4}\sum_{p,m}\Sigma^{ab}_{pm}\,d\theta_a\,d\theta_b + \frac{1}{2}\sum_{p\neq m} [z^{ba}_{mp} + z^{ba}_{pm}]\,d\theta_a\,dk_b,$$1
Mixed $$ds^2 = \sum_{p\neq m} g^{ab}_{mp}\,dk_a\,dk_b + \frac{1}{4}\sum_{p,m}\Sigma^{ab}_{pm}\,d\theta_a\,d\theta_b + \frac{1}{2}\sum_{p\neq m} [z^{ba}_{mp} + z^{ba}_{pm}]\,d\theta_a\,dk_b,$$2-wave altermagnet	$$ds^2 = \sum_{p\neq m} g^{ab}_{mp}\,dk_a\,dk_b + \frac{1}{4}\sum_{p,m}\Sigma^{ab}_{pm}\,d\theta_a\,d\theta_b + \frac{1}{2}\sum_{p\neq m} [z^{ba}_{mp} + z^{ba}_{pm}]\,d\theta_a\,dk_b,$$3	$$ds^2 = \sum_{p\neq m} g^{ab}_{mp}\,dk_a\,dk_b + \frac{1}{4}\sum_{p,m}\Sigma^{ab}_{pm}\,d\theta_a\,d\theta_b + \frac{1}{2}\sum_{p\neq m} [z^{ba}_{mp} + z^{ba}_{pm}]\,d\theta_a\,dk_b,$$4, $$ds^2 = \sum_{p\neq m} g^{ab}_{mp}\,dk_a\,dk_b + \frac{1}{4}\sum_{p,m}\Sigma^{ab}_{pm}\,d\theta_a\,d\theta_b + \frac{1}{2}\sum_{p\neq m} [z^{ba}_{mp} + z^{ba}_{pm}]\,d\theta_a\,dk_b,$$5, $$ds^2 = \sum_{p\neq m} g^{ab}_{mp}\,dk_a\,dk_b + \frac{1}{4}\sum_{p,m}\Sigma^{ab}_{pm}\,d\theta_a\,d\theta_b + \frac{1}{2}\sum_{p\neq m} [z^{ba}_{mp} + z^{ba}_{pm}]\,d\theta_a\,dk_b,$$6 broken	All IGMCs nonzero: $$ds^2 = \sum_{p\neq m} g^{ab}_{mp}\,dk_a\,dk_b + \frac{1}{4}\sum_{p,m}\Sigma^{ab}_{pm}\,d\theta_a\,d\theta_b + \frac{1}{2}\sum_{p\neq m} [z^{ba}_{mp} + z^{ba}_{pm}]\,d\theta_a\,dk_b,$$7, $$ds^2 = \sum_{p\neq m} g^{ab}_{mp}\,dk_a\,dk_b + \frac{1}{4}\sum_{p,m}\Sigma^{ab}_{pm}\,d\theta_a\,d\theta_b + \frac{1}{2}\sum_{p\neq m} [z^{ba}_{mp} + z^{ba}_{pm}]\,d\theta_a\,dk_b,$$8, $$ds^2 = \sum_{p\neq m} g^{ab}_{mp}\,dk_a\,dk_b + \frac{1}{4}\sum_{p,m}\Sigma^{ab}_{pm}\,d\theta_a\,d\theta_b + \frac{1}{2}\sum_{p\neq m} [z^{ba}_{mp} + z^{ba}_{pm}]\,d\theta_a\,dk_b,$$9, $g^{ab}_{mp}$0

Here, $g^{ab}_{mp}$1 is the general Hamiltonian, with Rashba SOC ($g^{ab}_{mp}$2) nonperturbatively included. The symmetry properties of $g^{ab}_{mp}$3 and the interplay with SOC control which ZQGT tensor components survive after Brillouin zone integration, dictating which IGMC components are physically observable (Chakraborti et al., 20 Aug 2025).

4. Analytical Structure in the Two-Band Model

The ZQGT in two-band Hamiltonians $g^{ab}_{mp}$4 admits compact forms. Defining the band eigenstates in terms of the normalized vector $g^{ab}_{mp}$5, with spherical angles $g^{ab}_{mp}$6, the key quantities are:

• Quantum metric: $g^{ab}_{mp}$7
• Berry curvature: $g^{ab}_{mp}$8
• Zeeman Berry curvature: $g^{ab}_{mp}$9
• Zeeman quantum metric: $\Sigma^{ab}_{pm}$0

These expressions clarify how the ZQGT is fully specified by the momentum-space spin texture and its gradients, and how it reduces to the standard quantum geometric tensor in the absence of spin rotations (Ezawa, 5 Dec 2025). The cross-couplings, such as the Zeeman metric and Berry curvature, underlie novel linear responses—including spin and charge cross-conductivities and field-driven spin polarizations.

5. Experimental Relevance and Diagnostics

The ZQGT provides a symmetry-sensitive probe into materials where traditional Berry curvature and quantum metric analysis fail to reveal active band geometry. In “zero-net-magnetization” unconventional magnets, where momentum-dependent spin splitting is protected by symmetry, the IGMC driven by the ZQGT yields measurable effects such as Hall voltages and conduction/displacement currents under oscillating magnetic fields. For instance, in $\Sigma^{ab}_{pm}$1 (a $\Sigma^{ab}_{pm}$2 altermagnet), parameters $\Sigma^{ab}_{pm}$3 eV, $\Sigma^{ab}_{pm}$4 eV, $\Sigma^{ab}_{pm}$5 eV, a $\Sigma^{ab}_{pm}$6 device with $\Sigma^{ab}_{pm}$7 G and $\Sigma^{ab}_{pm}$8 Hz yields an estimated Hall voltage of approximately $\Sigma^{ab}_{pm}$9 mV due to the conduction IGMC. Displacement IGMCs are similarly accessible in the THz regime. Additional candidate materials with relevant symmetries include $z^{ab}_{mp}$0 (mixed $z^{ab}_{mp}$1-wave) and $z^{ab}_{mp}$2 (potential $z^{ab}_{mp}$3- or mixed-wave) (Chakraborti et al., 20 Aug 2025).

6. Generalizations: Non-Hermitian and Mixed-State Quantum Geometry

The framework of the ZQGT extends to non-Hermitian Hamiltonians (with right and left eigenstates $z^{ab}_{mp}$4) via a non-Hermitian quantum geometric tensor

$z^{ab}_{mp}$5

where $z^{ab}_{mp}$6. The real and imaginary parts again encode generalized quantum metric and Berry curvature, and in two-band models the geometric content is fully determined by the right eigenstate spin-texture $z^{ab}_{mp}$7 (Ezawa, 5 Dec 2025).

For mixed states (finite temperature or open systems), quantum information geometry identifies the Uhlmann quantum geometric tensor and Fisher information metric as the mixed-state analogues: $z^{ab}_{mp}$8 with $z^{ab}_{mp}$9 the symmetric logarithmic derivative, $$z^{ab}_{mp} \equiv r^a_{mp} \sigma^b_{pm} = Q^{ab}_{mp} - \frac{i}{2}Z^{ab}_{mp}.$$0 the quantum Fisher information metric, and $$z^{ab}_{mp} \equiv r^a_{mp} \sigma^b_{pm} = Q^{ab}_{mp} - \frac{i}{2}Z^{ab}_{mp}.$$1 the mean Uhlmann curvature. In the pure-state limit, this reduces to the standard quantum metric and Berry curvature, ensuring continuity of geometric diagnostics (Ezawa, 5 Dec 2025).

7. Significance in Quantum Materials and Outlook

The ZQGT framework crystallizes how combined momentum and spin-space geometry drive electromagnetic cross-responses in unconventional magnets, offering access to hidden band-structure features not visible via conventional geometric methods. Its tensor structure, symmetry properties, and explicit role in linear response facilitate the identification and classification of novel magnetic phases—distinguishing, for example, between $$z^{ab}_{mp} \equiv r^a_{mp} \sigma^b_{pm} = Q^{ab}_{mp} - \frac{i}{2}Z^{ab}_{mp}.$$2, $$z^{ab}_{mp} \equiv r^a_{mp} \sigma^b_{pm} = Q^{ab}_{mp} - \frac{i}{2}Z^{ab}_{mp}.$$3, and mixed $$z^{ab}_{mp} \equiv r^a_{mp} \sigma^b_{pm} = Q^{ab}_{mp} - \frac{i}{2}Z^{ab}_{mp}.$$4-wave order. The generalization to non-Hermitian and mixed-state settings promotes quantum geometry as a unifying theme in topological materials, spintronics, and quantum information. Ongoing research continues to explore universal features in $$z^{ab}_{mp} \equiv r^a_{mp} \sigma^b_{pm} = Q^{ab}_{mp} - \frac{i}{2}Z^{ab}_{mp}.$$5-, $$z^{ab}_{mp} \equiv r^a_{mp} \sigma^b_{pm} = Q^{ab}_{mp} - \frac{i}{2}Z^{ab}_{mp}.$$6-, $$z^{ab}_{mp} \equiv r^a_{mp} \sigma^b_{pm} = Q^{ab}_{mp} - \frac{i}{2}Z^{ab}_{mp}.$$7-, $$z^{ab}_{mp} \equiv r^a_{mp} \sigma^b_{pm} = Q^{ab}_{mp} - \frac{i}{2}Z^{ab}_{mp}.$$8-, and $$z^{ab}_{mp} \equiv r^a_{mp} \sigma^b_{pm} = Q^{ab}_{mp} - \frac{i}{2}Z^{ab}_{mp}.$$9-wave magnets through analytically tractable two-band formulas and ab initio studies (Chakraborti et al., 20 Aug 2025, Ezawa, 5 Dec 2025).