Spin-Rotation Quantum Geometric Tensor

• Spin-rotation quantum geometric tensor is a measure that unifies quantum distance and Berry curvature under spin rotations, capturing essential geometric and topological features.
• It explains how spin fluctuations and symmetry operations yield measurable nonlinear gyrotropic responses and signature phase transitions in advanced materials.
• Experimental protocols such as spin-polarized STM, parametric qubit driving, and photonic lattice interferometry enable direct mapping of its metric and Berry curvature components.

The spin-rotation quantum geometric tensor (SRQGT) is a fundamental tensorial quantity encoding the geometric and topological properties of quantum states or operators under infinitesimal spin rotations. It unifies aspects of quantum distance (metric) and geometric phase (Berry curvature) in spin systems, and plays a central role in emergent nonlinear and gyrotropic phenomena in magnetoelectric and topological materials, quantum liquids, cold atoms, and photonic lattices. The SRQGT generalizes the standard quantum geometric tensor (QGT) to mixed parameter manifolds involving both spin rotation and momentum, giving rise to a rich interplay of symmetry, band topology, and measurable physical responses.

1. Formal Definition and Structure

The SRQGT extends the conventional (momentum-derivative) quantum geometric tensor to include infinitesimal spin rotations. For a family of Bloch spinors $|u_{mk}\rangle$, the Hilbert-space distance between $|u_{mk}\rangle$ and its neighbor under momentum translation $dk$ and spin rotation $d\theta$ is given by (Chakraborti et al., 29 Jan 2026): $$ds^2 = \sum_{p\neq m} r_{mp}^a r_{pm}^b\,dk_a\,dk_b + \tfrac{1}{4}\sum_{p\neq m} \sigma_{mp}^a \sigma_{pm}^b\,d\theta_a\,d\theta_b + \sum_{p\neq m} r_{mp}^a \sigma_{pm}^b\,dk_a\,d\theta_b$$ where $r_{mp}^a = \langle u_m|i\partial_{k_a}|u_p\rangle$ and $\sigma_{mp}^a = \langle u_m|\sigma^a|u_p\rangle$. The term

$S_{mp}^{ab} = \sigma_{mp}^a \sigma_{pm}^b$

defines the SRQGT. Its symmetric real part, $\mathscr{R}_{mp}^{ab}$, is the spin-rotation quantum metric; its antisymmetric imaginary part, $\Lambda_{mp}^{ab}$, is the spin-rotation Berry curvature: $$\mathscr{R}_{mp}^{ab} = \tfrac{1}{2}\big(\sigma_{mp}^a \sigma_{pm}^b + \sigma_{pm}^a \sigma_{mp}^b \big),\qquad \Lambda_{mp}^{ab} = i\big( \sigma_{mp}^a \sigma_{pm}^b - \sigma_{pm}^a \sigma_{mp}^b \big)$$ Thus, in direct analogy with the standard QGT $g_{ab}$, the SRQGT is $$g_{ab}^{sr} = \mathscr{R}_{ab} - (i/2) \Lambda_{ab}$$ (Chakraborti et al., 29 Jan 2026).

For a generic quantum state $|\psi(\lambda)\rangle$ smoothly parameterized by $\lambda^\mu$, the conventional QGT reads (Lu et al., 2024, Yu et al., 2018): $$Q_{\mu\nu}(\lambda) = \langle \partial_\mu\psi | (1 - |\psi\rangle\langle\psi|) | \partial_\nu\psi\rangle = g_{\mu\nu}(\lambda) - \tfrac{i}{2}\Omega_{\mu\nu}(\lambda)$$ where $g_{\mu\nu}$ is the Fubini-Study metric and $\Omega_{\mu\nu}$ is the Berry curvature.

2. Geometric Interpretation and Relation to Spin-Fluctuation Covariances

In spin systems, the SRQGT is directly connected to the second moments of spin operators for families of states or evolution operators parameterized by spin-rotation angles. If $\rho$ is a quantum state and $S_\alpha$ are spin operators, the operator quantum geometric tensor (OQGT) for a unitary evolution $U(\Omega) = \exp(-i\Omega\cdot S)$ is (Lu et al., 2010): $$Q_{\alpha\beta}(\Omega) = \langle S_\alpha S_\beta\rangle_\rho - \langle S_\alpha\rangle_\rho \langle S_\beta\rangle_\rho$$ with metric and curvature

$$g_{\alpha\beta} = \text{Re } Q_{\alpha\beta},\qquad F_{\alpha\beta} = 2\,\text{Im } Q_{\alpha\beta} = -2\varepsilon_{\alpha\beta\gamma} \langle S_\gamma\rangle_\rho$$

In spin-1 systems, the spin-fluctuation tensor $T_{ij}$ encodes both metric and Berry curvature contributions, and parallel transport of the spin vector acquires non-Abelian geometric phases associated with $SO(3)$ holonomy, corresponding to the “generalized solid angle” integrated from $T_{ij}$ (Bharath, 2017).

3. Physical Consequences: Nonlinear Gyrotropic Magnetotransport

The SRQGT manifests directly in nonlinear magnetic phenomena. In two-dimensional systems under time-periodic magnetic driving, the second-harmonic gyrotropic current is governed by (Chakraborti et al., 29 Jan 2026):

• The diagonal (intraband) channel dictated by the SRQGT $(\mathscr{R}, \Lambda)$:

$$\chi_{abc}^{(d)} \sim \sum_{mp} v_{mp}^a \left[ \mathscr{R}_{mp}^{bc} - \frac{i}{2}\Lambda_{mp}^{bc} \right]$$

The metric $\mathscr{R}^{bc}$ drives the “conduction” channel, i.e., the nonlinear magnetic analogue of the quantum metric dipole, while the curvature $\Lambda^{bc}$ drives the “displacement” or circular dichroic second-harmonic response.

• The off-diagonal (interband) channel governed by symplectic and metric Zeeman connections built from Berry and spin connections.

Symmetry analysis reveals that the activation or suppression of these channels depends on the underlying spatial and time-reversal symmetries. For example, in hexagonally warped TI surfaces, the displacement (Berry curvature) channel survives and is sharply peaked at the Dirac point, whereas in PT-symmetric antiferromagnets, only the conduction (metric) channel is active (Chakraborti et al., 29 Jan 2026).

4. Experimental Probing and Measurement Protocols

The SRQGT and the closely related QGT can be measured by various advanced techniques:

• Spin-polarized STM: By inducing Friedel oscillations with a single magnetic impurity and measuring spin-resolved local density of states, one can reconstruct the spin texture $\mathbf{S}(\mathbf{k})$, then numerically differentiate to obtain $g_{ij}(\mathbf{k})$ and $\Omega_{ij}(\mathbf{k})$ across the Brillouin zone (Zhang et al., 23 Jan 2025). This directly yields the QGT, with the spin-rotation content emerging from the structure of $\mathbf{S}(\mathbf{k})$.
• Parametric qubit driving: In NV-center and two-qubit systems, coherent Rabi oscillations induced by small amplitude and phase modulations allow measurement of both quantum metric and Berry curvature as a function of parameter-space angles $(\theta, \phi)$. The protocol involves extracting Rabi frequencies under linear and elliptical modulation and inferring the QGT components from the oscillation amplitudes (Yu et al., 2018).
• Photonic and polaritonic lattices: In planar cavities or coupled resonator arrays, Stokes/ pseudospin measurements combined with interferometry provide a full mapping of the QGT via momentum-space derivatives of the measured spin or polarization texture (Bleu et al., 2017).
• Quantum spin-fluctuation ellipsoid tomography: In spin-1 cold atom or solid-state platforms, quantum state tomography of the second moment tensor $T_{ij}$ before and after closed loops in the Bloch ball yields the holonomy corresponding to the geometric phase acquired by spin-rotation, including non-Abelian features (Bharath, 2017).

5. SRQGT and Topological Phase Transitions

The SRQGT provides unambiguous geometric and thermodynamic signatures of topological phase transitions:

• In the Kitaev honeycomb model, the $xy$-component of the Berry curvature tracks Chern number jumps and phase boundaries, while the $zz$-component of the Fubini-Study metric diverges at the multicritical point where three phases meet (Lu et al., 2024).
• The connection between curvature, susceptibility derivatives, and topological invariants allows the identification of topological transitions via changes in the SRQGT, robust to local perturbations.
• At finite temperature, the quantum geometric tensor generalizes to include the Uhlmann curvature and Fisher–Rao metric, enabling the study of quantum-to-classical crossover in geometric responses (Lu et al., 2024).

Physical Quantity	Geometric Origin	Observable Signature
Berry curvature $\Omega_{ij}$	SRQGT curvature ($\Lambda$)	Chern number jumps, nonlinear Hall/Chern signals
Quantum metric $g_{ij}$	SRQGT metric ($\mathscr{R}$)	Metric dips/peaks at phase boundaries
Uhlmann curvature	Mixed-state curvature	Peak/dip in thermally broadened response
Fisher–Rao metric	Classical fluctuation metric	Isotropic thermal broadening at finite $T$

6. Theoretical Extensions and Generalizations

The framework of the SRQGT is extensible to a variety of settings:

• Mixed parameter spaces: The inclusion of both momentum and spin rotation parameters leads to mixed QGTs, e.g., Zeeman QGT terms, which are essential in separating different contributions to linear and nonlinear magnetic responses (Chakraborti et al., 29 Jan 2026).
• Higher spin systems: In spin-1 and above, treatments via the spin-fluctuation tensor and its transport provide access to both Abelian (solid angle) and non-Abelian geometric phases, governed by holonomies in $SO(3)$ (Bharath, 2017).
• Operator-geometry perspective: The OQGT unifies the sensitivity of unitary evolution under multiparameter perturbations, with the SRQGT being a special case for rotation angle manifolds; this provides a Loschmidt-echo type metric for dynamical phase transitions (Lu et al., 2010).

7. Outlook and Material Design Implications

Isolation of the SRQGT as a distinct source of nonlinear and gyrotropic responses in condensed matter and photonic systems suggests practical avenues for device engineering:

• Selectively enhancing or suppressing SRQGT-driven effects by symmetry breaking (e.g., by hexagonal warping, tilt, or staggered magnetization).
• Maximizing nonlinear optical rotation or dichroic signals by chemical potential tuning to Dirac points or band extrema (Chakraborti et al., 29 Jan 2026).
• Utilizing the distinct geometric separation between Zeeman and spin-rotation QGTs as a “geometric dial” in magnetic photodetectors and THz devices, separate from conventional orbital Berry curvature channels. A plausible implication is that measurement of the SRQGT in engineered heterostructures may enable controlled nonreciprocal transport, quantum information protocols, and direct quantum state tomography of geometric invariants.

The SRQGT thus defines a unified, deeply geometric language for quantum phase transitions, mixed-state topology, nonlinear response, and spin-resolved quantum metrology across a wide range of spin, electronic, and photonic systems (Lu et al., 2024, Zhang et al., 23 Jan 2025, Bleu et al., 2017, Chakraborti et al., 29 Jan 2026, Yu et al., 2018, Bharath, 2017, Lu et al., 2010).