Function quantum_geometric_tensor_kernel 

pub fn quantum_geometric_tensor_kernel(
    eigenvalues: &CausalTensor<f64>,
    eigenvectors: &QuantumEigenvector,
    velocity_i: &QuantumVelocity,
    velocity_j: &QuantumVelocity,
    band_n: usize,
    regularization: f64,
) -> Result<Complex<f64>, PhysicsError>

Calculates the Quantum Geometric Tensor (QGT) component $Q_{ij}^n(\mathbf{k})$ for band $n$.

The QGT is a fundamental object in the geometry of quantum states, encapsulating both the distance (metric) and curvature (Berry curvature) in the parameter space (Brillouin zone).

Physical Model

Uses the sum-over-states (perturbative) formula derived from the Fubini-Study metric (see, e.g., Kang et al., arXiv:2412.17809 for recent experimental applications):

$$ Q_{ij}^n = \sum_{m \neq n} \frac{\langle n | v_i | m \rangle \langle m | v_j | n \rangle}{(E_n - E_m)^2 + \epsilon} $$

where:

• $|n\rangle, |m\rangle$: Eigenstates of the Hamiltonian.
• $v_i = \partial_{k_i} H$: Velocity operator components.
• $E_n, E_m$: Eigenenergies.

Returns

A complex scalar $Q_{ij}$.

• Real Part: Quantum Metric $g_{ij}$ (Symmetric).
• Imaginary Part: $-\frac{1}{2} \Omega_{ij}$ (Berry Curvature, Antisymmetric).

Arguments

• eigenvalues - Energy eigenvalues $E_n$ (Rank 1 Tensor).
• eigenvectors - Matrix of eigenstates $U$ (Rank 2 Tensor [basis, states]).
• velocity_i - Velocity matrix $v_i$ components in the eigenbasis (Rank 2).
• velocity_j - Velocity matrix $v_j$ components in the eigenbasis (Rank 2).
• band_n - The target band index $n$.
• regularization - Small $\epsilon$ parameter to regularize the denominator at band crossing points (degeneracies).

Errors

• DimensionMismatch - If input tensor shapes are inconsistent.