Module linear 

Linear response: Berry curvature and quantum metric

Theory

The quantum geometric tensor is

$$G^{ab}n(\mathbf{k}) = \sum{m=\not n} \frac{\langle\partial_a u_n|u_m\rangle\langle u_m|\partial_b u_n\rangle} {(E_n-E_m)^2} = g^{ab}_n - \frac{i}{2}\Omega^{ab}_n$$

In terms of velocity matrix elements $v^\alpha_{nm} = \langle u_n|\partial_\alpha H|u_m\rangle$:

$$G^{ab}n(\mathbf{k}) = \sum{m=\not n} \frac{v^a_{nm} v^b_{mn}}{(E_n-E_m)^2 + \eta^2}$$

where $\eta$ is a small regularisation width. The real and imaginary parts give the quantum metric $g^{ab}_n$ and Berry curvature $\Omega^{ab}_n$:

$$g^{ab}_n = \operatorname{Re} G^{ab}_n, \qquad \Omega^{ab}_n = -2\operatorname{Im} G^{ab}_n$$

The anomalous Hall conductivity (AHC) at $T=0$ is

$$\sigma^{xy}{\text{AHC}}(\mu) = -\frac{e^2}{\hbar}\sum_n \int{\text{BZ}} \Theta(\mu-E_n)\Omega^{xy}_n(\mathbf{k})d\mathbf{k}$$

API

Method	Path	Formula
BerryCurvature::berry_curvature_at	one k-point	band-resolved $\Omega_n^{ab}$
Model::hall_conductivity	direct sum or energy cut	$\sigma_{\text{AHC}}(\mu)$

Structs

HallConductivityResult

Hall conductivity evaluated on the requested chemical-potential grid.