berry_curvature¶

sisl.physics.electron.berry_curvature(state, energy, dHk, dSk=None, degenerate=None, complex=False)[source]¶

Calculate the Berry curvature matrix for a set of states (using Kubo)

The Berry curvature is calculated using the following expression (\(\alpha\), \(\beta\) corresponding to Cartesian directions):

\[\boldsymbol\Omega_{n,\alpha\beta} = - \frac2\hbar^2\Im\sum_{m\neq n} \frac{v_{nm,\alpha} v_{mn,\beta}} {[\epsilon_m - \epsilon_n]^2}\]

Note that this method optionally returns the complex valued equivalent of the above. I.e. \(\Im\) is not applied if complex is true.

For details see Eq. (11) in 1 or Eq. (2.59) in 2.

Parameters

state (array_like) – vectors describing the electronic states, 2nd dimension contains the states. In case of degenerate states the vectors may be rotated upon return.
energy (array_like, optional) – energies of the states. In case of degenerate states the eigenvalues of the states will be averaged in the degenerate sub-space.
dHk (list of array_like) – Hamiltonian derivative with respect to \(\mathbf k\). This needs to be a tuple or list of the Hamiltonian derivative along the 3 Cartesian directions.
dSk (list of array_like, optional) – \(\delta \mathbf S_k\) matrix required for non-orthogonal basis. Same derivative as dHk. NOTE: Using non-orthogonal basis sets are not tested.
degenerate (list of array_like, optional) – a list containing the indices of degenerate states. In that case a prior diagonalization is required to decouple them. This is done 3 times along each of the Cartesian directions.
complex (logical, optional) – whether the returned quantity is complex valued (i.e. not only the imaginary part is returned)