# sisl.physics.Bloch

class sisl.physics.Bloch(*bloch)

Bases: object

Bloch’s theorem object containing unfolding factors and unfolding algorithms

This class is a wrapper for expanding any matrix from a smaller matrix cell into a larger, using Bloch’s theorem. The general idea may be summarized in the following equation:

$\begin{split}\mathbf M_{K}^N =\frac1N \; \sum_{ \substack{j=0\\ k_j=\frac{K+j}N } }^{N-1} \quad \begin{bmatrix} 1 & \cdots & e^{i (1-N)k_j} \\ e^{i k_j} & \cdots & e^{i (2-N)k_j} \\ \vdots & \ddots & \vdots \\ e^{i (N-1)k_j} & \cdots & 1 \end{bmatrix} \otimes \mathbf M_{k_j}^1.\end{split}$
Parameters:

bloch ((3,) int) – Bloch repetitions along each direction

Examples

>>> bloch = Bloch([2, 1, 2])
>>> k_unfold = bloch.unfold_points([0] * 3)
>>> M = [func(*args, k=k) for k in k_unfold]
>>> bloch.unfold(M, k_unfold)


Methods

 unfold(M, k_unfold) Unfold the matrix list of matrices M into a corresponding k-point (unfolding k-points are k_unfold) Return a list of k-points to be evaluated for this objects unfolding bloch Number of Bloch expansions along each lattice vector
__call__(func, k: sisl.typing.KPoint, *args, **kwargs)[source]

Return a functions return values as the Bloch unfolded equivalent according to this object

Calling the Bloch object is a shorthand for the manual use of the Bloch.unfold_points and Bloch.unfold methods.

This call structure is a shorthand for:

>>> bloch = Bloch([2, 1, 2])
>>> k_unfold = bloch.unfold_points([0] * 3)
>>> M = [func(*args, k=k) for k in k_unfold]
>>> bloch.unfold(M, k_unfold)


Notes

The function passed must have a keyword argument k.

Parameters:
• func (callable) – method called which returns a matrix.

• k ((3, ) of float) – k-point to be unfolded

• *args (list) – arguments passed directly to func

• **kwargs (dict) – keyword arguments passed directly to func

Returns:

unfolded Bloch matrix

Return type:

ndarray

__init__(*bloch)[source]

Create Bloch object

property bloch

Number of Bloch expansions along each lattice vector

unfold(M, k_unfold: Sequence[sisl.typing.KPoint])[source]

Unfold the matrix list of matrices M into a corresponding k-point (unfolding k-points are k_unfold)

Parameters:
Returns:

unfolded matrix of size M[0].shape * k_unfold.shape[0] ** 2

Return type:

ndarray

unfold_points(k: sisl.typing.KPoint)[source]

Return a list of k-points to be evaluated for this objects unfolding

The k-point k is with respect to the unfolded geometry. The return list of k points are the k-points required to be sampled in the folded geometry.

Parameters:

k ((3,) of float) – k-point evaluation corresponding to the unfolded unit-cell

Returns:

a list of np.prod(self.bloch) k-points used for the unfolding

Return type:

ndarray