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TBTK
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#include <ReciprocalLattice.h>
Public Member Functions | |
| ReciprocalLattice (UnitCell *unitCell, std::initializer_list< int > size) | |
| ~ReciprocalLattice () | |
| Model * | generateModel (std::initializer_list< double > momentum) const |
| const std::vector< std::vector< double > > & | getReciprocalLatticeVectors () const |
The ReciprocalLattice provides methods for constructing momentum space Models from a UnitCell. Let the real space Hamiltonian be written on the form
![\[ H = \sum_{RiR'i'}a_{RiR'i'}c_{Ri'}^{\dagger}c_{R'i'}, \]](form_20.png)
where R and R' are UnitCell-indices and i and i' are intra UnitCell-indices. Expanding the operators in the momentum basis we have
![\[ H = \sum_{RkiR'k'i'}a_{RiR'i'}c_{ki}^{\dagger}c_{k'i'}e^{i(k\cdot R . k'\cdot R')}. \]](form_21.png)
Now assuming translational invaraince, such that the coefficients only depend on the relative UnitCell positions, we can write
![\[ H = \sum_{\bar{R}kiR'k'i'}a_{\bar{R}i0i'}c_{ki}^{\dagger}c_{k'i'}e^{ik\cdot\bar{R}}e^{i(k-k')\cdot R'}, \]](form_22.png)
where
![\[\bar{R} = R - R'\]](form_23.png)
. Carrying out the sum over R' we arrive at
![\[ H = \sum_{kik'i'}\left(\sum_{\bar{R}}a_{\bar{R}i0i'}e^{ik\cdot\bar{R}}\right)c_{ki}^{\dagger}c_{k'i'}. \]](form_24.png)
It therefore follows that the momentum space HoppingAmplitudes are given by
![\[ a_{kik'i'} = \sum_{\bar{R}}a_{\bar{R}i0i'}e^{ik\cdot\bar{R}}. \]](form_25.png)
The purpose of this class is to provide method for constructing Models for given k and k' using these coefficients. This is done by specifying a real space environment around a reference UnitCell, large enough to ensure that the sum can run over all relevant
![\[\bar{R}\]](form_26.png)
, and then using this to calcualte the coefficeints when Models with given k and k' is demanded.
| TBTK::ReciprocalLattice::ReciprocalLattice | ( | UnitCell * | unitCell, |
| std::initializer_list< int > | size | ||
| ) |
Constructor.
| TBTK::ReciprocalLattice::~ReciprocalLattice | ( | ) |
Destructor.
| Model * TBTK::ReciprocalLattice::generateModel | ( | std::initializer_list< double > | momentum | ) | const |
Genearates a Model for give momentum.
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inline |
Get reciprocal lattice vectors.
1.8.11
