triqs_modest.obe.DownfoldingProjector

class triqs_modest.obe.DownfoldingProjector

The projector that downfolds the energy bands onto a set of localized atomic-like orbitals.

A downfolding projector contains the projector, the kind of spin used in the projection, and the number of bands per k-point for cases when a band goes outside of the projection window.

The projectors \(P_{m\nu}^{\sigma}(\mathbf{k})\) connect the Bloch space \({\cal B}\) to \({\cal C}\). The projectors are obtained from DFT codes or Wannier90. They are defined by

\[P_{(a,m_{a})\nu}^{\sigma}(\mathbf{k})\equiv e^{-i \mathbf{k} R_a} \langle \chi_{m_{a}}^{R_a \sigma} | \psi_{\nu}^{\sigma}(\mathbf{k}) \rangle,\]

where \(| \chi_{m_{a}}^{R_a \sigma} \rangle\) is a Wannier function localized at atom \(a\) with index \(m_a\) at position \(R_a\) and \(| \psi_{\nu}^{\sigma}(\mathbf{k}) \rangle\) is the Kohn-Sham wavefunction. The relation between the Wannier and Bloch function is therefore

\[| \chi_{m_{a}}^{R_a \sigma} \rangle = \sum_{\mathbf{k} \nu} e^{-i \mathbf{k} R_a} \bigl(P^\sigma_{(a,m_{a})\nu} (\mathbf{k})\bigr)^* | \psi_{\nu}^{\sigma}(\mathbf{k}) \rangle.\]

Some properties:

• Basis change in \(\cal C\) space: They are given by a unitary matrix \(U\), the projector transforms as \(P^{'\sigma}_{m\nu}(\mathbf{k}) = U^{\dagger}_{m, m'} P^{\sigma}_{m'\nu}(\mathbf{k}).\)

• Partial unitarity property: In general \(P\) is not unitary as \(N_\nu^{\mathbf{k}} > M\). However, if the Wannier functions are reorthonormalized with respect to the truncated band basis, we have \(\sum_{ \nu} P^{\sigma}_{m\nu}(\mathbf{k}) P^{\dagger\sigma}_{\nu m'}(\mathbf{k}) = \delta_{mm'}\).

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Synthesized constructor with the following keyword arguments:

Parameters:

spin_kindstr {“Polarized”, “NonPolarized”, “NonColinear”}

P_kndarray[complex, 4]

n_bands_per_kndarray[int, 2]

Attributes

P_k	Projector \(P_{m\nu}^{\sigma}(\mathbf{k})\).
n_bands_per_k	Number of bands for each k-point and \(\sigma\).
spin_kind	Spin kind of the one-body data.

Methods

P	Get \(P_{m\nu}^{\sigma}(\mathbf{k})\) for a given \(\mathbf{k}\) and \(\sigma\).
rotate_local_basis	Rotates the local basis of the downfolding projector.