About k-integrationΒΆ

See also

Using SmearingKWeight.

RISB requires integrating many mean-field matrices as a function of \(k\). The way to do this that generalizes to many kinds of \(k\)-space integration methods is to find the weight of the integral at each \(k\)-point. This is, e.g., how linear tetrahedron works and smearing methods work.

The reference energy for the integration weights in RISB are the eigenenergies of the single-particle Hamiltonian \(\hat{H}^{\mathrm{qp}}\), which change at every iteration of the self-consistent process. The quasiparticle Hamiltonian is given by

All of the integrals are in the thermodynamic limit and take the form

\[ I = \lim_{\mathcal{N} \rightarrow \infty} \frac{1}{\mathcal{N}} \sum_k \mathbf{A}_k f(\mathbf{H}^{\mathrm{qp}}_k), \]

where \(\mathcal{N}\) is the number of unit cells, \(\mathbf{A}_k\) is a matrix of a generic function of \(k\), and \(f(\mathbf{H}^{\mathrm{qp}})\) is the Fermi-Dirac distribution. The meaning of \(f(\mathbf{H}^{\mathrm{qp}}_k)\) is specifically the operation

\[ \mathbf{U}^{}_k \mathbf{U}^{\dagger}_k f(\mathbf{H}^{\mathrm{qp}}_k) \mathbf{U}^{}_k \mathbf{U}^{\dagger}_k = \mathbf{U}^{}_k f(\xi^{\mathrm{qp}}_{kn}) \mathbf{U}^{\dagger}_k, \]

where \(\mathbf{U}_k\) is the matrix representation of the unitary that diagonalizes \(\hat{H}^{\mathrm{qp}}_k\), \(\xi^{\mathrm{qp}}_{kn}\) are the eigenenergies (bands) of \(\hat{H}^{\mathrm{qp}}\), and \(f(\mathbf{\xi}^{\mathrm{qp}}_{kn})\) is a diagonal matrix of the Fermi-Dirac distribution for each quasiparticle band \(n\).

The integral can be converted to a series of finite \(k\)-points, with an appropriate integration weight such that the integral now takes the form

\[ I = \sum_k A_k w(\xi^{\mathrm{qp}}_{kn}). \]

Most \(k\)-space integration methods can be reduced to different approximations to choose the weighting function \(w(\xi^{\mathrm{qp}}_{kn})\).

All of our Solver classes require a function that takes the quasiparticle eigenenergies \(\xi_{kn}^{\mathrm{qp}}\) and returns the weights \(w(\xi_{kn})\).