DMFT loop

Typical quantities computed at each DMFT iteration: local Green’s function, chemical-potential search, impurity levels, and hybridization function.

Local Green’s function

The local Green’s function on the correlated subspace \(\mathcal{C}\) is obtained by k-summing the lattice Green’s function projected with \(P(\mathbf{k})\):

\[G_{\mathcal{C},\,\mathrm{loc}}^{\sigma}(\omega) = \sum_{\mathbf{k}} P(\mathbf{k})\, \bigl[(\omega + \mu)\,\mathbb{1} - H^{\sigma}(\mathbf{k}) - P^{\dagger}(\mathbf{k})\, \bigl(\Sigma_{\mathrm{imp}}^{\sigma}(\omega) - \Sigma_{\mathrm{DC}}^{\sigma}\bigr)\, P(\mathbf{k})\bigr]^{-1}\, P^{\dagger}(\mathbf{k}).\]

ModEST evaluates this in \(\mathcal{C}\) via the Woodbury-reduced form (7).

triqs_modest.local_gf.gloc

Compute local Green's function on a \(M \times M\) mesh.

Chemical-potential search

The chemical potential is determined by a root-finding problem on the total electron density:

\[\mu_{\star} :\quad n(\mu_{\star}) - n_{\mathrm{target}} = 0, \qquad n(\mu) = \sum_{\sigma}\sum_{\mathbf{k}} n^{\sigma}(\mathbf{k};\,\mu),\]

where \(n^{\sigma}(\mathbf{k};\,\mu)\) is the lattice density of (8) (computed efficiently in \(\mathcal{C}\) through (11)) and \(n_{\mathrm{target}}\) is the target filling. ModEST exposes several algorithms (bisection, Brent, …) and is modular enough that an external root finder can be plugged in.

triqs_modest.rho_and_mu.find_chemical_potential	Find the chemical potenital from the local Green's function given a target density.
triqs_modest.rho_and_mu.density	Compute the density of the lattice Green's function with a self-energy using Woodbury.
triqs_modest.rho_and_mu.density_nk	Compute number of particles \(n = \sum f(\beta(\varepsilon(k) - μ))\).

Impurity levels and hybridization function

The impurity-level matrix \(E_{\mathrm{imp}}\) is read off from the high-frequency expansion of the local Green’s function and includes the double-counting correction:

\[E_{\mathrm{imp}} = \langle H^{\sigma}(\mathbf{k}) \rangle_{\mathbf{k}} - \mu - \Sigma_{\mathrm{DC}}^{\sigma}.\]

The hybridization function is then defined by the Dyson equation on the impurity,

\[\Delta^{\sigma}(\omega) = \omega\,\mathbb{1} - E_{\mathrm{imp}} - \bigl[G_{\mathcal{C},\,\mathrm{loc}}^{\sigma}(\omega)\bigr]^{-1} - \Sigma_{\mathrm{imp}}^{\sigma}(\omega).\]

The derivation of \(E_{\mathrm{imp}}\) and the role of \(\Sigma_{\mathrm{DC}}^{\sigma}\) are spelled out in Double counting.

triqs_modest.atomic_levels_and_delta.impurity_levels	Compute the atomic (impurity) levels from an obe.
triqs_modest.atomic_levels_and_delta.hybridization	Compute the hybridization function from the effective impurity levels, the local Green's function, and the impurity self-energy.