algorithm.md

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Algorithm overview

Self-consistent mean-field loop

To calculate the mean-field interaction in {eq}mf_infinite, we require the ground-state density matrix \rho_{mn}(R). However, {eq}density is a function of the mean-field interaction \hat{V}_{\text{MF}} itself. Therefore, we need to solve for both self-consistently.

A single iteration of this self-consistency loop is a function that computes a new mean-field correction from a given one:

\text{MF}(\hat{V}_{\text{init, MF}}) \to \hat{V}_{\text{new, MF}},

which is defined in {autolink}~meanfi.model.Model.mfield method. It performs the following steps:

1. Calculate the total Hamiltonian \hat{H}(R) = \hat{H_0}(R) + \hat{V}_{\text{init, MF}}(R) in real-space.
2. ({autolink}~meanfi.mf.density_matrix) Compute the ground-state density matrix \rho_{mn}(R):
   1. ({autolink}~meanfi.tb.transforms.tb_to_kgrid) Fourier transform the total Hamiltonian to momentum space \hat{H}(R) \to \hat{H}(k).
   2. ({autolink}numpy.linalg.eigh) Diagonalize the Hamiltonian \hat{H}(R) to obtain the eigenvalues and eigenvectors.
   3. ({autolink}~meanfi.mf.fermi_on_kgrid) Calculate the fermi level given the desired filling of the unit cell.
   4. ({autolink}~meanfi.mf.density_matrix_kgrid) Calculate the density matrix \rho_{mn}(k) using the eigenvectors and the fermi level.
   5. ({autolink}~meanfi.tb.transforms.kgrid_to_tb) Inverse Fourier transform the density matrix to real-space \rho_{mn}(k) \to \rho_{mn}(R).
3. ({autolink}~meanfi.mf.meanfield) Calculate the new mean-field correction \hat{V}_{\text{new, MF}}(R) using {eq}mf_infinite.

Self-consistency criteria

To define the self-consistency condition, we first introduce an invertible function f that uniquely maps \hat{V}_{\text{MF}} to a real-valued vector which minimally parameterizes it:

f : \hat{V}_{\text{MF}} \to f(\hat{V}_{\text{MF}}) \in \mathbb{R}^N.

In the code, f corresponds to the {autolink}~meanfi.params.rparams.tb_to_rparams function (inverse is {autolink}~meanfi.params.rparams.rparams_to_tb). Currently, f parameterizes the mean-field interaction by taking only the upper triangular elements of the matrix V_{\text{MF}, nm}(R) (the lower triangular part is redundant due to the Hermiticity of the Hamiltonian) and splitting it into real and imaginary parts to form a real-valued vector.

With this, we define the self-consistency criterion as a fixed-point problem:

f(\text{MF}(\hat{V}_{\text{MF}})) = f(\hat{V}_{\text{MF}}).

Instead of solving the fixed point problem, we rewrite it as the difference of the two successive self-consistent mean-field iterations in {autolink}~meanfi.solvers.cost_mf. That re-defines the problem into a root-finding problem which is more consistent with available numerical solvers such as {autolink}~scipy.optimize.anderson. That is exactly what we do in the {autolink}~meanfi.solvers.solver function, although we also provide the option to use a custom optimizer.