Algorithm overview

Algorithm overview#

Self-consistent mean-field loop#

To calculate the mean-field interaction in (5), we require the ground-state density matrix \(\rho_{mn}(R)\). However, (3) 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 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. (density_matrix) Compute the ground-state density matrix \(\rho_{mn}(R)\):

    1. (tb_to_kgrid) Fourier transform the total Hamiltonian to momentum space \(\hat{H}(R) \to \hat{H}(k)\).

    2. (numpy.linalg.eigh) Diagonalize the Hamiltonian \(\hat{H}(R)\) to obtain the eigenvalues and eigenvectors.

    3. (fermi_on_kgrid) Calculate the fermi level given the desired filling of the unit cell.

    4. (density_matrix_kgrid) Calculate the density matrix \(\rho_{mn}(k)\) using the eigenvectors and the fermi level.

    5. (kgrid_to_tb) Inverse Fourier transform the density matrix to real-space \(\rho_{mn}(k) \to \rho_{mn}(R)\).

  3. (meanfield) Calculate the new mean-field correction \(\hat{V}_{\text{new, MF}}(R)\) using (5).

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 tb_to_rparams function (inverse is 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 cost_mf. That re-defines the problem into a root-finding problem which is more consistent with available numerical solvers such as anderson. That is exactly what we do in the solver function, although we also provide the option to use a custom optimizer.

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