algorithm.md

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Algorithm overview

Self-consistency loop

In order to calculate the mean-field interaction in {eq}mf_infinite, we require the ground-state density matrix

$\rho_{mn}(R)$

. However, the density matrix in {eq}density is a functional of the mean-field interaction

$\hat{V}_{\text{MF}}$

itself. Therefore, we need to solve for both self-consistently.

We define a single iteration of a self-consistency loop:

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

such that it performs the following operations given an initial mean-field interaction

$\hat{V}_{\text{init, MF}}$

:

1. Calculate the total Hamiltonian
   $\hat{H}(R) = \hat{H_0}(R) + \hat{V}_{\text{init, MF}}(R)$
   in real-space.
2. Fourier transform the total Hamiltonian to the momentum space
   $\hat{H}(R) \to \hat{H}(k)$
   .
3. Calculate the ground-state density matrix
   $\rho_{mn}(k)$
   in momentum space.
   1. Diagonalize the Hamiltonian
      $\hat{H}(k)$
      to obtain the eigenvalues and eigenvectors.
   2. Calculate the fermi level
      $\mu$
      given the desired filling of the unit cell.
   3. Calculate the density matrix
      $\rho_{mn}(k)$
      using the eigenvectors and the fermi level
      $\mu$
      (currently we do not consider thermal effects so
      $\beta \to \infty$
      ).
4. Inverse Fourier transform the density matrix to real-space
   $\rho_{mn}(k) \to \rho_{mn}(R)$
   .
5. Calculate the new mean-field interaction
   $\hat{V}_{\text{new, MF}}(R)$
   via {eq}mf_infinite.

Self-consistency criterion

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:

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

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 a real and imaginary parts to form a real-valued vector.

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

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

To solve this fixed-point problem, we utilize a root-finding function scipy.optimize.anderson which uses the Anderson mixing method to find the fixed-point solution. However, our implementation also allows to use other custom fixed-point solvers by either providing it to solver or by re-defining the solver function.