create solvers and interface modules

Co-authored-by: Johanna Zijderveld

As summary from an online discussion, we have identified several things to try:

Applying the code

To see whether the library design makes sense, we should use it in different scenarios.

• One is the application of it to the graphene project.
• Another one, which can go into the demo notebooks, is plotting the free energy as a function of the mean field amplitude and confirming that the converged result is actually the minimum.

Short range interactions

I believe that the code right now doesn't exploit the interaction being short range (when it actually is short range). It seems that the k-space tensor contraction can be made into a smaller one by Fourier transforming the density matrix and doing the contraction in real space.

This is also a possible source of the lack of convergence of the minimizers.

Separation between mean field parameters and interaction range

Right now the fixed point algorithm requires that the mean field correction has the same range as the interaction, which is technically always the case. However, in many physical scenarios one may want to consider a long-range interaction, e.g. Coulomb, and only a short range symmetry breaking. I therefore propose to separate mean field correction generation from the matrix structure.

As a concrete example I propose to consider 1D model with Coulomb repulsion, and an onsite antiferromagnetic order parameter. Then I propose to define the free energy as a a k-space convolution and solve a 1D minimization problem for it. This will confirm that the code is usable for constrained problems and allows us to verify whether the concepts are clearly separated in the interface.

This is a dangerous step because we are potentially discarding nonzero contributions to a total energy, and biasing the solution. We should defined a constrained problem instead.

This _params={} has to be outside the loop between line 97 to 98. Currently this will only save the last parameter and not multiple.

Also here the _params={} has to be moved outside the loop.

added 4 commits

• a5ddef60 - remove dim from function arguments
• 1a3f357d - remove from redundant args
• 17a7647d - normalize total energy to kpoint number
• 4ed37a68 - push total energy check

Compare with previous version

added 1 commit

• 126e03cc - fix test of energy minimum

Compare with previous version

added 2 commits

• fa38b895 - fix x label
• e3f1ee69 - remove line label

Compare with previous version

added 1 commit

• 0db7facf - add constant term

Compare with previous version

added 1 commit

• 263c5960 - fix Bloch phase

Compare with previous version

added 5 commits

• 1934c936 - fix Hamiltonian
• 5cde05f0 - add constant correction
• 7a09e384 - shift Fermi level at the end
• 6d3bf614 - prototype for total energy solver
• d7e96d28 - cleanup notebook

Compare with previous version

merged

mentioned in commit 6f281e4f

mentioned in merge request !4 (merged)