(dual) Bethe-Salpeter equation and w2dynamics interface

[egcpvanloon]

Description

Users have reported issues when combining the Bethe-Salpeter equation and/or dual Bethe-Salpeter equation with the w2dynamics solver via the TRIQS interface. In particular, the results of the Bethe-Salpeter equation and dual Bethe-Salpeter equation appear to be inconsistent for certain tensorial components and even certain orbital-averages (for example, the density channel). For certain other combinations, the problems seem to be absent (e.g., S^z channel), possibly due to cancellation of errors. Debugging efforts are ongoing to see exactly where the problem is (this issue will be updated with additional information).

Steps to Reproduce

The problems are visible in the Hubbard-Zeeman example described here: https://zenodo.org/records/8203356

or paste a minimal code example to reproduce the issue.

Expected behavior:

1. The Bethe-Salpeter equation and the Dual Bethe-Salpeter equation should converge to the same result as the number of fermionic frequencies becomes large, up to Monte Carlo noise. This should hold for all tensor components.
2. The q=\Gamma, \Omega=0 charge and spin susceptibility should be consistent with linear response to applied fields.
3. The SxSx and SxSy susceptibilities at the \Gamma point should follow the analytical formula for Larmor precession.

Actual behavior:

1. Deviations between the Bethe-Salpeter equation and the Dual Bethe-Salpeter equation appear.
2. Testing is ongoing
3. The Dual Bethe-Salpeter equation based on w2dynamics data is consistent with the analytical formulas, the normal Bethe-Salpeter equation is currently not.

Versions

Currently testing on triqs 3.3.0 and tprf 3.3.0

[egcpvanloon]

After further testing, the code looks OK but improvements in the documentation are needed:

• Conventions regarding time-ordering of two-frequency and one-frequency objects
• Importance of DMFT self-consistency for equivalence of BSE and DBSE.
• The BSE tutorial (https://triqs.github.io/tprf/latest/user_guide/dmft_susceptibility/dmft_susceptibility.html) only calculates the magnetic susceptibility, for a single-orbital model, using blocks and cthyb. The DBSE example calculates the full spin-orbital-resolved susceptibility in a three-orbital model. There is some value in having an example that shows the spin-orbital-resolved susceptibility in a single-orbital model with w2dynamics, using impurity_irreducible_vertex_gamma.
• It would be instructive to show the convergence for finite \Omega, across the Brillouin Zone
• The first BSE tutorial is conveniently chosen to be at beta=1, and is not very explicit about the factors of beta that occur in some of the quantities.

[egcpvanloon]

A new Hubbard-Zeeman tutorial is now available on the Zeeman branch.