Make sdiag_inv_sqrt use pseudo-inverse

[pbrehmer]

Implements pseudo-inverse for the inverse square root inside of sdiag_inv_sqrt and its reverse-rule.

Closes #73 as suggested by @huangrzh.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Files with missing lines	Coverage Δ
src/utility/util.jl	57.30% <100.00%> (+1.48%)	⬆️

... and 6 files with indirect coverage changes

[pbrehmer]

I'm not super sure about the default inverse cutoff tolerance to choose here; any opinions on this? @lkdvos

[lkdvos]

I think TensorKit uses a tol which corresponds to eps(real(float(scalartype(T))))^(3/4) * svals[1] in the tsvd rrule, maybe we could just use the same?

[pbrehmer]

I like that TensorKit way of handling the singular value cutoff tolerance. I have decided to again cut the inv_sqrt_tol field from ProjectorAlg since that is probably anyway a too fine-grained parameter. Also there is not a good consistent way to tweak that same tolerance inside the rrule of tsvd! from within PEPSKit, so I think it's better to just leave that parameter away and define a reasonable default.

missed failing tests

[pbrehmer]

Also updated the p-wave tests a bit and again found it weird that apparently the p-wave differentiation in :fixed mode leads to very high gauge-sensitivity. (This is not the case for the :diffgauge mode.) My suspicion is that this is improved with the fix in #72, where the gauge-fixing contractions had daggers in the wrong place and perhaps these daggers matter in the fermionic case. I will investigate further with the Hubbard model...

[lkdvos]

Looks good to me. If you checked the gauge sensitivity again after the fixes, Im okay with merging this like this.

[lkdvos]

did you re check this after the gaugefixing fixes?

[pbrehmer]

Okay I checked again and the gauge-fixing fixes don't make a difference to the high gauge-sensitivity. I will have to investigate and fix that in a different PR. I just haven't played around that much with fermions yet.

I put back in nothing to the test, since I accidentally left that commented out as I was trying around.

[lkdvos]

at the risk of sounding like a broken record, I really don't like that we have to change the settings of all algorithms on every single update of the package. We really have to get some better defaults in, because this is not really feasible in the long run. (not something that has to be fixed in this pr)

[pbrehmer]

You definitely have a point in repeating that and I completely agree :-) Actually, for most of the tests we should already be able to replace all settings with the default ones I just haven't gotten around to change them. I just thought that I would wait until we anyway redesign our interface and code up some kind of default parameter selection thing.

[lkdvos]

OK if tests turn green.