band unfolding not good for finding "gamma" bands

[ExpHP]

Band unfolding does not seem to be working too well for identifying gamma bands in tblg.

Ideally, the probabilities associated with the unit cell gamma would be all nearly 0 or 1, but instead what I typically see are (using the primitive cell of one of the layers as the primitive cell):

• The 3 acoustics tend to have probabilities around 0.6. (yes, really. more on this below)
• The 3 lowest frequency layer acoustics have gamma probabilities around 0.5.
• Among the rest of the modes, there are some probabilities around ~0.5, maybe ~0.2, and then lots of small frequencies (many of which are "not that small"; like ~0.01).

If I'm lucky, there are 12 modes with probabilities above 0.2. I'm not always lucky.

How on earth is the gamma probability of the acoustics only 0.6? Here's what I think:

• The band unfolding algorithm requires sampling images of gamma in the primitive reciprocal lattice. (a simple example of why this is necessary is a breathing mode at gamma in the primitive cell; if you naively only sample the true gamma, then all the velocities will cancel and you get zero)
• Because we're using a layer as the primitive structure, one of our layers is periodic in our primitive reciprocal lattice (call this the primary layer), while the other is not (call this the secondary layer).
• Sampling breaks down like this:
  • Sampling the primary layer at the true PC gamma produces 1 or 0.
  • Sampling the primary layer at images of PC gamma produces 1 or 0.
  • Sampling the secondary layer at the true PC gamma produces 1 or 0.
  • Sampling the secondary layer at images of the PC gamma produces something unusual because the phase factors in the bra used to sample this q point do not line up across different unit cells.

[ExpHP]

the code was changed ages ago to do separate unfolding analyses for each layer. I think