Help with Definitions in Numerical Calculation of Multiband Berry Phase

Question

In the third chapter of Vanderbilt's book, they discuss the so-called multiband parallel transport and provide a scheme for numerical calculations that is similar to the single band case (where the Berry phase along a closed loop may be calculated using a discretized version of $-\text{Im ln} \langle \Phi_N | \Phi_0\rangle$, where $N$ and $0$ are the start and ending points of the loop corresponding to the same Hamiltonian - eq 3.100 on pg 106).

Now, consider the multiband case (described in pages 106 to 107), and which is usually applied to systems with point degeneracies somewhere. First, they consider two neighboring points $k_0$ and $k_1$, choose a set of eigenvectors to define an overlap matrix with, and use the results of its singular value decomposition to generate a unitary rotation matrix that will rotate the eigenvector(s?) at the next point to be "optimally aligned". Then they use a similar method to calculate the multiband Berry phase.

I do not understand several definitions in the problem. For ease of answering, let's consider 3-band problem in 2D (with bands labeled A,B,C), where the energy dispersion is degenerate at some points on the BZ. For example, let bands A and B touch at $K$, and bands A,B,C touch at $\Gamma$.

1. Now I want to calculate the phase due to $\langle A_N | B_0\rangle$. But I am stumped in the definition of the overlap matrix in eq 3.102 ($\tilde{M}_{mn}^{k_0,k_1}=\langle \tilde{A_{k_0}} | B_{k_1}\rangle$). I am lost on what exactly to choose for $|A\rangle,|B\rangle$. It seems as if these are NOT single $3\times 1$ column vectors, but more like matrices.

2. I become lost again in going from eq. 3.102 to eq. 3.105, because both the left hand side and the right hand side are concerning point $k_1$...

3. Additionally, is the difference between eqs. 3.109 and 3.110 that the latter skips the SVD and rotation entirely?

4. Finally, I want to work up to calculating the single "multiband Berry phases"/"Wilson loop eigenvalues" $\phi_m$ in eq. 3.108. I do not know what these are in bra-ket notation. Are these analogous to the usual Berry connection integral? That is, does $\oint \langle A | dA\rangle \rightarrow \oint \langle A | dB\rangle$?

My end goal is that I want all my wavefunctions in the region of interest to have a smooth gauge (no discontinuities in phase), with the intention of exploring the quantities $\langle A | dB\rangle,\langle A | dC\rangle,\langle B | dC\rangle$.

I asked a similar question for the everywhere-degenerate case here: https://physics.stackexchange.com/questions/618989/numerically-calculating-non-abelian-berry-curvature-definition-of-multiplet-in, but cannot seem to extrapolate from it. Apologies for the several questions, but they all seem to stem from a basic confusion in definitions. Thanks.

Answer 1

Question 1. eq. 3.102 defines a matrix in "band-space", e.g. it's NxN where N = number of bands being considered. The right hand side is a regular vector inner product for fixed m,n.

The vectors on the right hand side are in any basis you want, so long as they span the same Kohn-Sham/band subspace.

Question 2. I'm not sure where you're lost. You say,

First, they consider two neighboring points 𝑘0 and 𝑘1, choose a set of eigenvectors to define an overlap matrix with, and use the results of its singular value decomposition to generate a unitary rotation matrix that will rotate the eigenvector(s?) at the next point to be "optimally aligned".

and that's exactly what's happening.

Question 3. Answer: Yes. It's an approximation as the text says. I could be completely wrong about this, but this approximation seems like it's related to the random phase approximation. The reason they make that approximation? I guess it's because it's easier to multiply N matrices first then take ln det of that, instead of factorizing N matrices, at least if you have a large number of bands you're dealing with.

Question 4. They define the phases in 3.108. The multiband generalization here necessarily involves a diagonalization of the "interband unitary" matrix in 3.106. I'm not sure what other bra-ket quantities you need. I'm not sure at all what you're asking about when you write $\oint \langle A | dA\rangle \rightarrow \oint \langle A | dB\rangle$, but hopefully my previous comments cleared this up.