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‘Proof’ that non-Abelian Berry phase vanishes identically
For a degenerate system with Hamiltonian $H =H(\mathbf{R})$ and eigenstates $\left|n(\mathbf{R})\right\rangle$ the non-Abelian Berry connection is
$$A^{(mn)}_i=\mathrm{i}\left\langle m|\partial_in\...
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Why gauge-invariant Berry curvature commutator looks like torsion?
The Berry Curvature is defined as (for invariant gauge transformations)
$$F_{ij} = [\partial_i, A_j] - [\partial_j,A_i] + [A_i,A_j]$$
The gauge covariance satisfies the transformation
$$A_i \...
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