It may sound a bit awkward, but is there a Berry curvature in non parameter-dependent Hamiltonians? For example, consider the Bloch problem. There is a dual perspective when working in Bloch Solids, depending on the choice of eigenvectors: one may choose to work with the periodic cell functions unk (leading to a k-dependent Hamiltonian, H(k)) or choose the eigenstates of the translation operator as well (leading to a non k-dependent Hamiltonian H).
-Why is someone obliged to use unk so that Berry Physics comes into play?
-What about choosing the "normal" k-dependent eigenstates of the translation op.?
We know that in this case, H would not depend on any parameters, but does this necessarily mean that the Berry curvature vanishes? If the answer is negative, then, the extended Berry curvature formula (the one containing the summation over the states) becomes indeed problematic, simply because \nabla_k H=0. If the answer is positive, then, again, the initial definition is problematic, because it only involves differentiations of the eigenstates, which are obviously non-zero.
It is better to read the original paper given by Dirac in 1931, which indicates that non-integrable phase factor naturally induces the topological vector potential. Topological vector potential can bring non-trivial effective magnetic field (Berry curvature). Understanding the concept of them could be helpful for getting the essential picture of Berry curvature in parameters space.
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