When people study symmetry protected topological phases, certain symmetry constraints are enforced on the Hamiltonian. Specifically, for non-interacting fermionic systems, we could focus on the symmetry properties of the single particle Hamiltonians instead.

For example, a free fermion system could be written as:

$$ \hat{H} = \sum_{\alpha,\beta} c_\alpha ^\dagger \mathcal{H}_{\alpha,\beta} c_\beta$$

  • possibily respecting time reversal symmetry: $\mathcal{T} \mathcal{H} \mathcal{T}^{-1} = \mathcal{H} $

  • particle hole symmetry: $\mathcal{P} \mathcal{H} \mathcal{P}^{-1} = -\mathcal{H}$

  • and chiral symmetry: $\mathcal{C} \mathcal{H} \mathcal{C}^{-1} = -\mathcal{H}$

where $\mathcal{T}$ and $\mathcal{P}$ are anti-unitary while $\mathcal{C}$ is unitary.

Transforming to momentum basis, the constraints above can be written as \begin{align} \mathcal{T} h(k) \mathcal{T}^{-1} & = h(-k),\\ \mathcal{P} h(k) \mathcal{P}^{-1} & = -h(-k),\\ \mathcal{C} h(k)\mathcal{C}^{-1} & = -h(k) . \end{align} It is usually the case that these unitary operators in momentum basis are independent of the momentum. For example, $\mathcal{C}$ could be chosen as $\sigma_z$ while $\mathcal{P}$ could be chosen as $\sigma_x \mathcal{K}$.

My question is, is there any case where these symmetry operators are momentum dependent? Would there be physical relevance for such systems?


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