# About symmetry constraints in momentum space

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?