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For noninteracting superconductors (fermion hopping plus fermion pairing), there is a particle hole (PH) symmetry which is a redundancy. The redundancy says that the BdG Hamiltonian satisfies the equation $$C H_{k}^T C^{-1} = -H_{-k}$$ where $H_k$ and $C$ are $2n\times 2n$ matrices where $n$ is the number of orbitals in each unit cell. Depending on $$C C^*= \eta=\pm 1$$ we have different topological classifications.

However, for interacting superconductors, the Hamiltonian is not quadratic, and there is no BdG single-particle Hamiltonian. So I can't write down an similar equation like $C H^T C^{-1} = -H$. So I don't know how to define $C$ and, correspondingly, $\eta$. Could any one help to clarify how to define these?

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