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The requirement is that $\chi^{(2)}$ be non-centrosymmetric. That's a bit different than having a particular parity. The states involved must be neither odd nor even; the parity must be mixed. That way the dipole matrix element exists between all three intermediate states involved in calculation of the susceptibility.

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There is a mistake in your definition of time reversal as $x$ is fixed under that transformation, the remaining transformations being correct. With this correct version of T, the Hamiltonian you study is PT symmetric.

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In most of your derivations, you have used the symbol $\psi$ for the genuine field operator (operator distribution). But your equation $$\gamma^5 \psi = -\psi$$ clearly doesn't work for any Dirac field. This equation an operator equation equivalent to $$(1+\gamma^5) \psi = 0$$ which says that one-half of the components of $\psi$ are zero as operators. But ...

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A basic postulate in elementary particle theories is CPT invariance. Also the weak interaction is the only fundamental interaction that breaks parity-symmetry, and similarly, the only one to break CP-symmetry. ...... The laws of nature were long thought to remain the same under mirror reflection, the reversal of one spatial axis. The results of an ...

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