I'm trying to understand the topological properties of different systems and where they fall on the Periodic Table of Topological Phases. Such systems might include the Quantum Anomalous Hall Effect, Su-Schrieffer-Heeger model of trans-polyacetylene, Kitaev chain, or B-phase of He-3.

Once a resource gives me the right matrices, I can verify that indeed, there does indeed exist a U such that

\(U H U^{-1} = -H \qquad U U^{\dagger} = \mathbb{1}\)

(or respective for the T and C symmetries) .

But how do I go about showing that a system **doesn't **have **any **U such that this holds? Or go about finding such a U if I didn't read it in a paper?

My attempt using eigenvectors seems to be telling me that things I know aren't symmetric are.