Good morning everybody, I am facing a problem when calculating the topological invariant in a semi-dirac system, whose Hamiltonian is: $$ H=k_x^2\sigma_x+k_y\sigma_y $$ My question is that this Hamiltonian has time-reversal symmetry, of the form $H(k)=H(-k)^*$ and therefore should not have topological invariant. Instead I have calculated the edge states and they exist. Does anyone know why it has edge states, but in theory it does not have topological invariance. Is it because the model is developed to too low order? Can anyone think of a way to prove the topology?
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$\begingroup$ Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. $\endgroup$– Community BotCommented May 28, 2022 at 10:43
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$\begingroup$ It may be a topological crystalline insultator? $\endgroup$– Feynnman pilowsCommented Jun 1, 2022 at 8:34
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$\begingroup$ It's too quick to conclude there are no topological invariants for the model just because of time-reversal symmetry. After all, topological insulators are also time-reversal-invariant. Does the edge states still exist if a mass term $m\sigma_z$ is added? This mass preserves the time-reversal symmetry you mentioned. $\endgroup$– Meng ChengCommented Jun 1, 2022 at 13:14
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$\begingroup$ @MengCheng Yes they exist i have calculated them. The only thing I don't understand is that according to the classification that depends on symmetries, if you have T=+1 you should not have invariant for dim=2 ? $\endgroup$– Feynnman pilowsCommented Jun 2, 2022 at 15:18
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$\begingroup$ I have recently check that it also has quiral symmetry and particle-hole symmetry, and maybe mirror symmetry? $\endgroup$– Feynnman pilowsCommented Jun 4, 2022 at 11:23
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