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Assume I have the Hamiltonian for a 1D topological insulators as: $$H=\sin(P_x) \sigma_x+i \Delta \sigma_{y}+[1-m-\cos(P_x)] \sigma_z $$ where $m$ is the mass term, $P_x$ is the momentum and $\Delta$ is an extra term to make the Hamiltonian non-Hermitian. If I choose the time reversal operator $T=i \sigma_y K$ and the parity operator $P=\sigma_{y}$, will the Hamiltonian be $PT$-symmetric?

If you wonder what is a non-Hermitian Hamiltonian, see this question and references therein.

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    $\begingroup$ Why don't you start by hacking out $[H,PT]$. Should be easy enough, but I'm tired. $\endgroup$
    – wsc
    May 31, 2011 at 4:42
  • $\begingroup$ Sorry to reopen the debate, but what the hell is a non-Hermitian Hamiltonian? $\endgroup$
    – Dani
    Mar 11, 2012 at 12:35

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It is the Hamiltonian of the 1D PT-symmetric SSH model in real gauge. In that case, the operators P and T can be denoted as $[1,0;0,-1]$ and $[1,0;0,-1]K$, respectively. Then $PT=K$ is commutated with the real Bloch Hamiltonian, i.e., $(PT)H(k)(PT)^{-1}=H(k)$.

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