# 1D Topological insulator with PT symmetry

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.

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

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., $$(PT)H(k)(PT)^{-1}=H(k)$$.