Consider a pair of one dimensional four band model $H_1$ and $H_2$, which read as:

$$ H_1 = \begin{pmatrix}k\sigma_x-E_0&0\\0&k\sigma_x+E_0\end{pmatrix} + \alpha \begin{pmatrix}0&\sigma_x\\\sigma_x&0\end{pmatrix} $$

$$ H_2 = \begin{pmatrix}k\sigma_x-E_0&0\\0&k\sigma_x+E_0\end{pmatrix} + \alpha \begin{pmatrix}0&\sigma_y\\\sigma_y&0\end{pmatrix} $$

where $k\in[-\pi, \pi]$ and $E_0, \alpha>0$.

I have calculate the band structure analytically that $H_1$ is gapless while $H_2$ is gapped. Can you show this property by basic quantum mechanics principal or symmetry or commutation relations etc.

I think the main point is on the $[\sigma_x, \sigma_y]\neq0$, but I can't give a complete proof.

  • $\begingroup$ What did you calculate the band structure to be? $\endgroup$ – Greg.Paul Oct 2 at 13:11

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