# Can you argue without explicitly calculate the eigenenergies that one Hamiltonian is gapped and another is not?

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.

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