1 &  &  &  &  &  & \gamma_1 &\\
      & 1 & \varepsilon & 1 &  & 1 &  &  &  & \gamma_1 &  &\\
     1 &  & 1 & \varepsilon & 1 &  &  &  & \gamma_1 &  &  &\\
      & 1 &  & 1 & \varepsilon & 1 &  & \gamma_1 &  &  &  &\\
     1 &  & 1 &  & 1 & \varepsilon & \gamma_1 &  &  &  &  &\\
      &  &  &  &  & \gamma_1 & \varepsilon & 1 &  & 1 &  &1\\
      &  &  &  & \gamma_1 &  & 1 & \varepsilon & 1 &  & 1 &\\
      &  &  & \gamma_1 &  &  &  & 1 & \varepsilon & 1&  &1\\
      &  & \gamma_1 &  &  &  & 1 &  & 1 & \varepsilon& 1&\\
      & \gamma_1 &  &  &  &  &  & 1 &  & 1 & \varepsilon& 1 \\
      \gamma_1&  &  &  &  &  & 1 &  & 1 &  & 1 &\varepsilon \\
 
     \end{array}\right|=0.$$

Solving equation leads to following energy levels:

$$\label{eq:9}
\varepsilon = \pm (3+\gamma_1), \pm (3-\gamma_1), \pm \gamma_1, \pm \gamma_1, \pm \gamma_1, \pm \gamma_1.$$

The main result that ensues from is that there is an energy gap $\varepsilon_g 