# How to find dispersion relation for 1 d topological insulator?

Is it correct to write the dispersion relation for following Hamiltonian where $\sigma_{x}$ act in spin space and $\tau_{x}$ acts in pseudo spin particle hole spin $H_{BdG} (k)=(\xi_{k}+B\sigma_{x}+u sin(k)\sigma_{y})\tau_{z}+\Delta\sigma_{y}\tau_{y}$

$E(k)=\sqrt[2]{\xi_{k} ^{2}+B^{2} +(u Sin(k))^{2}+\Delta^{2}}$ because pauli matrices for spin and pseudo spin are acting in orthogonal spaces, but when I am trying to solve the $4\times4$ matrix of above hamiltonian to find eigenvalue I can't get to the above dispersion relation what I am doing wrong here?

• Your BdG Hamiltonian is wrong. – Everett You Jan 18 '15 at 21:29
• @EverettYou can you specify the mistake ? – user48826 Jan 18 '15 at 22:29
• @Everettyou I have checked it again and I think it is right. – user48826 Jan 18 '15 at 22:31
• Your Hamiltonian can not give the $E(k)$ you want, but the following Hamiltonian can $H_\text{BdG}=\xi_k\sigma_z+B\sigma_x+u\sin k\sigma_y\tau_z+\Delta\sigma_y\tau_y$ – Everett You Jan 19 '15 at 23:05