what is the energy dispersion of the transverse field Ising model looks like in the case of open boundary conditions?

In the case of periodic boundary, the energy takes the form of enter image description here

and the ground state energy is obtained by summing over all the q-modes. For h=0, the energy dispersion is q independent. Since there are N independent q-modes if the system has N sites, the ground state energy is readily obtained as N. For open boundary condition, if the energy dispersion still have the same form as above, I would expect the ground state energy still be N when h=0 as there are still N q-modes. However, the true ground state energy should be N-1 instead of N in the case of OBC. Am I right?

I wonder how is the dispersion relation looks like in OBC and what are form of the corresponding q modes?

  • $\begingroup$ OBC are studied in the following paper: G. G. Cabrera and R. Jullien Phys. Rev. B 35, 7062 (1987). $\endgroup$ – Christophe May 8 '18 at 8:45

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