# ground state of spin chain with $Z_i X_{i+1} Z_{i+2}$ interaction

the problem comes from transverse field Ising model, with an extra 3-spin interaction term $$H=H_0+H_1+H_2=-h\sum_{i=1}^{N}X_i -\lambda_1 \sum_{i=1}^{N-1}Z_i Z_{i+1}-\lambda_2 \sum_{i=1}^{N-2}Z_i X_{i+1} Z_{i+2}$$

now, I am only interested in the extreme case of $H_0, H_1, H_2$ separately.

the ground state of $H_0$ and $H_1$ is clear to us,

$H_0$ unique ground state $|\rightarrow\rightarrow\rightarrow\rightarrow\rightarrow\rightarrow\rightarrow>$

$H_1$ double degenerate ground states $|\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow>$ and $|\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow>$

my question is, what is the spin representation ground states of $H_2=-\lambda_2 \sum_{i=1}^{N-2}Z_i X_{i+1} Z_{i+2}$

what I know: the degeneracy is 4, the ground state energy is $E_{H_2}=-\lambda_2 (N-2)$, excitations all have energy of $2\lambda_2$.

although Jordan-Wigner transformation can diagonalize the entire problem in fermion picture, however, the ground state is not clear in spin presentation. I am expecting the ground state of $H_2$ has some non-trivial entanglement shape.

To answer your direct question, the ground state of the cluster state Hamiltonian is obtained by starting from $$|\phi_\ell\rangle|+\rangle|+\rangle\cdots|+\rangle|\phi_r\rangle$$ with arbitrary boundary conditions $|\phi_\bullet\rangle$, and applying a Controlled-Z operation between all adjacent sites.
• @wwwjjj Controlled-Z=$\mathrm{diag}(1,1,1,-1)$, so yes, they commute. If you want to understand the structure of the cluster state from a more CM point of view I suggest to read up on some papers discussing its SPT nature. Basically, the system (with the right symmetry) is in the Haldane phase. Jul 2, 2017 at 18:03