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

Question

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.

Answer 1

This Hamiltonian is known in quantum information as the cluster state Hamiltonian (though usually with some additional boundary terms). Its ground state, the cluster state, is a resource for measurement based quantum computing (in two dimensions). The cluster state exhibits symmetry protected (SPT) order, this is, it has a non-trivial entanglement structure and exhibits hidden long-range order witnessed by string order parameters, and it has been found that this SPT order protects the ability to perform measurement based computation. The SPT order also relates to the 4-fold degenerate ground state with open boundary conditions.

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.

You can find a lot of papers on the topic when searching for the corresponding keywords (cluster states, SPT, measurement based computation). If you ask more specific questions, I can try to answer them more specifically.