Classical Ising Hamiltonian operator

The classical Ising Hamiltonian on a chain of $n$ sites with nearest-neighbor coupling $J$ and longitudinal field $h$ is

$H = -J\sum_{i=1}^{n-1} Z_i Z_{i+1} - h\sum_{i=1}^{n} Z_i.$

Diagonal structure. Every term is a product of $Z$ operators, and $Z$ is diagonal in the computational basis with eigenvalues $\pm1$. Consequently $H$ itself is diagonal: for a basis state $|s_1\cdots s_n\rangle$ with $Z_i|s_i\rangle = (-1)^{s_i}|s_i\rangle$, writing $\sigma_i = (-1)^{s_i} \in \{+1,-1\}$,

$H|s_1\cdots s_n\rangle = \Big(-J\sum_i \sigma_i\sigma_{i+1} - h\sum_i \sigma_i\Big)|s_1\cdots s_n\rangle,$

i.e. every computational basis state is already an exact eigenstate, with eigenvalue equal to the classical Ising energy of that spin configuration. No diagonalization or simulation is required to find the spectrum — it is read off termwise.

Physical content. For $J>0$ (ferromagnetic) the ground state(s) align all spins ($\sigma_i$ all $+1$ or all $-1$ when $h=0$, degenerate); for $J<0$ (antiferromagnetic) on a bipartite lattice the ground state alternates. The field $h$ breaks the up/down degeneracy by favoring one alignment. Because $[H, Z_i]=0$ for every $i$, there is no term driving transitions between basis states — this is a classical Hamiltonian dressed in quantum notation, and running it alone on a quantum computer produces no dynamics beyond an overall phase per basis state.

Role in the catalog. This diagonal operator is the baseline against which the transverse-field Ising model (adding $-h_x\sum X_i$, which does not commute with $Z_i$) is compared: the transverse term is exactly what turns a classical statistical-mechanics model into a quantum many-body Hamiltonian with a genuine phase transition driven by quantum fluctuations rather than thermal fluctuations alone.