Transverse-field Ising model

Example

Let $L \ge 2$ and let $\mathcal{H} = (\C^2)^{\otimes L}$ be the tensor product space of $L$ spins $\tfrac12$. Let $\sigma^x_j, \sigma^z_j$ be the Pauli matrices acting on site $j$, with $\sigma^z_{L+1} = \sigma^z_1$. For $J > 0$ and $h \ge 0$, the transverse-field Ising chain is the lattice model with Hamiltonian $$H = -J \sum_{j=1}^{L} \sigma^z_j \sigma^z_{j+1} - h \sum_{j=1}^{L} \sigma^x_j.$$ The first term is the Ising model coupling and the second is a magnetic field perpendicular to the Ising axis. Check at the two ends. For $h = 0$ the ground states are the two aligned states $\lvert\uparrow\cdots\uparrow\rangle$ and $\lvert\downarrow\cdots\downarrow\rangle$, with energy $-JL$. For $J = 0$ the unique ground state has every spin in the $+1$ eigenstate of $\sigma^x$, with energy $-hL$.