Transverse-field Ising model (TFI) is a true quantum Ising model. In TFI, there is a transverse field term $\propto \hat{\sigma}^x$, while the remaining term is the classical Ising model term $\propto \hat{\sigma}^z\hat{\sigma}^z$. These two terms, $\hat{\sigma}^x$ and $\hat{\sigma}^z\hat{\sigma}^z$, are not commute, $[\hat{\sigma}^x,\hat{\sigma}^z\hat{\sigma}^z]\neq 0$, and it makes a real quantum effect.

$$ \hat{H}{\rm TFI} = -J \sum{(j,k)\in \rm{n.n}} \hat{\sigma}^z_j\hat{\sigma}^z_k - \Gamma \sum_j \hat{\sigma}^x_j \\ =-J \sum_{(j,k)\in \rm{n.n}} \hat{\sigma}^z_j\hat{\sigma}^z_k - Jg \sum_j \hat{\sigma}^x_j $$

where $g = \Gamma/J$

$J \ll \Gamma \quad (g \gg 1)$: External field dominates. Spins are aligned to the direction of the external field;

The ground state is paramagnet (PM)

$$ \hat{H}_{\rm TFI} \sim - \Gamma \sum_j \hat{\sigma}^x_j $$

$$ \left|\psi_g\right> = \left|\rightarrow\rightarrow\right> \\

\left|\rightarrow \right>=\frac{\left|\uparrow\right>+\left|\downarrow\right>}{\sqrt{2}}\\

\hat{\sigma}^x \left|\rightarrow\right> = \left|\rightarrow\right> $$

$J\gg \Gamma \quad(g\ll 1)$: Nearest neighbor (NN) interaction dominates. Spins are aligned to the internal ground state; If $J>0$, The ground state is ferromagnet (FM)

$$ \hat{H}{\rm TFI} \sim -J\sum{(j,k)\in \rm{n.n}} \hat{\sigma}^z_j\hat{\sigma}^z_k $$

$$ \left|\psi_g\right> = \left|\uparrow\uparrow\right> \\ \left|\psi_g\right> = \left|\downarrow\downarrow\right> $$

If $J>0$, The ground state is anti-ferromagnet (AFM)

$$ \left|\psi_g\right> = \left|\uparrow\downarrow\right> \\ \left|\psi_g\right> = \left|\downarrow\uparrow\right> $$

2nd order quantum phase transition

The order parameter: magnetization