Extending the \(N\)-vector model to a classical lattice field theory, it can be written as, for instance
\[ \mathcal{H} = -\beta\sum_{\langle i,j\rangle} \phi_i \phi_j+m\sum_{i}\phi_i^2+\lambda\sum_{i}\left(\phi_i^2-1\right)^2, \]
where \( \phi_i \) denotes an \(N\)-component real variable. Compared to the standard Ising Hamiltonian [compare here], an additional mass term as well as a fourth-order interaction have been added. For any positive \(\lambda\), this system undergoes a continuous phase transition which lies in the universality class of the \( O(N) \) model. The parameter \(\lambda\) can be particularly useful in a numerical analysis, as oftentimes it can be tuned such that leading scaling corrections approximately vanish. In this case one speaks of an improved Hamiltonian. For \(\lambda\to\infty\) the classical XY, Heisenberg and higher-symmetry vector-models are recovered, as the field is effectively forced to unit-length \(\phi_i^2=1\).
To be continued ...