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 ...