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 ...