model $$\label{Sorp}
S = - K \sum_{\langle n,n' \rangle} \phi_n \cdot
  \phi_{n'} + \sum_n \phi^2_n + \lambda \sum_n \left(
  \phi_n^2 - N \right)^2.$$ Each site on the hypercubic spacetime lattice is now specified by the vector $n = n_\mu = (n_1, n_2, \cdots , n_d)$ with integer components $n_\mu = x_\mu/a$ and the sum $\sum_{\langle n,n' \rangle}$ extends over all nearest neighbor pairs. In terms of these new dimensionless variables, the action is independent of the lattice spacing $a$. The partition function $Z$ now reads $$Z = \int \mathrm{D} \mu(\phi) \exp \left(K \sum_{\langle n,n'
    \rangle} \phi_n \cdot \phi_{n'} \right),$$ with the on-site measure $$\label{onsite} 
  \int \mathrm{D} \mu(\phi) = \int \prod_n \mathrm{d} \phi_n \;
      \mathrm{e}^{-\phi^2_n - \lambda \left( \phi_n^2 - N \right)^2}.$$ In the limit $\lambda \to \infty$, the field theory reduces to the standard O($N$) spin model, with a “spin” variable $\phi_n$ of fixed length, $\phi_n^2 = N$, located at each site of the spacetime lattice.

### Noninteracting Theory {#sec:noninteracting}

We first consider the limit $\lambda \to 0$ in Eq. (\[Sorp\]), corresponding to the noninteracting field theory. The rescaling factor $K$ introduced in Eqs. (\[K1\])-(\[K3\]) then becomes $$K = \frac{1}{2d + m^2 a^2}$$ and the action reduces to the quadratic 