coordinates, representing discretized spacetime, are specified by $x =
x_\mu= (x_1, x_2, \cdots , x_d)$, with $\mu=1,2,\cdots , d$ and $x_d=\tau$ denoting the imaginary time coordinate, while $\hat{\mu}$ denotes the unit vector pointing in the $\mu$-direction. In the continuum limit $a \to 0$, the lattice action reduces to $$\label{Scont}
S = \int \mathrm{d}^d x \left\{ \frac{1}{2} \left[\partial_\mu \varphi(x)
\right]^2 + \frac{m^2}{2} \varphi^2(x) + \frac{g}{4!}  \varphi^4(x)
\right\},$$ where $\varphi(x)$ stands for the field defined in continuous spacetime. Unless otherwise indicated, natural units $\hbar = c = 1$ are adopted throughout. The trace Tr in Eq. (\[Z\]) stands for the sum or integral over all possible field configurations: $$\label{Tr} 
\mathrm{Tr} =  \prod_x \int \mathrm{d} \varphi_x.$$ In the continuum limit, where the lattice spacing is taken to zero $a
\to 0$, the right hand defines the *functional* measure denoted by $\int\mathrm{D} \varphi$.

For numerical simulations, a more convenient form of the lattice action is obtained by casting Eq. (\[Sor\]) in terms of dimensionless fields and parameters defined via[@JanSmit] $$\begin{aligned}
\label{K1}
a^{d-2} \varphi^2_x &=& 2 K \, \phi_n^2 \\ 
\label{K2} 
a^{4-d} g &=& 6 \frac{\lambda}{K^2} \\ 
\label{K3}
m^2 a^2  &=& \frac{1-2\lambda N}{K} -2 d.\end{aligned}$$ The action then takes the form of an O($N$) spin 