the fermionic field operator that creates \[annihilates\] a $\sigma$ species fermion at position $x$. $|g^{(1)}_{\sigma}(x,x';t)|$ is bounded within the interval $[0,1]$ and measures the proximity of the MB state to a product state for a fixed set of coordinates $x$, $x'$. Two different spatial regions $R$, $R'$, with $R \cap R' = \varnothing$, possessing $|g^{(1)}_{\sigma}(x,x';t)|= 0$ with $x\in R$ and $x'\in R'$ are referred to as perfectly incoherent, whilst for $|g^{(1)}_{\sigma}(x,x';t)|= 1$, $x\in R$, $x'\in R'$ the regions are said to be fully coherent. When at least two distinct spatial regions are partially incoherent, i.e. $|g^{(1)}_{\sigma}(x,x';t)|<1$ this signifies the emergence of one-body intraspecies correlations, while their absence is designated by $|g^{(1)}_{\sigma}(x,x';t)|=1$ for every $x$, $x'$. Most importantly, the situation where a certain spatial region $R$ is fully coherent, i.e. $|g^{(1),\sigma}(x,x';t)|^2 \approx 1$ $x,x'\in R$, and perfect incoherence occurs between different spatial regions $R$, $R'$, i.e. $|g^{(1),\sigma}(x,x';t)|^2\approx 0$, $x\in R$, $x'\in R'$ with $R \cap R' = \varnothing$, indicates the appearance of Mott-like correlations [@sherson2010single; @larson2008mott; @katsimiga2017dark; @mistakidis2017correlation].

To estimate the degree of second order intra- and interspecies correlations in the course of the dynamics, we inspect the normalized two-body correlation function [@sakmann2008reduced; @mistakidis2017correlation] $$\begin{aligned}
g^{(2)}_{\sigma\sigma\prime}(x,x^\prime;t)=\frac{\rho_{\sigma\sigma\prime}^{(2)}(x,x^\prime;t)}{\rho_\sigma^{(1)}(x;t)\rho_{\sigma\prime}^{(1)}(x^\prime;t)}. \label{two_body_cor}\end{aligned}$$ In Eq. (\[two\_body\_cor\]), $\rho^{(2)}(x,x';t)=\langle\Psi(t)|\hat{\Psi}^{\sigma,\dagger}(x')\hat{\Psi}^{\dagger,\sigma^\prime}(x)\hat{\Psi}^{\sigma^\prime}
