of $\phi^{\sigma}_i(x;t)$ are the so-called natural populations $n^{\sigma}_i(t)$. If more than $N_\sigma$ natural populations, $n_i(t)$, possess a non-negligible occupation ($0<n_i(t)<1$ with $N_{\sigma}<i<m^{\sigma}$), the fermionic $\sigma$-species is termed intraspecies correlated, otherwise the MB state reduces to the HF ansatz [@pethick2002bose; @giorgini2008theory; @pitaevskii2016bose]. Indeed ML-MCTDHX enables us to operate within different approximation orders [@ML-MCTDHX], and we e.g. retrieve the HF ansatz [@pethick2002bose] in the limit of $D=1$ and $m^{\sigma}=N_{\sigma}$ $$\begin{split}
&\Psi_{HF}(\vec x^{A},\vec x^{B};t) =\\         &\prod_{\sigma=A,B}\sum_{i=1}^{N_{\sigma}!} {\rm sign}(\mathcal{P}_i) \mathcal{P}_i
\left[\varphi_1(x_1^{\sigma};t) \cdots \varphi_{N_{\sigma}}(x_{N_{\sigma}}^{\sigma};t) \right].  
\label{Eq:HF}
\end{split}$$ Furthermore, employing the Dirac-Frenkel variational principle [@frenkel1932wave; @dirac1930note] for the MB ansatz \[see Eqs. (\[Eq:WF\]), (\[Eq:SPFs\])\] we obtain the ML-MCTDHX equations of motion [@ML-MCTDHX] for the fermionic mixture. These equations correspond to $D^2$ linear differential equations of motion for the coefficients $\lambda_i(t)$ coupled to a set of $D$\[${m_A}\choose{N_A}$+$ {m_B}\choose{N_B}$\] non-linear integro-differential equations for the species functions and $m^A+m^B$ integro-differential equations for the SPFs.

Correlation Functions {#def_cor_functions}
---------------------

To unveil the degree of intraspecies correlations at the one-body level during the quench dynamics we employ the normalized spatial first order correlation function [@naraschewski1999spatial; @sakmann2008reduced; @mistakidis2017correlation] $$\begin{aligned}
g^{(1)}_\sigma(x,x^\prime;t)=\frac{\rho_\sigma^{(1)}(x,x^\prime;t)}{\sqrt{\rho_\sigma^{(1)}(x;t)\rho_\sigma^{(1)}(x^\prime;t)}}.\label{one_body_cor}\end{aligned}$$ Here, $\rho_\sigma^{(1)}(x,x^\prime;t)=\langle\Psi(t)|\hat{\Psi}^{\sigma,\dagger}(x)\hat{\Psi}^\sigma(x^\prime)|\Psi(t)\rangle$ refers to the one-body reduced density matrix of the $\sigma$ species and $\rho_\sigma^{(1)}(x;t)\equiv\rho_\sigma^{(1)}(x,x^\prime=x;t)$ is the one-body density. $\hat{\Psi}^{\sigma,\dagger}(x)$ \[$\hat{\Psi}^{\sigma}(x)$\] is 