## 3 Results
### The Miscibility Phase Diagram  
The ground-state configurations obtained in the previous section show the flexibility of the \({}^{23}\)Na-\({}^{39}\)K mixture in achieving different miscibility regimes with the change of the Feshbach magnetic field \(B\) and the atom number ratio \(\eta\). The construction of a phase diagram miscible-immiscible needs a more quantitative way of defining the miscibility region of a given set of parameters for the two-species system. Proposals to characterize the regime of such system include the calculation of the binder cumulant of the system's magnetization [56], the difference between the centers of mass of each atomic cloud [32], the study of the entropy of the mixture as defined in [44; 57] and the monitor of dipole oscillations of the atomic clouds in a harmonic trap [35]. Here, similar to the works presented in [29; 58], we follow the definition of the miscible and immiscible phases and propose the calculation of the spatial overlap between the atomic clouds to be an indicator of the phase transition. We define the spatial overlap of the atomic clouds as:  
\[O=\int_{-\infty}^{\infty}|\psi_{1}(\mathbf{r})|^{2}|\psi_{2}(\mathbf{r})|^{2} \mathrm{d}\mathbf{r}=\int_{-\infty}^{\infty}n_{1}(\mathbf{r})n_{2}(\mathbf{r })\mathrm{d}\mathbf{r}, \tag{11}\]  
where \(n_{1}(\mathbf{r})\) and \(n_{2}(\mathbf{r})\) are the atomic densities of species 1 and 2, respectively. In Figure 3, we show the spatial overlap normalized by the overlap, \(O_{0}=O(a_{12}=0)\), at each case as a function of the Feshbach field for different atom number ratio, \(\eta\). The normalization was performed considering that the case of vanishing inter-species interaction exhibits the maximum spatial overlap between the atomic clouds possible in each  
Figure 2: Density profiles along the \(z\)-axis of the simulated ground-state of \({}^{23}\)Na (in blue) and \({}^{39}\)K (in red): (**a**) \(B=100\) G with \(\delta=4.32\), (**b**) \(B=108\) G with \(\delta=0.22\) and (**c**) \(B=111\) G with \(\delta=-0.50\). In each case, we display the results of three atom number ratio \(\eta=N_{\text{Na}}/N_{\text{K}}\) equal to 50 (solid lines), 10 (dashed lines) and 5 (dotted lines). configuration. In the immiscible region (\(B<B_{0}=109.1\) G), \(O_{\rm norm}\) exhibit small values with \(O_{\rm norm}\to 0\) when reducing the magnetic field for all \(\eta\). Approaching \(B_{0}\), the spatial overlap increases differently for each \(\eta\) with larger \(\eta\) showing an earlier increase on the spatial overlap. The limit case of \(\eta=50\) shows a significant increase in \(O_{\rm norm}\) for \(B>104\) G, increasing over a broad range of magnetic fields. This increase occurs before the critical point estimated with the miscibility parameter, in accordance with was already observed in the density profiles of Figure 2. Reducing \(\eta\) increases the magnetic field for which the spatial overlap significantly increases. For the other limit with \(\eta=1\), the spatial overlap starts to increase for \(B>111\) G, a magnetic field larger than \(B_{0}\). The dashed vertical line in Figure 3 represents the critical point for the transition at \(B_{0}\) with \(\delta=0\). To define the transition from immiscible to miscible from the normalized overlap, we associate a threshold-like behaviour and identify the Feshbach field (\(B_{\rm peak}\)) for which \(O_{\rm norm}\) varies the most. This is performed performing the numerical second derivative of the normalized overlap and identifying its maximum value. The second derivatives as a function of the Feshbach field for all \(\eta\) are displayed in Figure 4. The maximum value of the curves drifts to larger magnetic fields as \(\eta\) decreases. In Figure 4b, we show \(B_{\rm peak}\) as a function of \(\eta\). The almost linear behaviour of the points in the semilog scale suggests a dependence of \(B_{\rm peak}(\eta)=B_{\rm peak}^{0}-\alpha\ln\eta\). We find \(B_{\rm peak}^{0}=113.5\) G and \(\alpha=1.70\).