## Appendix A Numerical Method SMPIMC
### Smpimc Algorithm for PCFs  
Calculation of PCFs with the SMPIMC method is based on the definition (5) and representation (26) of the diagonal density matrix \(\rho_{DG}\). Integration over the variables \(\mathbf{q}_{a}\) and \(\mathbf{z}_{a}^{k}\) is performed using the Monte Carlo approach. Electron \(a\) in the simulation box is represented as the closed poly-line with vertices \(\mathbf{q}_{a}^{k}=\mathbf{q}_{a}+\mathbf{z}_{a}^{k}\), so that the \"center\"-coordinate \(\mathbf{q}_{a}\) describes the position of the whole \(a\)-th electron, and the \"bead\"-coordinate \(\mathbf{z}_{a}^{k}\) -- the relative position of the \(k\)-th bead. Note that only coordinates \(\mathbf{q}_{a}\) are used in the calculations of PCFs directly, while the beads are \"inner\" coordinates and define the weight and probability density of the configuration. The detailed SMPIMC algorithm for the calculation of PCFs is presented below. 1. Set the run number \(l=0\) and the initial state \(\mathbf{x}_{0}\): the coordinates \(\mathbf{q}_{a}\) are uniformly distributed in the simulation box, the coordinates \(\mathbf{z}_{a}^{k}\) are equal to zero (\(a=1,\ldots,N\), \(k=0,\ldots,M\)). 2. Set the step number \(i=1\) and the first state \(\mathbf{x}_{1}=\mathbf{x}_{0}\). 3. Select the particle number \(a=1,\ldots,N\) randomly, then select the type of the step: \(\delta q\)-step with the probability \(P_{q}\) or \(\delta z\)-step with the probability \(P_{z}=1-P_{q}\). If \(\delta q\)-step has been chosen, modify \(\mathbf{q}_{a}\to\mathbf{q}_{a}+\Delta\mathbf{q}\) with \(\Delta\mathbf{q}\) uniformly distributed in the volume \(\sim L^{3}/N\). If \(\delta z\)-step has been chosen, select the bead number \(k=1,\ldots,M-1\) randomly and modify \(\mathbf{z}_{a}^{k}\to\mathbf{z}_{a}^{k}+\Delta\mathbf{z}\) with \(\Delta\mathbf{z}\) uniformly distributed in the volume \(\sim\lambda^{3}/(4\pi K)^{3/2}\). In both cases, take into account the PBC. The resulting state has to be set as the proposed state \(\mathbf{x}_{i}^{\prime}\). 4. Accept the proposed state \(\mathbf{x}_{i}^{\prime}\) with the probability \(A(\mathbf{x}_{i}\to\mathbf{x}_{i}^{\prime})\) (10) or reject it. In the case of acceptance -- set \(\mathbf{x}_{i+1}=\mathbf{x}_{i}^{\prime}\), in the case of rejection -- set \(\mathbf{x}_{i+1}=\mathbf{x}_{i}\). 5. Calculate the distances between each pair of electrons \(r_{ab}=|\mathbf{q}_{a}-\mathbf{q}_{b}|\) and build the related histograms: \(h_{uu}^{n}(\mathbf{x}_{i})\), \(h_{dd}^{n}(\mathbf{x}_{i})\) and \(h_{ud}^{n}(\mathbf{x}_{i})\), where \(n=1,\ldots,N_{q}\) is the cell number with the length \(\Delta\), so \(n=\lceil\tau_{ab}/\Delta_{r}\rceil+1\). 6. Repeat steps (3)--(5) for \(i=1,\ldots,N_{steps}\). 7. Calculate the average histograms for the obtained sample of \(N_{steps}\) states via averaging of \(h_{uu}^{n}\), \(h_{dd}^{n}\) and \(h_{ud}^{n}\) with \(g(\mathbf{x})\) as the weight function: \[\langle h_{s_{1}s_{2}}^{n}\rangle_{l}=\frac{\sum_{i=1}^{N_{steps}}h_{s_{1}s_{ 2}}^{n}(\mathbf{x}_{i})\,g(\mathbf{x}_{i})}{\sum_{i=1}^{N_{steps}}g(\mathbf{x }_{i})}\quad(s_{1},s_{2}=\{u,d\}).\]8. Repeat steps (2)-(7) for \(l=1,2,\ldots,N_{runs}\), but instead of initialization \(\mathbf{x}_{1}=\mathbf{x}_{0}\) use the last state from the previous run: \(\mathbf{x}_{1}\big{|}_{l}=\left(\mathbf{x}_{N_{steps}}\right)\big{|}_{l-1}\). 9. As a result, the sample of average histograms \(\langle h_{s_{1}s_{2}}^{n}\rangle_{l},l=0,1,\ldots,N_{runs}\) is obtained. Considering the 0-th run as idle and omitting it to eliminate the influence of the initial state, calculate the resulting average histograms over the sample and the statistical errors as follows: \[\langle h_{s_{1}s_{2}}^{n}\rangle=\frac{\sum_{l=1}^{N_{runs}}\langle h_{s_{1} s_{2}}^{n}\rangle_{l}}{N_{runs}},\quad\sigma(h_{s_{1}s_{2}}^{n})=\sqrt{\frac{ \sum_{l=1}^{N_{runs}}(\langle h_{s_{1}s_{2}}^{n}\rangle_{l}-\langle h_{s_{1} s_{2}}^{n}\rangle)}{N_{runs}}}.\]
10.