Tracking-of-selected-states.md

+**VERSION>=2021.09.18**
+
+# Tracking the states
+W-SLDA Toolkit allows to track observables arsing from selected states in time. Presently following quantities can be tracked:
+* `subset_rho_a`:  
+$`n_{\uparrow}^{\textrm{(subset)}}(r)= \sum_{E_{\textrm{min}}^{\textrm{(subset)}}<E_n<E_{\textrm{max}}^{\textrm{(subset)}}}|u_{n,\uparrow}(r)|^2 f_{\beta}(E_n)`$
+* `subset_rho_b`:  
+$`n_{\downarrow}^{\textrm{(subset)}}(r) = \sum_{E_{\textrm{min}}^{\textrm{(subset)}}<E_n<E_{\textrm{max}}^{\textrm{(subset)}}}|v_{n,\downarrow}(r)|^2 f_{\beta}(-E_n)`$
+* `subset_j_a_x`, `subset_j_a_y`, `subset_j_a_z`:  
+$`\vec{j}_{\uparrow}^{\textrm{(subset)}}(r) = -\sum_{E_{\textrm{min}}^{\textrm{(subset)}}<E_n<E_{\textrm{max}}^{\textrm{(subset)}}} \textrm{Im}[u_{n,\uparrow}(r)\nabla u_{n,\uparrow}^*(r)]f_{\beta}(E_n)`$  
+* `subset_j_b_x`, `subset_j_b_y`, `subset_j_b_z`:  
+$`\vec{j}_{\downarrow}^{\textrm{(subset)}}(r) = \sum_{E_{\textrm{min}}^{\textrm{(subset)}}<E_n<E_{\textrm{max}}^{\textrm{(subset)}}} \textrm{Im}[v_{n,\downarrow}(r)\nabla v_{n,\downarrow}^*(r)]f_{\beta}(-E_n)`$   
+
+To control interval range $`E_{\textrm{min}}^{\textrm{(subset)}}<E_n<E_{\textrm{max}}^{\textrm{(subset)}}`$ use `input` file variables:
+```bash
+subsetMinEn             0.0     # in eF units, deafault=0.0
+subsetMaxEn             0.0     # in eF units, deafault=0.0
+```
+
+In the case of spin-imbalanced systems, it is also convenient to introduce a shift of quasiparticle energies when selecting the states: $`E_n \rightarrow E_n+\frac{\mu_{\uparrow}-\mu_{\downarrow}}{2}`$. This can be done automatically via enabling 
+```bash
+subsetShiftDmu          1       # if 1 then apply extra shift of quasiparticle energies by (mu_a-mu_b)/2, default=0
+```
+
+# Example
+TODO: Andrea
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