VERSION>=2021.09.18

Tracking the states

W-SLDA Toolkit allows the tracking of observables arising from selected states in time. Presently, the 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:

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 by enabling

subsetShiftDmu          1       # if 1 then apply extra shift of quasiparticle energies by (mu_a-mu_b)/2, default=0

Limitations

The states to be tracked and contracted into subset_rho and subset_j are tagged at the beginning of the simulations. It means that we select states based on their energies at t=0. However, as the dynamics proceed energies of the states can change as well, so some of them can acquire energy that is out of the given range, and some that were not included initially can acquire energy within the considered energy range. These effects are not taken into account by the presented prescription.

Example

Below we present a snapshot from a simulation with SLDA functional for the spin-symmetric system. td-wslda-2d has been used. The unitary Fermi gas is confined in a tube, and the vortex dipole is imprinted in the initial state. Following options for subset tracking were used

# ---------------- SUBSET TRACKING ------------------
# See: Wiki -> Tracking of selected states
# observables (densities, currents) arising from state in the energy interval En in [subsetMinEn,subsetMaxEn]
# will be computed, if  subsetMinEn=subsetMaxEn=0  the functionality is disables
subsetMinEn             -0.35     # in eF units, deafault=0.0
subsetMaxEn             0.35     # in eF units, deafault=0.0
subsetShiftDmu          1       # if 1 then apply extra shift of quasiparticle energies by (mu_a-mu_b)/2, default=0

For the unitary Fermi gas \Delta/\varepsilon_{F}=0.5, thus as the subset we select only in-gap states, which in this case corresponds to Andreev vortex states. They are localized in the vortex core. It is visualized below: the left half shows \Delta(r)/\varepsilon_{F}, while the right half shows subset_rho_a.