VERSION>=2021.09.18

Tracking the states

W-SLDA Toolkit allows to track observables arising 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:

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

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.