The W-SLDA Toolkit allows calculations at finite temperature. Here we describe how the temperature effects are implemented in the toolkit. We emphasize that at the conceptual level some of the assumptions may be incorrect.

Static calculations

Finite temperature effects are introduced by adding quasiparticle occupation probabilities in the form of the Fermi-Dirac function:

f_{\beta}(E_n)=\dfrac{1}{\exp(\beta E_n)+1}

to definition of densities (see here from explicit formulas). Here \beta=1/T is the inverse of temperature. In the zero temperature limit, this function reduces to a step function. Quasiparticle energies E_n are taken from the solution of Bogoliubov-de Gennes type equations

H_{\textrm{BdG}} \begin{pmatrix}u_{n,a}(r) \\ v_{n,b}(r)\end{pmatrix}= E_n\begin{pmatrix}u_{n,a}(r) \\ v_{n,b}(r)\end{pmatrix}

See here from explicit form of H_{\textrm{BdG}}.

Presently, the definition of densities is the only place where temperature enters into the calculation process. In general, finite temperature DFT allows energy density functional \mathcal{E}_{\textrm{edf}} to be temperature dependent, which is not the case for either (A)SLDA or SLDAE functionals.

Below, we present a collection of predictions from ASLDA for functional when applied to 3D uniform and spin-symmetric systems. Figures are taken from the diploma thesis of Aleksandra Olejak, WUT, 2017 .

From these plots, it is clearly seen that the critical temperature is located around T_c\approx0.3\varepsilon_F, whereas the experimentally measured critical temperature is T_c^{(\textrm{exp})}\approx0.15\varepsilon_F. It clearly demonstrates that the current formulation of ASLDA can at best simulate finite-temperature effects at a qualitative level.

In order to activate finite temperature calculations, you need to uncomment in the input file:

temperature             0.1      # requested temperature in units of eF, default T=0

Recommendation: For numerical purposes, it is convenient to introduce a very small temperature (much smaller than any other energy scale in the system). It greatly improves the convergence properties of the algorithm, especially in spin-imbalanced systems.

Time-dependent calculations

Conceptually correct introduction of temperature effects into time-dependent calculations is still an open question. For integrity purposes of the W-SLDA Toolkit, the temperature effects are introduced to td-wslda codes as follows:

1. All densities are calculated with the included Fermi-Dirac function, for example:

n_{a}(\bm{r},t)= \sum_{|E_n|<E_c}|u_{n,a}(\bm{r},t)|^2 f_{\beta}(E_n)

2. values of E_n are taken from the initial (static) solution and kept frozen over the entire time evolution.