Finite temperature calculations

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

Static calculations

Finite temperature effects are introduced by adding quasiparticle occupation probabilities in form of 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 inverse of temperature. In zero temperature limit this function reduced to step function. Quasiparticle energies $E_n$ are taken from solution of Bogoliubov-de Gennes type equations

$$H_{\textrm{BdG}} \begin{pmatrix}u_{n\uparrow}(r) \\ v_{n\downarrow}(r)\end{pmatrix}= E_n\begin{pmatrix}u_{n\uparrow}(r) \\ v_{n\downarrow}(r)\end{pmatrix}$$

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

Presently, definition of densities is the only place where temperature enters into calculation process. In general finite temperature DFT allows energy density functional $\mathcal{E}_{\textrm{edf}}$ to be temperature dependent, which is not the case neither (A)SLDA or BdG functionals.

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

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

In order to activate finite temperature calculations you need to uncomment in 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 convergence properties of algorithm, especially in context of 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 W-SLDA Toolkit the temperature effects are introduced to td-wslda codes as follow:

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

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

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