Functionals.md

@@ -114,6 +114,32 @@ Below we provide predictions of BdG functional for total energy and paring gap a
 ![BdG-E](uploads/a3e8e87c80efd2c4a31d2ea50dec6191/BdG-E.png)  
 ![BdG-D](uploads/7529bf3d47e71594ef42d2b2b3a51865/BdG-D.png) 
 
+# SLDAE -  SLDA Extended
+SLDAE is extension of SLDA-type functional to arbitrary value of $`\lambda=|ak_F|`$. The functional has generic form:
+```math
+	\mathcal{E}  = A_\lambda \frac{\tau}{2}
+	        + \frac{3}{5}B_\lambda n \varepsilon_F
+	        + \frac{C_\lambda }{n^{1/3}} |\nu|^2
+                + (1 - A_\lambda) \frac{\vb{j}^2}{2n}
+``` 
+where functional parameters $`A_\lambda`$, $`B_\lambda`$ and $`C_\lambda`$ are constructed in such a way to reproduce properly quasiparticle properties in the whole regime of $`\lambda`$, and also satisfy constraints in limits $`\lambda\rightarrow 0`$ and $`\lambda\rightarrow \infty`$. Precisely, the SLDAE is constructed in such a way to reproduce data for the:
+* the ground-state energy per unit volume $`E = 3n\xi_\lambda \varepsilon_F/5`$, and thus corresponding chemical potential $`\mu/\varepsilon_F = \zeta_\lambda`$ which must verify  the thermodynamic relationship $`\zeta_\lambda = \xi_\lambda + (\lambda/5) \xi_\lambda^\prime`$,
+* the pairing gap function $`\Delta_\lambda=\eta_\lambda \varepsilon_F`$,
+* the effective mass of particle $`\alpha_\lambda=m/m^\star`$.
+The plot below shows $`\lambda=|ak_F|`$ dependence of these quantities in the SLDAE functional. 
+![sldae_qp_properties](uploads/bc7018319141055f9f032d7526a2dc9c/sldae_qp_properties.png)
+For more details see https://arxiv.org/abs/2201.07626.
+
+## SLDAE with $`m/m^\star=1`$
+The last term of SLDAE functional (depending on currents $`j`$) introduces a significant cost to the computation process. In many cases, it is sufficient to proceed with the functional that neglects corrections related to the effective mass (actually the effective mass is not known with high precision). The W-SLDA Toolkit provides a variant of the SLDAE functional where the constraint $`\alpha_\lambda = m/m^\star=1`$ is imposed. In order to activate it you need to use flag (in `predefines.h`):
+```c
+/**
+ * Meaningful only in case SLDAE.
+ * Sets effective mass to be equal, and speeds-up computation (approximately by a factor of two)
+ * */
+#define SLDAE_FORCE_A1
+```
+
 # Regularization scheme of pairing field
 By default, W-SLDA implements a regularization scheme of the pairing field known as Superfluid Local Density Approximation (SLDA). Precisely, the implemented formulas are (9.86)-(9.87) from paper [arXiv:1008.3933](https://arxiv.org/abs/1008.3933).