Functionals.md

@@ -70,6 +70,19 @@ There meaning is presented on the figure below.
 ![stabilization](uploads/740d294dbfb30a203cabec0b140fa326/stabilization.png)   
 Smooth transition between densities `ASLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY` and `ASLDA_STABILIZATION_RETAIN_ABOVE_DENSITY` is introduced in order to avoid discontinuities for quantities, that may lead to divergences when computing derivatives.
 
+## Chemical potential
+The spin-symmetric and uniform unitary Fermi is expected to be scale-invariant where the following relations for the total energy and the chemical potential satisfy: $`E=\xi E_\textrm{ffg}`$ and $`\mu=\xi \varepsilon_{\textrm{F}}`$, where $`\xi\approx 0.4`$ is the Bertch parameter. However, in numerical realization, the relation for the chemical potential is satisfied only in the limit of very low densities. It is demonstrated in the table below, where results for the uniform solution obtained on lattice `128x128x128` is presented, and $`k_{\textrm{F}}=\sqrt{2\varepsilon_{\textrm{F}}}=(6\pi^2 n_{\uparrow})^{1/3}`$.  
+
+|$`k_{\textrm{F}}`$| $`E/E_\textrm{ffg}`$ | $`\mu/\varepsilon_{\textrm{F}}`$|
+| -----------------|----------------------|---------------------------------|
+| 1.0 | 0.397 | 0.431 |
+| 0.8 | 0.397 | 0.423 |
+| 0.6 | 0.397 | 0.416 | 
+| 0.4 | 0.397 | 0.409 | 
+| 0.2 | 0.397 | 0.403 | 
+
+It is seen that the total energy $`E/E_\textrm{ffg}=\xi`$ satisfies the relation all the time, while the chemical potential $`\mu/\varepsilon_{\textrm{F}}\rightarrow\xi`$ only in low-density limit. 
+ 
 # SLDA - superfluid local density approximation
 The term in ASLDA functional that depends on the currents $`\bm{j}_{\sigma}`$ introduces a significant cost to the computation. This term is responsible for maintaining Gallilean invariance of the ASLDA theory. The effective mass was found to be consistent with the bare mass to within 10% for a large range of polarizations,  Therefore, in many applications, one can set $`\alpha_{\sigma}=1`$ without losing qualitative features of ASLDA theory. Under this ausumptin we obtain SLDA functional:
 ```math