Renormalization-with-cubic-cutoff-older-versions.md

+**!!! APPLIES TO VERSION<2026.06.05 !!!** 
+
+# Renormalization with cubic cutoff
+**!!! THIS FEATURE IS EXPERIMENTAL - NEEDS MORE TESTING !!!**  
+```c
+#define REGULARIZATION_SCHEME CUBIC_CUTOFF
+```
+The effective coupling constant is computed according to the prescription:
+```math
+\dfrac{1}{g_{\textrm{eff}}}=\dfrac{1}{g_0} -   \dfrac{m}{2\alpha_+}\dfrac{K}{2\hbar^2\pi dx},
+```
+where $`K=2.442 75`$ is a numerical constant. In this formula, we assume that all states contribute to the densities. Physically, it means that we take into account states up to the maximal value of energy set by the lattice, which is of the order $`E_c\approx 3\frac{\hbar^2\pi^2}{2mdx^2}`$ (assuming that $`dx=dy=dz`$).  
+*Note*: when working with this renormalization scheme value of tag `ec` will be ignored.
+
+# Impact of the regularization scheme on the quality of static results
+Below, we present results for the uniform unitary Fermi gas as a function of gas density measured by $`k_F=(3\pi^2 n)^{1/3}`$ and scaled with respect to the cut-off momentum $`k_c=\pi/dx`$. Note that the lattice spacing defines the value of the energy cut-off $` E_c\approx\frac{k_c^2}{2}=\frac{\pi^2}{2dx^2}`$. It is a technical parameter, so the results should not depend on its choice (assuming that it is chosen from a reasonable range). We also note that typical applications utilize $`k_F dx\approx 1`$ (or $`k_F dx/\pi\approx 0.32`$).
+
+For the test, we used ASLDA functional with the parameters provided in https://arxiv.org/abs/1008.3933, and which were adjusted in such a way to provide for the spin-symmetric case:
+```math
+\frac{E}{E_{\rm{FG}}}=0.40(1),\quad \frac{\Delta}{\varepsilon_F}=0.504(24)
+```
+
+The graph below shows the sensitivity of the energy and the pairing gap as a function of the $`k_F/k_c`$, for the calculations with spherical cut-off. In this case, the energy is almost not sensitive to the choice of the $`k_F/k_c`$, while the pairing gap shows some residual dependence, but at an acceptable level. 
+![spehrical-cutoff](uploads/0fa452388ed3b8f200e11bb3f5838e68/spehrical-cutoff.png)
+
+The next graph shows the results of the same test, but with the activated cubic cutoff scheme. In this case, we observe quantitative degradation of the quality of the results. This issue needs further investigation.  
+![cubic-cutoff](uploads/fc0506f7d0323822452fe00b7ca981d3/cubic-cutoff.png)
+ 
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