Regularization-schemes-of-the-pairing-field.md

@@ -5,13 +5,23 @@ procedure is required. There are two predefined regularization schemes that can
 ```c
 /**
  * Scheme of pairing field renormalization procedure. 
- * SPHERICAL_CUTOFF: use spherical momentum space cutoff, in this case, you need to set `ec` variable in the input file (default).
- * CUBIC_CUTOFF: use cubic momentum space cutoff, in this case `ec` will be set to infinity automatically.
+ * For more info see: Wiki -> Regularization schemes of the pairing field
+ * Select one:
+ * SPHERICAL_CUTOFF: use spherical momentum space cutoff; in this case, you need to set the `ec` variable in the input file (default).
+ * CUBIC_CUTOFF: use cubic momentum space cutoff; in this case, `ec` will be set to infinity automatically.
  * */
 #define REGULARIZATION_SCHEME SPHERICAL_CUTOFF
 // #define REGULARIZATION_SCHEME CUBIC_CUTOFF
+
+/**
+ * Enable this flag to include mean-field correction from the regularization scheme.
+ * For more info see: Wiki -> Regularization schemes of the pairing field
+ */
+#define INCLUDE_MF_CORRECTION_FROM_REG_SCHEME
 ```
 
+Note that the regularization scheme is a subtle issue in superfluid variants of DFT, and there is no unique prescription for it. To learn more about the regularization scheme, read Regularization of the functional in section [Theoretical framework](https://arxiv.org/html/2602.08982#S2) in our manuscript [arXiv:2602.08982 ](https://arxiv.org/abs/2602.08982).
+
 # Renormalization with spherical cutoff
 ```c
 #define REGULARIZATION_SCHEME SPHERICAL_CUTOFF
@@ -43,16 +53,12 @@ For more info, see [arXiv:1008.3933](https://arxiv.org/abs/1008.3933).
 *Note*: the spherical cut-off scheme results in a significant decrease in memory consumption and improved performance of `td` codes. 
 
 # Renormalization with cubic cutoff
-**!!! THIS FEATURE IS EXPERIMENTAL - NEEDS MORE TESTING !!!**  
+If you are using VERSION<2026.06.05, see [here](Renormalization-with-cubic-cutoff-older-versions).  
 ```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. 
+The overall procedure is the same as for spherical cutoff, with the difference that in this scheme all quasiparticle states are included. 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, the value of tag `ec` will be ignored. 
 
 # Custom renormalization scheme
 Static codes enable you to define your own renormalization scheme. You need to provide the formula in `void modify_potentials(...)` function. See [here](Strict 2D or 1D mode) for example. 
@@ -71,17 +77,16 @@ It is clearly visible that for evolution with the cubic cutoff, energy is conser
 
 In conclusion, we find that typically for trajectories of length $`te_F\approx1000`$ the spherical cutoff provides reasonable accuracy, while for the generation of long trajectories $`te_F\gg 1000`$ it is recommended to use the cubic cutoff. 
 
-# 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 $`k_F dx/\pi\approx 0.32`$).
+# Correction to the mean-field
+The regularization function may itself depend on the densities. This is the case for the spherical and cubic cutoff schemes implemented in the toolkit by default. As a consequence, the regularization procedure generates an additional contribution to the mean-field potential, which in density functional theory is defined as $`V_{\sigma}=\dfrac{\delta\mathcal{E}_{\textrm{edf}}}{\delta n_{\sigma}}`$. 
 
-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)
+To include this correction in the calculation, add the following definition to `predefines.h`:
+```c
+#define INCLUDE_MF_CORRECTION_FROM_REG_SCHEME
 ```
+The correction is usually small and does not typically affect the qualitative behavior of the solution. It may, however, produce noticeable quantitative changes, particularly in the values of the chemical potentials.
 
-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)
+Below, we show an example illustrating how the mean-field correction affects the total energy and chemical potential obtained with the SLDA functional for a uniform system. The results were generated using the spherical cutoff scheme and demonstrate the dependence on the cutoff momentum $`k_c=\sqrt{2E_c}`$. In principle, physical observables should be insensitive to this technical cutoff parameter. When the mean-field correction is included, both the energy and the chemical potential remain essentially independent of the cutoff. Moreover, the expected relation for a unitary Fermi gas is recovered: $`E/E_{FG}=\mu/\varepsilon_F`$. When the correction is disabled, the chemical potential begins to deviate from this relation, while the total energy remains largely unaffected.
+![mf-correction](uploads/fca7d14a783df7ee50a08f388d3e8823/mf-correction.png)
 
+**Note**: When using a custom regularization scheme, the mean-field correction should be disabled.
\ No newline at end of file