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

+[[_TOC_]]
+# Inroduction
+W-SLDA Toolkit utilizes a local pairing field $`\Delta(\bm{r})`$. In such case renormalization
+procedure is required. There are two predefined regularization schemes that can be selected in `predefines.h` file:
+```c
+/**
+ * Scheme of pairing field renormalization procedure. 
+ * SPHERICAL_CUTOFF: use spherical momentum space cutoff, in this case you need to set `ec` variable in 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
+```
+
+# Renormalization with spherical cutoff
+```c
+#define REGULARIZATION_SCHEME SPHERICAL_CUTOFF
+```
+The effective coupling constant is computed according to prescription:
+```math
+\dfrac{1}{g_{\textrm{eff}}}=\dfrac{1}{g_0} -   \dfrac{m}{2\alpha_+}\dfrac{k_c}{\hbar^2\pi^2},
+  \left\{
+    1 - \frac{k_0}{2k_c}
+    \ln\frac{k_c + k_0}{k_c - k_0}
+  \right\}
+```
+where 
+```math
+\begin{aligned}
+    \frac{\hbar^2}{2m}\alpha_{+}(\vec{r})k_0^2(\vec{r}) -
+    \mu_{+}(\vec{r})
+    &=0, \\
+    \frac{\hbar^2}{2m}\alpha_{+}(\vec{r})k_c^2(\vec{r}) -
+    \mu_{+}(\vec{r}) 
+    &= E_c.
+\end{aligned}
+```
+with $`\mu_{+} = (\mu_a - V_a + \mu_b - V_b)/2`$ being average local chemical potential and $`\frac{m}{2\alpha_+}=m_r`$ is reduced mass. $`E_c`$ stands for energy cutoff scale that can be controlled by tag:
+```bash
+# ec                      4.9348022 # energy cut-off for regularization scheme, default ec = 0.5*(pi/DX)^2
+```
+For more info see [arXiv:1008.3933](https://arxiv.org/abs/1008.3933). 
+
+# Renormalization with cubic cutoff
+```c
+#define REGULARIZATION_SCHEME CUBIC_CUTOFF
+```
+The effective coupling constant is computed according to prescription:
+```math
+\dfrac{1}{g_{\textrm{eff}}}=\dfrac{1}{g_0} -   \dfrac{m}{2\alpha_+}\dfrac{K}{2\hbar^2\pi^2 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 maximal value of energy set by 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. 
+
+# Custom renormalization scheme
+Static codes allow for defining your own renormalization scheme. You need to provide the formula  in `void modify_potentials(...)` function. See [here](Strict 2D or 1D mode) for example. 
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