• Introduction
• Renormalization with spherical cutoff
• Renormalization with cubic cutoff
• Custom renormalization scheme

Introduction

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:

/**
 * 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

#define REGULARIZATION_SCHEME SPHERICAL_CUTOFF

The effective coupling constant is computed according to prescription:

\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

\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:

# ec                      4.9348022 # energy cut-off for regularization scheme, default ec = 0.5*(pi/DX)^2

For more info see arXiv:1008.3933.

Renormalization with cubic cutoff

!!! THIS FEATURE IS EXPERIMENTAL - NEEDS MORE TESTING !!!

#define REGULARIZATION_SCHEME CUBIC_CUTOFF

The effective coupling constant is computed according to prescription:

\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 for example.