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Regularization schemes of the pairing field

Introduction

W-SLDA Toolkit utilizes a local pairing field \Delta(\bm{r}). In such a case, a renormalization procedure is required. There are two predefined regularization schemes that can be selected in predefines.h file:

/**
 * Scheme of pairing field renormalization procedure. 
 * 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 in our manuscript arXiv:2602.08982 .

Renormalization with spherical cutoff

#define REGULARIZATION_SCHEME SPHERICAL_CUTOFF

The effective coupling constant is computed according to the 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 the energy cutoff scale that can be controlled by the tag:

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

For more info, see arXiv: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

If you are using VERSION<2026.06.05, see here.

#define REGULARIZATION_SCHEME CUBIC_CUTOFF

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 for example.

Regularization scheme and the energy conservation in td calculations

In the publication arXiv:1606.02225, it was noted that TDBdG-like equations formally conserve energy only if all quasiparticle states are evolved; see the discussion of Eqs. (25)-(26). This situation corresponds to the cubic cutoff. If the space is truncated, eg, by introducing a spherical cutoff at some initial time, then in general energy may not be conserved.

In practical applications, we observe that the energy, when applying the spherical regularization scheme, is conserved only with some accuracy, which is not related to the integrator's accuracy. Below we provide an example of (3d calculation), where for the time interval te_F<170 we apply an external time-dependent potential (we pump energy into the system), and for te_F>170 the system evolves without any external perturbation. wiki1 wiki2 It is clearly visible that for evolution with the cubic cutoff, energy is conserved with high accuracy, whereas for the spherical cutoff, the quality of energy conservation is significantly lower.

wlog files for these runs:

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.

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}}.

To include this correction in the calculation, add the following definition to predefines.h:

#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.

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

Note: When using a custom regularization scheme, the mean-field correction should be disabled.