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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. 
 * 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.
 * */
#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 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

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

#define REGULARIZATION_SCHEME CUBIC_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}{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 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.

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.

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:

\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

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