• Introduction
• Renormalization with spherical cutoff
• Renormalization with cubic cutoff
• Custom renormalization scheme
• Regularization scheme and the energy conservation in td calculations
• Known issues

Introduction

W-SLDA Toolkit utilizes a local pairing field \Delta(\bm{r}). In such a 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 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 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.
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 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 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.

Regularization scheme and the energy conservation in td calculations

In publication arXiv:1606.02225 it was pointed that TDBdG like equations, formally conserve energy only if all quasiparticle states are evolved, see 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 maybe not conserved.

In practical applications, we observe that the energy when applied the spherical regularization scheme is conserved only with some accuracy, which is not related to the integrator 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. It is clearly visible, that for evolution with the cubic cutoff the energy is conserved up to high accuracy, while for spherical cutoff the quality of the energy conservation is significantly lower.

wlog files for these runs:

• cubic.wlog
• spherical.wlog

In conclusion, we find that typically for trajectories of length te_F\approx1000 the spherical cutoff provides reasonable accuracy, while for generation of long trajectories te_F\gg 1000 it is recommended to use the cubic cutoff.

Known issues

Below we present results for the uniform unitary Fermi gas as a function of lattice spacing, while keeping the fixed volume of the box. Note that the lattice spacing defines value of the energy cut-off E_c\approx\frac{p_c^2}{2}=\frac{\pi^2}{2DX^2}. Conditions of the test are as follow:

# VOLUME: 32 x 32 x 32
# ENERGY DENSITY FUNCTIONAL: SLDA
# UNIFORM_TEST_MODE: Setting number of particles to be: (554.000000,554.000000)

The result for the total energy is:

DX	Spherical cut-off	Cubic cut-off
1.0	0.39761	0.39926
0.8	0.39834	0.38539
0.5	0.39825	0.36422

In the case of the cubic cut-off, we observe the dependence of the lattice spacing. This issue needs further investigation.
For raw data see: test-spherical-vs-cubic.txt