Functionals

[[TOC]]

Selecting the functional

The functional must be selected at compilation stage in predefines.h file:

/**
 * Select functional:
 *  - SLDA: 
 *      for simulating unitary Fermi gas, 
 *      it sets effective mass of particles to 1.0 which assures better convergence properties,
 *      in case of time time-dependent calculations SLDA is about 2x faster than ASLDA.
 *  - ASLDA:
 *      for simulating unitary Fermi gas,
 *      at qualitative level it produces results compatible with SLDA, however it is more accurate,
 *      due to presence of current terms in the functional it has worse convergence properties.
 *  - BDG:
 *      for simulating systems in BCS regime,
 *      equations of motion are equivalent to Bogoliubov-de-Gennes equations,
 *      you MUST set aBdG value in input file when using this functional.
 * */
// #define FUNCTIONAL SLDA
#define FUNCTIONAL ASLDA
// #define FUNCTIONAL BDG

ASLDA - asymmetric superfluid local density approximation

The ASLDA functional has been designed in order to capture properties of strongly interacting unitary Fermi gas. The functional has the generic form:

$$\begin{aligned} \mathcal{E}_{\textrm{edf}}(n,\nu,\ldots) & = \alpha_{\uparrow}(n_{\uparrow},n_{\downarrow})\frac{\tau_{\uparrow}}{2} + \alpha_{\downarrow}(n_{\uparrow},n_{\downarrow})\frac{\tau_{\downarrow}}{2} \\\ &+ D(n_{\uparrow},n_{\downarrow}) + g(n_{\uparrow},n_{\downarrow})\nu^{\dagger}\nu \\ &+ [1-\alpha_{\uparrow}(n_{\uparrow},n_{\downarrow})]\dfrac{\bm{j}_{\uparrow}^2}{2n_{\uparrow}} + [1-\alpha_{\downarrow}(n_{\uparrow},n_{\downarrow})]\dfrac{\bm{j}_{\downarrow}^2}{2n_{\downarrow}} \end{aligned}$$

The functional is fitted to quantum Monte Carlo data, and for spin symmetric and uniform systems it provides:

$$E/E_{\textrm{ffg}}=\xi=0.40(1),\qquad\Delta/\varepsilon_F=0.504(24).$$

For more info about fitting procedure and explicit form of functional terms, see: arXiv:1008.3933.

Stabilization of ASLDA functional

In the case of calculations for trapped system term $\frac{\bm{j}_{\sigma}^2}{2n_{\sigma}}$ is source of numerical instabilities. Precisely, for small-density regions we have:

• $n_{\sigma}\rightarrow 0$,
• $\bm{j}_{\sigma}\rightarrow 0$,
• $\frac{\bm{j}_{\sigma}^2}{2n_{\sigma}}\rightarrow 0$.

However, the division of very small numbers is numerically not stable operation. For this reason, we introduce the stabilization procedure:

$$\dfrac{\bm{j}_{\sigma}^2}{2n_{\sigma}}\longrightarrow f_{\textrm{reg.}}(n_{\sigma})\dfrac{\bm{j}_{\sigma}^2}{2n_{\sigma}}$$

Role of $f_{\textrm{reg.}}$ is to exclude from computation regions of small density. The function is controlled by two parameters in predefines.h file:

/**
 * Meaningful only in case of ASLDA.
 * Parameters defining stabilization procedure of ASLDA functional. 
 * For regions with density smaller than SLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY 
 * contribution from current term j^2/2n is assumed to be zero. 
 * For regions with density above SLDA_STABILIZATION_RETAIN_ABOVE_DENSITY 
 * the contribution is assumed to be intact by the stabilization procedure. 
 * */
#define SLDA_STABILIZATION_RETAIN_ABOVE_DENSITY  1.0e-5
#define SLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY 1.0e-7

Their meaning is presented in the figure below.
A smooth transition between densities SLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY and SLDA_STABILIZATION_RETAIN_ABOVE_DENSITY is introduced to avoid discontinuities for quantities, that may lead to divergences when computing derivatives.

Chemical potential

The spin-symmetric and uniform unitary Fermi is expected to be scale-invariant where the following relations for the total energy and the chemical potential satisfy: $E=\xi E_\textrm{ffg}$ and $\mu=\xi \varepsilon_{\textrm{F}}$, where $\xi\approx 0.4$ is the Bertch parameter. However, in numerical realization, the relation for the chemical potential is satisfied only in the limit of very low densities. It is demonstrated in the table below, where results for the uniform solution obtained on lattice 128x128x128 with DX=1 is presented, and $k_{\textrm{F}}=\sqrt{2\varepsilon_{\textrm{F}}}=(6\pi^2 n_{\uparrow})^{1/3}$.

$k_{\textrm{F}}$	$E/E_\textrm{ffg}$	$\mu/\varepsilon_{\textrm{F}}$
1.0	0.397	0.431
0.8	0.397	0.423
0.6	0.397	0.416
0.4	0.397	0.409
0.2	0.397	0.403

It is seen that the total energy $E/E_\textrm{ffg}=\xi$ satisfies the relation all the time, while the chemical potential $\mu/\varepsilon_{\textrm{F}}\rightarrow\xi$ only in low-density limit. For raw data see energy-vs-kF.txt

SLDA - superfluid local density approximation

The term in ASLDA functional that depends on the currents $\bm{j}_{\sigma}$ introduces a significant cost to the computation. This term is responsible for maintaining Gallilean invariance of the ASLDA theory. The effective mass was found to be consistent with the bare mass to within 10% for a large range of polarizations, Therefore, in many applications, one can set $\alpha_{\sigma}=1$ without losing qualitative features of ASLDA theory. Under this assumption we obtain SLDA functional:

$$\mathcal{E}_{\textrm{edf}}(n,\nu,\ldots) = \frac{\tau_{\uparrow}}{2} + \frac{\tau_{\downarrow}}{2} + D(n_{\uparrow},n_{\downarrow}) + g(n_{\uparrow},n_{\downarrow})\nu^{\dagger}\nu$$

Note 1: SLDA functional exhibits much better convergence properties than ASLDA, i.e st-wslda codes typically converge in a significantly smaller number of iterations.

Note 2: SLDA functional provides by a factor about 2x better performance in case of td-wslda codes than ASLDA variant.

Below we provide a plot showing relative energy change (in %) between simulations of SLDA and ASLDA that compute the energy of N fermions (both even and odd particle numbers) in the spin-symmetric unitary Fermi gas trapped in an isotropic harmonic oscillator. For more info see supplemental material of this paper.

BdG - Bogoliubov de-Gennes functional

The BdG functional is equivalent to Bogoliubov de-Gennes mean-field approximation. This approximation is valid in BCS regime, under assumption $-ak_F<1$. The functional has form:

$$\mathcal{E}_{\textrm{edf}}(n,\nu,\ldots) = \alpha_\uparrow\frac{\tau_{\uparrow}}{2} + \alpha_\downarrow\frac{\tau_{\downarrow}}{2} + g\nu^{\dagger}\nu$$

The (bare) coupling constant $g$ is related to scattering length by the formula $g=4\pi\hbar^2a/m_r$, where $m_r$ is twice of reduced mass, i.e.: $\frac{1}{m_r}=\frac{1}{2}(\alpha_\uparrow+\alpha_\downarrow)$. Default values of effective masses are $\alpha_\uparrow=\alpha_\downarrow=1$. See mass imbalanced gas in a harmonic trap to learn how to modify effective masses.

Below we provide predictions of BdG functional for total energy and paring gap as a function of coupling constant $-ak_F$ computed for uniform and spin-symmetric system. Results are compared with predictions of BCS theory, and as expected the agreement is observed for $-ak_F<1$.

SLDAE - SLDA Extended

SLDAE is extension of SLDA-type functional to arbitrary value of $\lambda=|ak_F|$. The functional has generic form:

$$\mathcal{E} = A_\lambda \frac{\tau}{2} + \frac{3}{5}B_\lambda n \varepsilon_F + \frac{C_\lambda }{n^{1/3}} |\nu|^2 + (1 - A_\lambda) \frac{j^2}{2n}$$

where functional parameters $A_\lambda$, $B_\lambda$ and $C_\lambda$ are constructed in such a way to reproduce properly quasiparticle properties in the whole regime of $\lambda$, and also satisfy constraints in limits $\lambda\rightarrow 0$ and $\lambda\rightarrow \infty$. Precisely, the SLDAE is constructed in such a way to reproduce data for the:

• the ground-state energy per unit volume $E = 3n\xi_\lambda \varepsilon_F/5$, and thus corresponding chemical potential $\mu/\varepsilon_F = \zeta_\lambda$ which must verify the thermodynamic relationship $\zeta_\lambda = \xi_\lambda + (\lambda/5) \xi_\lambda^\prime$,
• the pairing gap function $\Delta_\lambda=\eta_\lambda \varepsilon_F$,
• the effective mass of particle $\alpha_\lambda=m/m^\star$. The plot below shows $\lambda=|ak_F|$ dependence of these quantities in the SLDAE functional. For more details see https://arxiv.org/abs/2201.07626.
  NOTE: the functional has been constructed for spin-symmetric systems, $N_a=N_b$.

SLDAE with $m/m^\star=1$

The last term of SLDAE functional (depending on currents $j$) introduces a significant cost to the computation process. In many cases, it is sufficient to proceed with the functional that neglects corrections related to the effective mass (actually the effective mass is not known with high precision). The W-SLDA Toolkit provides a variant of the SLDAE functional where the constraint $\alpha_\lambda = m/m^\star=1$ is imposed. In order to activate it you need to use flag (in predefines.h):

/**
 * Meaningful only in case SLDAE.
 * Sets effective mass to be equal, and speeds-up computation (approximately by a factor of two)
 * */
#define SLDAE_FORCE_A1

Custom SLDAE

The W-SLDA Toolkit allows for creating custom SLDAE-like functionals. For more info see [Custom SLDAE functional](Custom SLDAE functional).

Regularization scheme of pairing field

By default, W-SLDA implements a regularization scheme of the pairing field known as Superfluid Local Density Approximation (SLDA). Precisely, the implemented formulas are (9.86)-(9.87) from paper arXiv:1008.3933.

Note 1: The regularization scheme is only valid for 3D systems. If you want to execute pure 2D or 1D calculations, you need to provide your regularization procedure. For more info see [Strict 2D or 1D modes](Strict 2D or 1D mode).

Note 2: The regularization uses average effective mass $\alpha_{+}$ and average chemical potential $\mu_{+}$. Thus, the regularization is valid for spin-imbalanced and mass-imbalanced systems as well.