Beane et al. Phys. Rev. A, 107:043314 (2023)

Ref. [1]

tdg.references.PRA107043314.contact_by_kF4(alpha)

The contact density normalized by \(k_F^4\), an intensive and dimensionless quantity,

\[\begin{align} \frac{c}{k_F^4} &= \frac{1}{4} \alpha^2 \left[ 1 + \left(\frac{3}{2} - \ln 4\right) \alpha + 3 \left[0.16079 - (\ln 4 - 1) \right] \alpha^2 + \mathcal{O}(\alpha^3) \right] &&(98) \end{align}\]

where \(c\) is the contact density so that this may be compared to contact_by_kF4().

Parameters

alpha (torch.tensor) – The natural expansion parameters in the EFT.

Returns

\(c/k_F^4\) given above evaluated for alpha

Return type

torch.tensor

tdg.references.PRA107043314.Fermi_Liquid_Energy_by_Fermi_Gas_Energy(alpha)

Using the energy-per-particle definitions from Ref. [1],

\[\begin{split}\begin{align} \frac{E_{FG}}{N} &= \varepsilon_{FG} = \frac{k_F^2}{2M} &&(30) \\ \frac{E_{FL}}{N} &= \varepsilon_{FG} \left[ 1 + \alpha + \alpha^2 (0.05685) - \alpha^3 (0.22550) + \mathcal{O}(\alpha^4) \right] && (102) \end{align}\end{split}\]

this returns function returns the dimensionless ratio \(E_{FG} / E_{FL}\).

In Figs. 15-18 Ref. [1] plots the difference of this ratio and Mean_Field_Energy_by_Fermi_Gas_Energy().

tdg.references.PRA107043314.Mean_Field_Energy_by_Fermi_Gas_Energy(alpha)

Ref. [1] gives the mean-field energy per particle

\[\begin{split}\begin{align} \frac{E_{MF}}{N} &= \varepsilon_{FG} \left[ 1 + \alpha \right] && \text{just after (102) and} \\ \frac{E_{FG}}{N} &= \varepsilon_{FG} = \frac{k_F^2}{2M} &&(30) \end{align}\end{split}\]

In Figs. 15-18 Ref. [1] plots the difference of this ratio and Fermi_Liquid_Energy_by_Fermi_Gas_Energy().

tdg.references.PRA107043314.contact_comparison(ax, *, alpha, cutoff_variation=0.05, **kwargs)

Plots contact_by_kF4() as a function of alpha.

Error bars are produced by varying the cutoff; see Ref. [1] Fig. 12.

tdg.references.PRA107043314.energy_comparison(ax, *, alpha, cutoff_variation=0.05, **kwargs)

Plots the difference between Fermi_Liquid_Energy_by_Fermi_Gas_Energy() and Mean_Field_Energy_by_Fermi_Gas_Energy() as a function of alpha.

Error bars are produced by varying the cutoff; see Ref. [1] Fig. 15. Note that for α>0 they include hard-disk effective-range effects.