Bertaina, EPJ Special Topics, 217(1):153–162, (2013)

Ref. [21] provides some contact data.

Figure 2 provides the normalized, mean-field subtracted energy normalized directly but as a function of their \(g=-1/2 \log(k_F b)\) (just after (2)) where \(b\) sets the binding energy as \(-1/mb^2\) (just below Figure 1). Therefore, \(b\) is the scattering length in our convention. However, \(g=\alpha/2\).

tdg.references.EPJST217153.figure_2()

Returns two tensors, data representing their blue triangles (HD JS-DMC) and gray circles (SW JS-DMC), respectively.

The rows are particle density times the scattering length squared \(na_{2D}^2\), the uncorrected energy normalized by the ideal Fermi gas \(E/E_{FG}\), its uncertainty, the finite-size-corrected energy normalized by the ideal Fermi gas, and its uncertainty.

The finite-size correction is described at the end of Section 2; it amounts to using the energy difference for the infinite and finite-volume noninteracting case.

The data were provided by Gianluca Bertaina.

tdg.references.EPJST217153.conventional_figure_2()

Rather than \(na_{2D}^2\), gives data as a function of \(\alpha\).

tdg.references.EPJST217153.energy_comparison(ax, **kwargs)

import tdg, torch
import matplotlib.pyplot as plt

fig, ax = plt.subplots(1,1, figsize=(8,6))

tdg.references.EPJST217153.energy_comparison(ax)

alpha = torch.linspace(-0.9, 0.9, 1000)
# Show just the mean-field subtracted piece of eq (2)
ax.plot(alpha, (3-4 *torch.tensor(2.).log())*(alpha/2)**2, color='orange', label='eq. (2)')
ax.set_xlim([-0.9, 0.9])
ax.set_ylim([-0.025, 0.15])
ax.set_xlabel('α = 2g')
ax.set_ylabel('(E/N - MF)/E_FG')

inset = ax.inset_axes([0, 0.06, 0.8, 0.08], transform=ax.transData)
blue, gray = tdg.references.EPJST217153.conventional_figure_2()
inset.errorbar(blue[0], blue[3], yerr=blue[4], color='blue', marker='v', linestyle='none')
inset.errorbar(gray[0], gray[3], yerr=gray[4], color='gray', marker='o', linestyle='none')
inset.set_xlim([-0.9, 0.9])
inset.set_ylabel('E/N / E_FG')
# Show just the whole of eq (2)
inset.plot(alpha, 1+alpha + (3-4 *torch.tensor(2.).log())*(alpha/2)**2, color='orange')

ax.legend()

(Source code, png, hires.png, pdf)

tdg.references.EPJST217153.contact_by_kF4()

Returns a tensor whose rows are \(\alpha\) and \(c/kF^4\).

tdg.references.EPJST217153.contact_comparison(ax, **kwargs)

Plots the data in contact_by_kF4() as points in a style that matches [1].