0
$\begingroup$

I am trying to rederive the results presented in the paper, in particular equation (30). That is, I am trying to compute the correction to the ground-state energy of a dipolar condensate due to beyond-mean field effects. This involves evaluating the following integral:

$$\Delta E = \frac{1}{2} V \int \frac{d^3q}{(2\pi)^3} \left[\varepsilon_{\textbf{q}} - \frac{\hbar^2\textbf{q}^2}{2M} - nV_{\text{int}}(\textbf{q}) + \frac{V_{\text{int}}(\textbf{q})^2}{q^2}\frac{2Mn^2}{\hbar^2} \right] $$

where $\varepsilon_{\textbf{q}} = \sqrt{\frac{\hbar^2\textbf{q}^2}{2M}\left(\frac{\hbar^2\textbf{q}^2}{2M} + 2gn_0\left[1 + \epsilon_{\text{dd}}(3\cos^2{\theta} - 1) \right] \right)}$ is the Boglibov spectrum and $V_{\text{int}}(\textbf{q}) = g\left[1 + \epsilon_{\text{dd}}(3\cos^2{\theta} - 1)\right]$ is the Fourier Transform of the dipole-dipole interaction potential.

The integral is the inverse Fourier transform and $\theta$ is the Fourier polar angle. Taking $n_0 = n$, the paper finds the result to be

$$\Delta E = V \frac{2\pi\hbar^2 a_s n^2}{M} \frac{128}{15} \sqrt{\frac{a_s^3n}{\pi}}\mathcal{Q}_5(\epsilon)$$

where they have reexpressed the result in terms of $a_s$ by substituting $g = 4\pi\hbar^2a_s/M$ . Here,

$$\mathcal{Q}_5(x) = (1-x)^{5/2} {}_2F_1(-\frac{l}{2},\frac{1}{2};\frac{3}{2};\frac{3x}{x-1})$$

where ${}_2F_1(\alpha,\beta;\gamma;z)$ represents the hypergeometric function.

The last term in the integral is supposed to remove the divergence, however when I try to evaluate the integral I get still get a divergent result.

I would appreciate any attempts to explain how they reach their result or how to go about dealing with the divergence.

Thank you.

N.B: The argument in the final result should be $\epsilon_\text{dd}$ but for some reason the TeX will not compile properly.

$\endgroup$
3
  • 1
    $\begingroup$ Where is the divergence and what have you tried that "failed" ? $\endgroup$ Nov 14, 2023 at 4:10
  • $\begingroup$ @NinadMunshi Without the last term in the integral, the integral clearly diverges since the first term $\propto q^4$; however, the last term is supposed to regulate this divergence. I tried plugging in the expressions into the integral and simplifying and evaluating it but this got me nowhere; numerical integration also gives a divergent result. Thank you. $\endgroup$ Nov 14, 2023 at 4:20
  • $\begingroup$ You must be forgetting about the Jacobian for spherical coordinates, because I get that the integrand is finite near the origin and $q^{-2}$ for large $q$. $\endgroup$ Nov 14, 2023 at 7:36

0

You must log in to answer this question.