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"\n",
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"This cell contains some definitions for equations and some CSS for styling\n",
" the notebook. If things look a bit strange, please try the following:\n",
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\n",
" - Choose \"Trust Notebook\" from the \"File\" menu.
\n",
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"# The Unitary Fermi Gas: From Monte Carlo to Density Functionals\n",
"\n",
"Equation numbers correspond to the preprint [arXiv:1008.3933](http://arXiv.org/abs/1008.3933) but match the book without the prefix. Thus (9.72a) in the preprint corresponds to (72a) in the published book [Bulgac:2011].\n",
"\n",
"\\begin{gather}\n",
"\\begin{aligned}\n",
" \\nu(\\vect{r}) &= \\frac{1}{2}\\sum_{n} u_n(\\vect{r})v_n^*(\\vect{r})\\Bigl(f_\\beta(E_n) - f_\\beta(-E_n)\\Bigr),\\\\\n",
" \\vect{j}_a(\\vect{r}) &= \\frac{-\\I}{2}\\sum_{n} [u_n^*(\\vect{r})\\nabla u_n(\\vect{r}) - u_n(\\vect{r})\\nabla u_n^*(\\vect{r})]f_\\beta(E_n),\\\\\n",
" \\vect{j}_b(\\vect{r}) &= \\frac{-\\I}{2}\\sum_{n} [v_n(\\vect{r})\\nabla v_n^*(\\vect{r}) - v_n^*(\\vect{r})\\nabla v_n(\\vect{r})]f_\\beta(-E_n),\n",
"\\end{aligned}\n",
"\\tag{9.72b}\n",
"\\end{gather}\n",
"\n",
"The correct formula should have $g < 0$ for attractive interactions. In this case, $\\Delta$ and $\\nu$ have opposite signs:\n",
"\n",
"\\begin{gather}\n",
" \\Delta = g\\nu = g_{\\mathrm{eff}} \\nu_c\\tag{9.74}\n",
"\\end{gather}\n",
"\n",
"\\begin{gather}\n",
" \\mathcal{E} = \\frac{\\hbar^2}{m}\\left(\\frac{\\tau_a+\\tau_b}{2}\\right) + \\Delta^\\dagger \\nu\\tag{9.75}\n",
"\\end{gather}\n",
"\n",
"\\begin{gather}\n",
" \\tau_+(k) = \\tau_a(k) + \\tau_b(k) \\rightarrow \\frac{2(m^*)^2\\Delta^\\dagger \\Delta}{\\hbar^4 k^2}, \\qquad\n",
" \\nu(k) \\rightarrow - \\frac{m^*\\Delta}{\\hbar^2k^2}\\tag{9.81}\n",
"\\end{gather}\n",
"\n",
"*and the following equation:*\n",
"\n",
"$$\n",
" \\frac{\\hbar^2 \\tau_+}{2m^*} + \\Delta^\\dagger \\nu \n",
" = \\frac{\\hbar^2}{m}\\left(\\frac{\\alpha_a\\tau_a}{2} +\\frac{\\alpha_b\\tau_b}{2}\\right) + g\\nu^\\dagger\\nu\n",
"$$\n",
"\n",
"\\begin{gather}\n",
" \\Delta = g\\nu = g_{\\mathrm{eff}} \\nu_c\\tag{9.82}\n",
"\\end{gather}\n",
"\n",
"\n",
"\\begin{gather}\n",
" \\tilde{C}(n_a, n_b) = \\frac{\\alpha_+ \\nu}{\\Delta} + \\frac{1}{2}\\int\\frac{\\d^3{\\vect{k}}}{(2\\pi)^3}\n",
" \\frac{1}{\\hbar^2k^2}{2m} - \\frac{\\mu_+}{\\alpha+} + \\I 0^+\n",
" = \\frac{\\alpha_+}{g_{\\mathrm{eff}}} + \\Lambda\n",
" \\tag{9.84}\n",
"\\end{gather}\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"[Bulgac:2011]: http://dx.doi.org/10.1007/978-3-642-21978-8_9 'Aurel Bulgac, Michael McNeil Forbes, and Piotr Magierski, \"The Unitary Fermi Gas: From Monte Carlo to Density Functionals\", 836, 305 -- 373 (2012) [1008.3933](http://arXiv.org/abs/1008.3933)'"
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"nikola": {
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"date": "2019-02-19 02:58:20 UTC-08:00",
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![](\"/images/nathan-dumlao-263787-unsplash.jpg\")
\n",
"Nathan Dumlao\n",
"