The Unitary Fermi Gas: From Monte Carlo to Density Functionals Equation numbers correspond to the preprint arXiv:1008.3933 but match the book without the prefix. Thus (9.72a) in the preprint corresponds to (72a) in the published book Bulgac:2011 .
ν ( r ⃗ ) = 1 2 ∑ n u n ( r ⃗ ) v n ∗ ( r ⃗ ) ( f β ( E n ) − f β ( − E n ) ) , j ⃗ a ( r ⃗ ) = − i 2 ∑ n [ u n ∗ ( r ⃗ ) ∇ u n ( r ⃗ ) − u n ( r ⃗ ) ∇ u n ∗ ( r ⃗ ) ] f β ( E n ) , j ⃗ b ( r ⃗ ) = − i 2 ∑ n [ v n ( r ⃗ ) ∇ v n ∗ ( r ⃗ ) − v n ∗ ( r ⃗ ) ∇ v n ( r ⃗ ) ] f β ( − E n ) , \begin{gather}
\begin{aligned}
\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),\\
\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),\\
\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),
\end{aligned}
\tag{9.72b}
\end{gather} ν ( r ) j a ( r ) j b ( r ) = 2 1 n ∑ u n ( r ) v n ∗ ( r ) ( f β ( E n ) − f β ( − E n ) ) , = 2 − i n ∑ [ u n ∗ ( r ) ∇ u n ( r ) − u n ( r ) ∇ u n ∗ ( r )] f β ( E n ) , = 2 − i n ∑ [ v n ( r ) ∇ v n ∗ ( r ) − v n ∗ ( r ) ∇ v n ( r )] f β ( − E n ) , ( 9.72b ) The correct formula should have g < 0 g < 0 g < 0 for attractive interactions. In this case, Δ \Delta Δ and ν \nu ν have opposite signs:
Δ = g ν = g e f f ν c \begin{gather}
\Delta = g\nu = g_{\mathrm{eff}} \nu_c\tag{9.74}
\end{gather} Δ = gν = g eff ν c ( 9.74 ) E = ℏ 2 m ( τ a + τ b 2 ) + Δ † ν \begin{gather}
\mathcal{E} = \frac{\hbar^2}{m}\left(\frac{\tau_a+\tau_b}{2}\right) + \Delta^\dagger \nu\tag{9.75}
\end{gather} E = m ℏ 2 ( 2 τ a + τ b ) + Δ † ν ( 9.75 ) τ + ( k ) = τ a ( k ) + τ b ( k ) → 2 ( m ∗ ) 2 Δ † Δ ℏ 4 k 2 , ν ( k ) → − m ∗ Δ ℏ 2 k 2 \begin{gather}
\tau_+(k) = \tau_a(k) + \tau_b(k) \rightarrow \frac{2(m^*)^2\Delta^\dagger \Delta}{\hbar^4 k^2}, \qquad
\nu(k) \rightarrow - \frac{m^*\Delta}{\hbar^2k^2}\tag{9.81}
\end{gather} τ + ( k ) = τ a ( k ) + τ b ( k ) → ℏ 4 k 2 2 ( m ∗ ) 2 Δ † Δ , ν ( k ) → − ℏ 2 k 2 m ∗ Δ ( 9.81 ) and the following equation:
ℏ 2 τ + 2 m ∗ + Δ † ν = ℏ 2 m ( α a τ a 2 + α b τ b 2 ) + g ν † ν \frac{\hbar^2 \tau_+}{2m^*} + \Delta^\dagger \nu
= \frac{\hbar^2}{m}\left(\frac{\alpha_a\tau_a}{2} +\frac{\alpha_b\tau_b}{2}\right) + g\nu^\dagger\nu 2 m ∗ ℏ 2 τ + + Δ † ν = m ℏ 2 ( 2 α a τ a + 2 α b τ b ) + g ν † ν Δ = g ν = g e f f ν c \begin{gather}
\Delta = g\nu = g_{\mathrm{eff}} \nu_c\tag{9.82}
\end{gather} Δ = gν = g eff ν c ( 9.82 ) C ~ ( n a , n b ) = α + ν Δ + 1 2 ∫ d 3 k ⃗ ( 2 π ) 3 1 ℏ 2 k 2 2 m − μ + α + + i 0 + = α + g e f f + Λ \begin{gather}
\tilde{C}(n_a, n_b) = \frac{\alpha_+ \nu}{\Delta} + \frac{1}{2}\int\frac{\d^3{\vect{k}}}{(2\pi)^3}
\frac{1}{\hbar^2k^2}{2m} - \frac{\mu_+}{\alpha+} + \I 0^+
= \frac{\alpha_+}{g_{\mathrm{eff}}} + \Lambda
\tag{9.84}
\end{gather} C ~ ( n a , n b ) = Δ α + ν + 2 1 ∫ ( 2 π ) 3 d 3 k ℏ 2 k 2 1 2 m − α + μ + + i 0 + = g eff α + + Λ ( 9.84 )
