= Solution
Normalize the ring modes as $\varphi_j(x)=L^{-1/2}e^{ik_jx}$, with distinct $k_j=2\pi\ell_j/L$ and integers $\ell_j$. Inserting $\hat\psi(x)=\varphi_1(x)a_1+\varphi_2(x)a_2$ into the <contact interaction> gives
$$
H_{\rm int}=\frac g2\sum_{i,j,k,l=1}^2 I_{ijkl}a_i^\dagger a_j^\dagger a_k a_l,
\qquad I_{ijkl}=\int_0^L\varphi_i^*\varphi_j^*\varphi_k\varphi_l\,dx.
$$
<Orthogonality> makes $I_{ijkl}=1/L$ when $k_i+k_j=k_k+k_l$, and zero otherwise: this is <momentum conservation> in the projected <Hamiltonian operator>. There is one all-mode-1 term, one all-mode-2 term, and four mixed terms with one particle in each mode. A pair conversion between distinct ring modes would require $2k_1=2k_2$ and therefore vanishes.
Using $a_j^{\dagger2}a_j^2=n_j(n_j-1)$ and $a_1^\dagger a_2^\dagger a_1a_2=n_1n_2$ yields the <contact interaction between two ring modes>,
$$
\boxed{H_{\rm int}=\frac{g}{2L}\left[n_1(n_1-1)+n_2(n_2-1)+4n_1n_2\right],\qquad
\alpha=\beta=\frac g{2L},\quad\gamma=\frac{2g}{L}.}
$$
The factor four comes from two independent choices of ordering for the <creation operators> and <annihilation operators>. This is the two-mode projected model; a full ring <contact interaction> can also scatter particles into other modes while conserving total <momentum>.
Back to article page
