Contact interaction between two ring modes ($H_{\mathrm{int}}=\frac{g}{2L}[n_1(n_1-1)+n_2(n_2-1)+4n_1{n}_2]$)

Two distinct normalized plane waves on a periodic one-dimensional ring of length $L$ give the displayed projected contact interaction. Momentum conservation excludes pair conversion between these two modes. Four mixed operator orderings give the coefficient of $n_1{n}_2$. At fixed particle number $N$, the energy is $g[N(N-1)+2n_1(N-n_1)]/(2L)$. For repulsive coupling, its minima are the two endpoint Fock states with all particles in one mode; a one-particle energy difference selects the lower mode. This ground-state statement concerns the two-mode projection.