= Contact interaction between two ring modes
{title2=$H_{\rm int}=\frac g{2L}[n_1(n_1-1)+n_2(n_2-1)+4n_1n_2]$}
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 $n_1n_2$. 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.
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