Fragmented and symmetry-restored condensates in a three-site Bose ring: exchange versus umklapp

We study $N$ bosons on a three-site ring with nearest-neighbour hopping, density-assisted hopping of the opposite sign, and an on-site interaction, written equivalently as three momentum modes $q=0,\pm 1$ with a channel-dependent quartic vertex. As the density-assisted term grows it inverts the band, and exact diagonalisation up to $N=400$ shows a first-order ground-state transition from the $q=0$ condensate to a state with $\text{avg}{n}_{+1}=\text{avg}{n}_{-1}\simeq N\mathrm{/}2$; the avoided-crossing gap closes faster than exponentially with $N$. In its symmetric eigenstates this state is fragmented in the Penrose-Onsager sense, and at leading order in $N$ it has the energy of a Gross-Pitaevskii (GP) standing wave, as expected for a few-mode mean-field system. We distinguish two cases by switching off umklapp scattering, which is available because $2\cdot (+1)\equiv -1$ on three sites, and by varying the on-site interaction. Without umklapp, $n_{+1}-n_{-1}$ is conserved, the eigenstates form a rotor tower whose lowest member is a fragmented Fock state, and the transition leads to fragmentation exactly when the $\pm 1$ interaction is attractive, as predicted by Nozières’ exchange argument with the sign of the interaction reversed; otherwise the system forms a single current-carrying condensate. With umklapp, the standing wave is pinned to three positions, the three momentum sectors become degenerate exponentially in $N$, and coherent admixture of the $q=0$ mode stabilises the standing wave over a wider range; the exact ground state is then a number-squeezed, momentum-projected standing-wave condensate rather than a robustly fragmented state.