Fragmented and symmetry-restored condensates in a three-site Bose ring: exchange versus umklapp
Abstract
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,±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 avgn+1=avgn−1≃ N/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·(+1)≡−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 ±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.
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BibTeX
@misc{siddiki2026fragmented,
title = {{Fragmented and symmetry-restored condensates in a three-site Bose ring: exchange versus umklapp}},
author = {A. Siddiki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05469},
eprint = {2610.05469},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05469},
}
RIS
TY - UNPB
TI - Fragmented and symmetry-restored condensates in a three-site Bose ring: exchange versus umklapp
AU - A. Siddiki
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05469
UR - https://arxiv.org/abs/2610.05469
AB - 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,±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 avgn_+1=avgn_−1≃ N/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·(+1)≡−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 ±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.
ER -