发表机构
Atlas University Vocational School（阿特拉斯大学职业学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过精确对角化研究三格点玻色环，发现密度辅助跳跃导致一级相变，产生碎裂或驻波凝聚体，并阐明倒逆散射与交换作用对碎裂稳定性的关键影响。
AI 中文摘要
我们研究了三格点环上的$N$个玻色子，该系统具有最近邻跳跃、反号密度辅助跳跃以及格点内相互作用，等价地可写为三个动量模式$q=0,\pm1$，并带有依赖于通道的四次顶点。随着密度辅助项的增强，能带发生反转，精确对角化（至$N=400$）显示从$q=0$凝聚体到具有$\avg{n_{+1}}=\avg{n_{-1}}\simeq N/2$的态的一级基态相变；避免交叉能隙随$N$的增大以快于指数的方式闭合。在其对称本征态中，该态在彭罗斯-昂萨格意义上是碎裂的，且在$N$的领头阶上具有格罗斯-皮塔耶夫斯基（GP）驻波的能量，正如少模平均场系统所预期的那样。我们通过关闭倒逆散射（在三格点上因$2\cdot(+1)\equiv-1$而可行）以及改变格点内相互作用来区分两种情况。无倒逆散射时，$n_{+1}-n_{-1}$守恒，本征态形成转子塔，其最低成员为碎裂的福克态，且当$\pm1$相互作用为吸引时（如诺齐埃交换论证所预测，但相互作用符号相反），转变恰好导致碎裂；否则系统形成单一载流凝聚体。有倒逆散射时，驻波被钉扎在三个位置，三个动量扇区在$N$上指数简并，$q=0$模式的相干混合在更宽范围内稳定驻波；此时精确基态是数压缩、动量投影的驻波凝聚体，而非稳健的碎裂态。
英文摘要
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 $q=0,\pm1$ 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 $\avg{n_{+1}}=\avg{n_{-1}}\simeq N/2$; 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 $2\cdot(+1)\equiv-1$ 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 $\pm1$ 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.
