First-hand information for everyone
Researchers studied a simple one‑dimensional lattice model in which hard‑core bosons (particles that cannot sit two on the same site, equivalent to spinless fermions) feel an attractive interaction only on every second bond. They found that as the attraction grows the system changes from a fluid of individual atoms to a fluid made of bound pairs (dimers), a ‘‘molecular’’ superfluid. Between these two regimes the system shows a surprising phase‑separated or ‘‘absorbing’’ region where particles cluster into a small, almost incompressible puddle with local half‑filling.
To reach these results the authors combined numerical and analytical tools. They used density‑matrix renormalization group (DMRG), a standard numerical method for one‑dimensional quantum systems, and wrote effective low‑energy models by treating strong interactions perturbatively. They tracked changes in ground‑state energy, correlation functions, momentum occupations, and a quantity called the fidelity susceptibility that signals phase changes.
The data show two crossovers as the attraction U increases. For weak attraction single‑particle correlations decay slowly with distance, as expected for a one‑dimensional atomic superfluid. For strong attraction single‑particle correlations become short ranged while pair (dimer) correlations decay slowly, signaling a dimer superfluid. The authors identify a lower critical value Uc1≈3.1 where bound pairs first appear and a larger scale (seen in examples near U≈5.8) where single particles are gapped and pairs dominate.
The paper also explains why these dimers behave the way they do. When pairs form, their effective hopping from one composite site to the next is very small — perturbation theory gives a hopping scale that falls rapidly with U (roughly proportional to t^4/U^3, where t is the single‑particle hopping). Virtual quantum processes produce a dominant repulsive interaction between neighboring dimers of order 2t^2/U. These competing small hopping and strong effective repulsion shape the phases: away from half‑filling the dimers form a mobile paired fluid, while at half‑filling other ordered, gapped states appear (a bond‑ordered phase at small U and a dimer charge‑density wave at large U).
At intermediate attractions the authors find a phase‑separated region. There, dimers develop an emergent attraction between themselves and the system nucleates a local charge‑density‑wave puddle with effective half‑filling. The system is unusually sensitive to single unpaired atoms: adding one lone particle can tip the balance and promote local charge order. In practice this sensitivity appears as jumps in particle number when the chemical potential is varied, and as narrow compressible regions in the phase diagram.
There are important caveats. The effective hopping scales are perturbatively small, so the gaps and energy differences can be extremely tiny. This makes the numerical work challenging and limits reliable DMRG results to modest system sizes (typically below about 100 sites in their study). Some features are therefore narrow and hard to resolve in the chemical‑potential versus interaction plane. The paper supports its claims with both numerics and low‑energy theory, but the small energy scales mean that finite‑size and precision effects must be handled with care.