Heilmann–Lieb nematic-order conjecture

Consider the monomer–dimer model on the $d$ -dimensional hypercubic lattice, with $d\ge 2$ , chemical potential $\mu$ , hard-core exclusion, and an attractive interaction of strength $b>0$ between parallel dimers. The Heilmann–Lieb conjecture asserts that, at sufficiently low temperature and whenever $\mu+2(d-1)b>0$ , the model exhibits liquid-crystalline (nematic) order: orientational symmetry is spontaneously broken, while there is no long-range translational (positional) order.

References

Primary source

arXiv

Additional references

Nematic order in monomer-dimer models of Heilmann and Lieb with lateral attraction in dimensions d≥2 — arXiv — Qidong He

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed progress

A new preprint reports the conjectured orientational ordering in a restricted parameter range in every dimension at least two, but the full conjecture remains open.

The Heilmann–Lieb conjecture predicts orientational order without positional order throughout the attractive regime. Heilmann and Lieb introduced the model in 1979 and conjectured this liquid-crystalline behavior.

Known results

A two-dimensional cluster-expansion proof established orientational order and absence of long-range positional order in a restricted regime (2015).
A Pirogov–Sinai argument proved the conjecture for strongly attractive dimer interactions at sufficiently large dimer activity (2017).
In two dimensions, nematic order, phase-wise Gibbs uniqueness, and correlation decay were proved under $\lambda+a>0$ and $3a>\lambda$ (He, 2025).

October 2026 all-dimensional extension

Qidong He’s preprint reports orientational order, uniqueness within each oriented phase, and anisotropic exponential correlation decay for all $d\ge 2$ when $(4d-6)b>\mu$ . The broader conjectured condition $\mu+2(d-1)b>0$ remains untreated; this latest proof claim is unverified.

Current status (as of October 2026): Restricted regimes are established, while the full condition $\mu+2(d-1)b>0$ remains open and the October all-dimensional extension is unverified.

Sources

No solutions have been posted yet.