发表机构
Northeastern University（东北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Heilmann-Lieb二聚体模型在$d\ge 2$维下，当$(4d-6)b>\mu$时存在向列序，并推导出Gibbs测度唯一性与各向异性指数衰减。
AI 中文摘要
我们研究了Heilmann-Lieb (1979)引入的液晶模型II ($d=2$)和模型V ($d=3$)的$d$维推广，该模型由超立方晶格上化学势为$\mu$的二聚体组成，二聚体之间通过硬核排斥和耦合常数为$b>0$的平行二聚体之间的侧向吸引相互作用。对于$d\in\{2,3\}$，这些模型被推测在低温下表现出液晶序，前提是$\mu+2(d-1)b>0$。我们在附加条件$(4d-6)b>\mu$下，对所有$d\ge 2$建立了向列序，该条件对应于物理上错位二聚体在一维参考系统相关长度相当的尺度上稀少的区域。虽然此处对取向序的处理遵循我们近期关于模型I的工作方法，但平移序缺失的证明依赖于Sheffield (2005)意义上的簇交换的定制实现。因此，我们推导出每个取向相内Gibbs测度的唯一性和相关性的各向异性指数衰减。
英文摘要
We study the $d$-dimensional generalization of Models II ($d=2$) and V ($d=3$) for liquid crystals introduced by Heilmann--Lieb (1979), which consists of dimers on the hypercubic lattice at chemical potential $μ$, interacting via a hard-core repulsion and a lateral attraction between parallel dimers with coupling constant $b>0$. For $d\in\{2,3\}$, these models were conjectured to exhibit liquid-crystalline order at low temperatures provided that $μ+2(d-1)b>0$. We establish nematic order for all $d\ge 2$ under the additional condition $(4d-6)b>μ$, which corresponds to the physical regime in which misaligned dimers are rare on scales comparable to the correlation length of the one-dimensional reference system. While the treatment of orientational order here follows the approach of our recent work on Model I, the proof of the absence of translational order relies on a custom implementation of cluster swapping in the sense of Sheffield (2005). Consequently, we deduce uniqueness of Gibbs measure and anisotropic exponential decay of correlations within each oriented phase.
Comments31 pages, one figure
