硬矩形气体柱状相的高活度展开
High-Activity Expansion for the Columnar Phase of the Hard Rectangle Gas
摘要：
我们研究了尺寸为$m \times d$的单分散硬矩形系统，其中$d\geq m$在二维正方形晶格上。对于足够大的纵横比，已知该系统经历三个熵驱动的相变与增加的活动$z$：第一从无序到无序，第二从无序到柱状和第三从柱状到亚晶格相。我们研究向列相柱过渡开发一个高活性的扩展在整数幂的$z^{-1/d}$的柱状相在一个模型中的矩形只允许在一个方向上取向。通过推导展开式中第一个$d+2$项的精确表达式，我们得到了临界密度和活性的下界。对于$m$、$k\gg 1$，这些界限随着$k$的增加和$m$的减小而减小。
英文摘要：
We study a system of monodispersed hard rectangles of size $m \times d$, where $d\geq m$ on a two dimensional square lattice. For large enough aspect ratio, the system is known to undergo three entropy driven phase transitions with increasing activity $z$: first from disordered to nematic, second from nematic to columnar and third from columnar to sublattice phases. We study the nematic-columnar transition by developing a high-activity expansion in integer powers of $z^{-1/d}$ for the columnar phase in a model where the rectangles are allowed to orient only in one direction. By deriving the exact expression for the first $d+2$ terms in the expansion, we obtain lower bounds for the critical density and activity. For $m$, $k\gg 1$, these bounds decrease with increasing $k$ and decreasing $m$.
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