论文标题
克莱因瓶上的二聚体和伊辛模型
The dimer and Ising models on Klein bottles
论文作者
论文摘要
我们在有限的平面加权图上研究二聚体和伊辛模型，并具有周期性抗凝聚的边界条件，即klein瓶中的$γ$ $ k $。令$γ_{mn} $表示通过粘贴$ m $行和$ n $ compies of $γ$获得的图表，$ n $ odd的$ k $嵌入了$ k $，并且在torus $ \ mathbb {t}^2 $中均嵌入$ n $。我们计算$γ_{mn} $ $γ_{mn} $ od $ n $奇数的二聚体分区$ z_ {mn} $，就众所周知的特征多项式$ p $ $γ_{12} \ subset \ subset \ mathbb {t}^2 $与新的特征性polynomial $ r $ r $ r $ g s $γ$ k $ k $ k $ k $ K $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $。使用此结果以及Kenyon，Sun和Wilson [Arxiv：1310.2603]的工作，我们表明，在两部分的情况下，此分区功能具有渐近扩展$ \ log Z_ {mn f_0/2 +\ f_0/2 +\ \ \ \ \ \ \ \ \ \ \ m { $ f_0 $是$γ_{12} \ subset \ mathbb {t}^2 $和$ \ mathrm {fsc} $显式有限大小校正项的批量自由能。该术语的显着特征是它的普遍性：它不取决于图$γ$，而仅取决于单位圆环和明确（纯粹想象中的）形状参数的$ p $的零。假设在$ p $的零上有猜想的状态，在非双方情况下也获得了类似的扩展。然后，我们表明，这种渐近分区功能也具有$ \ mathrm {fsc} $，采用特别简单的形式：它在亚临界体制中消失了，等于超临界状态中的$ \ log（2）$，以及对形状的明确函数的显着性函数。这些结果与Blöte，Cardy和Nightingale的保形场理论预测完全一致。
We study the dimer and Ising models on a finite planar weighted graph with periodic-antiperiodic boundary conditions, i.e. a graph $Γ$ in the Klein bottle $K$. Let $Γ_{mn}$ denote the graph obtained by pasting $m$ rows and $n$ columns of copies of $Γ$, which embeds in $K$ for $n$ odd and in the torus $\mathbb{T}^2$ for $n$ even. We compute the dimer partition function $Z_{mn}$ of $Γ_{mn}$ for $n$ odd, in terms of the well-known characteristic polynomial $P$ of $Γ_{12}\subset\mathbb{T}^2$ together with a new characteristic polynomial $R$ of $Γ\subset K$. Using this result together with work of Kenyon, Sun and Wilson [arXiv:1310.2603], we show that in the bipartite case, this partition function has the asymptotic expansion $\log Z_{mn}=mn f_0/2 +\mathrm{fsc}+o(1)$, for $m, n$ tending to infinity and $m/n$ bounded below and above, where $f_0$ is the bulk free energy for $Γ_{12}\subset\mathbb{T}^2$ and $\mathrm{fsc}$ an explicit finite-size correction term. The remarkable feature of this later term is its universality: it does not depend on the graph $Γ$, but only on the zeros of $P$ on the unit torus and on an explicit (purely imaginary) shape parameter. A similar expansion is also obtained in the non-bipartite case, assuming a conjectural condition on the zeros of $P$. We then show that this asymptotic expansion holds for the Ising partition function as well, with $\mathrm{fsc}$ taking a particularly simple form: it vanishes in the subcritical regime, is equal to $\log(2)$ in the supercritical regime, and to an explicit function of the shape parameter at criticality. These results are in full agreement with the conformal field theory predictions of Blöte, Cardy and Nightingale.
