arXiv 1612.094772016-12math-phcond-mat.stat-mechhep-thmath.MP

二聚体的自由费米子六顶点模型的Yang-Baxter解

Yang-Baxter Solution of Dimers as a Free-Fermion Six-Vertex Model

Paul A. Pearce, Alessandra Vittorini-Orgeas

发表信息：: arXiv:1612.09477 [math-ph] 30 Dec 2016
DOI：: 10.1088/1751-8121/aa86bc

摘要：

证明了二聚体作为六顶点模型在自由费米子点处具有交叉参数$λ=\tfracπ{2}$是Yang-Baxter可积的。顶点到二聚体配置的一对多映射允许自由费米子解被应用于正方形晶格上的各向异性二聚体模型，其中二聚体与它们通常的取向相比旋转45度。这个二聚体模型在任意有限尺寸的几何中是完全可解的。本文建立并求解了具有周期边界条件的二聚体在圆柱面上的反演恒等式。在粒子表示中，局部面瓦算子给出了费米子代数的表示，费米子粒子轨迹起着非局部（对数）自由度的作用。在合适的规范下，二聚体模型由具有圈逸度$β=2\cosλ=0$的Temperley-Lieb代数描述。在各向同性的点，精确的解决方案允许明确计数的45度旋转二聚体配置上的周期性$M\times N$矩形晶格。我们证明了环面上的模不变配分函数与辛费米子和临界稠密聚合物是相同的。我们还表明，非平凡的约旦细胞出现的二聚体哈密顿量的带真空边界条件。因此，我们认为，在连续标度极限下，二聚体模型产生了一个{ \em对数}共形场论，中心荷c=-2，最小共形重量$Δ_{\text{min}}=-1/8$和有效的中心荷$c_{\text{eff}}=1$。

英文摘要：

It is shown that dimers is Yang-Baxter integrable as a six-vertex model at the free-fermion point with crossing parameter $λ=\tfracπ{2}$. A one-to-many mapping of vertex onto dimer configurations allows the free-fermion solutions to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by 45 degrees compared to their usual orientation. This dimer model is exactly solvable in geometries of arbitrary finite size. In this paper, we establish and solve inversion identities for dimers with periodic boundary conditions on the cylinder. In the particle representation, the local face tile operators give a representation of the fermion algebra and the fermion particle trajectories play the role of nonlocal (logarithmic) degrees of freedom. In a suitable gauge, the dimer model is described by the Temperley-Lieb algebra with loop fugacity $β=2\cosλ=0$. At the isotropic point, the exact solution allows for the explicit counting of 45 degrees rotated dimer configurations on a periodic $M\times N$ rectangular lattice. We show that the modular invariant partition function on the torus is the same as symplectic fermions and critical dense polymers. We also show that nontrivial Jordan cells appear for the dimer Hamiltonian on the strip with vacuum boundary conditions. We therefore argue that, in the continuum scaling limit, the dimer model gives rise to a {\em logarithmic} conformal field theory with central charge c=-2, minimal conformal weight $Δ_{\text{min}}=-1/8$ and effective central charge $c_{\text{eff}}=1$.

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