Kosaki-Longo指数与二维量子自旋模型中的电荷分类
Kosaki-Longo index and classification of charges in 2D quantum spin models
摘要：
我们考虑二维量子自旋模型中对应于锥可定域电荷的电荷超选择扇区，并证明了这种电荷的等价类的个数受包含某些可观测代数的Kosaki-Longo指数的约束.为了证明这一结果的力量，我们将理论应用于二维无限晶格上的复曲面代码。对于这个模型，我们可以计算这个包含的指数，并得出结论，有四个不同的不可约电荷在这个模型中，根据分析的环面代码模型在紧凑的表面。我们还给出了电荷扇区非简并性的充分准则，即Verlinde矩阵S是可逆的。
英文摘要：
We consider charge superselection sectors of two-dimensional quantum spin models corresponding to cone localisable charges, and prove that the number of equivalence classes of such charges is bounded by the Kosaki-Longo index of an inclusion of certain observable algebras. To demonstrate the power of this result we apply the theory to the toric code on a 2D infinite lattice. For this model we can compute the index of this inclusion, and conclude that there are four distinct irreducible charges in this model, in accordance with the analysis of the toric code model on compact surfaces. We also give a sufficient criterion for the non-degeneracy of the charge sectors, in the sense that Verlinde's matrix S is invertible.
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