Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras
Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras
复制标题
基于Hopf代数的广义Kitaev量子双模型中的带算子
DOI：
10.1088/1751-8121/ac552c
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Cui, Shawn X
中科院分区：
文献类型：
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作者：
Yan, Bowen;Chen, Penghua;Cui, Shawn X
基塔耶夫的量子双重模型是一组精确可解的晶格模型，实现了物质的二维拓扑相。该模型最初基于有限群，后来推广到半简单Hopf代数。我们严格地定义和研究了广义量子双模型中的带状算子。这些带算子是理解准粒子激发的重要工具。事实证明，在定义运算符时，有一些微妙之处，而不是人们天真地认为的那样。特别地，人们必须区分两类丝带，我们称之为局部顺时针和局部逆时针的丝带。此外，我们指出，这个问题在基于有限非阿贝尔群的原始模型中已经存在，但在文献中似乎没有注意到。我们表明，如果我们不区分这两类色带，即使在原始模型中，某些共同的性质也会失效。也许并不奇怪，在新的定义下，功能区操作符满足所有期望的属性。例如，它们只在带的末端产生准粒子激发，准粒子的类型对应于输入Hopf代数的德林菲尔德双元的不可约表示。然而，这些性质的证明要比有限群的证明复杂得多。这部分是由于处理一般Hopf代数比处理群代数更复杂。
Kitaev's quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. The model was originally based on finite groups, and was later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think of. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, we point out that the issue already exists in the original model based on finite non-abelian groups, but it seems to not have been noticed in the literature. We show how certain common properties would fail even in the original model if we were not to distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than group algebras.