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arXiv 1809.01504cond-mat.mes-hallmath-phmath.MP
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量子霍尔系统中电子相互作用的代数方法

An Algebraic Approach to Electron Interactions in Quantum Hall Systems

S. B. Mulay, J. J. Quinn, M. A. Shattuck

摘要:

令$m$表示包含全部$N$粒子的量子霍尔系统中的准电子(QE)的数目。我们表明,在几个一般情况下,对于系统包含$m$ QE在一个单一的角动量壳层以上的$N-m$费米子在一个不可压缩的量子液体(IQL)状态具有填充因子$ν=\frac{1}{3}$,总是存在一个配置的对称关联函数$G$是非零的。这扩展了最近关于IQL状态的可比结果。因此,对于包含任意数量QE的所有$N=8$粒子系统,可以(明确地)获得具有非零$G$的构型。为了建立我们的结果,我们构造了一个家庭的多重图上的$N$顶点满足一定的限制的程度的顶点,并拥有的财产,每当一个计算的线性对称化的图单项式的任何家庭成员,其结果总是非零的。在每种情况下得到的非零线性对称化实际上是一个被称为度$N$的(通用)二进制形式的相对半不变量的例子。因此,除了提供新的关联函数的相互作用费米子系统包含QE,我们的建设可能是感兴趣的不变量和图论的观点。

英文摘要:

Let $m$ denote the number of quasielectrons (QEs) in a quantum Hall system containing $N$ particles altogether. We show in several general cases that for systems containing $m$ QEs in a single angular momentum shell above $N-m$ Fermions in an incompressible quantum liquid (IQL) state having filling factor $ν=\frac{1}{3}$ that there always exists a configuration whose symmetric correlation function $G$ is nonzero. This extends recent comparable results concerning the IQL state. As a consequence, one can obtain (explicitly) a configuration having a nonzero $G$ for all $N=8$ particle systems containing any number of QEs. To establish our result, we construct a family of multi-graphs on $N$ vertices satisfying certain restraints on the degrees of the vertices and possessing the property that whenever one computes the linear symmetrization of the graph monomial of any member of the family, the result is always nonzero. The nonzero linear symmetrization that is obtained in each case is in fact an example of what is called a relative semi-invariant of a (generic) binary form of degree $N$. Thus, in addition to providing new correlation functions for systems of interacting Fermions containing QEs, our construction could be of interest from both the invariant and graph theoretic standpoints.

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