Consider a system of N electrons projected onto the lowest Landau level with filling factor of the form n2pn±1<12$\frac {n}{2pn\pm 1}<\frac {1}{2}$ and N a multiple of n. We show that there always exists a two-dimensional symmetric correlation function (arising as a nonzero symmetrization) for such systems and hence one can always write a variational wave function. This extends an earlier observation of Laughlin for an incompressible quantum liquid (IQL) state with filling factor equal to the reciprocal of an odd integer ≥$\geqslant $ 3. To do so, we construct a family of d-regular multi-graphs on N vertices for all N whose graph-monomials have nonzero linear symmetrization and obtain, as special cases, the aforementioned nonzero correlations for the IQL state. The linear symmetrization that is obtained is in fact an example of what is called a binary invariant of type (N,d). Thus, in addition to supplying new variational wave functions for systems of interacting Fermions, our construction is of potential interest from both the graph and invariant theoretic viewpoints.

中文翻译：

---

FQHE变分波函数的代数方法

考虑投影到最低朗道能级的 N 个电子系统，填充因子为 n2pn±1<12$\frac {n}{2pn\pm 1}<\frac {1}{2}$ 和 N 的倍数n. 我们表明，对于这样的系统，总是存在一个二维对称相关函数（作为非零对称化产生），因此我们总是可以写出一个变分波函数。这扩展了 Laughlin 早期对不可压缩量子液体 (IQL) 状态的观察，其填充因子等于奇整数的倒数≥$\geqslant $ 3。为此，我们在其图单项式具有非零线性对称化的所有 N 的 N 个顶点，并在特殊情况下获得上述 IQL 状态的非零相关性。获得的线性对称化实际上是所谓的 (N,d) 类型的二元不变量的一个例子。因此，除了为相互作用的费米子系统提供新的变分波函数之外，我们的构造从图形和不变理论的观点来看都具有潜在的意义。