Abstract
In this chapter, our focus is on providing a comprehensive and mathematically rigorous treatment of the correlation diagrams and their associated correlation functions. From the previous chapter, recall that a correlation diagram for N (where N is tacitly assumed to be at least 3) indistinguishable Fermions graphically exhibits the potencies of their mutual correlations. In purely mathematical terms, such a diagram is an undirected, loopless multi-graph on N vertices. Here, the term multi-graph simply means a graph in which there may be multiple (albeit, finitely many) edges between a vertex-pair. In what follows, we regard correlation diagram and multi-graph as equivalent terms. Let \(\Gamma \) be a multi-graph on N vertices with some chosen labeling of its vertices by numbers \(1, 2, \ldots , N\). Then, to \(\Gamma \) corresponds a product of the terms \((z_{i} - z_{j})^{p_{ij}}\), denoted by \(\mu (\Gamma )\), in which \(z_{1}, \ldots , z_{N}\) are indeterminates and for \(1 \le i < j \le N\), the nonnegative integer \(p_{ij}\) is the number of edges between the vertices labeled i and j (in \(\Gamma \)). In the classical theory of invariants, \(\mu (\Gamma )\) is called the graph-monomial of \(\Gamma \). Note that since our N Fermions are indistinguishable, we must consider each of the possible choices of vertex-labelings, for the correlation diagram under consideration, on an equal footing. Two multi-graphs \(\Gamma _{1}\) and \(\Gamma _{2}\), each with N labeled vertices, are said to be isomorphic provided one can be obtained from the other by a relabeling of its vertices (see Fig. 2.1 for an example of isomorphic multi-graphs).
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Mulay, S., Quinn, J.J., Shattuck, M. (2018). Correlation Functions. In: Strong Fermion Interactions in Fractional Quantum Hall States. Springer Series in Solid-State Sciences, vol 193. Springer, Cham. https://doi.org/10.1007/978-3-030-00494-1_2
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