Abstract
Consider a system of N electrons projected onto the lowest Landau level with filling factor of the form \(\frac {n}{2pn\pm 1}<\frac {1}{2}\) 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.
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References
Alexandersson, P., Shapiro, B.: Discriminants, symmetrized graph monomials, and sums of squares. Experiment. Math. 21, 353–361 (2012)
Brouwer, A.E., Draisma, J., Popoviciu, M.: The degrees of a system of parameters of the ring of invariants of a binary form. arXiv:1404.5722
Dixmier, J.: Quelques résultats et conjectures concernant les séries de Poincaré des invariants des formes binaires. In: d’algèbre, S., Dubreil, P., Malliavin, M.-P. (eds.) 36ème année (Paris, 1983-1984). Lecture Notes in Math., 1146, pp 127–160. Springer, Berlin (1985)
Elliot, E.B.: An Introduction to the Algebra of Quantics, 2nd edn. Chelsea Publishing Company, New York. (1913), reprint (1964)
Grace, J.H., Young, A.: The Algebra of Invariants. Chelsea Publishing Company, New York (1964). (1903), reprint
Greenhill, C., McKay, B.D.: Asymptotic enumeration of sparse multigraphs with given degrees. SIAM J. Discret. Math. 27, 2064–2089 (2013)
Hansson, T.H., Hermanns, M., Simon, S.H., Viefers, S.F.: Quantum Hall physics: Hierarchies and conformal field theory techniques. Rev. Mod. Phys. 89, 025005 (2017)
Jain, J.K.: Composite-Fermion approach for the fractional quantum Hall effect. Phys. Rev. Lett. 63, 199–202 (1989)
Jain, J.K.: Theory of the fractional quantum Hall effect. Phys. Rev. B 41, 7653–7665 (1990)
Kung, J.P.S., Rota, G. -C.: The invariant theory of binary forms. Bull. Amer. Math. Soc. 10, 27–85 (1984)
Laughlin, R.B.: Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398 (1983)
Mulay, S.B., Quinn, J.J., Shattuck, M.A.: Correlation diagrams: an intuitive approach to correlations in quantum Hall systems. J. Phys.: Conf. Ser. 702, 1–10 (2016)
Quinn, J.J., Wójs, A., Yi, K. -S., Simion, G.: The hierarchy of incompressible fractional quantum Hall states. Phys. Rep. 481(3–4), 29–81 (2009)
Zariski, O., Samuel, P.: Commutative Algebra, vol. I and II. Springer, New York (1976). Graduate Texts in Mathematics
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Mulay, S., Quinn, J.J. & Shattuck, M. An Algebraic Approach to FQHE Variational Wave Functions. Math Phys Anal Geom 22, 14 (2019). https://doi.org/10.1007/s11040-019-9311-y
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DOI: https://doi.org/10.1007/s11040-019-9311-y