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Parametrization and thermodynamic scaling of pair correlation functions for the Fractional Quantum Hall Effect
by Jørgen Fulsebakke, Mikael Fremling, Niall Moran, J. K. Slingerland
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|Authors (as registered SciPost users):||Mikael Fremling|
|Preprint Link:||https://arxiv.org/abs/2211.00371v1 (pdf)|
|Date submitted:||2022-11-02 10:03|
|Submitted by:||Fremling, Mikael|
|Submitted to:||SciPost Physics|
The calculation of pair correlations and density profiles of quasiholes are routine steps in the study of proposed fractional quantum Hall states. Nevertheless, the field has not adopted a standard way to present the results of such calculations in an easily reproducible form. We develop a polynomial expansion that allows for easy quantitative comparison between different candidate wavefunctions, as well as reliable scaling of correlation and quasihole profiles to the thermodynamic limit. We start from the well-known expansion introduced by Girvin [PRB, 30 (1984)] (see also [Girvin, MacDonald and Platzman, PRB, 33 (1986)]), which is physically appealing but, as we demonstrate, numerically unstable.0 We orthogonalize their basis set to obtain a new basis of modified Jacobi polynomials, whose coefficients can be stably calculated. We then apply our expansion to extract pair correlation expansion coefficients and quasihole profiles in the thermodynamic limit for a wide range of fractional quantum Hall wavefunctions. These include the Laughlin series, composite fermion states with both reverse and direct flux attachment, the Moore-Read Pfaffian state, and BS hierarchy states. The expansion procedure works for both abelian and non-abelian quasiholes, even when the density at the core is not zero. We find that the expansion coefficients for all quantum Hall states considered can be fit remarkably well using a cosine oscillation with exponentially decaying amplitude. The frequency and the decay length are related in an intuitive, but not elementary way to the filling fraction. Different states at the same filling fraction can have distinct values for these parameters. Finally, we also use our scaled correlation functions to calculate estimates for the magneto-roton gaps of the various states.
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Reports on this Submission
Report 1 by Koyena Bose on 2023-1-4 (Invited Report)
- Cite as: Koyena Bose, Report on arXiv:2211.00371v1, delivered 2023-01-04, doi: 10.21468/SciPost.Report.6448
1. The authors have come up with a process to stably parameterize the pair-correlation functions for fractional quantum Hall states. They also propose a way to scale the pair-correlation function to the thermodynamic limit. This allows one to obtain various quantities directly in the thermodynamic limit as opposed to extrapolating results to the thermodynamic limit from evaluations on finite systems.
2. The authors have discussed and provided necessary details to explain why their parameterization of the pair correlation function is more stable than procedures deployed in the past. They have also discussed multiple techniques and showed how their parameters are convergent and robust to different choices of calculation.
3. The authors have presented $g(r)$ results for many different wave functions like Laughlin, Jain and Bonderson-Slingerland states where the thermodynamic scaling gives better results in comparison to previous calculations where computations were done for finite systems.
1. The employed method still results in instabilities in the orthogonal coefficients, $G_n(\eta)$, of the parameterized pair correlation function for large $n$. A resolution of this would be required to describe the long range behaviour for more involved states like parton states.
2. More applications such as evaluating the Fermi wave vector of composite fermions (which has been measured in experiments [Phys. Rev. Lett. 113, 196801]) from the pair-correlation function could be discussed/mentioned.
The study is well formulated with substantial material to support that the parameterization used for pair correlation function $g(r)$ for fractional quantum Hall states is stable in comparison to past efforts [Phys. Rev. B, 30:558–560, 1984]. The authors have also successfully applied their technique to find the pair correlation functions of some well known wave functions like Laughlin, Jain and Bonderson-Slingerland states. One of the interesting parts of this work is the scaling of $g(r)$ to the thermodynamic limit which was earlier calculated only for finite systems. This opens a path to explore various quantities in this limit which was earlier not doable , i.e. Fermi wave vector, or done for a finite system and then extrapolated to the thermodynamic limit , i.e. the evaluation of per-particle energies. It would also be interesting to see if the instability in the calculation of $G_n(\eta)$ for higher $n$ introduces considerable error when dealing with more complicated wave-functions like parton states.
1. For Moore-Read state with non-abelian quasi-holes [5.2; Figure: 11(c)], we see $g(r) \neq 0$ even at $r=0$. What does this signify? For a fermionic state, why is there a finite probability to find two overlapping fermions? Why does this only show up when non-abelian quasi-holes are introduced?
2. Discuss some more applications of the tools developed such as the above mentioned evaluation of the Fermi wave vector of composite fermions.