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arXiv:2211.00371 · SciPost Physics

Parametrization and thermodynamic scaling of pair correlation functions for the fractional quantum Hall effect

Jørgen Fulsebakke, Mikael Fremling, Niall Moran, Johannes K. Slingerland

Published Jun 9, 2023• 5 citations
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Abstract

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. 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.

DOI
10.21468/scipostphys.14.6.149

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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. 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.
Motivated by the structure of the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor [Formula: see text] (m odd), I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin's quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory, without a priori assumption of the presence of CFs. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for 1/m state. Each of these wave functions is interpreted as expelling an electron into a different Hilbert subspace from the original Hilbert space of the Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions [Formula: see text], respectively, can be constructed as coherent superposition of n coupled Laughlin condensates and their 'conjugates', formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for [Formula: see text] sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transition between two topologically distinct phases each corresponding to a family of states: one consists of large number of coupled Laughlin condensates of filling factor 1/3 and the other corresponds to large number of coupled conjugate Laughlin condensates of filling factor 1, which may be distinguished, respectively, by the absence and presence of upstream edge modes.
Model wavefunctions are essential for studying fractional quantum Hall (FQH) phases. They reveal the intricate connections between model FQH states, conformal field theory, Jack polynomials, and pseudopotential Hamiltonians. However, most model wavefunctions studied so far are in the continuum. In this work, we construct model wavefunctions and the corresponding exact parent Hamiltonians for the entire ZkZ_kZk​ Read--Rezayi series on a lattice. Our model wavefunctions obey a generalized clustering rule implemented on the lattice, which we find is encoded in the Macdonald polynomials providing a unified algebraic description of our lattice-specific model states. This ensures that the model states are exact zero-energy ground states of local density interactions and exhibit an infinitely large gap in the particle entanglement spectrum. Numerical studies of the Laughlin and Moore--Read series find the expected zero-mode and one-flux quasihole counting and positive energy gaps at the system sizes studied, and the available finite-size scaling suggests nonzero thermodynamic gaps. Tuning the model interaction range can drive a phase transition out of the FQH phase. Our work not only advances the theoretical understanding of the underlying mathematical structure of lattice model FQH states but also is potentially useful for designing exotic many-body phases in engineered quantum platforms.
Determining correlated ground states directly in the thermodynamic limit remains a central challenge in quantum many-body physics. Here we develop a quantum many-body bootstrap that yields a rigorous and tight lower bound on the thermodynamic ground-state energy of a fractional quantum Hall system together with an optimized reduced density matrix ρW⋆ρ_W^\starρW⋆​ on a spatial window. For sufficiently tight bounds, we find that ρW⋆ρ_W^\starρW⋆​ accurately reproduces the ground-state reduced density matrix, as directly tested by its fidelity with known model states. For exactly solvable Laughlin and Moore--Read parent Hamiltonians, the bootstrap also reproduces pair correlations and orbital entanglement spectra (OES). For Landau-level-projected Coulomb interactions across multiple fillings, the energy bounds closely track established thermodynamic estimates, while the window density matrix recovers the familiar ν=1/3ν=1/3ν=1/3 Laughlin and ν=5/2ν=5/2ν=5/2 Moore--Read states and the composite Fermi liquid at ν=1/2ν=1/2ν=1/2. More importantly, the same framework also resolves unsettled regimes. At ν=1/4ν=1/4ν=1/4, the OES and real-space correlations identify a composite Fermi liquid, while at ν=12/5ν=12/5ν=12/5 the combined diagnostics suggest RR‾3\overline{\mathrm{RR}}_3RR3​ as the ground-state order. Our bootstrap therefore provides direct access to both ground-state energies and many-body structure in the thermodynamic limit.
Fractional quantum Hall (FQH) states are prototypical examples of strongly interacting topologically ordered systems. In this work, we obtain thermodynamic fits on the plane for the pair correlation function, and its Fourier transform, the static structure factor, of two-component spin-singlet Halperin and Jain FQH fluids by expanding them in the recently introduced basis of the orthogonal associated Laguerre polynomials [Fulsebakke et al., SciPost Phys. 14, 149 (2023), https://doi.org/10.21468/SciPostPhys.14.6.149 ] and ascertaining the expansion coefficients by fitting them to large-system Monte Carlo data evaluated using their trial wavefunctions. In this fitting procedure, aside from constraining the exact short-distance behavior of the wavefunction, we also derive and enforce the sum rules that the long-wavelength expansion of the static structure factor must adhere to. We show that incorporating these constraints is crucial for obtaining numerically stable and accurate values of the long-wavelength Girvin-MacDonald-Platzman (GMP)/symmetric density-wave excitation gap. We further extend this approach to spin-resolved density-correlation functions, enabling the evaluation of the gap of the antisymmetric density-wave mode for these spin-singlet FQH states. Finally, we use the density-correlators to compute variational energies of the states and construct phase diagrams for bilayer FQH systems. These could be relevant for understanding recent bilayer FQH experiments that map out the phase diagram by tuning the interlayer separation and density-imbalance/layer-polarization.
The fractional quantum Hall effect (FQHE) observed at half filling of the second Landau level is believed to be caused by a pairing of composite fermions captured by the Moore-Read Pfaffian wave function. The generating Hamiltonian for the Moore-Read Pfaffian is a purely three-body model that breaks particle-hole symmetry and lacks other properties, such as dominate two-body repulsive interactions, expected from a physical model of the FQHE. We use exact diagonalization to study the low-energy states of a more physical two-body generator model derived from the three-body model. We find that the two-body model exhibits the essential features expected from the Moore-Read Pfaffian: pairing, non-Abelian anyon excitations, and a neutral fermion mode. The model also satisfies constraints expected for a physical model of the FQHE at half-filling because it is short range, spatially decaying, particle-hole symmetric, and supports a roton mode with a robust spectral gap in the thermodynamic limit. Hence, this two-body model offers a bridge between artificial three-body generator models for paired states and the physical Coulomb interaction and can be used to further explore properties of non-Abelian physics in the FQHE.
We develop a hybrid Monte Carlo (HMC) method to efficiently compute the physical observables from the samplings of the Laughlin and the Moore-Read (MR) wave functions of fractional quantum Hall (FQH) systems. With the advancements in methodology, including global updates and double stereographic projection on spherical geometry, our HMC simulation is significantly faster than the widely used Metropolis Monte Carlo scheme. As a result, we can readily simulate systems with electron numbersN>1000on both disk and sphere geometries. We apply this method to investigate the topological shift obtained from the edge dipole moment, computed from the wave function density on the disk. We also numerically computed the non-Abelian braiding matrices for different braiding schemes of the MR quasiholes on the sphere. Results with much better quality compared with previous works have been achieved. With the thermodynamic limit results obtained at ease, we also discuss the future usage of our method to clarify the questions on the instability of FQH states in an ideal Chern band setting or under quantum decoherence.