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Physical review. B, Condensed matter

Density-density response function in the fractional quantum Hall effect

M. Saarela

Published Jan 15, 1987• 4 citations
Publisher

Abstract

We derive the density-density response function in the fractional quantum Hall effect using the Jastrow theory. The approximation where the two-particle correlation function is kept fixed gives as a result the single-mode approximation and the excitation spectrum derived by Girvin, MacDonald, and Platzman [Phys. Rev. Lett. 54, 581 (1985); Phys. Rev. B 33, 2481 (1986)]. The continuity equation satisfied by the system is also obtained. It is speculated that the quasiexcitons could be described by allowing also the two-particle correlation factor to change.

DOI
10.1103/physrevb.35.854

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We derive the density-density response function in the fractional quantum Hall effect using the Jastrow theory. The approximation where the two-particle correlation function is kept fixed gives as a result the single-mode approximation and the excitation spectrum derived by Girvin, MacDonald, and Platzman [Phys. Rev. Lett. 54, 581 (1985); Phys. Rev. B 33, 2481 (1986)]. The continuity equation satisfied by the system is also obtained. It is speculated that the quasiexcitons could be described by allowing also the two-particle correlation factor to change.
Spectral response functions are central quantities in the analysis of quantum many-body states, since they describe the response of many-body systems to external perturbations and hence directly correspond to observables in experiments. In this paper, we evaluate a momentum-averaged dynamical density structure factor for the fermionic ν = 1 / 3 fractional quantum Hall (FQH) state on a torus, using the continued fraction method to compute the dynamical correlation function. We establish the scaling behaviour of the screened Coulomb structure factor with respect to interaction range, and expose an inherent self-similarity of structure factors in the frequency domain. These results highlight the statistical properties of spectral response functions for FQH states and show how they can be efficiently approximated in numerical models.
We study fractional quantum Hall states at filling fractions in the Jain sequences using the framework of composite Dirac fermions. Synthesizing previous work, we write an effective field theory consistent with all symmetry requirements, including Galilean invariance and particle-hole symmetry. Employing a Fermi-liquid description, we demonstrate the appearance of the Girvin-Macdonald-Platzman algebra and compute the dispersion relation of neutral excitations and various response functions. Our results satisfy requirements of particle-hole symmetry. We show that while the dispersion relation obtained from the modified random-phase approximation (MRPA) of the Halperin-Lee-Read (HLR) theory is particle-hole symmetric, correlation functions obtained from this scheme are not. The results of the Dirac theory are shown to be consistent with the Haldane bound on the projected structure factor, while those of the MPRA of the HLR theory violate it.
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.
In this paper we consider the three-particle density correlation function for a fractional quantum Hall liquid. The study of this object is motivated by recent experimental studies of fractional quantum Hall systems using inelastic light scattering and phonon absorption techniques. Symmetry properties of the correlation function are noted. An exact sum rule is derived which this quantity must obey. This sum rule is used to assess the convolution approximation that has been used to estimate the matrix elements for such experiments.
In this paper we consider the three-particle density correlation function for a fractional quantum Hall liquid. The study of this object is motivated by recent experimental studies of fractional quantum Hall systems using inelastic light scattering and phonon absorption techniques. Symmetry properties of the correlation function are noted. An exact sum-rule is derived which this quantity must obey. This sum-rule is used to assess the convolution approximation that has been used to estimate the matrix elements for such experiments. PACS Numbers: 73.40.Hm, 73.20.Mf, 72.10.Di 1