Three-Point Density Correlation Functions in the Fractional Quantum Hall Regime

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.

arXiv:cond-mat/0003458Journal of Physics A Mathematical and GeneralJun 2, 20001 citation

Density-density response function in the fractional quantum Hall effect

M. Saarela

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.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.

Physical review. B, Condensed matterJan 15, 19874 citations

Developments in the applications of density functional theory to fractional quantum Hall systems

Yi Yang, Zi-Xiang Hu, Zi-Xiang Hu

Abstract The fractional quantum Hall effect remains a captivating area in condensed matter physics, characterized by strongly correlated topological order, which manifests as fractionalized excitations and anyonic statistics. Numerical simulations, such as exact diagonalization, density matrix renormalization groups, matrix product states, and Monte Carlo methods are essential for examining the properties of strongly correlated systems. Recently, density functional theory has been employed in this field within the framework of composite fermion theory. This paper systematically evaluates how density functional theory approaches have addressed fundamental challenges in fractional quantum Hall systems, including ground state and low-energy excitations. Special attention is given to the insights provided by density functional theory regarding composite fermion behavior, edge effects, and the nature of fractional charge and magnetoroton excitations. The discussion critically examines both the advantages and limitations of these approaches, while highlighting the productive interplay between numerical simulations and theoretical models. Future directions are explored, particularly the promising potential of time-dependent density functional theory for modeling non-equilibrium dynamics in quantum Hall systems.Abstract The fractional quantum Hall effect remains a captivating area in condensed matter physics, characterized by strongly correlated topological order, which manifests as fractionalized excitations and anyonic statistics. Numerical simulations, such as exact diagonalization, density matrix renormalization groups, matrix product states, and Monte Carlo methods are essential for examining the properties of strongly correlated systems. Recently, density functional theory has been employed in this field within the framework of composite fermion theory. This paper systematically evaluates how density functional theory approaches have addressed fundamental challenges in fractional quantum Hall systems, including ground state and low-energy excitations. Special attention is given to the insights provided by density functional theory regarding composite fermion behavior, edge effects, and the nature of fractional charge and magnetoroton excitations. The discussion critically examines both the advantages and limitations of these approaches, while highlighting the productive interplay between numerical simulations and theoretical models. Future directions are explored, particularly the promising potential of time-dependent density functional theory for modeling non-equilibrium dynamics in quantum Hall systems.

arXiv:2504.14558Communications in Theoretical PhysicsSep 25, 20250 citations

Universal properties of the wave functions of fractional quantum Hall systems

Ana Lopez, Eduardo Fradkin

We show that the wave functions of the fluid ground states of fractional quantum Hall systems, in the thermodynamic limit, are universal at long distances and that they have a generalized Laughlin form. This universality is a consequence of the analytic properties of the equal-time density correlation functions at long distances. The correlation functions calculated from the field theoretic approach to the fractional quantum Hall effect have the correct analytic properties and the wave function calculated in the Gaussian approximation becomes exact in the asymptotic limit.We show that the wave functions of the fluid ground states of fractional quantum Hall systems, in the thermodynamic limit, are universal at long distances and that they have a generalized Laughlin form. This universality is a consequence of the analytic properties of the equal-time density correlation functions at long distances. The correlation functions calculated from the field theoretic approach to the fractional quantum Hall effect have the correct analytic properties and the wave function calculated in the Gaussian approximation becomes exact in the asymptotic limit.

Physical Review LettersOct 5, 199264 citations

Sum rules and density-wave modes in spin-singlet fractional quantum Hall fluids

Ritajit Kundu, Rakesh K. Dora, Dung Xuan Nguyen +1 more

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.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.

arXiv:2608.11133Aug 11, 20260 citations

Generalized Cooperative-Ring-Exchange Theory of the Fractional Quantum Hall Effect

D. H. Lee, G. Baskaran, Steven Allan Kivelson

Through the calculations of the diagonal and off-diagonal elements of the full $N$ -particle density matrix we show that all essential features of the Laughlin variational wave functions can be derived in a generalized cooperative-ring-exchange theory of the fractional quantum Hall fluid.Through the calculations of the diagonal and off-diagonal elements of the full $N$-particle density matrix we show that all essential features of the Laughlin variational wave functions can be derived in a generalized cooperative-ring-exchange theory of the fractional quantum Hall fluid.

Physical Review LettersNov 23, 198725 citations

Through the calculations of the diagonal and off-diagonal elements of the full

N

-particle density matrix we show that all essential features of the Laughlin variational wave functions can be derived in a generalized cooperative-ring-exchange theory of the fractional quantum Hall fluid.