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 1In 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

Jan 1, 20002 citations

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

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

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

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

Fractional quantum Hall states in two-dimensional electron systems with anisotropic interactions

Hao Wang, Rajesh Narayanan, Xin Wan +1 more

We study the anisotropic effect of the Coulomb interaction on a 1/3-filling fractional quantum Hall system by using an exact diagonalization method on small systems in torus geometry. For weak anisotropy the system remains to be an incompressible quantum liquid, although anisotropy manifests itself in density correlation functions and excitation spectra. When the strength of anisotropy increases, we find the system develops a Hall-smectic-like phase with a one-dimensional charge density wave order and is unstable towards the one-dimensional crystal in the strong anisotropy limit. In all three phases of the Laughlin liquid, Hall-smectic-like, and crystal phases the ground state of the anisotropic Coulomb system can be well described by a family of model wave functions generated by an anisotropic projection Hamiltonian. We discuss the relevance of the results to the geometrical description of fractional quantum Hall states proposed by Haldane [Phys. Rev. Lett. 107, 116801 (2011)].We study the anisotropic effect of the Coulomb interaction on a 1/3-filling fractional quantum Hall system by using an exact diagonalization method on small systems in torus geometry. For weak anisotropy the system remains to be an incompressible quantum liquid, although anisotropy manifests itself in density correlation functions and excitation spectra. When the strength of anisotropy increases, we find the system develops a Hall-smectic-like phase with a one-dimensional charge density wave order and is unstable towards the one-dimensional crystal in the strong anisotropy limit. In all three phases of the Laughlin liquid, Hall-smectic-like, and crystal phases the ground state of the anisotropic Coulomb system can be well described by a family of model wave functions generated by an anisotropic projection Hamiltonian. We discuss the relevance of the results to the geometrical description of fractional quantum Hall states proposed by Haldane [Phys. Rev. Lett. 107, 116801 (2011)].

arXiv:1203.1982Physical Review BJul 16, 201257 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.