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arXiv:2504.14558 · Communications in Theoretical Physics

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

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

Published Sep 25, 2025• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1572-9494/ae0b47

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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.
Multicomponent quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, leads us to new emergent topological orders. Here, we report the theoretical discovery of fractional quantum hall effect of strongly correlated Bose-Fermi mixtures classified by the K=(m11n)\mathbf{K}=\left(\begin{array}{cc}m& 1\\ 1& n\end{array}\right)K=(m1​1n​) matrix (even mmm for boson and odd nnn for fermion), using topological flat band models. Utilizing the state-of-the-art exact diagonalization and density-matrix renormalization group methods, we build up the topological characterization based on three inherent aspects: (i) topological $(mn\ensuremath{-}1)$-fold ground-state degeneracy equivalent to the determinant of the K\mathbf{K}K matrix; (ii) fractionally quantized topological Chern number matrix equivalent to the inverse of the K\mathbf{K}K matrix; and (iii) two parallel-propagating chiral edge branches with level counting 1,2,5,10 consistent with the conformal field theory description.
The quantum Hall effect observed in a two-dimensional electron gas exposed to a perpendicular magnetic field is one of the most important modern discoveries in condensed matter physics. Advances in technology have allowed both experimental and computer-based theoretical study of miniature-size counterparts of conventional quantum Hall devices. Due to their versatile tunable electronic and magnetic properties, these systems show great promise for future technological applications. This work investigates two-dimensional semiconductor quantum dots and extended quantum rings in the quantum Hall regime. Besides being interesting on purely theoretical grounds as experimentally reachable extreme quantum-mechanical interacting many-body systems, the systems considered have major applicational interest in the realm of quantum information processing. Our emphasis lies in the strongly correlated regime where the enhanced electron-electron interactions lead to a computationally hard problem. For the systems with a few electrons, we employ the in principle exact configuration interaction method, which allows accurate study of the effects of the confinement potential and the effective form of the electron-electron interaction. Larger systems are modeled by Monte Carlo and density functional based methods. Moreover, we develop a computational method based on the reduced density-matrix functional theory to study the physics at the strongly correlated fractional quantum Hall regime with large particle numbers. The main results are related to the interplay of electron correlation and magnetism as well as the effects of tuning the electron correlation through sample engineering. The results on magnetism may find applications in future spintronics devices and spin qubits. Meanwhile, the results on electron correlation are relevant for the attempts to utilize fractional quantum Hall states in topologically protected quantum computing.
We use the dynamical structure factors of the quantum Hall states at $\ensuremath{\nu}=1/3$ and 1/21/21/2 in the lowest Landau level to study their excitation spectrum. Using the density matrix renormalization group in combination with the time-dependent variational principle on an infinite cylinder geometry, we extract the low-energy properties. At $\ensuremath{\nu}=1/3$, a sharp magnetoroton mode and the two-roton continuum are present and the finite-size effects can be understood using the fractional charge of the quasiparticle. At $\ensuremath{\nu}=1/2$, we find low-energy modes with linear dispersion and the static structure factor $\overline{s}(q)\ensuremath{\sim}{(q\ensuremath{\ell})}^{3}$ in the limit $q\ensuremath{\ell}\ensuremath{\rightarrow}0$. The properties of these modes agree quantitatively with the predictions of the composite-fermion theory placed on the infinite cylinder.
Strong correlation among electrons under high magnetic field gives rise to an entirely new arena of emergent physics, namely fractional quantum Hall effect. Such systems have entirely different elementary degrees of freedom and generally, demand non-perturbative approaches to develop a better understanding. In the literature, there are several analytical methodologies and numerical toolkits available to study such a system. Clustering of zeros, parent Hamiltonian, off-diagonal order parameter, parton construction, matrix product states are to be named among a few of those popular methodologies in the existing literature. Most of these methods work well in the lowest Landau level or holomorphic wavefunction framework. It is, however, imperative to develop such methodology to study systems with Landau levels mixing to study more exotic as well as experimentally relevant states. In this work, we have developed particular methodologies, which denounce the traditional importance of the analytic properties of first quantized model wavefunction thereby extend the existing parent Hamiltonian, topological order-parameter, matrix product states descriptions to mixed Landau level systems. Such extension produces a deeper, compact and holistic understanding of universal physics of exotic phases in strongly correlated systems from the microscopic viewpoint, as well as produces interesting new results. Our second quantized/ non-analytic approach allows us to construct the ``entangled Pauli principle", a guidebook to extract universal/topological properties such as braiding statistics, fractional charge quantization, topological degeneracy of the ground states starting from a relatively simple many-body wavefunction, ``root pattern" of fractional quantum Hall state. Such an entangled Pauli principle can be derived from a microscopic parent Hamiltonian setting, thereby provide us a potential tool to probe the non-universal physics in quantum Hall fluids as well. Essentially, entangled Pauli principle is the ``DNA" of fractional quantum Hall states. Using this guiding principle, we have shown ground states with non-abelian excitations, such as Majorana fermion or Fibonacci fermion can be stabilized for two-particle interaction. Fibonacci fermion supports universal quantum gates, thereby a potential candidate for the topologically protected universal quantum computer. Entangled Pauli principle, along with a recently developed topological order parameter for composite fermions, gives rise to Parent Hamiltonian description for composite fermions as well.
This proceedings presents recent advancements in the realization of topological states of matter using ultracold atomic gases. We focus on the simulation of the quantum Hall effect, originally discovered in two-dimensional electron gases subjected to a magnetic field. Given that ultracold atoms are neutral, simulating the effect of a magnetic field requires an ad-hoc technique. We review various methods developed for this purpose, highlighting their respective advantages and limitations. Furthermore, we explore the extension of these techniques to interacting quantum gases within topological bands, aiming to create analogs of fractional quantum Hall states. These many-body states are distinguished by their topological order, which is characterized by long-range entanglement among particles and the emergence of low-lying excitations exhibiting fractional quantum statistics. The discussion will encompass the current state of research and potential future directions in the field.
The Fermi-hypernetted-chain (FHNC) theory and the effective hypernetted-chain method are applied to study the composite-fermion (CF) states of the fractional quantum Hall effect. Using this theory we compute, in the thermodynamic limit, the correlation energy, radial distribution function, and static structure factor for all unprojected CF wave functions. The unprojected excitation gaps for $\ensuremath{\nu}=1/3,1/5$ were obtained by adopting in the FHNC a scheme previously used to compute nuclear matter excitation spectra. The results obtained so far are consistent with Monte Carlo simulations and small-number exact diagonalizations.