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

TL;DR

The paper advances density functional theory for fractional quantum Hall systems by embedding the composite fermion picture into a KS-DFT framework, enabling efficient treatment of ground-state densities and edge effects. It develops CF-based KS equations with kinetic, Hartree, external, and exchange-correlation contributions, and validates the approach against Laughlin-like states at $\nu=1/3$ by reproducing accurate densities and a consistent ground-state energy per particle $(-0.41026\pm 0.00003)$. It then shows that DFT can capture fractional charge, anyonic statistics via Berry phases, and the magnetoroton dispersion, while identifying limitations such as the lack of explicit LLL projection and the need for better functionals. Looking forward, the work highlights TDDFT for non-equilibrium dynamics, LLL-projected functionals, and ML-assisted functionals as promising directions to enhance accuracy and applicability to more complex FQH states.

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 group, matrix product states, and Monte Carlo methods, are essential to examine 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.

Figures (11)

• Figure 1: Schematic illustration of the mapping between $\nu = 1/3$ FQH state and $\nu = 1$ IQH state after binding two Chern-Simons flux to each electron in CF theory.
• Figure 2: Density profile of the $\nu=1/3$ FQH state: DFT ground state vs. Laughlin state. The edge density oscillations are more pronounced in DFT due to Coulomb interactions.
• Figure 3: The ground state energy per CF at $\nu=1/3$. The extrapolated value in thermodynamic limit obtained ($-0.41026 \pm 0.00003$) agrees well with previous reported value $\epsilon \approx -0.41015$backpotentialgsenergy2gsenergy1.
• Figure 4: Charge accumulation for (a) a quasihole and (b) a quasiparticle at the origin in the $\nu = 1/3$ FQH state. (a) Quasihole with charge $e/3$ at position $(n=0,~m=0)$, and (b) Quasiparticle with charge $-e/3$ at position $(n=1,~m=-1)$. The inset shows the corresponding electron density distribution for each case.
• Figure 5: (a) The total angular momentum of the global ground state as tuning the magnitude of $Q/h$ for 100 electrons at $\nu = 1/3$. The inset schematic diagram shows the CF occupations. (b) The charge accumulation $\mathcal{C}$ for all kinds of excitations.