Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Europhysics Letters (EPL)

Exact Two-Body Solutions for Two-Dimensional Interacting Electrons in a Magnetic Field

Y. F Ge

Published Sep 1, 1991• 3 citations
Publisher

Abstract

The Schrödinger equation for the two-body problem of two-dimensional interacting electrons in a magnetic field is exactly solved. It is found that the degeneracy of the Landau levels for noninteracting electrons is removed by the Coulomb interaction, and that the ground state is nondegenerate and possesses an energy gap from the first excited state. Our result suggests that the interacting electrons in a strong magnetic field form a new condensate which may be identified as a vortex.

DOI
10.1209/0295-5075/16/1/003

Related papers

Interacting electrons in two-dimensional Landau levels: Results for small clusters

S. M. Girvin, Terrence Jach

We study the two-dimensional electron gas in a quantizing magnetic field for the cases of Coulomb and harmonic interactions among the electrons. Numerical solutions for the quantum states of clusters of up to five electrons show that the strength of the excitation gap is a strongly oscillating function of the density not unlike what is observed in the anomalous quantum Hall effect. We present analytic results for the case of harmonic interactions and show that the variational wave function recently proposed by Laughlin for the Coulomb problem is in fact an exact eigenstate of the harmonic problem.

Physical review. B, Condensed matter•Oct 15, 1983•133 citations

Interacting electrons in magnetic fields: Tracking potentials and Jastrow-product wave functions

Gábor Fáth, Stephen B. Haley

The Schr\"odinger equation for an interacting spinless electron gas in a nonuniform magnetic field admits an exact solution in Jastrow product form when the fluctuations in the magnetic field track the fluctuations in the scalar potential. For tracking realizations in a two-dimensional electron gas, the degeneracy of the lowest Landau level persists and the ``tracking'' solutions span the ground state subspace. In the context of the fractional quantum Hall problem, the Laughlin wave function is shown to be a tracking solution. Tracking solutions for screened Coulomb interactions are also constructed. The resulting wave functions are proposed as variational wave functions with potentially lower energy in the case of non-negligible Landau level mixing than the Laughlin function.

arXiv:cond-mat/9810328•Physical review. B, Condensed matter•Jul 15, 1998•4 citations

Exactly Solvable Model for Strongly Interacting Electrons in a Magnetic Field

Abhishek Anand, Jainendra K. Jain, G. J. Sreejith

States of strongly interacting particles are of fundamental interest in physics and can produce exotic emergent phenomena and topological structures. We consider here two-dimensional electrons in a magnetic field, and, departing from the standard practice of restricting to the lowest LL, introduce a model short-range interaction that is infinitely strong compared to the cyclotron energy. We demonstrate that this model lends itself to an exact solution for the ground as well as excited states at arbitrary filling factors ν<1/2p and produces a fractional quantum Hall effect at fractions of the form ν=n/(2pn+1), where n and p are integers. The fractional quantum Hall states of our model share many topological properties with the corresponding Coulomb ground states in the lowest Landau level, such as the edge physics and the fractional charge of the excitations.

arXiv:2010.12594•Physical Review Letters•Mar 29, 2021•6 citations

Sub-Landau levels in two-dimensional electron system in magnetic field

G.‐Q. Hai

We study two interacting electrons in a two-dimensional system under a strong magnetic field and show that their numerically exact solutions organize into a set of {\em sub-Landau levels} characterized by relative angular momentum quantum number $m$. These sub-levels define correlation-resolved subspaces of the Landau-level Hilbert space, while retaining the full degeneracy associated with center-of-mass motion. Within this structure, the accessible states in each correlation channel are effectively reduced, leading to a natural organization of guiding-center states consistent with a fractional occupancy. We further analyze the role of electron correlation, Zeeman splitting, and disorder in stabilizing spin-polarized electron-pair states. Building on the two-electron states, we construct a class of many-electron trial wavefunctions based on correlated electron pairs with fixed $m$, which encode short-range correlations through the vanishing of the pair wavefunction at small separation. Our results establish a direct connection between exact two-body physics and the organization of correlated many-electron states in the lowest Landau level, providing a microscopic perspective on how relative angular momentum structures can underpin the emergence of correlated phases in quantum Hall systems.

arXiv:2406.06212•Jun 10, 2024•1 citation

Effect of the Landau Level Mixing on the Ground State of Two-Dimensional Electrons

D. Yoshioka

Two dimensional electrons in a strong magnetic field show the fractional quantum Hall effect at low temperatures. This effect is explained successfully by a discovery of a new liquid type ground state. However the infinitely strong magnetic field has been assumed in existing theories. Here the ground state around one third filling of the lowest Landau level is investigated at a finite magnetic field by numerical diagonalization of the Hamiltonian. When the cyclotron energy is not too small compared to a typical Coulomb energy, no qualitative change of the ground state is found: A natural generalization of the liquid state at the infinite magnetic field describes the ground state. Excitation energies of quasiparticles decrease as the magnetic field decreases.

Journal of the Physical Society of Japan•Nov 15, 1984•80 citations

Chapter 6 Microscopic Theory of a Strongly Correlated Two-Dimensional Electron Gas

MIKHAIL V. ZVEREV, V. A. Khodel

This chapter presents the microscopic theory of a two-dimensional (2D) electron gas based on an ab initio functional approach to the many-body problem developed by V. A. Khodel et al. and V. R. Shaginyan. It discusses the detailed structure of the theory and scheme of the numerical solution of relevant equations. The chapter also discusses the method of evaluation of the single-particle spectrum of a 2D electron gas. It examines the behavior of a 2D electron gas in external magnetic fields. The chapter focuses on the disappearance of magnetic oscillations in electron systems of metal-oxide-semiconductor field-effect transistors due to merging of spin- and valley-split Landau levels in silicon, being the topological rearrangement of the Landau state, analogous to the formation of the fermion condensate. It explores the impact of the violation of the equidistance of electron spectra in strong magnetic fields on the structure of magnetic oscillations near the critical density.

•May 25, 2017•0 citations
The Schrödinger equation for the two-body problem of two-dimensional interacting electrons in a magnetic field is exactly solved. It is found that the degeneracy of the Landau levels for noninteracting electrons is removed by the Coulomb interaction, and that the ground state is nondegenerate and possesses an energy gap from the first excited state. Our result suggests that the interacting electrons in a strong magnetic field form a new condensate which may be identified as a vortex.
We study the two-dimensional electron gas in a quantizing magnetic field for the cases of Coulomb and harmonic interactions among the electrons. Numerical solutions for the quantum states of clusters of up to five electrons show that the strength of the excitation gap is a strongly oscillating function of the density not unlike what is observed in the anomalous quantum Hall effect. We present analytic results for the case of harmonic interactions and show that the variational wave function recently proposed by Laughlin for the Coulomb problem is in fact an exact eigenstate of the harmonic problem.
The Schr\"odinger equation for an interacting spinless electron gas in a nonuniform magnetic field admits an exact solution in Jastrow product form when the fluctuations in the magnetic field track the fluctuations in the scalar potential. For tracking realizations in a two-dimensional electron gas, the degeneracy of the lowest Landau level persists and the ``tracking'' solutions span the ground state subspace. In the context of the fractional quantum Hall problem, the Laughlin wave function is shown to be a tracking solution. Tracking solutions for screened Coulomb interactions are also constructed. The resulting wave functions are proposed as variational wave functions with potentially lower energy in the case of non-negligible Landau level mixing than the Laughlin function.
States of strongly interacting particles are of fundamental interest in physics and can produce exotic emergent phenomena and topological structures. We consider here two-dimensional electrons in a magnetic field, and, departing from the standard practice of restricting to the lowest LL, introduce a model short-range interaction that is infinitely strong compared to the cyclotron energy. We demonstrate that this model lends itself to an exact solution for the ground as well as excited states at arbitrary filling factors ν<1/2p and produces a fractional quantum Hall effect at fractions of the form ν=n/(2pn+1), where n and p are integers. The fractional quantum Hall states of our model share many topological properties with the corresponding Coulomb ground states in the lowest Landau level, such as the edge physics and the fractional charge of the excitations.
We study two interacting electrons in a two-dimensional system under a strong magnetic field and show that their numerically exact solutions organize into a set of {\em sub-Landau levels} characterized by relative angular momentum quantum number mmm. These sub-levels define correlation-resolved subspaces of the Landau-level Hilbert space, while retaining the full degeneracy associated with center-of-mass motion. Within this structure, the accessible states in each correlation channel are effectively reduced, leading to a natural organization of guiding-center states consistent with a fractional occupancy. We further analyze the role of electron correlation, Zeeman splitting, and disorder in stabilizing spin-polarized electron-pair states. Building on the two-electron states, we construct a class of many-electron trial wavefunctions based on correlated electron pairs with fixed mmm, which encode short-range correlations through the vanishing of the pair wavefunction at small separation. Our results establish a direct connection between exact two-body physics and the organization of correlated many-electron states in the lowest Landau level, providing a microscopic perspective on how relative angular momentum structures can underpin the emergence of correlated phases in quantum Hall systems.
Two dimensional electrons in a strong magnetic field show the fractional quantum Hall effect at low temperatures. This effect is explained successfully by a discovery of a new liquid type ground state. However the infinitely strong magnetic field has been assumed in existing theories. Here the ground state around one third filling of the lowest Landau level is investigated at a finite magnetic field by numerical diagonalization of the Hamiltonian. When the cyclotron energy is not too small compared to a typical Coulomb energy, no qualitative change of the ground state is found: A natural generalization of the liquid state at the infinite magnetic field describes the ground state. Excitation energies of quasiparticles decrease as the magnetic field decreases.
This chapter presents the microscopic theory of a two-dimensional (2D) electron gas based on an ab initio functional approach to the many-body problem developed by V. A. Khodel et al. and V. R. Shaginyan. It discusses the detailed structure of the theory and scheme of the numerical solution of relevant equations. The chapter also discusses the method of evaluation of the single-particle spectrum of a 2D electron gas. It examines the behavior of a 2D electron gas in external magnetic fields. The chapter focuses on the disappearance of magnetic oscillations in electron systems of metal-oxide-semiconductor field-effect transistors due to merging of spin- and valley-split Landau levels in silicon, being the topological rearrangement of the Landau state, analogous to the formation of the fermion condensate. It explores the impact of the violation of the equidistance of electron spectra in strong magnetic fields on the structure of magnetic oscillations near the critical density.