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Theoretical studies of the two-dimensional interacting electron system in high magnetic field

Matthew Brownlie

Published Dec 10, 2013• 0 citations
OpenAlex

Abstract

This is a mathematical study of certain aspects of the interacting electron system in very high perpendicular magnetic field.\nWe analyse restrictions imposed upon the density correlation functions of this system and propose a set of sum rules which they must obey.\nWe study the possibility of building a bosonisation scheme for the projected density operators in the lowest Landau level. We suggest a second order bosonisation, along with an approximation scheme, which may be useful for carrying out calculations in the lowest Landau level.\nWe analyse the possible ground states of the system. We suggest a set of variational wavefunctions which can have lower energy than the Laughlin state for sufficiently soft interaction potentials.\nWe study the collective excitations of the system, paying particular attention to its symmetries. We suggest a set of variational excited states and discuss their applicability to finite as well as infinite systems.

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This is a mathematical study of certain aspects of the interacting electron system in very high perpendicular magnetic field.\nWe analyse restrictions imposed upon the density correlation functions of this system and propose a set of sum rules which they must obey.\nWe study the possibility of building a bosonisation scheme for the projected density operators in the lowest Landau level. We suggest a second order bosonisation, along with an approximation scheme, which may be useful for carrying out calculations in the lowest Landau level.\nWe analyse the possible ground states of the system. We suggest a set of variational wavefunctions which can have lower energy than the Laughlin state for sufficiently soft interaction potentials.\nWe study the collective excitations of the system, paying particular attention to its symmetries. We suggest a set of variational excited states and discuss their applicability to finite as well as infinite systems.
Various methods and approximation schemes are used to study many-electron interacting systems. Two important many-particle models, the Anderson model and the Hubbard model, and their electromagnetic properties have been investigated in many parameter regimes, and applied to physical systems. An Anderson single-impurity model Hamiltonian based calculation of the magnetic susceptibility is performed for YbN in the presence of crystal fields using an alteration of the Non-Crossing Approximation proposed by Zwicknagl et.al., incorporating parameters obtained from ab initio band structure calculations. It yields good agreement with experimental data. For the Anderson lattice model, a variational scheme which uses specific many-electron wavefunctions as basis is applied to both one- and two-dimensional systems represented by symmetric Anderson lattice Hamiltonians. Without much computational effort, the ground state energy is well approximated, especially in strong-coupling limit. Some electronic properties are examined using the variational ground state wavefunction. The one-dimensional Hubbard model has been solved exactly for small-size clusters by diagonalizing the Hamiltonian in the basis of many-electron Bloch states. The results for the energy spectrum and eigenfunctions of the ground state and low-lying excited states are presented. Also, mean field calculations of the two-dimensional single-band Hubbard model and Cu-O lattice model (three-band Hubbard model) are carried out for various physical quantities including the energy, occupation probability, staggered magnetization, momentum distribution Fermi surface and density of states, by using a projection operator formalism. To develop a systematic approach to solving many-electron problems, the many-particle partition function for the free electron gas system is explored using a cumulant expansion scheme. Starting from the ground state, the partition function can be approximated to any order in terms of excitation energy. Its application to interacting systems such as the Anderson model and the Hubbard model is briefly discussed.
A system of N interacting bosons or fermions in a two-dimensional harmonic potential (or, equivalently, magnetic field) whose states are projected onto the lowest Landau level is considered. Generic expressions are derived for matrix elements of any interaction, in the basis of angular momentum eigenstates. For the fermion "ground state" (N=1 Laughlin state), this makes it possible to exactly calculate its energy all the way up to the mesoscopic regime N ~ 1000. It is also shown that for N = 3 and Coulomb interaction, several rational low-lying values of energy exist, for bosons and fermions alike.
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.
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.