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Physical review. B, Condensed matter

Density matrices for states in the lowest Landau level of a two-dimensional electron gas

Allan H. MacDonald, S. M. Girvin

Published Sep 15, 1988• 58 citations
Publisher

Abstract

We show that possible forms for the density matrices of states of a two-dimensional electron gas (2D EG) are profoundly limited when the electrons are constrained to lie in the lowest Landau level. In particular for any number of particles the matrices may be expressed in terms of their diagonal elements in a position representation. The simplification which is known to occur for Hartree-Fock calculations in a single Landau level of the 2D EG is a specific consequence of this general property.

DOI
10.1103/physrevb.38.6295

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We show that possible forms for the density matrices of states of a two-dimensional electron gas (2D EG) are profoundly limited when the electrons are constrained to lie in the lowest Landau level. In particular for any number of particles the matrices may be expressed in terms of their diagonal elements in a position representation. The simplification which is known to occur for Hartree-Fock calculations in a single Landau level of the 2D EG is a specific consequence of this general property.
There exists a widely believed opinion, that the many-body ground state of a two-dimensional electron system at a completely filled lowest Landau level (the filling factor ν=1ν=1ν=1) is described by the so-called Hartree-Fock wave function, and that this solution is the unique, exact eigenstate of the system at ν=1ν=1ν=1. I show that this opinion is erroneous, construct an infinite number of other variational many-body wave functions, and discuss the properties of a few states which have the energy substantially lower than the energy of the Hartree-Fock state.
We investigate the ground state properties of a two-dimensional electron gas in the lowest Landau level using the Density Matrix Renormalization Group. The electron gas is confined to a cylinder with a strong magnetic field perpendicular to the surface. For a thin cylinder, the ground state is for generic filling factors a charge density wave state, however, at ν=1/2ν=1/2ν=1/2 a homogeneous and apparently gapless ground state is found and particle and hole excitations are studied.
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.
Abstract A partial trace over the occupation numbers of all but k states in the density matrix of an ensemble with an arbitrary number of single‐particle states is defined as the (reduced) k‐state density matrix. This matrix is used to obtain a complete, practical solution to the problem of determining the representability of the diagonal elements of the one‐ and two‐particle (reduced) density matrices. This solution is expressed as a series of linear inequalities involving the density‐matrix elements; the inequalities are identical with those derived previously by Davidson and McCrae by a different method. In addition, our method is used to obtain nonlinear, matrix inequalities on the off‐diagonal elements of the density matrices.
In this work we introduce a bosonization scheme for the low-energy excitations of a two-dimensional interacting electron gas in the presence of a uniform magnetic field under conditions where a large integral number of Landau levels are filled. We give an explicit construction for the electron operator in terms of the bosons. We show that the elementary neutral excitations, known as the magnetic excitons or magnetoplasma modes, can be described within a bosonic language and that it provides a quadratic bosonic Hamiltonian for the interacting electron system that can be easily diagonalized.
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.