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arXiv:cond-mat/9611004 · Physical review. B, Condensed matter

Landau level bosonization of a two-dimensional electron gas

Harry Westfahl, A. H. Castro Neto, A. O. Caldeira

Published Mar 15, 1997• 16 citations
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Abstract

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.

DOI
10.1103/physrevb.55.r7347

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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.
We present an effective theory describing the low-energy properties of an interacting two-dimensional electron gas at large noninteger filling factors \ensuremath{\nu}\ensuremath{\gg}1. Assuming that the interaction is sufficiently weak, ${\mathit{r}}_{\mathit{s}}$1, we integrate out all the fast degrees of freedom, and derive the effective Hamiltonian acting in the Fock space of the partially filled Landau level only. This theory enables us to find two energy scales controling the electron dynamics at energies less than \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$. The first energy scale, (\ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$/\ensuremath{\nu})ln(\ensuremath{\nu}${\mathit{r}}_{\mathit{s}}$), appears in the one-electron spectral density as the width of a pseudogap. The second scale, rs{\mathit{r}}_{\mathit{s}}rs​\ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$, is parametrically larger, it characterizes the exchange-enhanced spin splitting and the thermodynamic density of states.
A solvable microscopic model of interacting electrons in a half-filled Landau level is presented. The Hamiltonian is a well-defined limit of the electron-electron interaction. The low energy sector consists of a set of independent, spatially interlaced, one-dimensional systems of non-interacting fermionic dipoles, each system with a Fermi sea ground state. The total number of dipoles in the set of Fermi seas forming the two-dimensional ground state equals the number of electrons in the system and the density of states at the Fermi level is that of a two-dimensional Fermi gas. The model is obtained by generalizing, or rather adopting, the existing exact solution on a thin, but not too thin, torus to the two-dimensional bulk case.
Novel controlled nonperturbative techniques are a must in the study of strongly correlated systems, especially near quantum criticality. One of these techniques, bosonization, has been extensively used to understand one-dimensional, as well as higher-dimensional electronic systems at finite density. In this paper, we generalize the theory of two-dimensional bosonization of Fermi liquids, in the presence of a homogeneous weak magnetic field perpendicular to the plane. Here, we extend the formalism of bosonization to treat free spinless fermions at finite density in a uniform magnetic field. We show that particle-hole fluctuations of a Fermi surface satisfy a covariant Schwinger algebra, allowing to express a fermionic theory with forward scattering interactions as a quadratic bosonic theory representing the quantum fluctuations of the Fermi surface. By means of a coherent-state path integral formalism, we compute the fermion propagator as well as particle-hole bosonic correlations functions. We analyze the presence of de Haas-van Alphen oscillations and show how the quantum oscillations of the orbital magnetization, the Lifshitz-Kosevich theory, are obtained by means of the bosonized theory. We also study the effects of forward scattering interactions. In particular, we obtain oscillatory corrections to the Landau zero sound collective mode.
We apply the diagram-technique formalism beyond the Hartree-Fock approximation to a two-dimensional nearly ideal electron gas in a weak perpendicular magnetic field. The case of an almost completely filled upper Landau level [for its filling factor ${\ensuremath{\nu}}_{0}$ holds $(1\ensuremath{-}{\ensuremath{\nu}}_{0})\ensuremath{\ll}1]$ with a quantum number ${N}_{0}\ensuremath{\gg}1$ is considered. We uncover two regimes of renormalization by electron-electron interactions. In the first regime, where ${N}_{0}^{1/2}(1\ensuremath{-}{\ensuremath{\nu}}_{0})\ensuremath{\ll}1,$ these interactions lead to a splitting of the Landau levels. In the second regime, where ${N}_{0}^{1/2}(1\ensuremath{-}{\ensuremath{\nu}}_{0})\ensuremath{\gg}1,$ apart of the splitting, a renormalization of the bare Zeeman splitting occurs. The intermediate case ${N}_{0}^{1/2}(1\ensuremath{-}{\ensuremath{\nu}}_{0})\ensuremath{\approx}1$ cannot be studied within our approach. The applicability of the Fermi-liquid description is investigated for both regimes.
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.