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arXiv:cond-mat/9505026 · Physical review. B, Condensed matter

Two-dimensional electron liquid in a weak magnetic field

I. L. Aleǐner, L. I. Glazman

Published Oct 15, 1995• 118 citations
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Abstract

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, ${\mathit{r}}_{\mathit{s}}$\ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$, is parametrically larger, it characterizes the exchange-enhanced spin splitting and the thermodynamic density of states.

DOI
10.1103/physrevb.52.11296

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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, rs{\mathit{r}}_{\mathit{s}}rs​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}\ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​. The first energy scale, (\ensuremath{\Elzxh}\ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​/\ensuremath{\nu})ln(\ensuremath{\nu}rs{\mathit{r}}_{\mathit{s}}rs​), appears in the one-electron spectral density as the width of a pseudogap. The second scale, rs{\mathit{r}}_{\mathit{s}}rs​\ensuremath{\Elzxh}\ensuremathωc{\mathrm{\ensuremath{\omega}}}_{\mathit{c}}\ensuremathωc​, is parametrically larger, it characterizes the exchange-enhanced spin splitting and the thermodynamic density of states.
We present the effective theory for the low-energy dynamics of two-dimensional interacting electrons in the presence of a weak short-range disorder and a weak perpendicular magnetic field, with the filling factor γ ≫ 1. We investigate the exchange enhancement of the g factor, the effective mass, and the decay rate of the simplest spin wave excitations at γ=2 N +1. We obtain the enhancement of the field-induced gap in the tunneling density of states and the dependence of the tunneling conductivity on the applied bias.
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.
A many-body theory is given to show that the magnetic properties of a two-dimensional electron system are dependent on the filling factor ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}$ defined as the ratio of the ideal Fermi energy against the field energy. Explicit results are obtained for absolute zero and large values of this factor. The internal energy is evaluated explicitly in consideration of the ideal, first-order exchange and ring contributions. The result converges for all rs{r}_{s}rs​ because for the ring contribution no small-momentum-transfer assumption is made and for low densities a rigorous analytical continuation is adopted. When plotted against ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}$, the internal energy shows parabolic variations in a periodic way until it approaches the average kinetic energy for large ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}$. The ideal susceptibility shows zigzag changes characteristic of absolute zero when it is plotted against ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}$. It is zero at ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}=2m+1$, where it jumps from $\ensuremath{-}2m$ to 2(m+1)2(m+1)2(m+1), mmm being an integer. The Coulomb interaction suppresses the zigzag variations for small rs{r}_{s}rs​ and inverts the oscillating pattern at rs{r}_{s}rs​ \ensuremath{\sim}0.7. The oscillations thereafter may be considered characteristic of a liquid state rather than a gaseous state as they are somewhat different from those for small rs{r}_{s}rs​. The susceptibility is convergent for all rs{r}_{s}rs​.
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 investigate the properties of the one-electron Green's function in an interacting two-dimensional electron system in a strong magnetic field, which describes an electron tunneling into such a system. From finite-size diagonalization, we find that its spectral weight is suppressed near zero energy, reaches a maximum at an energy of about 0.2e2{\mathit{e}}^{2}e2/\ensuremath{\epsilon}lc{\mathit{l}}_{\mathit{c}}lc​, and decays exponentially at higher energies. We propose a theoretical model to account for the low-energy behavior. For the case of Coulomb interactions between the electrons, at even-denominator filling factors such as \ensuremath{\nu}=1/2, we predict that the spectral weight varies as ${\mathit{e}}_{0}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\omega}}}$/\ensuremath{\Vert}\ensuremath{\omega}\ensuremath{\Vert}, for \ensuremath{\omega}\ensuremath{\rightarrow}0.
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.