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arXiv:cond-mat/0203022 · Journal of Experimental and Theoretical Physics

Two-dimensional electron liquid with disorder in a weak magnetic field

I. S. Burmistrov

Published Jul 1, 2002• 7 citations
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Abstract

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.

DOI
10.1134/1.1499911

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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 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.
The conductance of a weakly interacting electron gas in the presence of a single scatterer is found at arbitrary strength of the scattering potential. At weak interaction, a simple renormalization group approach can be used instead of the standard bosonization procedure. Our technique allows us to take into account the backscattering of electrons that leads to deviations of the low-temperature conductance from the behavior typical for Luttinger liquid. In the presence of magnetic field, the backscattering may give rise to a peak in differential conductance at bias equal to the Zeeman energy.
We consider a clean two-dimensional interacting electron gas subject to a random perpendicular magnetic field h(r)h(\mathbf{r})h(r). The field is nonquantizing in the sense that Nh{\mathcal{N}}_{h}Nh​, a typical flux into the area ${\ensuremath{\lambda}}_{F}^{2}$ in the units of the flux quantum (${\ensuremath{\lambda}}_{F}$ is the de Broglie wavelength), is small, Nh⪡1{\mathcal{N}}_{h}⪡1Nh​⪡1. If the spatial scale $\ensuremath{\xi}$ of change of h(r)h(\mathbf{r})h(r) is much larger than ${\ensuremath{\lambda}}_{F}$, the electrons move along semiclassical trajectories. We demonstrate that a weak-field-induced curving of the trajectories affects the interaction-induced electron lifetime in a singular fashion: it gives rise to the correction to the lifetime with a very sharp energy dependence. The correction persists within the interval $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\omega}}_{0}={E}_{F}{\mathcal{N}}_{h}^{2∕3}$ much smaller than the Fermi energy EF{E}_{F}EF​. It emerges in the third order in the interaction strength; the underlying physics is that a small phase volume $\ensuremath{\sim}{(\ensuremath{\omega}∕{E}_{F})}^{1∕2}$ for scattering processes involving two electron-hole pairs is suppressed by curving. An even more surprising effect that we find is that disorder-averaged interaction correction to the density of states $\ensuremath{\delta}\ensuremath{\nu}(\ensuremath{\omega})$ exhibits oscillatory behavior periodic in ${(\ensuremath{\omega}∕{\ensuremath{\omega}}_{0})}^{3∕2}$. In our calculations of interaction corrections, a random field is incorporated via the phases of the Green functions in the coordinate space. We discuss the relevance of the new low-energy scale for realizations of a smooth random field in composite fermions and in disordered phase of spin-fermion model of ferromagnetic quantum criticality.
In this paper, we study the effect of a weak magnetic field and the charging energy on electron conduction in a Luttinger liquid with two impurities. We consider the situation in which the two potential barriers are different in the symmetric tunneling and asymmetric tunneling limits. We focus on the dependence of the charge conductance on the temperature, the charging energy and the magnetic field. It is shown that the asymmetry of the potential barrier obstructs the resonant conduction in the symmetric tunneling limit. On the other hand, in the asymmetric tunneling limit, we treat the cases of excitation both with and without a gap. For excitation without a gap, the charging energy changes the exponent of the temperature dependence of the conductance, and there appears resonant conduction under zero magnetic field. Contrastingly, for excitation with a gap, the charging energy does not change the exponent of the conductance, and there appears resonant conduction unrelated to the magnetic field. Our results suggest that there is no spin-filter phase in the asymmetric tunneling limit.
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​.
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.