Electron-electron interactions, quantum Coulomb gap, and dynamical scaling near integer quantum Hall transitions

Abstract

The effects of electron-electron interactions on tunneling into the bulk of a two-dimensional electron system are studied near the integer quantum Hall transitions. Taking into account the dynamical screening of the interactions in the critical conducting state, we show that the behavior of the tunneling density of states (TDOS) is significantly altered at low energies from its noninteracting counterpart. For the long-range Coulomb interaction, we demonstrate that the TDOS vanishes linearly at the Fermi level according to a quantum Coulomb gap form $\ensuremath{\nu}(\ensuremath{\omega}{)=C}_{Q}|\ensuremath{\omega}|{/e}^{4},$ with ${C}_{Q}$ a nonuniversal coefficient of a quantum-mechanical origin. In the case of short-range or screened Coulomb interactions, the TDOS is found to follow a power law $|\ensuremath{\omega}{|}^{\ensuremath{\alpha}},$ with \ensuremath{\alpha} proportional to the bare interaction strength. Since short-range interactions are known to be irrelevant perturbations at the noninteracting critical point, we predict that, upon scaling, the power law is smeared, leading to a finite zero-bias TDOS $\ensuremath{\nu}(\ensuremath{\omega})/\ensuremath{\nu}(0)=1+(|\ensuremath{\omega}|/{\ensuremath{\omega}}_{0}{)}^{\ensuremath{\gamma}},$ where \ensuremath{\gamma} is a universal exponent determined by the scaling dimension of short-ranged interactions. We also consider the case of quasi-one-dimensional (1D) samples with edges, i.e., the long Hall bar geometry, and find that the TDOS becomes dependent on the Hall conductance due to an altered boundary condition for diffusion. For short-range interactions, the TDOS of a quasi-1D strip with edges is linear near the Fermi level, with a slope inversely proportional to ${\ensuremath{\rho}}_{\mathrm{xx}}$ in the perturbative limit. These results are in qualitative agreement with the findings of bulk tunneling experiments. We discuss recent developments in understanding the role played by electron-electron interactions at the integer quantum Hall transitions and the implications of these results on the dynamical scaling of the transition width. We argue that for long-range Coulomb interactions, the existence of the quantum Coulomb gap in the quantum critical regime of the transition gives rise to the observed dynamical exponent $z=1.$