Interaction effects in a two-dimensional electron gas in a random magnetic field: Implications for composite fermions and the quantum critical point

Abstract

We consider a clean two-dimensional interacting electron gas subject to a random perpendicular magnetic field $h(\mathbf{r})$. The field is nonquantizing in the sense that ${\mathcal{N}}_{h}$, 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, ${\mathcal{N}}_{h}⪡1$. If the spatial scale $\ensuremath{\xi}$ of change of $h(\mathbf{r})$ 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 ${E}_{F}$. 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.