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

We consider a clean two-dimensional interacting electron gas subject to a random perpendicular magnetic field h (r). The field is nonquantizing in the sense that Nh, a typical flux into the area λF2 in the units of the flux quantum (λF is the de Broglie wavelength), is small, Nh 1. If the spatial scale ξ of change of h (r) is much larger than λF, 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 ω∼ ω0 = EF Nh 2 3 much smaller than the Fermi energy EF. It emerges in the third order in the interaction strength; the underlying physics is that a small phase volume ∼ (ω EF) 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 δν (ω) exhibits oscillatory behavior periodic in (ω ω0) 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.