Ransom Magnetic Field and the Dirac Fermi Surface

We study a single 2d Dirac fermion at finite density, subject to a quenched random magnetic field. Applications of this theory include graphene when restricted to a single valley, the gapless surface states of a 3d time-reversal topological insulator, an integer quantum Hall plateau transition, and the (mean-field) Dirac composite fermion description of the half-filled lowest Landau level. At low energies and sufficiently weak disorder, the theory maps onto an infinite collection of 1d chiral fermions (associated to each point on the Fermi surface) coupled by a random vector potential. This low-energy theory exhibits an exactly solvable random fixed line, along which we directly compute various disorder-averaged observables without the need for the usual replica, supersymmetry, or Keldysh techniques. We find the longitudinal dc conductivity in the collisionless ω/T → ∞ limit to be nonuniversal and to vary continuously along the fixed line.