Two-Dimensional Non-Fermi-Liquid Metals: A Solvable Large-$N$ Limit

Significant effort has been devoted to the study of "non-Fermi-liquid" (NFL) metals: gapless conducting systems that lack a quasiparticle description. One class of NFL metals involves a finite density of fermions interacting with soft order parameter fluctuations near a quantum critical point. The problem has been extensively studied in a large-$N$ limit ($N$ corresponding to the number of fermion flavors) where universal behavior can be obtained by solving a set of coupled saddle-point equations. However, a remarkable study by Lee revealed the breakdown of such approximations in two spatial dimensions. We show that an alternate approach, in which the fermions belong to the fundamental representation of a global SU($N$) flavor symmetry, while the order parameter fields transform under the adjoint representation (a "matrix large-$N$" theory), yields a tractable large $N$ limit. At low energies, the system consists of an overdamped boson with dynamical exponent z = 3 coupled to a non-Fermi-liquid with self-energy Σ(ω)~ω^2/3, consistent with previous studies.