Infrared catastrophe and tunneling into strongly correlated electron systems: Exact x-ray edge limit for the one-dimensional electron gas and two-dimensional Hall fluid

Abstract

In previous work [K. R. Patton and M. R. Geller, Phys. Rev. B 72, 125108 (2005)] we have proposed that the non-Fermi-liquid spectral properties in a variety of low-dimensional and strongly correlated electron systems are caused by the infrared catastrophe, and we used an exact functional integral representation for the interacting Green's function to map the tunneling problem onto the x-ray edge problem, plus corrections. The corrections are caused by the recoil of the tunneling particle, and, in systems where the method is applicable, are not expected to change the qualitative form of the tunneling density of states (DOS). Qualitatively correct results were obtained for the DOS of the one-dimensional electron gas and two-dimensional Hall fluid when the corrections to the x-ray edge limit were neglected and when the corresponding Nozi\`eres-De Dominicis integral equations were solved by resummation of a divergent perturbation series. Here we reexamine the x-ray edge limit for these two models by solving these integral equations exactly, finding the expected modifications of the DOS exponent in the one-dimensional case but finding no changes in the DOS of the two-dimensional Hall fluid with short-range interaction. Our analysis provides an exact solution of the Nozi\`eres-De Dominicis equation for the two-dimensional electron gas in the lowest Landau level.