Low-energy effective theory of Fermi surface coupled with U(1) gauge field in2+1dimensions

Abstract

We study the low-energy effective theory for a non-Fermi-liquid state in $2+1$ dimensions, where a transverse U(1) gauge field is coupled with a patch of Fermi surface with $N$ flavors of fermion in the large $N$ limit. In the low-energy limit, quantum corrections are classified according to the genus of the two-dimensional surface on which Feynman diagrams can be drawn without a crossing in a double line representation and all planar diagrams are important in the leading order. The emerging theory has the similar structure to the four-dimensional SU(N) gauge theory in the large $N$ limit. Because of strong quantum fluctuations caused by the abundant low-energy excitations near the Fermi surface, low-energy fermions remain strongly coupled even in the large $N$ limit. As a result, there are infinitely many quantum corrections that contribute to the leading frequency dependence of the Green's function of fermion on the Fermi surface. On the contrary, the boson self-energy is not modified beyond the one-loop level and the theory is stable in the large $N$ limit. The nonperturbative nature of the theory also shows up in correlation functions of gauge-invariant operators.