Low energy effective theory of Fermi surface coupled with U(1) gauge field in 2+1 dimensions

TL;DR

This paper develops a low energy effective theory for a non-Fermi liquid state in 2+1 dimensions with a U(1) gauge field coupled to a Fermi surface, revealing complex quantum corrections and stability features.

Contribution

It introduces a novel large N effective theory for fermions coupled to a U(1) gauge field in 2+1D, highlighting the importance of planar diagrams and the non-perturbative nature of the system.

Findings

All planar diagrams are leading order in the large N limit.

Fermion Green's function has infinitely many quantum corrections.

Boson self energy remains unchanged beyond one-loop.

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 2d 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 non-perturbative nature of the theory also shows up in correlation functions of gauge invariant operators.