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

TL;DR

The work addresses the emergence of a non-Fermi liquid in 2+1D by coupling a Fermi surface patch to a transverse U(1) gauge field in the large $N$ limit. It introduces a minimal local action and uses a genus-based $1/N$ expansion to show that planar diagrams govern the leading dynamics, yielding infinitely many leading corrections to fermion propagation while the boson self-energy remains fixed at one-loop, ensuring stability. Gauge-invariant correlators reveal nonperturbative structure despite perturbative stability in the boson sector, highlighting a robust yet strongly coupled low-energy theory. The results illuminate how non-Fermi liquid behavior can arise from gauge interactions in two spatial dimensions and set the stage for further exploration of multi-patch extensions and potential connections to broader strong-coupling frameworks.

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.

Figures (20)

• Figure 1: The parabolic Fermi surface of the model in Eq. (\ref{['a1']}). The shaded region includes negative energy states.
• Figure 2: The one-loop boson self energy.
• Figure 3: The one-loop fermion self energy. Here the boson propagator is a dressed propagator which include the one-loop self energy correction in Fig. \ref{['fig:RPA']}.
• Figure 4: A two-loop vertex correction.
• Figure 5: A three-loop fermion self energy correction.