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

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.

• The paper develops a systematic framework that classifies quantum corrections using a genus expansion of Feynman diagrams.
• The paper reveals that even at large N, strong non-perturbative effects modify the fermion Green’s function while leaving boson self-energy largely unchanged.
• The paper confirms the stability of the effective theory by demonstrating the absence of higher-order renormalization, offering insights into quantum critical phenomena.

Analysis of Low Energy Effective Theory of Fermi Surfaces and U(1) Gauge Fields in 2+1 Dimensions

The paper "Low energy effective theory of Fermi surface coupled with U(1) gauge field in 2+1 dimensions" by Sung-Sik Lee addresses significant theoretical challenges in understanding non-Fermi liquid (NFL) states. These NFL states arise in systems where a transverse U(1) gauge field interacts with a patch of Fermi surface populated by $N$ fermion flavors in two spatial dimensions. Examining this interaction in the limit of large $N$, the paper explores the classification of quantum corrections and elucidates the intricacies of achieving a stable, interacting quantum field theory under such conditions.

Key Contributions

1. Theory Construction and Diagram Classification: The author develops a comprehensive framework within which quantum corrections to the effective low-energy theory are systematically categorized. By employing a topological consideration, the paper classifies Feynman diagrams based on the genus of a 2D surface that supports the lattice structure of these diagrams without crossing. This technique is reminiscent of the genus expansion approach used in the SU(N) gauge theories, where the structure of Feynman diagrams can be analyzed by considering surfaces in their double-line representation.
2. Non-Perturbative Insights: Notably, even in the large $N$ limit, low-energy fermions remain strongly coupled due to the persistent quantum fluctuations engendered by proximity to the Fermi surface. This feature foregrounds the non-perturbative nature of the theory. As a consequence, the fermion Green's function experiences substantial quantum corrections, while intriguingly, the boson self-energy remains unaltered beyond the one-loop level. Planar diagrams contribute significantly to the effective action, reflecting the inadequacy of truncated Dyson equations or traditional summation methods.
3. Stability and Absence of Higher-Order Divergences: The paper asserts the stability of the theory in the large $N$ limit by confirming that the coupling constants do not undergo renormalization beyond the leading order. Hence, the absence of perturbative expansion challenges not only the understanding of critical phenomena but also motivates deeper exploration into strongly correlated systems.
4. Implications for Gauge Invariant Observables: The study underscores how the non-perturbative interactions manifest in gauge-invariant observables, such as correlation functions of density operators. Particularly, the three-point correlation functions acquire enhancement due to possible systemic particle-hole excitations that remain on the Fermi surface.

Implications and Speculations

The theoretical findings of this work have critical implications for characterizing quantum critical behaviors, particularly those involving complex materials like high-temperature superconductors and quantum spin liquids. The distinct nature of the emerging quantum field theory promotes the fundamental understanding of materials that fall beyond conventional Fermi liquid descriptions. Additionally, the insights gathered provide potent avenues for exploring failure modes of perturbative expansions in non-relativistic quantum field theories.

Looking forward, the study invites further investigation into dual models or alternative computational tools that could adequately capture non-perturbative features in such stark NFL states. The analogies drawn with SU(N) gauge theories beckons for explorations in holographic correspondences that might furnish a more tractable landscape for addressing the problems encased in the work. As research progresses, these theoretical advancements may illuminate unexplored depths of quantum correlations in condensed matter systems, driving technological innovations through novel superconducting or quantum phase materials.