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.
Recent work has used a U(1) gauge theory to describe the physics of Fermi pockets in the presence of fluctuating spin density wave order. We generalize this theory to an arbitrary band structure and ordering wave vector. The transition to the large Fermi-surface state, without pockets induced by local spin density wave order, is described by embedding the U(1) gauge theory in a SU(2) gauge theory. The phase diagram of the SU(2) gauge theory shows that the onset of spin density wave order in the Fermi liquid occurs either directly, in the framework discussed by Hertz, or via intermediate non-Fermi-liquid phases with Fermi surfaces of fractionalized excitations. We discuss application of our results to the phase diagram of the cuprates.
On the basis of the self-consistent calculation scheme for the electron self-energy Σ with the use of the three-point vertex function Γ always satisfying the Ward identity, we find that the obtained quasi-particle dispersion in the normal state in gapped systems, such as semiconductors, insulators, and molecules, is well reproduced by that in the one-shot GW (or G0W0) approximation. In calculating the superconducting transition temperature Tc, we also find a similar situation; the result for Tc in the gauge-invariant self-consistent framework including the effect of Γ satisfying the Ward identity is different from that in the conventional Eliashberg theory (which amounts to the GW approximation for superconductivity) but is close to that in the G0W0 approximation. Those facts indicate that the G0W0 approximation actually takes proper account of both vertex and high-order self-energy corrections in a mutually cancelling manner and thus we can understand that the G0W0 approximation is better than the fully self-consistent GW one in obtaining some of physical quantities.
A nonequilibrium Ward identity (NE WI) connecting the scalar transport vertex correction with one-particle self-energy is derived using the global
U(1) symmetry of the fermion nonequilibrium Green's functions (NGF). The nonperturbative derivation does not depend on the details of the many-body system. A renormalized multiplicative composition rule for the NGF, reflecting time coherence, is obtained and related to the NE WI. Applications involve (i) testing the consistency of approximations shown in the example of a self-consistent Born approximation for disorder scattering, and (ii) in the general quantum transport theory, the formalism permits one to assess routes to generalized master equations, in particular those based on the generalized Kadanoff-Baym ansatz.