First-Matsubara-frequency rule in a Fermi liquid. I. Fermionic self-energy

We analyze in detail the fermionic self-energy Σ(ω,T) in a Fermi liquid (FL) at finite temperature T and frequency ω. We consider both canonical FLs, i.e., systems in spatial dimension D2, where the leading term in the fermionic self-energy is analytic [the retarded ImΣR(ω, T)=C(ω2+π2T2)], and noncanonical FLs in 1<D<2, where the leading term in ImΣR(ω,T) scales as TD or ωD. We relate the ω2+π2T2 form to a special property of the self-energy, "the first-Matsubara-frequency rule," which stipulates that ΣR(iπT,T) in a canonical FL contains an O(T) but no T2 term. We show that in any D>1, the next term after O(T) in ΣR(iπT,T) is of order TD (T3lnT in D=3). This TD term comes from only forward and backward scattering, and is expressed in terms of fully renormalized amplitudes for these processes. The overall prefactor of the TD term vanishes in the "local approximation," when the interaction can be approximated by its value for the initial and final fermionic states right on the Fermi surface. The local approximation is justified near a Pomeranchuk instability, even if the vertex corrections are non-negligible. We show that the strength of the first-Matsubara-frequency rule is amplified in the local approximation, where it states that not only the TD term vanishes, but also that ΣR(iπT,T) does not contain any terms beyond O(T). This rule imposes two constraints on the scaling form of the self-energy: upon replacing ω by iπT, ImΣR(ω,T) must vanish and ReΣR(ω,T) must reduce to O(T). These two constraints should be taken into consideration in extracting scaling forms of ΣR(ω,T) from experimental and numerical data.