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

We analyze in detail the fermionic self-energy Sigma(omega,T) in a Fermi liquid (FL) at finite temperature T and frequency omega. We consider both canonical FLs, i.e., systems in spatial dimension D > 2, where the leading term in the fermionic self-energy is analytic [the retarded Im Sigma(R)(omega,T) = C(omega(2) + pi(2) T-2)], and noncanonical FLs in 1 < D < 2, where the leading term in Im Sigma(R)(omega, T) scales as T-D or omega(D). We relate the omega(2) + pi T-2(2) form to a special property of the self-energy, the first-Matsubara-frequency rule, which stipulates that Sigma(R)(i pi T,T) in a canonical FL contains an O(T) but no T-2 term. We show that in any D > 1, the next term after O(T) in Sigma(R)(i pi T,T) is of order T-D (T-3 ln T in D = 3). This T-D 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 T-D 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 T-D term vanishes, but also that Sigma(R)(i pi T,T) does not contain any terms beyond O(T). This rule imposes two constraints on the scaling form of the self-energy: upon replacing omega by i pi T, Im Sigma(R)(omega,T) must vanish and Re Sigma(R)(omega,T) must reduce to O(T). These two constraints should be taken into consideration in extracting scaling forms of Sigma(R)(omega,T) from experimental and numerical data.