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

TL;DR

This paper investigates the fermionic self-energy in Fermi liquids at finite temperature and frequency, revealing a special 'first-Matsubara-frequency rule' that constrains the self-energy's temperature and frequency dependence, especially near a Pomeranchuk instability.

Contribution

It establishes the first-Matsubara-frequency rule in Fermi liquids and analyzes how the self-energy's temperature dependence is constrained, especially in the local approximation near instabilities.

Findings

The T^D term in self-energy arises from forward and backward scattering.

In the local approximation, the T^D term vanishes, amplifying the first-Matsubara-frequency rule.

The rule constrains the self-energy's scaling form, affecting experimental and numerical analysis.

Abstract

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 -- 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 non-canonical 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^2 T^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.