Landau Fermi liquid theory is a fixed-point theory of metals that includes the forward-scattering amplitudes as exact marginal couplings. However, the fixed-point theory that only includes the strict forward scatterings is nonlocal in real space. In this paper, we revisit the Fermi liquid theory for charge-neutral fermions using the field-theoretic functional renormalization group formalism and show how the scale-invariant fixed point emerges as a local theory, which includes not only the forward scatterings but also nonforward scatterings with small but nonzero momentum transfers. In the low-energy limit, the near-forward scattering and pairing interactions take scale-invariant forms if the momentum transfer and the center-of-mass momentum of Cooper pairs, respectively, are comparable to the energy. The coupling functions fully capture the universal low-energy dynamics of the collective modes and instabilities of Fermi liquids. A runaway flow of the coupling function in the particle-hole channel beyond a critical interaction suggests an instability toward an ordered phase with a wave vector that depends on the interaction strength. At the critical interaction, the instability corresponds to the uniform Pomeranchuk or Stoner instability, but the momentum of the leading instability becomes nonzero for stronger attractive interaction. In the particle-particle channel, the coupling function reveals the dynamics of the unstable mode associated with the BCS instability. When an unstable normal metal evolves into the superconducting state, there exists a period in which a superconducting state with spatially nonuniform phase appears due to the presence of unstable Cooperon modes with nonzero momenta.
Forward scattering and backscattering play an exceptional role in the physics of two-dimensional interacting fermions. In a Fermi liquid, both give rise to a nonanalytic ${\ensuremath{\omega}}^{2}ln(\ensuremath{\omega})$ form of the fermionic scattering rate at second order in the interaction. Here we argue that higher powers of $ln(\ensuremath{\omega})$ appear in the backscattering contribution at higher orders. We show that these terms come from ``planar'' processes, which are effectively one-dimensional. This is explicitly demonstrated by extending a Fermi liquid to the limit of $N\ensuremath{\gg}1$ fermionic flavors, when only planar processes survive. We sum the leading logarithms for the case of a two-dimensional Fermi liquid near a nematic transition, and we obtain an expression for the scattering rate at
T=0 to all orders in the interaction. For a repulsive interaction, the resulting rate is logarithmically suppressed, and the result is valid down to $\ensuremath{\omega}=0$. For an attractive interaction, the ground state is an
s-wave superconductor with a gap ${\mathrm{\ensuremath{\Delta}}}_{0}$. We show that in this case the scattering rate increases as $\ensuremath{\omega}$ is reduced toward ${\mathrm{\ensuremath{\Delta}}}_{0}$. At $\ensuremath{\omega}\ensuremath{\ge}{\mathrm{\ensuremath{\Delta}}}_{0}$, the behavior of the scattering rate is rather unconventional as many pairing channels compete near a nematic critical point, and the
s-wave wins only by a narrow margin. We take superconductivity into consideration and obtain the scattering rate also at smaller $\ensuremath{\omega}\ensuremath{\simeq}{\mathrm{\ensuremath{\Delta}}}_{0}$.
We present a class of one-dimensional, generic, spinless fermion lattice Hamiltonians that express quasi-Fermi liquid physics, manifesting both Luttinger and Fermi liquid features solely due to irrelevant interactions. Using infinite matrix product state techniques, we unveil its universal structure by calculating static and dynamic responses. Key features include a finite discontinuity in the momentum distribution at the Fermi level, despite power-law singularities in the spectral function protected by particle-hole symmetry. Away from half filling, Landau quasiparticles emerge. Charge dynamics show either high-energy bound states or a concentration of spectral weight within the continuum for attractive or repulsive interactions, respectively. These universal features are realized across multiple models and energy scales, thus reifying the quasi-Fermi liquid as a unique paradigm for one-dimensional fermions.