Transport equation of interacting fermions is re-considered in the light of the conservation law. We show that the exact equation for the particle-hole excitation is derived directly from the Ward identity, by simply transforming it. It is almost identical to that derived diagrammatically by Éliashberg, but the collision integral is written in the symmetric form. We further discuss the hidden ultraviolet divergence in the both treatments, claiming that the microscopic justification of the Landau-Silin equation has not yet been completed. The divergence is removed by incorporating the particle-particle and hole-hole pairs of excitations, without sacrificing the rigour. This requires re-identification of the distribution function, but does not alter the basic structure of the fermi liquid theory.
We analyze the quasiparticle interaction function (the fully dressed and antisymmetrized interaction between fermions) for a two-dimensional Fermi liquid at zero temperature close to a q=0 charge quantum critical point (QCP) in the $s\text{\ensuremath{-}}\mathrm{wave}$ channel (the one leading to phase separation). By the Ward identities, this vertex function must be related to quasiparticle residue
Z, which can be obtained independently from the fermionic self-energy. We show that to satisfy these Ward identities, one needs to go beyond the standard diagrammatic formulation of Fermi-liquid theory and include a series of additional contributions to the vertex function. These contributions are not present in a conventional Fermi liquid, but do emerge near a QCP, where the effective 4-fermion interaction is mediated by a soft dynamical boson. We demonstrate explicitly that including these terms restores the Ward identity. Our analysis is built on previous studies of the vertex function near an antiferromag netic QCP [Phys. Rev. B 89, 045108 (2014)] and a
d-wave charge-nematic QCP [Phys. Rev. B 81, 045110 (2010)]. We show th at for $s\text{\ensuremath{-}}\mathrm{wave}$ charge QCP the analysis is more straightforward and allows one to obtain the full quasiparticle interaction function (the Landau function) near a QCP. We show that all partial components of this function (Landau parameters) diverge near a QCP, in the same way as the effective mass
m∗, except for the
s-wave charge component, which approaches $\ensuremath{-}1$. Consequently, the susceptibilities in all channels, except for the critical one, remain finite at a QCP, as they should.
The applicability of Eliashberg theory, for fermions interacting with fluctuations of a soft collective boson near a quantum-critical point (QCP) in two dimensions, remains a subject of debate. The velocity of the soft boson is comparable to the Fermi velocity. Hence, Migdal's justification of Eliashberg theory is formally not applicable. The authors argue here that some elements of Migdal's reasoning remain valid because the collective boson is overdamped, yet vertex corrections to Eliashberg theory are
O(1). At the same time, they show that these corrections are numerically small, at least near an Ising-nematic QCP.
Quantum criticality plays a central role in understanding non-Fermi liquid behavior and unconventional superconductivity in strongly correlated systems. In this review, we explore the quantum critical Eliashberg theory, which extends conventional Eliashberg approaches to non-Fermi liquid regimes governed by critical fluctuations. We discuss the theoretical foundations and recent developments in the field, focusing on the interplay between electronic interactions and bosonic modes near quantum phase transitions as described in the Yukawa-coupled version of the Sachdev-Ye-Kitaev model. Special emphasis is placed on the breakdown of quasiparticle coherence, anomalous scaling behaviour, Cooper pairing without quasiparticles, and emergent universality in different physical settings. Starting from a zero-dimensional "quantum-dot" model, we discuss the generalization to higher spatial dimensions and demonstrate the connection between quantum-critical Eliashberg theory and holographic superconductivity. Our analysis provides a perspective on how quantum criticality shapes the dynamics of strongly correlated metals and superconductors.
Quantum criticality plays a central role in understanding non-Fermi liquid behavior and unconventional superconductivity in strongly correlated systems. In this review, we explore the quantum critical Eliashberg theory, which extends conventional Eliashberg approaches to non-Fermi liquid regimes governed by critical fluctuations. We discuss the theoretical foundations and recent developments in the field, focusing on the interplay between electronic interactions and bosonic modes near quantum phase transitions as described in the Yukawa-coupled version of the Sachdev–Ye–Kitaev model. Special emphasis is placed on the breakdown of quasiparticle coherence, anomalous scaling behavior, Cooper pairing without quasiparticles, and emergent universality in different physical settings. Starting from a zero-dimensional quantum dot model, we discuss the generalization to higher spatial dimensions and demonstrate the connection between quantum critical Eliashberg theory and holographic superconductivity. Our analysis provides a perspective on how quantum criticality shapes the dynamics of strongly correlated metals and superconductors.