Shear Viscosity at the van Hove singularity
The applicability of the semiclassical Boltzmann transport theory is fundamentally challenged in strongly correlated systems where quasiparticle excitations are ill-defined. When the fermion spectral broadening becomes much larger than the boson broadening, the Boltzmann approach to transport is not always valid, particularly in the dirty limit of the critical regime. Using a diagrammatic Kubo formalism, we compute several critical transport coefficients at a van Hove singularity and show that, while the conductivity happens to agree with the Boltzmann result, the dc shear viscosity exhibits qualitatively different behavior. The diagrammatic Kubo results are more reliable because, under the fermion sharp peak approximation--an assumption that strictly breaks down in the dirty critical limit--we demonstrate that the leading order Feynman diagrams reduce to the Boltzmann equation. The same critical model, which can also account for strange metal, makes experimentally testable predictions for the optical and dc shear viscosities, ${\rm Re}[\eta(\Omega)]\sim (|\Omega|^{3/2}+T^{3/2})/\Omega^2$ and ${\rm Re}[\eta(\Omega=0)]\sim T$, providing further opportunities to assess the validity of our theoretical framework.