Quantum-critical transport in marginal Fermi liquids

TL;DR

This paper calculates electrical and thermal transport properties in quantum-critical metals with marginal Fermi-liquid behavior, revealing the effects of collective fluctuations and vertex corrections on resistivity, thermal conductivity, and Seebeck coefficient.

Contribution

It provides an exact calculation of transport coefficients in a quantum-critical model with collective fluctuations, highlighting the absence of vertex renormalizations and the specific role of mass renormalization effects.

Findings

Electrical resistivity is linear in temperature due to Umklapp scattering.

Thermal conductivity is unaffected by vertex corrections and mass renormalization.

Seebeck coefficient exhibits a logarithmic temperature dependence from mass renormalization.

Abstract

We use the Kubo response functions to calculate the electrical and thermal conductivity and Seebeck coefficient at low temperatures and frequencies in the quantum-critical region for fermions on a lattice. The theory uses scattering of the fermions with the previously derived collective fluctuations due to topological defects of the quantum XY model coupled to fermions. The microscopic model is applicable to the fluctuations of the loop-current order in cuprates as well as to a class of quasi-two-dimensional heavy-fermion and other metallic antiferromagnets, and proposed recently also for the possible loop-current order in Moir\'{e} twisted bi-layer graphene and bilayer WSe$_2$. All these metals have a linear-in-temperature electrical resistivity in the quantum-critical region of their phase diagrams, often termed ``Planckian" resistivity. The solution of the Kubo equation for transport shows that vertex renormalizations to the external fields, beside those caused by Aslamazov-Larkin (A-L) processes, are absent. A-L appears as an Umklapp scattering matrix, which gives a temperature-independent multiplicative factor for the electrical resistivity but does not affect the thermal conductivity. We also show that the mass renormalization which gives a logarithmic enhancement of the marginal Fermi-liquid specific heat does not appear in the electrical resistivity and, more remarkably, in the thermal conductivity. On the other hand the mass renormalization $\propto \ln \omega_c/T$ appears in the Seebeck coefficient. We also discuss in detail the conservation laws which play a crucial role in all transport properties. We calculate exactly, the numerical coefficients of the transport properties for a circular Fermi surface. The leading temperature dependences is shown to remain the same for a general Fermi surface, but it is too messy to calculate the numerical coefficient.