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 caused by topological defects of the quantum XY model coupled to fermions. As in the marginal Fermi-liquid phenomenology, the fluctuations are a scale invariant function $\mathcal{F}(\omega /T)$, but unlike it, they have both a momentum dependence and a momentum-dependent coupling to fermions. They have the unusual form of being products of $\mathcal{F}(\omega /T)$ and a function of the momentum transfer. This gives that there is no vertex correction to the single-particle self-energy over a range of momenta and frequencies near the Fermi-surface, unlike Migdal's theorem. The imaginary part of the single-particle self-energy then continues to be linear in $\max (\omega ,T)$ with a weak momentum dependence for a range of momentum near the Fermi surface, as in the phenomenology. This is used to solve the vertex equation in the Kubo formula for the transport properties at low temperatures with multiplicative corrections of $O(T/E_F)$. 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 was proposed recently also for the possible loop-current order in Moiré twisted bilayer graphene and bilayer ${\mathrm{WSe}}_2$. 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, besides 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 $\propto ln{\omega }_c/T$ (${\omega }_c$ is the upper cutoff of the fluctuations), appears in the Seebeck coefficient. We also discuss in detail the conservation laws that 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 dependence is shown to remain the same for a general Fermi surface, but it is too messy to calculate the numerical coefficient.

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• Electrical conductivity
• Electron thermal conductivity
• Quantum criticality
• Seebeck effect
• Strongly correlated systems
• Non-Fermi-liquid theory