DC resistivity near a nematic quantum critical point: Effects of weak disorder and acoustic phonons

We calculate the resistivity associated with an Ising-nematic quantum critical point in the presence of disorder and acoustic phonons in the lattice model. We use the memory-matrix transport theory, which has a crucial advantage compared to other methods of not relying on the existence of well-defined quasiparticles in the low-energy effective theory. As a result, we obtain that by including an inevitable interaction between the nematic fluctuations and the elastic degrees of freedom of the lattice (parametrized by the nemato-elastic coupling $\kappa_{\text{latt}}$), the resistivity $\rho(T)$ of the system as a function of temperature obeys a universal scaling form described by $\rho(T)\sim T\ln (1/T)$ at high temperatures, reminiscent of the paradigmatic strange metal regime observed in many strongly correlated compounds. For a window of temperatures comparable with $\kappa^{3/2}_{\text{latt}}\varepsilon_F$ (where $\varepsilon_F$ is the Fermi energy of the microscopic model), the system displays another regime in which the resistivity is consistent with a description in terms of $\rho(T)\sim T^{\alpha}$, where the effective exponent roughly satisfies the inequality $1\lesssim\alpha\lesssim 2$. However, in the low-temperature limit (\textit{i.e.}, $T\ll\kappa^{3/2}_{\text{latt}}\varepsilon_F$), the properties of the quantum critical state change in an important way depending on the type of disorder present in the system: It can either recover a conventional Fermi liquid described by $\rho(T)\sim T^2$ or it could exhibit yet another non-Fermi liquid regime characterized by the scaling form $\rho(T)-\rho_0\sim T^2\ln T$. Our results emphasize the key role played by both phonon and disorder effects in the scenario of nematic quantum criticality and might be fundamental for addressing recent transport experiments in some iron-based superconductors.