Umklapp correction to Landau damping and conditions for non-trivial modifications to quantum critical transport

Abstract

We compute the particle--hole bubble for an Ising-nematic metal when the critical Fermi surface approaches the Brillouin zone boundary for $d=2$ dimensions. We find two qualitatively distinct contributions: i)~the standard antipodal piece, which gives ${\Pi }_{\mathrm{ATP}}(\mathbf{q},i\Omega )\propto \Omega /q$ and ii)~an additional umklapp piece from electrons near the zone boundary, which gives ${\Pi }_{\mathrm{U}}(\mathbf{q},i\Omega )\propto {\Omega }^{\alpha }$ at the minimum umklapp momentum $q\approx {\Delta }_q$ with $\alpha =2/3$ or $1/2$ depending on the temperature $T$. At high $T$ when $\alpha =1/2$, the minimum $T$ for the activation of linear/quasi-linear in $T$ resistivity, which is expected to be $T_U\propto {\Delta }_q^3$ from $z=3$ criticality, could potentially get reduced to $T_U\propto {\Delta }_q^4$ due to the $\sqrt{\Omega }$ term and discuss why we find only one hyper-specific scenario where this possibility might be realized. For $d=3$ the umklapp contribution gives ${\Pi }_{\mathrm{U}}\sim \Omega$ irrespective of $T$ therefore $T_U$ is not modified in this case.