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

TL;DR

This paper analyzes how umklapp processes affect Landau damping and quantum critical transport, revealing distinct contributions in 2D and 3D systems and their implications for resistivity behavior.

Contribution

It introduces a detailed computation of the particle-hole bubble considering umklapp effects near the Brillouin zone boundary in quantum critical metals.

Findings

In 2D, umklapp processes modify the damping with a temperature-dependent exponent.

In 3D, umklapp contributions do not alter the damping behavior.

The study discusses potential reductions in the activation temperature for linear resistivity due to umklapp effects.

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 $\Pi_{\rm{ATP}}(\mathbf{q}, i\Omega)\propto\Omega/q$ and ii)~an additional umklapp piece from electrons near the zone boundary, which gives $\Pi_{\rm{U}}(\mathbf{q}, i\Omega)\propto \Omega^\alpha$ 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 $\Pi_{\rm{U}}\sim \Omega$ irrespective of $T$ therefore $T_U$ is not modified in this case.