Scale-invariant hyperscaling-violating holographic theories and the resistivity of strange metals with random-field disorder

Abstract

We compute the direct-current resistivity of a scale-invariant, $d$-dimensional strange metal with dynamic critical exponent $z$ and hyperscaling-violating exponent $\ensuremath{\theta}$, weakly perturbed by a scalar operator coupled to random-field disorder that locally breaks a ${\mathbb{Z}}_{2}$ symmetry. Independent calculations via Einstein-Maxwell dilaton holography and memory matrix methods lead to the same results. We show that random-field disorder has a strong effect on resistivity and leads to a short relaxation time for the total momentum. In the course of our holographic calculation, we use a nontrivial dilaton coupling to the disordered scalar, allowing us to study a strongly coupled scale-invariant theory with $\ensuremath{\theta}\ensuremath{\ne}0$. Using holography, we are also able to determine the disorder strength at which perturbation theory breaks down. Curiously, for locally critical theories, this breakdown occurs when the resistivity is proportional to the entropy density, up to a possible logarithmic correction.