Homes' Law and Universal Planckian Relaxation

Abstract

According to Zaanen's interpretation of Homes' empirical law~[Zaanen, {\it Nature} {\bf 430}, 512 (2004)], the superconducting transition temperatures in the cuprates are high because their metallic states are as viscous as quantum mechanics permits. Here, we show that Homes' law in fact implies three key points: (i) the resistivity is linear in temperature in the normal state near the transition temperature; (ii) the dimensionless coefficient of proportionality of the relaxation rate with temperature is of order unity -- the so-called universal Planckian relaxation rate; and (iii) the logarithmically broad applicability of this law arises from an unusually wide range of effective masses throughout the cuprate phase diagram. In fact, a universal Planckian relaxation rate implies Homes' law only if the mechanism of mass renormalization is independent of the Planckian relaxation.