Current Correlations and Conductivity in SYK-Like Systems: An Analytical Study

We present a functional-based approach to compute thermal expectation values for actions expressed in the $G-\Sigma$ formalism, applicable to any time sequence ordering. Utilizing this framework, we analyze the linear response to an electric field in various Sachdev-Ye-Kitaev (SYK) chains. We consider the SYK chain where each dot is a complex $q/2$-body interacting SYK model, and we allow for $r/2$-body nearest-neighbor hopping where $r=\kappa q$. We find exact analytical expressions in the large-$q$ limit for conductivities across all temperatures at leading order in $1/q$ for three cases, namely $\kappa = \{ 1/2, 1, 2\}$. When $\kappa = \{1/2, 1\}$, we observe linear-in-temperature $T$ resistivities at low temperatures, indicative of strange metal behavior. Conversely, when $\kappa = 2$, the resistivity diverges as a power law at low temperatures, namely as $1/T^2$, resembling insulating behavior. As $T$ increases, there is a crossover to Fermi liquid behavior ($\sim T^2$) at the minimum resistivity. Beyond this, another crossover occurs to strange metal behavior ($\sim T$). In comparison to previous linear-in-$T$ results in the literature, we also show that the resistivity behavior exists even below the MIR bound, indicating a true strange metal instead of a bad metal. In particular, we find for the $\kappa = 2$ case a smooth crossover from an insulating phase to a Fermi liquid behavior to a true strange metal and eventually becoming a bad metal as temperature increases. We extend and generalize previously known results on resistivities to all temperatures, do a comparative analysis across the three models where we highlight the universal features and invoke scaling arguments to create a physical picture out of our analyses. Remarkably, we find a universal maximum DC conductivity across all three models when the hopping coupling strength becomes large.