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-Σ 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=κq. 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 κ= \{ 1/2, 1, 2\}. When κ= \{1/2, 1\}, we observe linear-in-temperature T resistivities at low temperatures, indicative of strange metal behavior. Conversely, when κ= 2, the resistivity diverges as a power law at low temperatures, namely as 1/T², resembling insulating behavior. As T increases, there is a crossover to Fermi liquid behavior $sim T²$ 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 κ= 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.