We present a functional-based approach to compute thermal expectation values for actions expressed in the $G\text{\ensuremath{-}}\mathrm{\ensuremath{\Sigma}}$ 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 $r=\ensuremath{\kappa}q$. 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, $\ensuremath{\kappa}={1/2,1,2}$. When $\ensuremath{\kappa}={1/2,1}$, we observe linear-in-temperature
T resistivities at low temperatures, indicative of strange metal behavior. Conversely, when $\ensuremath{\kappa}=2$, 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 ($\ensuremath{\sim}{T}^{2}$) at the minimum resistivity. Beyond this, another crossover occurs to strange metal behavior ($\ensuremath{\sim}T$). 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 $\ensuremath{\kappa}=2$ 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.
The quantum world is allegedly strange. But is it? What if there’s a simple mathematical explanation, and a simple solution? The success of quantum mechanics (QM) arises from the accuracy of its probability predictions, which are obtained by squaring amplitudes (the Born rule). Suppose for a moment that nature uses the negative of QM’s equations. When squared they would yield the same probabilities, confirmed by the same experiments and technological triumphs. If that were true, if nature uses the negative of QM’s equations, then the quantum world would become transparent, easy to understand. No more Schrödinger’s-cat. No quantum-eraser. No backwards-in-time cause-and-effect. No paradoxes nor enigmas. But, what’s a negative quantum equation? It could mean that particles follow zero-energy waves backwards, instead of forwards. That’s still an eccentric idea, a residual strangeness. Overall, it’s a bargain. We could swap one odd idea for another, because wave-particle duality is odd. We are accustomed to wave-particle duality. But we’ll show that experiments support the other arrangement: quantum particles follow zero-energy waves backwards. How could that possibly be true? We present substantial experimental evidence, new mathematics, and six dozen colorful illustrations. This is the Theory of Elementary Waves (TEW).