Physical laws? Have fun with that. Vanadium dioxide (VO2) will be over here conducting electricity without conducting heat in brazen defiance of one of the most fundamental rules of conductors: the Wiedemann–Franz law. There are a few other materials that can conduct electricity better than they conduct heat, but they require such frigid temperatures to do it that they're pretty impractical. VO2, meanwhile, conducts electricity exclusively when it's above room temperature, while producing virtually no heat.
How? Junqiao Wu, a researcher on the study that was published in the January 2017 issue of Science, explains: "For electrons, heat is a random motion. Normal metals transport heat efficiently because there are so many different possible microscopic configurations that the individual electrons can jump between. In contrast, the coordinated, marching-band-like motion of electrons in vanadium dioxide is detrimental to heat transfer as there are fewer configurations available for the electrons to hop randomly between." Even better, you can control how much electricity and heat VO2 conducts by mixing it with other materials. That means we might be able to make it into an insulator that only dissipates heat once the temperature gets to a certain point—a very handy thing for building engines, for instance.