The nuts and bolts of gauge invariance of heat transport

In this study, I revisit the notion of gauge invariance in thermal transport and show, in the simplest and most general possible terms, why heat conductivity is unaffected by the specific choice of energy density. I provide the minimal and general conditions under which any two energy densities, though differing locally, lead to the same heat conductivity within the Green-Kubo framework. The relevance of gauge invariance in heat-transport simulations performed with machine-trained neural-network potentials is also briefly highlighted.