Thermodynamic limit for the magnetic uniform electron gas and representability of density-current pairs

Although the concept of the uniform electron gas is essential to quantum physics, it has only been defined recently in a rigorous manner by Lewin, Lieb and Seiringer. We extend their approach to include the magnetic case, by which we mean that the vorticity of the gas is also held constant. Our definition involves the grand-canonical version of the universal functional introduced by Vignale and Rasolt in the context of current-density-functional theory. Besides establishing the existence of the thermodynamic limit, we derive an estimate on the kinetic energy functional that also gives a convenient answer to the (mixed) current-density representability problem.