The breakdown of the zeroth law of thermodynamics and the definition of temperature in small quantum systems

We study two small quantum systems coupled to the same reservoir which is in thermal equilibrium. By studying the particle density and the energy density in the two systems before and after they contact each other, we find that the two systems are not in thermal equilibrium with each other. Our result shows that the zeroth law of thermodynamics is broken in small quantum systems at low temperatures. Therefore, the traditional way of defining temperature fails due to the breakdown of the transitive relation of thermal equilibrium. Then we show a different way of defining temperature by attaching an auxiliary site, which plays the role of a thermometer, to the small quantum system.