  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.
