  Phase transitions are commonly held to occur only in the thermodynamical
limit of large number of system components. Here we exemplify at the hand of
the exactly solvable Jaynes-Cummings (JC) model and its generalization to
finite JC-lattices that finite component systems of coupled spins and bosons
may exhibit quantum phase transitions (QPT). For the JC-model we find a
continuous symmetry-breaking QPT, a photonic condensate with a macroscopic
occupation as the ground state and a Goldstone mode as a low-energy excitation.
For the two site JC-lattice we show analytically that it undergoes a
Mott-insulator to superfluid QPT. We identify as the underlying principle of
the emergence of finite size QPT the combination of increasing atomic energy
and increasing interaction strength between the atom and the bosonic mode which
allows for the exploration of an increasingly large portion of the infinite
dimensional Hilbert space of the bosonic mode. This suggests that finite system
phase transitions will be present in a broad range of physical systems.
