  Equilibrium phase transitions usually emerge from the microscopic behavior of
many-body systems and are associated to interesting phenomena such as the
generation of long-range order and spontaneous symmetry breaking. They can be
defined through the non-analytic behavior of thermodynamic potentials in the
thermodynamic limit. This limit is obtained when the number of available
configurations of the system approaches infinity, which is conventionally
associated to spatially-extended systems formed by an infinite number of
degrees of freedom (infinite number of particles or modes). Taking previous
ideas to the extreme, we argue that such a limit can be defined even in
non-extended systems, providing a specific example in the simplest form of a
single-mode bosonic Hamiltonian. In contrast to previous non-extended models,
the simplicity of our model allows us to find approximate analytical
expressions that can be confronted with precise numerical simulations in all
the parameter space, particularly as close to the thermodynamic limit as we
want. We are thus able to show that the system undergoes a change displaying
all the characteristics of a second-order phase transition as a function of a
control parameter. We derive critical exponents and scaling laws revealing the
universality class of the model, which coincide with that of more elaborate
non-extended models such as the quantum Rabi or Lipkin-Meshkov-Glick models.
Analyzing our model, we are also able to offer insights into the features of
this type of phase transitions, by showing that the thermodynamic and classical
limits coincide. In other words, quantum fluctuations must be tamed in order
for the system to undergo a true phase transition.
