  Since Hund's work on the ammonia molecule, the double well potential has
formed a key paradigm in physics. Its importance is further underlined by the
central role it plays in the Landau theory of phase transitions. Recently, the
study of entanglement properties of many-body systems has added a new angle to
the study of quantum phase transitions of discrete and continuous degrees of
freedom, i.e., spin and harmonic chains. Here we show that control of the
radial degree of freedom of trapped ion chains allows for the simulation of
linear and non-linear Klein-Gordon fields on a lattice, in which the parameters
of the lattice, the non-linearity and mass can be controlled at will. The
system may be driven through a phase transition creating a double well
potential between different configurations of the ion crystal. The dynamics of
the system are controllable, local properties are measurable and tunnelling in
the double well potential would be observable.
