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

Double wells, scalar fields and quantum phase transitions in ions traps