  A string of repulsively interacting particles exhibits a phase transition to
a zigzag structure, by reducing the transverse trap potential or the
interparticle distance. The transition is driven by transverse, short
wavelength vibrational modes. Based on the emergent symmetry Z_2 it has been
argued that this instability is a quantum phase transition, which can be mapped
to an Ising model in transverse field. We perform an extensive Density Matrix
Renormalization Group analysis of the behaviour at criticality and evaluate the
critical exponents and the central charge with high precision. We thus provide
strong numerical evidence confirming that the quantum linear-zigzag transition
belongs to the critical Ising model universality class. These results show that
structural instabilities of one-dimensional interacting atomic arrays can
simulate quantum critical phenomena typical of ferromagnetic systems.

Statistical Mechanics;Quantum Physics