  A quantum phase transition is an unequivocal signature of strongly correlated
many-body physics. Signatures of such phenomena are yet to be observed in
ballistic transport through quantum wires. Recent developments in quantum wires
have made it possible to enhance the interaction between the electrons. Here we
show that hitherto unexplained anticrossing between conduction energy
sub-bands, observed in such experiments, can be explained through a simple yet
effective discretised model which undergoes a second order quantum phase
transition within the Ising universality class. Accordingly, we observe how the
charge distribution, transverse to the direction of the wire will vary across
the phase transition. We show that data coming from three different samples
with differing electron densities and gate voltages show a remarkable universal
scaling behavior, determined by the relevant critical exponent, which is only
possible near a quantum phase transition.
