Quantum field theory for the chiral clock transition in one spatial dimension*

We describe the quantum phase transition in the N-state chiral clock model in spatial dimension d=1. With particular couplings, such a model is in the universality class of recent experimental studies of the ordering of pumped Rydberg states in a one-dimensional chain of trapped ultracold alkali atoms. For N=3, the model is expected to have a direct phase transition from a gapped phase with a broken global Z_N symmetry, to a gapped phase with the Z_N symmetry restored. The transition has dynamical critical exponent z≠1, and so cannot be described by a relativistic quantum field theory. We map the transition onto that of a Bose gas in d=1, involving the onset of a single boson condensate in the background of a higher-dimensional N-boson condensate. We present a renormalization group analysis of the strongly coupled field theory for the Bose gas transition in an expansion in 2-d, with 4-N chosen to be of order 2-d. At two-loop order, we find a renormalization group fixed point which can describe a direct phase transition.