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

We describe the quantum phase transition in the -state chiral clock model in spatial dimension . With couplings chosen to preserve time-reversal and spatial inversion symmetries, 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 such couplings and , the clock model is expected to have a direct phase transition from a gapped phase with a broken global symmetry, to a gapped phase with the symmetry restored. The transition has dynamical critical exponent , and so cannot be described by a relativistic quantum field theory. We use a lattice duality transformation to map the transition onto that of a Bose gas in , involving the onset of a single boson condensate in the background of a higher-dimensional -boson condensate. We present a renormalization group analysis of the strongly coupled field theory for the Bose gas transition in an expansion in , with chosen to be of order . At two-loop order, we find a regime of parameters with a renormalization group fixed point which can describe a direct phase transition. We also present numerical density-matrix renormalization group studies of lattice chiral clock and Bose gas models for , finding good evidence for a direct phase transition, and obtain estimates for and the correlation length exponent .