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 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 $N=3$, the clock model is expected to have a direct phase transition from a gapped phase with a broken global ${\mathbb{Z}}_{N}$ symmetry, to a gapped phase with the ${\mathbb{Z}}_{N}$ symmetry restored. The transition has dynamical critical exponent $z\ensuremath{\ne}1$, 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 $d=1$, 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 $2\ensuremath{-}d$, with $4\ensuremath{-}N$ chosen to be of order $2\ensuremath{-}d$. 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 $N=3$, finding good evidence for a direct phase transition, and obtain estimates for $z$ and the correlation length exponent $\ensuremath{\nu}$.

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