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

TL;DR

This paper investigates the quantum phase transition in the one-dimensional N-state chiral clock model, revealing a non-relativistic critical point and mapping it to a Bose gas, supported by renormalization group analysis and numerical studies.

Contribution

It introduces a lattice duality mapping of the chiral clock transition to a Bose gas and provides a renormalization group analysis for the non-relativistic critical point.

Findings

Identification of a non-relativistic critical point with z ≠ 1.

Evidence of a direct phase transition from numerical DMRG studies.

Estimates of critical exponents for the transition.

Abstract

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 $z \neq 1$, 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 $2-d$, with $4-N$ chosen to be of order $2-d$. 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 $\nu$.