Numerical study of the chiral $\mathbb{Z}_3$ quantum phase transition in one spatial dimension

TL;DR

This paper investigates the quantum phase transition in a one-dimensional system exhibiting $ ext{Z}_3$ symmetry breaking, using numerical methods to analyze critical behavior and uncover nonconformal criticality.

Contribution

It provides the first detailed numerical analysis of the $ ext{Z}_3$ chiral clock model's transition, revealing a strongly-coupled nonconformal critical point.

Findings

Critical exponents $ u$ and $z$ determined via finite-size scaling.

Identification of a nonconformal critical point with $z e 1$.

Numerical evidence supporting the $ ext{Z}_3$ universality class.

Abstract

Recent experiments on a one-dimensional chain of trapped alkali atoms [arXiv:1707.04344] have observed a quantum transition associated with the onset of period-3 ordering of pumped Rydberg states. This spontaneous $\mathbb{Z}_3$ symmetry breaking is described by a constrained model of hard-core bosons proposed by Fendley $et\, \,al.$ [arXiv:cond-mat/0309438]. By symmetry arguments, the transition is expected to be in the universality class of the $\mathbb{Z}_3$ chiral clock model with parameters preserving both time-reversal and spatial-inversion symmetries. We study the nature of the order-disorder transition in these models, and numerically calculate its critical exponents with exact diagonalization and density-matrix renormalization group techniques. We use finite-size scaling to determine the dynamical critical exponent $z$ and the correlation length exponent $\nu$. Our analysis presents the only known instance of a strongly-coupled transition between gapped states with $z \ne 1$, implying an underlying nonconformal critical field theory.