Quantum structural phase transition in chains of interacting atoms

TL;DR

This paper models a quantum phase transition in a chain of trapped atoms, mapping it to the Ising model, and estimates the critical parameters for experimental observation in ion traps.

Contribution

It introduces a mapping of the quantum structural transition to the Ising model, providing a way to estimate critical parameters for experiments.

Findings

Quantum critical point estimated in terms of trap frequency.

Finite temperature effects and measurement feasibility analyzed.

Experimental observation of quantum critical behavior deemed possible.

Abstract

A quasi one--dimensional system of trapped, repulsively interacting atoms (e.g., an ion chain) exhibits a structural phase transition from a linear chain to a zigzag structure, tuned by reducing the transverse trap potential or increasing the particle density. Since it is a one dimensional transition, it takes place at zero temperature and therefore quantum fluctuations dominate. In [Fishman, et al., Phys. Rev. B 77, 064111 (2008)] it was shown that the system close to the linear-zigzag instability is described by a $\phi^4$ model. We propose a mapping of the $\phi^4$ field theory to the well known Ising chain in a transverse field, which exhibits a quantum critical point. Based on this mapping, we estimate the quantum critical point in terms of the system parameters. This estimate gives the critical value of the transverse trap frequency for which the quantum phase transition occurs, and which has a finite, measurable deviation from the critical point evaluated within the classical theory. A measurement is suggested for atomic systems which can probe the critical trap frequency at sufficiently low temperatures T. We focus in particular on a trapped ion system, and estimate the implied limitations on T and on the interparticle distance. We conclude that the experimental observation of the quantum critical behavior is in principle accessible.