Quantum structural phase transition in chains of interacting atoms

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 ${\varphi }^4$ model. We propose a mapping of the ${\varphi }^4$ 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.

Figure 1

Sketch of the phase diagram exhibiting a linear-zigzag transition, according to the mapping to the one-dimensional model of a quantum Ising transition. Here $T$ is the temperature of the sample and the dimensionless parameter $\epsilon$ is tuned by the confining potential or the interparticle distance. The value $\epsilon =0$ corresponds to the value of the transverse frequency, where the linear chain is classically unstable [12]. The quantum critical point, at ${\epsilon }_c>0$, separates the linear chain from the zigzag phase at $T=0$. For $0<\epsilon <{\epsilon }_c$ quantum fluctuations dominate, and the crystal is in the linear (disordered) phase. The dashed lines indicate the boundaries of the quantum critical region, where thermal fluctuations dominate.

Figure 2

Structure form factor $S(k_y)/N$ as a function of $k_y$ (in units of the transverse length ${\varphi }_{0}a/\pi$). (a) For an ion string ($\epsilon <{\epsilon }_c$). The red dashed curve corresponds to the classically stable regime (i), where $S(k_y)$ is given by $S_{\mathrm{lin}}$ [Eq. (53)]; the blue solid curve corresponds to the ion string in the quantum disordered phase regime (ii) [$S_{\mathrm{dis}}$ in Eq. (56) with ${\varphi }_0/l=1.5$]. (b) For a zigzag structure, regime (iii) (with ${\varphi }_0=10l$). The dashed and solid lines correspond to $S_{\mathrm{zz}}(k_y)$ [Eq. (55)] when $k_x=\pi n/a$ with $n$ even and odd, respectively.