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arXiv:1012.4332 · Physical Review A

Quantum structural phase transition in chains of interacting atoms

Efrat Shimshoni, Giovanna Morigi, Shmuel Fishman

Published Mar 14, 2011• 37 citations
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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 ${\ensuremath{\phi}}^{4}$ model. We propose a mapping of the ${\ensuremath{\phi}}^{4}$ 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.

DOI
10.1103/physreva.83.032308

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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 \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 model. We propose a mapping of the \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 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 TTT. We focus in particular on a trapped-ion system and estimate the implied limitations on TTT and on the interparticle distance. We conclude that the experimental observation of the quantum critical behavior is, in principle, accessible.
A string of trapped ions at zero temperature exhibits a structural phase transition to a zigzag structure, tuned by reducing the transverse trap potential or the interparticle distance. The transition is driven by transverse, short wavelength vibrational modes. We argue that this is a quantum phase transition, which can be experimentally realized and probed. Indeed, by means of a mapping to the Ising model in a transverse field, we estimate the quantum critical point in terms of the system parameters, and find a finite, measurable deviation from the critical point predicted by the classical theory. A measurement procedure is suggested which can probe the effects of quantum fluctuations at criticality. These results can be extended to describe the transverse instability of ultracold polar molecules in a one-dimensional optical lattice.
A string of repulsively interacting particles exhibits a phase transition to a zigzag structure, by reducing the transverse trap potential or the interparticle distance. Based on the emergent symmetry it has been argued that this instability is a quantum phase transition, which can be mapped to an Ising model in transverse field. An extensive Density Matrix Renormalization Group analysis is performed, resulting in an high‐precision evaluation of the critical exponents and of the central charge of the system, confirming that the quantum linear‐zigzag transition belongs to the critical Ising model universality class. Quantum corrections to the classical phase diagram are computed, and the range of experimental parameters where quantum effects play a role is provided. These results show that structural instabilities of one‐dimensional interacting atomic arrays can simulate quantum critical phenomena typical of ferromagnetic systems.
A chain of singly charged particles confined by a harmonic potential exhibits a structural transition to a zigzag configuration when the radial potential frequency is at a critical value, which depends on the particle number. This structural change is a phase transition of second order, whose order parameter is the ions' transverse displacement from the chain axis. The transition is driven by transverse, short-wavelength vibrational modes. At ultra-low temperatures the linear-zigzag instability is a quantum phase transition of the same universality class of Ising models.[GRAPHICS].
We study the crossover from classical to quantum phase transitions at zero temperature within the framework of ${\ensuremath{\phi}}^{4}$ theory. The classical transition at zero temperature can be described by the Landau theory, turning into a quantum Ising transition with the addition of quantum fluctuations. We perform a calculation of the transition line in the regime where the quantum fluctuations are weak. The calculation is based on a renormalization group analysis of the crossover between classical and quantum transitions, and is well controlled even for space-time dimensionality DDD below 4. In particular, for D=2D=2D=2 we obtain an analytic expression for the transition line which is valid for a wide range of parameters, as confirmed by numerical calculations based on the density matrix renormalization group. This behavior could be tested by measuring the phase diagram of the linear-zigzag instability in systems of trapped ions or repulsively interacting dipoles.
We study the ground-state phase diagram of ultracold dipolar gases in highly anisotropic traps. Starting from a one-dimensional geometry, by ramping down the transverse confinement along one direction, the gas reaches various planar distributions of dipoles. At large linear densities, when the dipolar gas exhibits a crystal-like phase, critical values of the transverse frequency exist below which the configuration exhibits transverse patterns. These critical values are found by means of a classical theory, and are in full agreement with classical Monte Carlo simulations. The study of the quantum system is performed numerically with Monte Carlo techniques and shows that the quantum fluctuations smoothen the transition and make it completely disappear in a gas phase. These predictions could be experimentally tested and would allow one to reveal the effect of zero-point motion on self-organized mesoscopic structures of matter waves, such as the transverse pattern of the zigzag chain.
Ultracold polar molecules, in highly anisotropic traps and interacting via a repulsive dipolar potential, may form one-dimensional chains at high densities. According to classical theory, at low temperatures there exists a critical value of the density at which a second order phase transition from a linear to a zigzag chain occurs. We study the effect of thermal and quantum fluctuations on these self-organized structures using classical and quantum Monte Carlo methods, by means of which we evaluate the pair correlation function and the static structure factor. Depending on the parameters, these functions exhibit properties typical of a crystalline or of a liquid system. We compare the thermal and the quantum results, identifying analogies and differences. Finally, we discuss experimental parameter regimes where the effects of quantum fluctuations on the linear - zigzag transition can be observed.