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arXiv:1405.6685 · Physical Review A

Structural transitions of nearly second order in classical dipolar gases

Florian Cartarius, Giovanna Morigi, Anna Minguzzi

Published Nov 3, 2014• 9 citations
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Abstract

Particles with repulsive power-law interactions undergo a transition from a single to a double chain (zigzag) by decreasing the confinement in the transverse direction. We theoretically characterize this transition when the particles are classical dipoles, polarized perpendicularly to the plane in which the motion occurs, and argue that this transition is of first order, even though weakly. The nature of the transition is determined by the coupling between transverse and axial modes of the chain and contrasts with the behavior found in Coulomb systems, where the linear-zigzag transition is continuous and belongs to the universality class of the ferromagnetic transition. Our results hold for classical systems with power-law interactions $1/{r}^{\ensuremath{\alpha}}$ when $\ensuremath{\alpha}>2$ and show that structural transitions in dipolar systems and Rydberg atoms can offer a test bed for simulating the critical behavior of magnets with lattice coupling.

DOI
10.1103/physreva.90.053601

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Particles with repulsive power-law interactions undergo a transition from a single to a double chain (zigzag) by decreasing the confinement in the transverse direction. We theoretically characterize this transition when the particles are classical dipoles, polarized perpendicularly to the plane in which the motion occurs, and argue that this transition is of first order, even though weakly. The nature of the transition is determined by the coupling between transverse and axial modes of the chain and contrasts with the behavior found in Coulomb systems, where the linear-zigzag transition is continuous and belongs to the universality class of the ferromagnetic transition. Our results hold for classical systems with power-law interactions 1/r\ensuremathα1/{r}^{\ensuremath{\alpha}}1/r\ensuremathα when \ensuremathα>2\ensuremath{\alpha}>2\ensuremathα>2 and show that structural transitions in dipolar systems and Rydberg atoms can offer a test bed for simulating the critical behavior of magnets with lattice coupling.
Dipolar particles in an elongated trap are expected to undergo a quantum phase transition from a linear to a zigzag structure with decreasing transverse confinement. We derive the low-energy effective theory of the transition showing that in the presence of quantum fluctuations the zigzag phase can be characterized by a long-ranged string order, while the local Ising correlations decay as a power law. This is also confirmed using density matrix renormalization group calculations on a microscopic model. The nonlocal order in the bulk gives rise to zero energy states localized at the interface between the ordered and disordered phases. Such an interface naturally arises when the particles are subject to a weak harmonic confinement along the tube axis. We compute the signature of the edge states in the single-particle tunneling spectra pointing to differences between a system with bosonic versus fermionic particles. Finally we assess the magnitude of the relevant quantum fluctuations in realistic systems of dipolar particles, including ultracold polar molecules as well as alkali atoms weakly dressed by a Rydberg excitation.
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.
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 theoretically consider ultracold polar molecules in a wave guide. The particles are bosons: They experience a periodic potential due to an optical lattice oriented along the wave guide and are polarized by an electric field orthogonal to the guide axis. The array is mechanically unstable by opening the transverse confinement in the direction orthogonal to the polarizing electric field and can undergo a transition to a double-chain (zigzag) structure. For this geometry we derive a multimode generalized Bose-Hubbard model for determining the quantum phases of the gas at the mechanical instability, taking into account the quantum fluctuations in all directions of space. Our model limits the dimension of the numerically relevant Hilbert subspace by means of an appropriate decomposition of the field operator, which is obtained from a field theoretical model of the linear-zigzag instability. We determine the phase diagrams of small systems using exact diagonalization and find that, even for tight transverse confinement, the aspect ratio between the two transverse trap frequencies controls not only the classical but also the quantum properties of the ground state in a nontrivial way. Convergence tests at the linear-zigzag instability demonstrate that our multimode generalized Bose-Hubbard model can catch the essential features of the quantum phases of dipolar gases in confined geometries with a limited computational effort.
We present a mean-field description of the zigzag phase transition of a quasi-one-dimensional system of strongly interacting particles, with interaction potential ${r}^{\ensuremath{-}n}{e}^{\ensuremath{-}r/\ensuremath{\lambda}}$, that are confined by a power-law potential (${y}^{\ensuremath{\alpha}}$). The parameters of the resulting one-dimensional Ginzburg-Landau theory are determined analytically for different values of $\ensuremath{\alpha}$ and nnn. Close to the transition point for the zigzag phase transition, the scaling behavior of the order parameter is determined. For $\ensuremath{\alpha}=2$, the zigzag transition from a single to a double chain is of second order, while for $\ensuremath{\alpha}>2$, the one-chain configuration is always unstable and, for $\ensuremath{\alpha}<2$, the one-chain ordered state becomes unstable at a certain critical density, resulting in jumps of single particles out of the chain.
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 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.