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arXiv:1111.0778 · Physical Review B

Nonlocal order in elongated dipolar gases

Jonathan Ruhman, Emanuele G. Dalla Torre, Sebastian D. Huber, Ehud Altman

Published Mar 22, 2012• 43 citations
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Abstract

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.

DOI
10.1103/physrevb.85.125121

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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.
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.
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.
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 theoretically investigate the ground-state properties and breathing-mode collective excitations of three-dimensional dipolar Bose gases in anisotropic harmonic traps incorporating quantum fluctuations. Combining a Gaussian variational ansatz with a non-perturbative sum-rule analysis, we derive explicit analytical expressions for both axial and radial breathing-mode frequencies, which are validated by numerical solutions of the time-dependent extended Gross-Pitaevskii equation. Our theoretical predictions show excellent agreement with existing experimental data for 166^{166}166Er and 162^{162}162Dy gases. By constructing comprehensive phase diagrams across the parameter space of the sss-wave scattering length, atom number, and trap aspect ratio, we reveal both discontinuous first-order phase transitions and smooth crossovers between the dilute Bose-Einstein condensate and dense quantum droplet phases. We confirm that the enhanced incompressibility induced by quantum fluctuations significantly elevates the breathing-mode frequencies in the droplet phase compared to conventional weakly interacting Bose gases. Furthermore, the system undergoes a phase transition and a crossover over the scattering length under the quasi-two-dimensional and quasi-one-dimensional confinements, characterized by discontinuous jumps and continuous crossovers in peak density and atomic cloud sizes, respectively. Our work offers a rigorous and highly accurate framework to characterize collective excitations in dipolar quantum gases, providing quantitative insights for forthcoming ultracold atom experiments in lanthanide atoms and polar molecules.
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.