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arXiv:cond-mat/0306162 · Optics Express

Variational ansatz for the superfluid Mott-insulator transition in optical lattices

Juan José García‐Ripoll, J. I. Cirac, P. Zoller, Corinna Kollath, U. Schollw�ck, Jan von Delft

Published Jan 1, 2004• 26 citations
arXivHTMLPDF

Abstract

We develop a variational wave function for the ground state of a one-dimensional bosonic lattice gas. The variational theory is initially developed for the quantum rotor model and later on extended to the Bose- Hubbard model. This theory is compared with quasi-exact numerical results obtained by Density Matrix Renormalization Group (DMRG) studies and with results from other analytical approximations. Our approach accurately gives local properties for strong and weak interactions, and it also describes the crossover from the superfluid phase to the Mott-insulator phase.

DOI
10.1364/opex.12.000042

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We develop a variational wave function for the ground state of a one-dimensional bosonic lattice gas. The variational theory is initially developed for the quantum rotor model and later on extended to the Bose- Hubbard model. This theory is compared with quasi-exact numerical results obtained by Density Matrix Renormalization Group (DMRG) studies and with results from other analytical approximations. Our approach accurately gives local properties for strong and weak interactions, and it also describes the crossover from the superfluid phase to the Mott-insulator phase.
Abstract The physics of the Bose–Hubbard (BH) model is the subject of intensive studies in recent years, since it has been realized that BH Hamiltonian can be applied to systems of cold atoms confined in periodic optical lattice potential, where the effects on interactions are strongly enhanced. Our aim is to extend the widely used mean‐field treatment of the superfluid–Mott insulator (SF–MI) transition in the BH model. Our method also improves the strong‐coupling expansion that works well only for sufficiently large insulating gap. The key point of the approach is to consider the representation of strongly interacting bosons as particles with attached U(1) gauge group flux tubes, which constitutes the quantum rotor description. The effective action formalism allows us to cast the problem in terms of the phaseonly action and obtain an analytical formulas for critical lines. Finally, we calculate the zero‐temperature SF–MI phase diagrams for three‐dimensional BH model and compare our results with the outcome of numerical Monte Carlo simulations. We found a very good agreement for the quantitative results regarding the details of the lobe in the phase diagrams showing the quantum transition from superfluid to the Mott insulating phase (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
We use the density-matrix renormalization group method to investigate ground-state and dynamic properties of the one-dimensional Bose-Hubbard model, the effective model of ultracold bosonic atoms in an optical lattice. For fixed maximum site occupancy n b = 5, we calculate the phase boundaries between the Mott insulator and the "superfluid" phase for the lowest two Mott lobes. We extract the Tomonaga-Luttinger parameter from the density-density correlation function and determine accurately the critical interaction strength for the Mott transition. For both phases, we study the momentum distribution function in the homogeneous system, and the particle distribution and quasi-momentum distribution functions in a parabolic trap. With our zero-temperature method we determine the photoemission spectra in the Mott insulator and in the "superfluid" phase of the one-dimensional Bose-Hubbard model. In the insulator, the Mott gap separates the quasi-particle and quasi-hole dispersions. In the "superfluid" phase the spectral weight is concentrated around zero momentum.
We determine the ground-state properties of a gas of interacting bosonic atoms in a one-dimensional optical lattice. The system is modeled by the Bose-Hubbard Hamiltonian. We show how to apply the time-evolving block decimation method to systems with periodic boundary conditions and employ it as a reference to find the ground state of the Bose-Hubbard model. Results are compared with recently proposed approximate methods, such as Hartree-Fock-Bogoliubov (HFB) theories generalized for strong interactions and the variational Bijl-Dingle-Jastrow method. We find that all HFB methods do not bring any improvement to the Bogoliubov theory and therefore provide correct results only in the weakly interacting limit, where the system is deeply in the superfluid regime. On the other hand, the variational Bijl-Dingle-Jastrow method is applicable for much stronger interactions but is essentially limited to the superfluid regime as it reproduces the superfluid--Mott-insulator transition only qualitatively.
Convergence properties of the variational cluster approach with respect to the variational parameter space, cluster size, and boundary conditions of the reference system are investigated and discussed for bosonic many-body systems. Specifically, the variational cluster approach is applied to the one-dimensional Bose-Hubbard model, which exhibits a quantum phase transition from Mott to superfluid phase. In order to benchmark the variational cluster approach, results for the phase boundary delimiting the first Mott lobe are compared with essentially exact density matrix renormalization group data. Furthermore, static quantities, such as the ground state energy and the one-particle density matrix are compared with high-order strong coupling perturbation theory results. For reference systems with open boundary conditions the variational parameter space is extended by an additional variational parameter which allows for a more uniform particle density on the reference system and thus drastically improves the results. It turns out that the variational cluster approach yields accurate results with relatively low-computational effort for both the phase boundary as well as the static properties of the one-dimensional Bose-Hubbard model, even at the tip of the first Mott lobe where correlation effects are most pronounced.
We describe the ground state of a gas of bosonic atoms with two coherently coupled internal levels in a deep optical lattice in a one-dimensional geometry. In the single-band approximation this system is described by a Bose-Hubbard Hamiltonian. The system has a superfluid and a Mott insulating phase that can be either paramagnetic or ferromagnetic. We characterize the quantum phase transitions at unit filling by means of a density-matrix renormalization-group technique and compare the results with a mean-field approach and an effective spin Hamiltonian. The presence of the ferromagnetic Ising-like transition modifies the Mott lobes. In the Mott insulating region the system maps to the ferromagnetic spin-1/2 XXZXXZXXZ model in a transverse field and the numerical results compare very well with the analytical results obtained from the spin model. In the superfluid regime quantum fluctuations strongly modify the phase transition with respect to the well-established mean-field three-dimensional classical bifurcation.
We study the quantum phase transition from a superfluid to a Mott insulator in optical lattices using a Bose-Hubbard Hamiltonian. For this purpose we develop a field theoretical approach in terms of path integral formalism to calculate the second-order quantum corrections to the energy density as well as to the superfluid fraction in cubic optical lattices. Using the present approach, the condensate fraction and ground-state energy are calculated as functions of the sss-wave scattering length. In contrast to the Bogoliubov model, which is, technically speaking, a one-loop approximation, we carry the calculation up to two loops and improve the result further by variational perturbation theory. The result suggests that the quantum phase transition exists.