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arXiv:1505.06592

Unavoidable Gapless Boundary State and Boundary Superfluidity of Trapped Bose Mott States in Two-Dimensional Optical Lattices

Takafumi J. Suzuki, Masahiro Sato

Published May 25, 2015• 1 citation
arXivHTMLPDF

Abstract

We study the boundary nature of trapped bosonic Mott insulators in optical square lattices, by performing quantum Monte Carlo simulation. We show that a finite superfluid density generally emerges in the incommensurate-filling (IC) boundary region around the bulk Mott state, irrespectively of the width of the IC region. Both off-diagonal and density correlation functions in the IC boundary region exhibit a nearly power-law decay. The power-law behavior and superfluidity are well developed below a characteristic temperature. These results indicate that a gapless boundary mode always emerges in any atomic Mott insulators on optical lattices. This further implies that if we consider a topological insulating state in Bose or Fermi atomic systems, its boundary possesses at least two gapless modes (or coupled modes) of an above IC edge state and the intrinsic topologically-protected edge state.

DOI
10.48550/arxiv.1505.06592

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We study the boundary nature of trapped bosonic Mott insulators in optical square lattices, by performing quantum Monte Carlo simulation. We show that a finite superfluid density generally emerges in the incommensurate-filling (IC) boundary region around the bulk Mott state, irrespectively of the width of the IC region. Both off-diagonal and density correlation functions in the IC boundary region exhibit a nearly power-law decay. The power-law behavior and superfluidity are well developed below a characteristic temperature. These results indicate that a gapless boundary mode always emerges in any atomic Mott insulators on optical lattices. This further implies that if we consider a topological insulating state in Bose or Fermi atomic systems, its boundary possesses at least two gapless modes (or coupled modes) of an above IC edge state and the intrinsic topologically-protected edge state.
We use quantum Monte Carlo (QMC) simulations to study the combined effects of harmonic confinement and temperature for bosons in a two-dimensional(2D) optical lattice. The scale-invariant, finite temperature state diagram is presented for the Bose-Hubbard model in terms of experimental parameters -- the particle number, confining potential and interaction strength. To distinguish the nature of the spatially separated superfluid, Mott insulator, and normal Bose liquid phases, we examine the local density, compressibility, superfluid density, and Green's function. In the annular superfluid rings, as the width of the ring decreases, the long-range superfluid correlations start to deviate from an equivalent homogeneous 2D system. At zero temperature, the correlation decay is intermediate between one-dimension and two-dimensions, while at a finite temperature, the decay is similar to that in one-dimension at a much lower temperature. The calculations reveal shortcomings of the local density approximation in describing superfluid properties of trapped bosons. We also present the finite-temperature phase diagram for the homogeneous two dimensional Bose-Hubbard model. We compare our state diagram with the results of a recent experiment at NIST on a harmonically trapped 2D lattice [Phys. Rev. Lett. 105, 110401 (2010)], and identify a finite-temperature effect in the experiment.
We study the quantum (zero-temperature) critical behaviors of confined particle systems described by the one-dimensional (1D) Bose-Hubbard model in the presence of a confining potential, at the Mott insulator to superfluid transitions, and within the gapless superfluid phase. Specifically, we consider the hard-core limit of the model, which allows us to study the effects of the confining potential by exact and very accurate numerical results. We analyze the quantum critical behaviors in the large trap-size limit within the framework of the trap-size scaling (TSS) theory, which introduces a new trap exponent $\ensuremath{\theta}$ to describe the dependence on the trap size. This study is relevant for experiments of confined quasi-1D cold atom systems in optical lattices. At the low-density Mott transition, TSS can be shown analytically within the spinless fermion representation of the hard-core limit. The trap-size dependence turns out to be more subtle in the other critical regions, when the corresponding homogeneous system has a nonzero filling fff, showing an infinite number of level crossings of the lowest states when increasing the trap size. At the n=1n=1n=1 Mott transition, this gives rise to a modulated TSS: The TSS is still controlled by the trap-size exponent $\ensuremath{\theta}$, but it gets modulated by periodic functions of the trap size. Modulations of the asymptotic power-law behavior is also found in the gapless superfluid region, with additional multiscaling behaviors.
We study distinctive features of local Mott insulator of trapped Bose-Fermi mixture systems on an optical lattice by using Monte-Carlo simulations of one-dimensional Bose-Fermi-Hubbard model. It was found that each species of the bosons and the fermions in a Mott insulator state had a finite stiffness but the density correlation function exhibited a quasi-long-range-order. This strongly suggested that in actual experiments the bosons and fermions of a commensurate filling with the lattice sites would form a Mott insulator state under strong interatomic interactions and have an order in their positional configuration.
In the absence of a confining potential, the boson-Hubbard model exhibits a superfluid to Mott insulator quantum phase transition at commensurate fillings and strong coupling. We use quantum Monte Carlo simulations to study the ground state of the one-dimensional bosonic Hubbard model in a trap. Some, but not all, aspects of the Mott insulating phase persist. Mott behavior occurs for a continuous range of incommensurate fillings, very different from the unconfined case, and the establishment of the Mott phase does not proceed via a traditional quantum phase transition. These results have important implications for interpreting experiments on ultracold atoms on optical lattices.
Confinement of atoms by laser technology has opened up a new field of physics. Following the realization of Bose– Einstein condensation (BEC) of bosonic atoms, fermionic atom systems and even boson–fermion mixture systems were successfully trapped and cooled. Application of a periodic potential, so called optical lattice, could also unveil interesting nature of the atoms. One of the interesting phenomena of the atoms on optical lattice is Mott transition. Although there is a long history of the study on bosons or fermions on lattice, coexistence of bosons and fermions on lattice is a rather new topic to study. In particular, we have little knowledge about strongly-correlated bosons and fermions on lattice. In our previous works we performed numerical simulations of Bose–Fermi mixtures to clarify the phase diagram and the effect of the confinement potential. When the total number of bosonic and fermionic atoms per site takes an integer value and interatomic interactions are sufficiently strong, the system would undergo Mott transition where the local atom density is fixed to the integer value. A characteristic of this Mott insulating phase is that a boson and fermion located next to each other can exchange their positions by going through a virtual doubly-occupied state with the atom exchange energy t ’ tbtf=Ubf where tb and tf are the hopping energy of the bosons and fermions respectively and Ubf denotes the on-site interaction between the boson and the fermion. Although there still remain a number of open questions on the Mott state of Bose–Fermi mixtures, the property of superfluid state of the strongly-correlated bosons and fermions has yet to be revealed. In this short note we would like to focus on strongly-correlated Bose–Fermi mixtures at commensurate filling in a three-dimensional periodical potential and see characteristic behaviors, if any, of the superfluid phase close to the phase boundary to the Mott insulating phase. In the following analysis we assume uniformity of the system and thus neglect the possibility of the emergence of so-called supersolid where superfluidity and structural long-range order coexist. For simplicity we ignore all the internal degree of freedom of the atoms and the influence of the trap potential which in fact creates a nonuniform density profile of the bosons and fermions. For the system at commensurate filling on optical lattice, we assume that the interatomic interactions are weak enough to avoid formation of Mott state but strong enough to allow us to expect that there are only few empty or doublyoccupied sites and most sites are singly occupied. Namely we analyze the system that stays in a superfluid phase but lies close to the boundary to the Mott phase. Since a boson and fermion located next to each other can exchange their positions as stated above, the energy of the superfluid state of the mixture system is given mostly by the kinetic energy of the bosons and fermions exchanging their positions. We therefore express the Hamiltonian in the following form, ignoring the kinetic energy of the atoms at the empty sites and the interaction energy at the doubly-occupied sites:
We describe weakly interacting atomic or molecular bosons trapped in harmonically confined optical lattices, which exhibit a wedding cake structure consisting of insulating (Mott) shells. We show that superfluid regions emerge between Mott shells as a result of fluctuations due to finite hopping and find that the order parameter equation in the superfluid regions is not of the standard Gross-Pitaevskii type except near the insulator-to-superfluid boundaries. We obtain the excitation spectra in the Mott and superfluid regions, and we show that the superfluid shells possess low-energy sound modes with spatially dependent sound velocity described by a local index of refraction directly related to the local superfluid density. Furthermore, we discuss the Berezinskii-Kosterlitz-Thouless transition and vortex-antivortex pairs in thin (wide) superfluid shells (rings) limited by three- (two-) dimensional Mott regions. Last, we propose an experiment to detect superfluid shells using Laguerre-Gaussian beams and Bragg spectroscopy.