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arXiv:0902.4442 · Physical Review A

Attractively bound pairs of atoms in the Bose-Hubbard model and antiferromagnetism

Bernd G Schmidt, Michael Bortz, Sebastian Eggert, Michael Fleischhauer, David Petrosyan

Published Jun 30, 2009• 22 citations
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Abstract

We consider a periodic lattice loaded with pairs of bosonic atoms tightly bound to each other via strong attractive on-site interaction that exceeds the intersite tunneling rate. An ensemble of such lattice dimers is accurately described by an effective Hamiltonian of hard-core bosons with strong nearest-neighbor repulsion, which is equivalent to the $XXZ$ model with Ising-like anisotropy. We calculate the ground-state phase diagram for a one-dimensional system, which exhibits incompressible phases, corresponding to an empty and a fully filled lattice (ferromagnetic phases) and a half-filled alternating density crystal (antiferromagnetic phase), separated from each other by compressible phases. In a finite lattice the compressible phases show characteristic oscillatory modulations on top of the antiferromagnetic density profile and in density-density correlations. We derive a kink model that provides simple quantitative explanation of these features. To describe the long-range correlations of the system, we employ the Luttinger-liquid theory with the relevant Luttinger parameter $K$ obtained exactly using the Bethe-ansatz solution. We calculate the density-density as well as first-order correlations and find excellent agreement with numerical results obtained with density-matrix renormalization-group methods.

DOI
10.1103/physreva.79.063634

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We consider a periodic lattice loaded with pairs of bosonic atoms tightly bound to each other via strong attractive on-site interaction that exceeds the intersite tunneling rate. An ensemble of such lattice dimers is accurately described by an effective Hamiltonian of hard-core bosons with strong nearest-neighbor repulsion, which is equivalent to the XXZXXZXXZ model with Ising-like anisotropy. We calculate the ground-state phase diagram for a one-dimensional system, which exhibits incompressible phases, corresponding to an empty and a fully filled lattice (ferromagnetic phases) and a half-filled alternating density crystal (antiferromagnetic phase), separated from each other by compressible phases. In a finite lattice the compressible phases show characteristic oscillatory modulations on top of the antiferromagnetic density profile and in density-density correlations. We derive a kink model that provides simple quantitative explanation of these features. To describe the long-range correlations of the system, we employ the Luttinger-liquid theory with the relevant Luttinger parameter KKK obtained exactly using the Bethe-ansatz solution. We calculate the density-density as well as first-order correlations and find excellent agreement with numerical results obtained with density-matrix renormalization-group methods.
We study the limit of large on-site repulsion of the one-dimensional Bose-Hubbard model at low densities, and derive a strong-coupling effective Hamiltonian. By taking the lattice parameter to zero, the Hamiltonian becomes a continuum model of fermions with attractive interactions. The leading corrections to the internal energy of a hard-core-boson (Tonks) gas as well as the (finite temperature) pair correlations of a strongly interacting Bose gas are calculated. We explore the possibility of realizing, in an optical lattice, a Luttinger liquid with stronger density correlations than the Tonks gas. A quantum phase transition to a charge-density-wave Mott insulator is also discussed.
We consider the extended Hubbard model with attractive on-site interaction U and nearest-neighbor repulsions V. We construct an effective Hamiltonian Heff{H}_{\mathrm{eff}}Heff​ for hopping $t\ensuremath{\ll}V$ and arbitrary U<0.U<0.U<0. Retaining the most important terms, Heff{H}_{\mathrm{eff}}Heff​ can be mapped onto two XXZ\mathrm{XXZ}XXZ models, solved by the Bethe ansatz. The quantum phase diagram shows two Luttinger liquid phases and a region of phase separation between them. For density n<0.422n<0.422n<0.422 and $U<\ensuremath{-}4,$ singlet superconducting correlations dominate at large distances. For some parameters, the results are in qualitative agreement with experiments in ${\mathrm{Ba}}_{1\ensuremath{-}x}{\mathrm{K}}_{x}{\mathrm{BiO}}_{3}.$
The ground-state phase diagram of mixtures of spin polarized fermions and bosons in a 1D periodic lattice is discussed in the limit of large fermion hopping and half filling of the fermions. Numerical simulations performed with the density matrix renormalization group (DMRG) show in addition to bosonic Mott insulating (MI), superfluid (SF), and charge density-wave phases (CDW) a yet unreported phase with spatial separation of MI and CDW regions. We derive an effective bosonic theory which allows for a complete understanding and quantitative prediction of the bosonic phase diagram. In particular the origin of CDW phase and the MI-CDW phase separation is revealed as an effective fermion-mediated long-range interaction between bosons with alternating sign.
The attractive Hubbard model is investigated in the framework of lattice density-functional theory (LDFT). The ground-state energy E=T+WE=T+WE=T+W is regarded as a functional of the single-particle density matrix ${\ensuremath{\gamma}}_{ij}$ with respect to the lattice sites, where $T[\ensuremath{\gamma}]$ represents the kinetic and crystal-field energies and $W[\ensuremath{\gamma}]$ the interaction energy. Aside from the exactly known functional $T[\ensuremath{\gamma}]$, we propose a simple scaling approximation to $W[\ensuremath{\gamma}]$, which is based on exact analytic results for the attractive Hubbard dimer and on a scaling hypothesis within the domain of representability of $\ensuremath{\gamma}$. As applications, we consider one-, two-, and three-dimensional finite and extended bipartite lattices having homogeneous or alternating onsite energy levels. In addition, the Bethe lattice is investigated as a function of coordination number. Results are given for the kinetic, Coulomb, and total energies, as well as for the density distribution ${\ensuremath{\gamma}}_{ii}$, nearest-neighbor bond order ${\ensuremath{\gamma}}_{ij}$, and pairing energy $\ensuremath{\Delta}{E}_{p}$, as a function of the interaction strength ∣U∣/t|U|/t∣U∣/t, onsite potential $\ensuremath{\varepsilon}/t$, and band filling n=Ne/Nan={N}_{e}/{N}_{a}n=Ne​/Na​. Remarkable even-odd and super-even oscillations of $\ensuremath{\Delta}{E}_{p}$ are observed in finite rings as a function of band filling. Comparison with exact Lanczos diagonalizations and density-matrix renormalization-group calculations shows that LDFT yields a very good quantitative description of the properties of the model in the complete parameter range, thus providing a significant improvement over the mean-field approaches. Goals and limitations of the method are discussed.
The half-filled Hubbard model on the Bethe lattice with coordination number z=3z=3z=3 is studied using the density-matrix renormalization group (DMRG) method. Ground-state properties such as the energy EEE, average local magnetization $ $, its fluctuations $ - ^2$ and various spin correlation functions $ - $ are determined as a function of the Coulomb interaction strength U/tU/tU/t. The calculated local magnetic moments $ $ increase monotonically with increasing Coulomb repulsion U/tU/tU/t forming an antiferromagnetic spin-density-wave state which matches the two sublattices of the bipartite Bethe lattice. At large U/tU/tU/t, $ $ is strongly reduced with respect to the saturation value 1/2 due to exchange fluctuations between nearest neighbors (NN) spins (∣∣≃0.35| |\simeq 0.35∣∣≃0.35 for U/t→+∞U/t\to +\inftyU/t→+∞). $ - ^2$ shows a maximum for U/t=2.4U/t=2.4U/t=2.4--2.9 which results from the interplay between the usual increase of $ $ with increasing U/tU/tU/t and the formation of important permanent moments $ $ at large U/tU/tU/t. NN sites show antiferromagnetic spin correlations which increase with increasing Coulomb repulsion. In contrast next NN sites are very weakly correlated over the whole range of U/tU/tU/t. The accuracy of the DMRG results is discussed by comparison with tight-binding exact results, independent DMRG calculations for the Heisenberg model and simple first-order perturbation estimates.
Ground-state properties of the attractive Hubbard model in one dimension are studied by means of both the exact Bethe-ansatz formalism and the self-consistent field (SCF) approach with renormalized chemical potential for general band fillings n and a wide range of coupling strength U/t.U/t.U/t. The energy, the concentration of double occupied sites, the kinetic energy and the chemical potential of the ground state are in a good numerical agreement with the exact results over a wide range of parameters U/tU/tU/t and n. The concentration of local pairs or double occupied sites in the Bethe-ansatz solution serves as a suitable parameter measuring the electron pairing correlations. The SCF theory provides a simple analytical relationship between the concentration of double occupied sites, the band filling and the BCS order parameter, valid for arbitrary U/tU/tU/t and n. The calculated energy gap, the BCS order parameter, the phase diagram and the compressibility are also discussed. The SCF theory in one dimension distinguishes the order parameter from the excitation gap and suggests a smooth crossover away from half-filling (at $n\ensuremath{\ne}1)$ from the BCS pairing to the Bose condensation regime under the variation of U/tU/tU/t and n.