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arXiv:cond-mat/0106571 · Physical review. B, Condensed matter

Strongly correlated hopping and many-body bound states

Julien Vidal, Benoît Douçot

Published Dec 28, 2001• 11 citations
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Abstract

We study a system in which the quantum dynamics of electrons depend on the particle density in their neighborhood. For any on-site repulsive interaction, we show that the exact two-body and three-body ground states are bound states. We also discuss the finite density case in a mean-field framework and we show that the system can undergo an unusual transition from an effective attractive interaction to a repulsive one, when varying the electron density.

DOI
10.1103/physrevb.65.045102

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We study a system in which the quantum dynamics of electrons depend on the particle density in their neighborhood. For any on-site repulsive interaction, we show that the exact two-body and three-body ground states are bound states. We also discuss the finite density case in a mean-field framework and we show that the system can undergo an unusual transition from an effective attractive interaction to a repulsive one, when varying the electron density.
The ground-state instabilities for a one-dimensional lattice system of electrons with onsite (Hubbard) and bond-site (hopping) interactions are analyzed in a perturbative approach. The zero temperature phase diagram at different band fillings is drawn; an attractive (repulsive) bond-site interaction favors the appearance of a superconductor state at low concentrations of electrons (holes). A comparison with the exact results for the Hubbard model and previous works for particular cases is also discussed.
A density-functional theory for many-body lattice models is considered in which the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is expressed as the sum of Hartree-Fock energy ${E}_{\mathrm{HF}}[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ and the correlation energy ${E}_{C}[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ Exact results are obtained for ${E}_{C}({\ensuremath{\gamma}}_{12})$ of the Hubbard model on various periodic lattices, where ${\ensuremath{\gamma}}_{\mathrm{ij}}={\ensuremath{\gamma}}_{12}$ for all nearest neighbors i and j. The functional dependence of ${E}_{C}({\ensuremath{\gamma}}_{12})$ is analyzed by varying the number of sites Na,{N}_{a},Na​, band filling Ne,{N}_{e},Ne​, and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of ${E}_{C}({\ensuremath{\gamma}}_{12})$ are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, and in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}<~{\ensuremath{\gamma}}_{12}<~{\ensuremath{\gamma}}_{12}^{0}).$ Using an appropriate scaling we observe that ${\ensuremath{\varepsilon}}_{C}{(g}_{12}{)=E}_{C}{/E}_{\mathrm{HF}}$ has a pseudo-universal behavior as a function of ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}).$ The fact that ${\ensuremath{\varepsilon}}_{C}{(g}_{12})$ depends weakly on Na,{N}_{a},Na​, Ne,{N}_{e},Ne​, and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of ${E}_{C}({\ensuremath{\gamma}}_{12})$ for repulsive (U>0)(U>0)(U>0) and attractive (U<0)(U<0)(U<0) interactions are contrasted.
The nonequilibrium quantum dynamics of trapped bosons interacting by strong interparticle interaction of finite range in one, two, and three spatial dimensions is investigated on an accurate many-body level. We use different time-dependent processes to destabilize the systems' ground states: A sudden quench of the strength of the interparticle repulsion is accompanied by displacement of the trap. Two qualitatively different but otherwise generic dynamical quantum many-body behaviors are discovered. In the first, the overall ``topology'' of the ground-state density is preserved, whereas in the second the density totally ``explodes.'' An intuitive many-body time-dependent model is devised to interpret and explain the observations. The generality of the discovered scenarios is explicitly confirmed in traps of various shapes and dimensionality, and interparticle interactions of different forms and ranges. Implications are briefly discussed.
It is shown that a quasi-one-dimensional system with strong attractive or repulsive coupling between electrons on the same site is equivalent to a Bose gas with hard core and longer-range interactions and to a Heisenberg-Ising magnet. Interchain and intrachain hopping and coupling are included and the equivalences are derived by elementary degenerate perturbation theory. Electronic correlation functions are derived from known pseudomagnetic ones and are used to discuss the circumstances in which superconductivity, charge- or spin-density wave transitions occur in the coupled chains.
By the bosonization and renormalization-group schemes, a half-filled Hubbard chain with density-dependent hopping is investigated at weak coupling. The model involved has an electron–hole symmetry and an SO(4) symmetry. The phase diagram consists of three different phases divided by a charge transition at on-site repulsion U = U c and a spin transition at U = U s , with U s = − U c = 16 π ( t c − 1 ) , including a Mott insulator with the SDW correlation, a dimerized insulator with the BCDW correlation, and a Luttinger metal with the coexisting TS and BSDW correlations. The effective three-body repulsion ( t c − 1 > 0 ) and attraction ( t c − 1 < 0 ) induce the dimerized and metallic phases, respectively. • We study a 1D Hubbard model with symmetric density-dependent hopping analytically. • We provide a theoretical scheme of realization of the BCDW and TS instabilities. • The results supplement further understanding of quantum properties of 1D correlated systems.