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Physical Review Letters

Hubbard model with nearest-neighbor and bond-charge interaction: Exact ground-state solution in a wide range of parameters

Rainer Strack, D. Vollhardt

Published Apr 26, 1993• 127 citations
Publisher

Abstract

We present the exact ground-state wave function and energy of a general Hubbard model with kinetic energy t and on-site, nearest-neighbor, and bond-charge interaction U,V,X, respectively, for a wide, physically relevant range of parameters with X=t on lattices with arbitrary coordination numbers Z at half filling.
DOI
10.1103/physrevlett.70.2637

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We obtain the exact solution of a general Hubbard chain with kinetic energy ttt, bond-charge interaction XXX and on-site interaction UUU with the only restriction t=Xt = Xt=X. At zero temperature and half filling, the model exhibits a Mott transition at U=4tU = 4tU=4t. In the metallic phase and near half filling, superconducting states are part of the degenerate ground state and are favored for small UUU if the system is slightly perturbed.
We obtain the exact solution of a general Hubbard chain with kinetic energy ttt, bond-charge interaction XXX, and on-site repulsion UUU with the only restriction t=Xt=Xt=X. At zero temperature and half filling, the model exhibits a Mott transition at U=4tU=4tU=4t. In the metallic phase and near half filling, superconducting states are part of the degenerate ground state and are favored for small UUU if the system is slightly perturbed.
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 recently introduced concept of exact mean-field theories is applied to the extended simplified Hubbard model, which is a one-band extended Hubbard model (including nearest-neighbor interaction) where only the \ensuremath{\downarrow} electrons can hop. An exact mean-field Hamiltonian is derived for the model on a Bethe lattice in the limit of a large coordination number (Z\ensuremath{\rightarrow}\ensuremath{\infty}). The mean-field Hamiltonian is used to calculate the Green functions and the free energy at half-filling. This exact solution is analyzed analytically at small and large values of the on-site (U) and nearest-neighbor (V) interaction strengths. One finds that a phase transition occurs at sufficiently low temperatures for all U,V>0. The low-temperature phase is a charge-density wave for V>1/2U and a spin-density wave for V1/2U. Special attention is paid to (i) the critical temperature as a function of U and V, (ii) the order parameter as a function of temperature, and (iii) the density of states. The density of states shows that the low-temperature phase is an insulator for all U,V provided that V/U\ensuremath{\ne}1/2, while the high-temperature phase is a paramagnetic metal for all V if U2, and a paramagnetic insulator if U>2. Many detailed results are given.
We have modified an existing real space renormalization group scheme to remove some physically inconsistent features and used the rectified method to study the ground state of the one-dimensional Hubbard model with nearest-neighbour repulsion and the bond-charge interaction for half filling. For the nearest-neighbour interaction we find a phase diagram which agrees nicely with the Monte Carlo results. For the case of bond-charge interaction we have reproduced some available exact results and found some interesting information in the parameter regime where no exact solution is available.