Variational Ground State of the One Dimensional Bose-Hubbard Modelel

Recently Eichenberger and Baeriswyl [PRB \textbf{76}, 180504R, (2007)] have introduced a novel variational ansatz to study the two dimensional Hubbard model. Their scheme involves choosing a trial ket which consists of an exponential operator constructed from the Hamiltonian which then operates on a mean field ground state $\left\vert \psi _{0}\right\rangle $. In this study, we wish to extend this ansatz by combining it with a second ansatz in which a variational basis is constructed by systematically taking derivatives with respect to a (set) of variational parameters. The model we will study is the one dimensional Bose-Hubbard Hamiltonian which is used to investigate the properties of interacting bosonic atoms in a one-dimensional optical lattice. The Hamiltonian is given by \[ H_{\mathrm{bh}}=-J\sum_{l=1}^{M}(a_{l}^{\dag }a_{l+1}+{\mathrm{h.c.}})+\frac{U% }{2}\sum_{l=1}^{M}n_{l}\left( n_{l}-1\right) , \]% where $a_{l}$ ($a_{l}^{\dag }$) creates a boson at the lowest level localized on the $l$-th site, $J$ is the hopping energy and $U>0$ is the onsite repulsion. Our results are then compared with other approximation methods such as Hartree-Fock-Bogoliubov theories and the variational Bijl-Dingle-Jastrow method.