Iterative Variational Ansatz for the Hubbard Model

A number of years ago Eichenberger and Baeriswyl [Phys.~Rev. B \textbf{76}, 180504(R) (2007)] (EB) introduced a novel variational ansatz for the study of the (repulsive) Hubbard model on a square lattice. Taking the Hubbard Hamiltonian to be $\hat{H}=t\hat{T}+U\hat{D}$ (where $\hat{T}$ and $\hat{D}$ are the usual Hubbard hopping and Coulomb terms, respectively), EB chose their variational trial function to be $\left\vert \psi\right\rangle =e^{-h\hat{T}% }e^{-g\hat{D}}\left\vert \psi_{0}\right\rangle $ where $h$ and $g$ are variational parameters. In this work we will consider moments of the Hamiltonian $h_{n}=\left\langle \psi_{0}\right\vert H^{n}\left\vert \psi _{0}\right\rangle =\left\langle 0\right\vert e^{-\alpha\hat{\Gamma}}% H^{n}e^{-\alpha\hat{\Gamma}}\left\vert 0\right\rangle \approx\left\langle 0\right\vert \left( 1-\alpha\hat{\Gamma}\right) H^{n}\left( 1-\alpha \hat{\Gamma}\right) \left\vert 0\right\rangle $, where $\alpha$ is a real parameter. Following EB we choose $\hat{\Gamma}=\hat{T}+\hat{D}$. Sequentially we minimize $h_{n}$ with respect to $\alpha$ for increasing values of $n$ in order to optimize the Hamiltonian moments. Preliminary results are given.