A perturbative and variational approach to quantum lattice Hamiltonians

Abstract

We propose a method to construct the ground state \ensuremath{\psi}(\ensuremath{\lambda}) of local lattice Hamiltonians with the generic form ${\mathit{H}}_{0}$+\ensuremath{\lambda}${\mathit{H}}_{1}$, where \ensuremath{\lambda} is a coupling constant and ${\mathit{H}}_{0}$ is a Hamiltonian with a nondegenerate ground state ${\mathrm{\ensuremath{\psi}}}_{0}$. The method is based on the choice of an exponential ansatz \ensuremath{\psi}(\ensuremath{\lambda})=exp[U(\ensuremath{\lambda})]${\mathrm{\ensuremath{\psi}}}_{0}$, which is a sort of generalized lattice version of a Jastrow wave function. We combine perturbative and variational techniques to get successive approximations of the operator U(\ensuremath{\lambda}). Perturbation theory is used to set up a variational method which in turn produces nonperturbative results. The computation with this kind of ansatz leads us to associate to the original quantum-mechanical problem a statistical-mechanical system defined in the same spatial dimension. In some cases these statistical-mechanical systems turn out to be integrable, which allows us to obtain exact upper bounds to the energy. The general ideas of our method are illustrated in the example of the Ising model in a transverse field.