Thetexpansion: A nonperturbative analytic tool for Hamiltonian systems

A systematic nonperturbative scheme is developed to calculate the ground-state expectation values of arbitrary operators for any Hamiltonian system. Quantities computed in this way converge rapidly to their true expectation values. The method is based upon the use of the operator $e^{-tH}$ to contract any trial state onto the true ground state of the Hamiltonian $H$. We express all expectation values in the contracted state as a power series in $t$, and reconstruct $t\to \infty$ behavior by means of Padé approximants. The problem associated with factors of spatial volume is taken care of by developing a connected graph expansion for matrix elements of arbitrary operators taken between arbitrary states. We investigate Padé methods for the $t$ series and discuss the merits of various procedures. As examples of the power of this technique we present results obtained for the Heisenberg and Ising models in 1+1 dimensions starting from simple mean-field wave functions. The improvement upon mean-field results is remarkable for the amount of effort required. The connection between our method and conventional perturbation theory is established, and a generalization of the technique which allows us to exploit off-diagonal matrix elements is introduced. The bistate procedure is used to develop a $t$ expansion for the ground-state energy of the Ising model which is, term by term, self-dual.