Operator Renormalization Group.

A powerful new hybrid method for calculating ground -state energies for lattice Hamiltonians is presented. This method combines the t-expansion with the real space renormalization group approach. The t-expansion is a nonperturbative method that calculates ground-state expectation values of many-body systems in Quantum Mechanics as a power series in the parameter t. The idea behind this method is that for any variational state |psi_0> , we can construct a new state | psi_{t}>, which is a better approximation to the true ground state for any value of t. As long as the initial state |psi_0> has an overlap with the true ground state, matrix elements are guaranteed to converge to their true vacuum expectation values in the limit t to infty . The renormalization group approach is an algorithm for constructing an appropriate choice of the class of trial states in a variational calculation. The idea behind the method is as follows. We begin by dissecting the lattice into small blocks containing a few sites which are coupled together via the gradient terms in the Hamiltonian. The Hamiltonian for the resulting few degrees of freedom is diagonalized and the degrees of freedom are "thinned" by a truncation procedure that keeps only an appropriate set of low-lying states. An effective Hamiltonian is then constructed by computing the matrix elements of the original Hamiltonian in the space of states spanned by the lowest -lying states in each block. This procedure is then repeated for this effective Hamiltonian until it goes to a fixed form so that the resulting Hamiltonian can be solved exactly. The operator renormalization group method (ORG) combines these two procedures with the hope of effectively extracting infinite volume physics at t to infty from calculations of only a few powers of t. We apply this method to the 1 + 1 dimensional Ising model and 1 + 1 dimensional anisotropic Heisenberg model. By comparing our results with the exact results we find that the combined method is more accurate than either method alone.