  We investigate an approach for studying the ground state of a quantum
many-body Hamiltonian that is based on treating the correlation functions as
variational parameters. In this approach, the challenge set by the
exponentially-large Hilbert space is circumvented by approximating the
positivity of the density matrix, order-by-order, in a way that keeps track of
a limited set of correlation functions. In particular, the density-matrix
description is replaced by a correlation matrix whose dimension is kept linear
in system size, to all orders of the approximation. Unlike the conventional
variational principle which provides an upper bound on the ground-state energy,
in this approach one obtains a lower bound instead. By treating several
one-dimensional spin $1/2$ Hamiltonians, we demonstrate the ability of this
approach to produce long-range correlations, and a ground-state energy that
converges to the exact result. Possible extensions, including to higher-excited
states are discussed.
