Abstract
A variational upper bound on the ground state energy E_ gs of a quantum system, E_ gs Ψ|H| Ψ , is well-known (here H is the Hamiltonian of the system and Ψ is an arbitrary wave function). Much less known are variational {\it lower} bounds on the ground state. We consider one such bound which is valid for a many-body translation-invariant lattice system. Such a lattice can be divided into clusters which are identical up to translations. The Hamiltonian of such a system can be written as H=Σ_i=1^M H_i, where a term H_i is supported on the i'th cluster. The bound reads E_ gs M inf_ρ_cl ∈ S_cl^G tr_clρ_cl H_cl , where S_cl^G is some wisely chosen set of reduced density matrices of a single cluster. The implementation of this latter variational principle can be hampered by the difficulty of parameterizing the set M, which is a necessary prerequisite for a variational procedure. The root cause of this difficulty is the nonlinear positivity constraint ρ>0 which is to be satisfied by a density matrix. The squaring parametrization of the density matrix, ρ=τ²/ trτ², where τ is an arbitrary (not necessarily positive) Hermitian operator, accounts for positivity automatically. We discuss how the squaring parametrization can be utilized to find variational lower bounds on ground states of translation-invariant many-body systems. As an example, we consider a one-dimensional Heisenberg antiferromagnet.