A variational lower bound on the ground state of a many-body system and the squaring parametrization of density matrices

A variational upper bound on the ground state energy \(E_{\rm gs}\) of a quantum system, \(E_{\rm gs} \leqslant \langle \Psi|H| \Psi \rangle\), is well-known (here \(H\) is the Hamiltonian of the system and \(\Psi\) 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 \(H=\sum_{i=1}^M H_i\), where a term \(H_i\) is supported on the \(i\)'th cluster. The bound reads \(E_{\rm gs}\geqslant M \inf\limits_{\rho_{cl} \in {\mathbb S_{cl}^G}} {\rm tr}_{cl}\rho_{cl} \, H_{cl} \), where \({\mathbb S_{cl}^G}\) 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 \(\mathbb M\), which is a necessary prerequisite for a variational procedure. The root cause of this difficulty is the nonlinear positivity constraint \(\rho>0\) which is to be satisfied by a density matrix. The squaring parametrization of the density matrix, \(\rho=\tau^2/{\rm tr}\,\tau^2\), where \(\tau\) 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.