Covariance-based method for eigenstate factorization and generalized singlets

We derive a general method for determining the necessary and sufficient conditions for exact factorization $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}={\ensuremath{\bigotimes}}_{p}|{\ensuremath{\psi}}_{p}\ensuremath{\rangle}$ of an eigenstate of a many-body Hamiltonian $H$, based on the quantum covariance matrix of the relevant local operators building the Hamiltonian. The ``site'' $p$ can be either a single component or a group of subsystems. The formalism is then used to derive exact dimerization and clusterization conditions in spin systems, covering from spin-$s$ singlets and clusters coupled to 0 total spin, to general nonmaximally entangled spin-$s$ dimers (generalized singlets). New results for field-induced dimerization in anisotropic $XYZ$ arrays under a magnetic field are obtained.