Covariance-based method for eigenstate factorization and generalized singlets

Abstract

We derive a general method for determining the necessary and sufficient conditions for exact factorization $|\hat{I}¨⟩={\otimes }_p|\ddot{I}{ˆ}_p⟩$ 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.

18 pages, 7 figures