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 |Ψ⟩=⊗_𝑝|𝜓_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 𝑋⁢𝑌⁢𝑍 arrays under a magnetic field are obtained.