Conserved operators and exact conditions for pair condensation

Abstract

We determine the necessary and sufficient conditions which ensure that an N=2m-particle fermionic or bosonic state has the form of an exact pair condensate (A^+)^m|0>, where A^+ is a general pair creation operator. These conditions can be cast as an eigenvalue equation for a modified two-body density matrix, and enable an exact reconstruction of the operator A^+, providing as well a measure of the proximity of a given state to an exact pair condensate. Through a covariance-based formalism, it is also shown that such states are fully characterized by a set of L "conserved" one-body operators which have the state as exact eigenstate, with L determined just by the single particle space dimension involved. The whole set of two-body Hamiltonians having such state as exact eigenstate is in this way determined, while a general subset having it as nondegenerate ground state is also identified. Extension to states f(A^+)|0>, with f an arbitrary function, is also discussed.