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 $|\hat{I}¨⟩\propto (A^{†}{)}^m|0⟩$, where $A^{†}=\frac{1}{2}{\sum }_{i,j}{A}_{ij}{c}_i^{†}{c}_j^{†}$ 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 $A^{\hat{a}}€$, 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 $|\hat{I}¨⟩$ as exact eigenstate, with $L$ determined just by the single particle space dimension involved. The whole set of two-body Hamiltonians having $|\hat{I}¨⟩$ as exact eigenstate is in this way determined, while a general subset having $|\hat{I}¨⟩$ as nondegenerate ground state is also identified. Extension to states $\propto f(A^{\hat{a}}€)|0⟩$ with $f$ an arbitrary function is also discussed.

17 pages, 5 figures, final version