Particle-Hole Matrix: Its Connection with the Symmetries and Collective Features of the Ground State

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Abstract
A detailed study is made of the properties of the ``particle‐hole matrix'' G abcd ≡〈g| (a†b−ρ ab )†(c†d−ρ cd ) |g〉 ,where |g> is a many‐fermion ground state and ρ ab is the 1‐body density matrix. It is shown that the zero eigenvalues of the particle‐hole matrix are intimately and simply related to the one‐body symmetries of the ground state. It is also shown that the large eigenvalues of Gabcd are closely related to the collective features of the ground state.