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Recently, within the framework of the Composite Operator Method, it has beenproposed a three-pole solution for the two-dimensional Hubbard model [Eur.Phys. J. B 87, 45 (2014)], which is still considered one of the best candidatemodel to microscopically describe high-$T_{c}$ cuprate superconductors. Theoperatorial basis comprise the two Hubbard operators (complete fermionic localbasis) and the electronic operator dressed by the nearest-neighbor spinfluctuations. The effectiveness of the approximate solution has been provedthrough a positive comparison with different numerical methods for variousquantities. In this article, after recollecting the main analytical expressionsdefining the solution and the behavior of basic local quantities (doubleoccupancy and chemical potential) and of the quasi-particle energy dispersions,we resolve and analyze the momentum components of relevant quantities: filling(i.e. the momentum distribution function), double occupancy andnearest-neighbor spin correlation function. The analysis is extended to COM(2p)solutions that will be used as primary reference. Thanks to this, the roleplayed by the third field, with respect to the two Hubbard ones, in determiningthe behavior of many relevant quantities and in allowing the extremely goodcomparison with numerical results is better understood giving a guideline tofurther improve and, possibly, optimize the application of the COM to theHubbard model.

COM(3p) solution of the 2D Hubbard model: momentum resolved quantities