Recently, within the framework of the Composite Operator Method, it has been proposed 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 candidate model to microscopically describe high-$T_{c}$ cuprate superconductors. The operatorial basis comprise the two Hubbard operators (complete fermionic local basis) and the electronic operator dressed by the nearest-neighbor spin fluctuations. The effectiveness of the approximate solution has been proved through a positive comparison with different numerical methods for various quantities. In this article, after recollecting the main analytical expressions defining the solution and the behavior of basic local quantities (double occupancy 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 and nearest-neighbor spin correlation function. The analysis is extended to COM(2p) solutions that will be used as primary reference. Thanks to this, the role played by the third field, with respect to the two Hubbard ones, in determining the behavior of many relevant quantities and in allowing the extremely good comparison with numerical results is better understood giving a guideline to further improve and, possibly, optimize the application of the COM to the Hubbard model.
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