  The microscopical analysis of the unconventional and puzzling physics of the
underdoped cuprates, as carried out lately by means of the Composite Operator
Method (COM) applied to the 2D Hubbard model, is reviewed and systematized. The
2D Hubbard model has been adopted as it has been considered the minimal model
capable to describe the most peculiar features of cuprates held responsible for
their anomalous behavior. COM is designed to endorse, since its foundations,
the systematic emergence in any SCS of new elementary excitations described by
composite operators obeying non-canonical algebras. In this case (underdoped
cuprates - 2D Hubbard model), the residual interactions - beyond a 2-pole
approximation - between the new elementary electronic excitations, dictated by
the strong local Coulomb repulsion and well described by the two Hubbard
composite operators, have been treated within the Non Crossing Approximation.
Given this recipe and exploiting the few unknowns to enforce the Pauli
principle content in the solution, it is possible to qualitatively describe
some of the anomalous features of high-Tc cuprate superconductors such as large
vs. small Fermi surface dichotomy, Fermi surface deconstruction (appearance of
Fermi arcs), nodal vs. anti-nodal physics, pseudogap(s), kinks in the
electronic dispersion. The resulting scenario envisages a smooth crossover
between an ordinary weakly-interacting metal sustaining weak, short-range
antiferromagnetic correlations in the overdoped regime to an unconventional
poor metal characterized by very strong, long-but-finite-range
antiferromagnetic correlations leading to momentum-selective non-Fermi liquid
features as well as to the opening of a pseudogap and to the striking
differences between the nodal and the anti-nodal dynamics in the underdoped
regime.
