The purpose of this work is twofold.In the first part we describe superfluidity/superconductivity as an emergent phenomenon resulting from two-body correlations in presence of the Bose-Einstein condensation of particles.We briefly discuss the underlying mechanism for bosons as well as fermion pairs and illustrate various realizations of superfluidity emphasizing the recent examples.In the second part we study the glassy liquid of incoherent pairs which might exist above the transition temperature T c in the underdoped regime of cuprate superconductors.In particular, we explore the angular variation of pseudogap within two-dimensional version of the boson-fermion model using for a quantitative analysis the projective method.We find that above T c the pseudogap closes first near the nodal areas restoring there pieces (arcs) of the Fermi surface whereas remaining parts of the large Fermi surface around the antinodal points are still absent due to incoherent pairs.Upon increasing temperature the length of the Fermi arcs enlarges because the superconducting correlations are gradually suppressed.An intriguing death of Fermi surface can thus be closely related to the Bogoliubov quasiparticles whose existence in the pseudogap state has been predicted by us and confirmed recently by the angle resolved photoemission spectroscopy measurements on Bi 2 Sr 2 CaCu 2 O 8 and La 1.895 Sr 0.105 CuO 4 compounds.
We model the underdoped cuprates using fermions moving in a background with local antiferromagnetic order. The antiferromagnetic order fluctuates in orientation, but not in magnitude, so that there is no long-range antiferromagnetism, but a ``topological'' order survives. The normal state is described as a fractionalized Fermi liquid (FL*), with electronlike quasiparticles coupled to the fractionalized excitations of the fluctuating antiferromagnet. The electronic quasiparticles reside near pocket Fermi surfaces enclosing total area
x (the dopant density), centered away from the magnetic Brillouin zone boundary. The violation of the conventional Luttinger theorem is linked to a ``species doubling'' of these quasiparticles. We describe phenomenological theories of the pairing of these quasiparticles, and show that a large class of mean-field theories generically displays a nodal-antinodal ``dichotomy'': The interplay of local antiferromagnetism and pairing leads to a small gap near the nodes of the
d-wave pairing along the Brillouin zone diagonals, and a large gap in the antinodal region.