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In the framework of Euclidean QCD on a torus, we study the spectrum of the Dirac operator through inverse moments of its eigenvalues, averaged over topological sets of gluonic configurations. The large-volume dependence of these sums is related to chiral order parameters. We sketch how these results may be applied to lattice simulations in order to investigate the chiral phase transitions occurring when N_f increases. In particular, we demonstrate how Dirac inverse moments at different volumes could be compared to detect in a clean way the phase transition triggered by the suppression of the quark condensate and by the enhancement of the Zweig-rule violation in the vacuum channel., Comment: 46 pages, 7 figures, RevTex, published version

We argue that light quark loops produce a paramagnetic suppression of infrared-sensitive order parameters such as , as the number N_f of light fermions increases. The possibly strong dependence of on N_f is related to the observed Zweig rule violation in the scalar channel. Presuming the existence of a chiral phase transition for not too large N_f, we discuss the phenomenological possibilities of separately determining the two-flavour and three-flavour condensates and the quark mass ratio r=2m_s/(m_u+m_d). The issue is closely related to the interpretation of new forthcoming precise pi-pi data at low energy., Comment: 15 pages, Latex, using JHEP.cls (included), 1 PS figure

Considering QCD in an Euclidean box, the mechanism of spontaneous breaking of chiral symmetry (SB$\chi$S) is analyzed in terms of average properties of lowest eigenstates of the Dirac operator. A formal analogy between the pion decay constant and conductivity in disordered systems is established. It follows that SB$\chi$S results from a subtle balance between the density of Euclidean quark states and their mobility. SB$\chi$S can be realized either with $<\bar q q > =0$, provided the low density of states is compensated by a high mobility, or with a non-vanishing condensate, provided the mobility is suppressed. It is conjectured that the first case corresponds to extended whereas the latter case to (weakly) localized quark states., Comment: 4 pages, LaTeX, submitted to Physical Review Letters