  In this paper, we examine the phase diagram of quenched QCD with two flavors
of Wilson fermions, proposing the following microscopic picture. The
super-critical regions inside and outside the Aoki phase are characterized by
the existence of a density of near-zero modes of the (hermitian) Wilson-Dirac
operator, and thus by a non-vanishing pion condensate. Inside the Aoki phase,
this density is built up from extended near-zero modes, while outside the Aoki
phase, there is a non-vanishing density of exponentially localized near-zero
modes, which occur in "exceptional" gauge-field configurations. Nevertheless,
no Goldstone excitations appear outside the Aoki phase, and the existence of
Goldstone excitations may therefore be used to define the Aoki phase in both
the quenched and unquenched theories. We show that the density of localized
near-zero modes gives rise to a divergent pion two-point function, thus
providing an alternative mechanism for satisfying the relevant Ward identity in
the presence of a non-zero order parameter. This divergence occurs when we take
a "twisted" quark mass to zero, and we conclude that quenched QCD with Wilson
fermions is well-defined only with a non-vanishing twisted mass. We show that
this peculiar behavior of the near-zero-mode density is special to the quenched
theory by demonstrating that this density vanishes in the unquenched theory
outside the Aoki phase. We discuss the implications for domain-wall and overlap
fermions constructed from a Wilson-Dirac kernel. We argue that both methods
work outside the Aoki phase, but fail inside because of problems with locality
and/or chiral symmetry, in both the quenched and unquenched theories.

High Energy Physics - Lattice;Disordered Systems and Neural Networks;High Energy Physics - Phenomenology;High Energy Physics - Theory