Is there an Aoki phase in quenched QCD ? ∗

This talk summarizes our efforts to understand aspects of the properties of Wilson-like fermions propagating on background quenched gauge configurations. We refer to this loosely as the “phase structure” of quenched QCD (QQCD), although this is not conventional thermodynamics as the fermions do not enter the partition function. We are interested in the supercritical region (bare quark mass m0 satisfying −8 < m0a < 0) in which the hermitian Wilson-Dirac operator HW can have zero eigenvalues. The key question here is whether there are ranges within this region in which the infinite volume spectral density of HW , ρ(λ), is non-vanishing all the way down to zero eigenvalue, and in which the near-zero modes have physical extent (rather than having a size comparable to the lattice spacing). This issue is of direct relevance to domain-wall or overlap fermions (DW/OF). As explained in in Refs. [1,2], long-distance near-zero modes lead to a loss of chiral symmetry with DWF and of locality for OF. We stress that in the application to DW/OF the gauge fields are always quenched with respect to HW , irrespective of whether they are generated with dynamical DW/OF loops or not. Thus, when using such fermions, one wants to know the quenched Wilson phase structure so as to avoid regions with such near-zero modes. Lack of space allows us only to outline our work here; technical details will be given in Ref. [3].