Wilson fermions, random matrix theory and the Aoki phase

The QCD partition function for the Wilson Dirac operator, D_W, at nonzero lattice spacing a can be expressed in terms of a chiral Lagrangian as a systematic expansion in the quark mass, the momentum and a². Starting from this chiral Lagrangian we obtain an analytical expression for the spectral density of γ₅(D_W +m) in the microscopic domain. It is shown that the γ₅-Hermiticity of the Dirac operator necessarily leads to a coefficient of the a² term that is consistent with the existence of an Aoki phase. The transition to the Aoki phase is explained, and the interplay of the index of D_W and nonzero a is discussed. We formulate a random matrix theory for the Wilson Dirac operator with index ν (which, in the continuum limit, becomes equal to the topological charge of gauge field configurations).