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 $a^2$. Starting from this chiral Lagrangian we obtain an analytical expression for the spectral density of $γ_5 (D_W+m)$ in the microscopic domain. It is shown that the $γ_5$-Hermiticity of the Dirac operator necessarily leads to a coefficient of the $a^2$ 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). It is shown by an explicit calculation that this random matrix theory reproduces the $a^2$-dependence of the chiral Lagrangian in the microscopic domain, and that the sign of the $a^2$-term is directly related to the $γ_5$-Hermiticity of $D_W$.