Wilson Fermions, Random Matrix Theory and the Aoki Phase

The QCD partition function for the Wilson Dirac operator, $D_W$, at nonzerolattice spacing $a$ can be expressed in terms of a chiral Lagrangian as asystematic expansion in the quark mass, the momentum and $a^2$. Starting fromthis chiral Lagrangian we obtain an analytical expression for the spectraldensity of $\gamma_5 (D_W+m)$ in the microscopic domain. It is shown that the$\gamma_5$-Hermiticity of the Dirac operator necessarily leads to a coefficientof the $a^2$ term that is consistent with the existence of an Aoki phase. Thetransition 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 theWilson Dirac operator with index $\nu$ (which, in the continuum limit, becomesequal to the topological charge of gauge field configurations). It is shown byan explicit calculation that this random matrix theory reproduces the$a^2$-dependence of the chiral Lagrangian in the microscopic domain, and thatthe sign of the $a^2$-term is directly related to the $\gamma_5$-Hermiticity of$D_W$.