Lattice2010

Wilson Fermions, Random Matrix Theory and the Aoki Phase

14 Jun 2010, 14:50

20m

Room6 (Villasimius, Sardinia)

Chiral symmetry Parallel 11: Chiral symmetry

Speaker

Jacobus Verbaarschot (Stony Brook University)

Description

The QCD partition function for the Wilson Dirac operator, $D_W$, at finite lattice spacing $a$ can be expressed in terms of a chiral Lagrangian as a systematic expansion in the mass, the momentum and $a^2$. Starting from this chiral Lagrangian we obtain an analytical expression  for the spectral density of ${\gamma }_5{D}_W$ in the microscopic domain (also known as the $]epsilon$-domain). It is shown that the ${\gamma }_5$-Hermiticity of the Dirac operator necessarily leads to the sign of  the coefficient of the $a^2$ term that allows an Aoki phase. The transition to the Aoki phase is explained in detail,  and the interplay of topological charge and finite $a$ is discussed. Finally, we formulate a random matrix theory for the Wilson Dirac operator in the sector of topological charge $\nu$. 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 the $a^2$-term is directly related to the ${\gamma }_5$-hermiticity of $D_W$.

Presentation materials

Slides

verbaarschot.pdf