# Wilson Fermions, Random Matrix Theory and the Aoki Phase

Akemann G, Damgaard PH, Splittorff K, Verbaarschot JJM (2010)
In: PoS (Lattice 2010)., 92. .

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Abstract
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$D_W$.
Publishing Year
Conference
The XXVIII International Symposium on Lattice Field Theory, Lattice2010
Location
Villasimius, Italy
Conference Date
2010-06-14
ISSN
PUB-ID

### Cite this

Akemann G, Damgaard PH, Splittorff K, Verbaarschot JJM. Wilson Fermions, Random Matrix Theory and the Aoki Phase. In: PoS (Lattice 2010). Vol 92. 2010.
Akemann, G., Damgaard, P. H., Splittorff, K., & Verbaarschot, J. J. M. (2010). Wilson Fermions, Random Matrix Theory and the Aoki Phase. PoS (Lattice 2010), 92
Akemann, G., Damgaard, P. H., Splittorff, K., and Verbaarschot, J. J. M. (2010). “Wilson Fermions, Random Matrix Theory and the Aoki Phase” in PoS (Lattice 2010), vol. 92,.
Akemann, G., et al., 2010. Wilson Fermions, Random Matrix Theory and the Aoki Phase. In PoS (Lattice 2010). no.92
G. Akemann, et al., “Wilson Fermions, Random Matrix Theory and the Aoki Phase”, PoS (Lattice 2010), vol. 92, 2010.
Akemann, G., Damgaard, P.H., Splittorff, K., Verbaarschot, J.J.M.: Wilson Fermions, Random Matrix Theory and the Aoki Phase. PoS (Lattice 2010). 92, (2010).
Akemann, Gernot, Damgaard, P. H., Splittorff, K., and Verbaarschot, J. J. M. “Wilson Fermions, Random Matrix Theory and the Aoki Phase”. PoS (Lattice 2010). 2010.Vol. 92.
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