Thursday, November 24, 2005

Dirac eigenvalues

In a recent talk entitled "Fun with Dirac eigenvalues", Michael Creutz discusses some issues arising in the study of the Dirac spectrum. The discussion involves a number of deceptively simple arguments on a rather complicated matter, and you should read it (and think about it) for yourself. The chiral condensate and the Banks-Casher relation, in particular, are discussed in a way that is obviously intended to first confuse, then astonish and finally enlighten the reader. Other points which I never thought about before are how the number of flavours influences the density of low-lying eigenvalues via the effects of the high eigenvalues on the gauge fields, and why topologically non-trivial configurations' contributions to correlation functions can be a problem in numerical simulations.

The discussion is kept in the context of the overlap operator, which makes sense for an analytical discussion of chiral properties. For an investigation of many of these issues in the context of the more widely used staggered quarks, see this paper by members of the HPQCD and UKQCD collaborations, where they show that, with improvement, staggered quarks exhibit all the properties expected of the Dirac spectrum, including obeying the Atiyah-Singer index theorem.