Fun with Dirac Eigenvalues

Amongst the lattice gauge community it has recently become quite popular to study the distributions of eigenvalues of the Dirac operator in the presence of the background gauge fields generated in simulations. There are a variety of motivations for this. First, in a classic work, Banks and Casher1 related the density of small Dirac eigenvalues to spontaneous chiral symmetry breaking. Second, lattice discretizations of the Dirac operator based the Ginsparg-Wilson relation2 have the corresponding eigenvalues on circles in the complex plane. The validity of various approximations to such an operator can be qualitatively assessed by looking at the eigenvalues. Third, using the overlap method3 to construct a Dirac operator with good chiral symmetry has difficulties if the starting Wilson fermion operator has small eigenvalues. This can influence the selection of simulation parameters, such as the gauge action.4 Finally, since low eigenvalues impede conjugate gradient methods, separating out these eigenvalues explicitly can potentially be useful in developing dynamical simulation algorithms.5 Despite this interest in the eigenvalue distributions, there are some dangers inherent in interpreting the observations. Physical results come from the full path integral over both the bosonic and fermionic fields. Doing these integrals one at a time is fine, but trying to interpret the intermediate results is inherently dangerous. While the Dirac eigenvalues depend on the given gauge field, it is important to remember that in a dynamical simulation the gauge field distribution itself depends on the eigenvalues. This circular behavior gives a highly non-linear system, and such systems are notoriously hard to interpret. Given that this is a joyous occasion, I will present some of this issues in terms of an amusing set of puzzles arising from naive interpretations of Dirac eigenvalues on the lattice. The discussion is meant to be a mixture of thought provoking and confusing. It is not necessarily particularly deep or new.