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 Casher related the density of small Dirac eigenvalues to spontaneous chiral symmetry breaking. Second, lattice discretizations of the Dirac operator based the Ginsparg-Wilson relation 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 method 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. Finally, since low eigenvalues impede conjugate gradient methods, separating out these eigenvalues explicitly can potentially be useful in developing dynamical simulation algorithms.
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