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We investigate general properties of the eigenvalue spectrum for improvedstaggered quarks. We introduce a new chirality operator $[\gamma_5 \otimes 1]$and a new shift operator $[1 \otimes \xi_5]$, which respect the same recursionrelation as the $\gamma_5$ operator in the continuum. Then we show that matrixelements of the chirality operator sandwiched between two eigenstates of thestaggered Dirac operator are related to those of the shift operator by the Wardidentity of the conserved $U(1)_A$ symmetry of staggered fermion actions. Weperform a numerical study in quenched QCD using HYP staggered quarks todemonstrate the Ward identity. We introduce a new concept of leakage patternswhich collectively represent the matrix elements of the chirality operator andthe shift operator sandwiched between two eigenstates of the staggered Diracoperator. The leakage pattern provides a new method to identify zero modes andnon-zero modes in the Dirac eigenvalue spectrum. This method is as robust asthe spectral flow method but requires much less computing power. Analysis usinga machine learning technique confirms that the leakage pattern is universal,since the staggered Dirac eigenmodes on normal gauge configurations respect it.In addition, the leakage pattern can be used to determine a ratio ofrenormalization factors as a by-product. We conclude that it might be possibleand realistic to measure the topological charge $Q$ using the Atiya-Singerindex theorem and the leakage pattern of the chirality operator in thestaggered fermion formalism.

Chiral symmetry and taste symmetry from the eigenvalue spectrum of staggered Dirac operators