  We investigate general properties of the eigenvalue spectrum for improved
staggered quarks. We introduce a new chirality operator $[\gamma_5 \otimes 1]$
and a new shift operator $[1 \otimes \xi_5]$, which respect the same recursion
relation as the $\gamma_5$ operator in the continuum. Then we show that matrix
elements of the chirality operator sandwiched between two eigenstates of the
staggered Dirac operator are related to those of the shift operator by the Ward
identity of the conserved $U(1)_A$ symmetry of staggered fermion actions. We
perform a numerical study in quenched QCD using HYP staggered quarks to
demonstrate the Ward identity. We introduce a new concept of leakage patterns
which collectively represent the matrix elements of the chirality operator and
the shift operator sandwiched between two eigenstates of the staggered Dirac
operator. The leakage pattern provides a new method to identify zero modes and
non-zero modes in the Dirac eigenvalue spectrum. This method is as robust as
the spectral flow method but requires much less computing power. Analysis using
a 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 of
renormalization factors as a by-product. We conclude that it might be possible
and realistic to measure the topological charge $Q$ using the Atiya-Singer
index theorem and the leakage pattern of the chirality operator in the
staggered fermion formalism.
