\section{Introduction}

In the finite-volume scaling regime $L \to \infty$ with $L << 1/m_{\pi}$
there are detailed analytical predictions for the rescaled microscopic
Dirac operator spectrum in gauge field sectors of fixed topological
charge $\nu$ \cite{SV,D,OTV}. We confront these predictions with quenched
si\-mu\-la\-tions on staggered fermions, with lattice size $8^4$ and
$\beta=5.1$. More details can be found in our paper \cite{DHNR}.

Random Matrix Theory, or equivalently finite-volume partition
functions, can be used to compute \emph{exactly} the microscopic Dirac
operator spectral density
\begin{equation}
\rho_{s}(\zeta)=\lim_{V \rightarrow \infty} \frac{1}{V} \rho(\frac{\zeta}
{V\Sigma})
\end{equation}
where
\begin{equation}
\Sigma=\lim_{m \rightarrow 0} \lim_{V \rightarrow \infty}<\bar{\psi}\psi>=
\pi\rho(0)
\end{equation}
is the infinite-volume chiral condensate.

At finite lattice spacing, staggered fundamental fermions of SU(3)
gauge theory lead to a microscopic Dirac spectrum in the universality
class known as the chiral unitary ensemble (chUE).
In a quenched gauge field sector of topological charge $\nu$, this
microscopic spectral density reads
\begin{equation}
\rho_{s}^{(\nu)}(\zeta)=\pi \rho(0) \frac{\zeta}{2}
\{J_{\nu}(\zeta)^{2}-J_{\nu-1}(\zeta)J_{\nu+1}(\zeta)\}
\end{equation}
As first observed by Verbaarschot \cite{V}, staggered fermions give good
agreement with
analytical prediction of just the $\nu$=0 sector, even when
\emph{all} gauge field configurations are summed over.
To test whether staggered fermions at similar $\beta$-values
\emph{are} sensitive to topology at all, we have divided a large number of
configurations ($\sim$17,000) into different topological
sectors based on a variant of APE-smearing (see ref. \cite{DHNR} for
the details).

\section{Classification scheme \& Analysis}

Using the naively latticized topological charge
\begin{equation}
\nu ~=~ \frac{1}{32\pi^2}\int\! d^4x {\mbox{\rm Tr}}[F_{\mu\nu}F_{\rho\sigma}]
\epsilon_{\mu\nu\rho\sigma}
\end{equation}
a given gauge field configuration is assigned an integer topological
charge, $\nu$, if the rounded-off value of the `naive' $\nu$ is stable
between 200 and 300 APE-smearing steps. Other configurations
($\sim37\%$) are simply rejected.

\begin{figure}[ht]
\includegraphics[scale=0.4]{Q_vs_smearing1.eps}
\caption{An example of the smearing evolution
of the naive $\nu$-value.}
\label{fig1}
\end{figure}

The rounding-off of the naive $\nu$-values makes good sense, as can be seen
in figure \ref{fig2}. It shows the distribution of the measured
topological charge after 200 smearing steps. The distribution is
strongly peaked around quantized values, and our measured $\nu$-values
are rounded off to the obvious integer assignment.
We see some
``renormalization'' of the topological charge as measured in this way,
but this is of no concern for us here. We shall only use the
measured $\nu$-values to make a classification of the {\em original},
un-smeared, configurations.

\begin{figure}[!htb]
\includegraphics[scale=0.4]{q200.eps}
\caption{The distribution of the naively measured
topological charge after 200 smearing steps.}
\label{fig2}
\end{figure}

The staggered Dirac operator spectrum has a $\pm$ symmetry, and when we
trace a few small eigenvalues as a function of smearing in figure
\ref{fig3}, we show only the positive ones.

\begin{figure}[ht]
\includegraphics[scale=0.4]{eigenv_smearing.eps}
\caption{The smearing history of the first few eigenvalues.}
\label{fig3}
\end{figure}

As expected, we find that $4\cdot\nu$ eigenvalues become small
compared with the rest after very many smearing steps.  These are the
$\nu$ ``would be'' zero modes of 4 (continuum) flavors on the very
smooth configurations.

All measurements of the microscopic Dirac ope\-ra\-tor spectrum are
performed on the original gauge field
configurations, classified into different
$\nu$-sectors according to the smeared values of $\nu$.
Shown in figure \ref{fig4} are the spectral densities obtained on
configurations classified by $\nu$ in that way.

\begin{figure}[ht]
\includegraphics[scale=0.4]{rho.eps}
\caption{Spectral densities configurations classified by $\nu$}
\label{fig4}
\end{figure}

Dashed lines are the analytical predictions for $\nu$=1 and 2 in
the appropriate graphs, and the solid curve is the analytical
prediction for $\nu$=0. At this $\beta$-value($\beta=5.1$) there is no
discernible deviation from the $\nu$=0 prediction even on
configurations that have been classified as $\nu=\pm 1,\pm 2.$

We have also compared the exact
predictions
\begin{eqnarray}
P_{min}^{\nu=0}(\zeta) & = &
\pi\rho(0)\frac{\zeta}{2}e^{-\frac{\zeta^2}{4}} \nonumber \\
\nonumber \\
P_{min}^{\nu=1}(\zeta) & = &
\pi\rho(0)\frac{\zeta}{2}I_2(\zeta)e^{-\frac{\zeta^2}{4}} \nonumber \\
\nonumber \\
P_{min}^{\nu=2}(\zeta) & = &
\pi\rho(0)\frac{\zeta}{2}\{I_2^2(\zeta)-I_1(\zeta)I_3(\zeta)\}e^{-\frac{\zeta^2}{4}}
\nonumber
\end{eqnarray}
for the distribution of just the smallest eigenvalue in the different
topological sectors. Even here we see no
deviation at all from the $\nu$=0 prediction.

It has recently been shown that it is possible to
recover the correct sensitivity of staggered fermions to topology
at very weak gauge coupling
(in the Schwinger model) \cite{Graz}. With fermions sensitive to gauge field
topology, nice agreement with the analytical predictions has been seen even
far from the continuum limit, as it should be \cite{EHKN}.

\begin{figure}[ht]
\includegraphics[scale=0.4]{Pmin.eps}
\caption{Distributions of the smallest eigenvalue.}
\label{fig5}
\end{figure}

The work of P.H.D. and K.R.
was partially supported by EU TMR grant ERBFMRXCT97-0122 and the work
of U.M.H by DOE contracts DE-FG05-85ER250000 and DE-FG05-96ER40979.
In addition P.H.D.
and U.M.H. acknowledge support by NATO Science Collaborative Research
Grant CRG 971487.
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