Universal fluctuations in spectra of the lattice Dirac operator
Abstract
Recently, Kalkreuter obtained the complete Dirac spectrum for an
SU(2)
lattice gauge theory with dynamical staggered fermions on a
12
4
lattice for
β=1.8
and
β=2.8
. We performed a statistical analysis of his data and found that the eigenvalue correlations can be described by the Gaussian Symplectic Ensemble. In particular, long range fluctuations are strongly suppressed: the variance of a sequence of levels containing
n
eigenvalues on average is given by
Σ
2
(n)∼
1
2
π
2
(logn+const.)
instead of
Σ
2
(n)=n
for a random sequence of levels. Our findings are in agreement with the anti-unitary symmetry of the lattice Dirac operator for
N
c
=2
with staggered fermions which differs from the continuuum theory. For
N
c
=3
we predict that the eigenvalue correlations are given by the Gaussian Unitary Ensemble.