Clover improvement, spectrum and Atiyah-Singer index theorem for the Diracoperator on the lattice

We study the role of the O(a)-improving clover term for the spectrum of the lattice Dirac oper ator using cooled and thermalized SU(2) gauge field con figurations. For cooled configurations we observe improvement of the spectr al properties when adding the clover term. For the thermalized case (12(4), beta = 2.4) without clover term we find a rather bad separation of physica l and doubler branches making a probabilistic interpretation of the Atiyah- Singer index theorem on the lattice questionable for this beta and lattice size. Adding the clover term leads to the creation of additional real eigen values which come in pairs of opposite chirality thus further worsening the situation for the index theorem. (C) 1999 Elsevier Science B.V.