  We calculate the complex phase of chiral determinant by the vacuum overlap
formula with configurations of two-dimensional U(1) gauge field fixed in Landau
and Laplacian gauge. The complex phase fluctuates over the Gribov copies, which
appear in the process of Landau gauge fixing and contain vortex-like
singularities. In the Laplacian gauge, the fluctuation can be reduced and the
phase can be determined uniquely. If it is used as a preconditioning for Landau
gauge fixing, the most smooth configuration is obtained among the copies
generated.
