  In the Monte Carlo study of QCD at finite baryon density based upon the phase
reweighting method, the pion condensation in the phase-quenched theory and
associated zero-mode prevent us to go to the low-temperature high-density
region. We propose a method to circumvent them by a simple modification of the
density of state method. We first argue that the standard version of the
density of state method, which is invented to solve the overlapping problem, is
effective only for a certain `good' class of observables. We then modify it so
as to solve the overlap problem for `bad' observables as well. While, in the
standard version of the density of state method, we usually constrain an
observable we are interested in, we fix a different observable in our new
method which has a sharp peak at some particular value characterizing the
correct vacuum of the target theory. In the finite-density QCD, such an
observable is the pion condensate. The average phase becomes vanishingly small
as the value of the pion condensate becomes large, hence it is enough to
consider configurations with small values of pion condensate, where the zero
mode does not appear. We demonstrate an effectiveness of our method by using a
toy model (the chiral random matrix theory) which captures the properties of
finite-density QCD qualitatively. We also argue how to apply our method to
other theories including finite-density QCD. Although the example we study
numerically is based on the phase reweighting method, the same idea can be
applied to more general reweighting methods and we show how this idea can be
applied to find a possible QCD critical point.
