  In many interesting physical systems, the determinant which appears from
integrating out fermions becomes complex, and its phase plays a crucial role in
the determination of the vacuum. An example of this is QCD at low temperature
and high density, where various exotic fermion condensates are conjectured to
form. Another example is the Euclidean version of the type IIB matrix model for
10d superstring theory, where spontaneous breaking of the SO(10) rotational
symmetry down to SO(4) is expected to occur. When one applies the complex
Langevin method to these systems, one encounters the singular-drift problem
associated with the appearance of nearly zero eigenvalues of the Dirac
operator. Here we propose to avoid this problem by deforming the action with a
fermion bilinear term. The results for the original system are obtained by
extrapolations with respect to the deformation parameter. We demonstrate the
power of this approach by applying it to a simple matrix model, in which
spontaneous symmetry breaking from SO(4) to SO(2) is expected to occur due to
the phase of the complex fermion determinant. Unlike previous work based on a
reweighting-type method, we are able to determine the true vacuum by
calculating the order parameters, which agree with the prediction by the
Gaussian expansion method.
