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In many interesting systems, the fermion determinant becomes complex and itsphase plays a crucial role in the determination of the vacuum. For instance, infinite density QCD at low temperature and high density, exotic fermioncondensates are conjectured to form due to such effects. When one applies thecomplex Langevin method to such a complex action system naively, one cannotobtain the correct results because of the singular-drift problem associatedwith the appearance of small eigenvalues of the Dirac operator. Here we proposeto avoid this problem by adding a fermion bilinear term to the action andextrapolating its coefficient to zero. We test this idea in an SO(4)-invariantmatrix model with a Gaussian action and a complex fermion determinant, whosephase is expected to induce the spontaneous breaking of the SO(4) symmetry. Ourresults agree well with the previous results obtained by the Gaussian expansionmethod.

Spontaneous symmetry breaking induced by complex fermion determinant --- yet another success of the complex Langevin method