  In recent years the complex Langevin method (CLM) has proven a powerful
method in studying statistical systems which suffer from the sign problem. Here
we show that it can also be applied to an important problem concerning why we
live in four-dimensional spacetime. Our target system is the type IIB matrix
model, which is conjectured to be a nonperturbative definition of type IIB
superstring theory in ten dimensions. The fermion determinant of the model
becomes complex upon Euclideanization, which causes a severe sign problem in
its Monte Carlo studies. It is speculated that the phase of the fermion
determinant actually induces the spontaneous breaking of the SO(10) rotational
symmetry, which has direct consequences on the aforementioned question. In this
paper, we apply the CLM to the 6D version of the type IIB matrix model and show
clear evidence that the SO(6) symmetry is broken down to SO(3). Our results are
consistent with those obtained previously by the Gaussian expansion method.
