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QCD is expected to have a rich phase structure. It is empirically known to be difficult to access low temperature and nonzero chemical potential µ regions in lattice QCD simulations. We address this issue in a lattice QCD with the use of a dimensional reduction formula of the fermion determinant.We investigate spectral properties of a reduced matrix of the reduction formula. Lattice simulations with different lattice sizes show that the eigenvalues of the reduced matrix follow a scaling law for the temporal size Nt. The properties of the fermion determinant are examined using the reduction formula. We find that as a consequence of the Nt scaling law, the fermion determinant becomes insensitive to µ as T decreases, and µ-independent at T = 0 for µ < mπ/2.The Nt scaling law provides two types of the low temperature limit of the fermion determinant: (i) for low density and (ii) for high-density. The fermion determinant becomes real and the theory is free from the sign problem in both cases. In case of (ii), QCD approaches to a theory, where quarks interact only in spatial directions, and gluons interact via the ordinary Yang-Mills action. The partition function becomes exactly Z3 invariant even in the presence of dynamical quarks because of the absence of the temporal interaction of quarks.The reduction formula is also applied to the canonical formalism and Lee-Yang zero theorem. We find characteristic temperature dependences of the canonical distribution and of Lee-Yang zero trajectory. Using an assumption on the canonical partition function, we discuss physical meaning of those temperature dependences and show that the change of the canonical distribution and Lee-Yang zero trajectory are related to the existence/absence of µ-induced phase transitions.

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