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We study a longstanding problem in lattice QCD at low temperature and nonzeroquark chemical potential on an onset of the quark number density at$\mu=m_\pi/2$. We introduce a physical parametrization of the eigenvalues inthe reduction formula of the fermion determinant. It is shown that theparametrization reduces the quark number density operator to an expression withthe Fermi distribution of the quark. For each configuration, the eigenvalues ofthe reduced matrix correspond to one-particle energy states of a quark. The gapof the eigenspectrum of the reduced matrix corresponds to the gap of the energystates, which causes the $\mu$-independence of the fermion determinant forsmall $\mu$ at T=0. Once $\mu$ exceeds the gap, the quark number densitybecomes nonzero for each configuration, which causes the early onset of thequark number density.

On early onset of quark number density at zero temperature