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On early onset of quark number density at zero temperature

We study a longstanding problem in lattice QCD at low temperature and nonzero quark chemical potential on an onset of the quark number density at $μ=m_π/2$. We introduce a physical parametrization of the eigenvalues in the reduction formula of the fermion determinant. It is shown that the parametrization reduces the quark number density operator to an expression with the Fermi distribution of the quark. For each configuration, the eigenvalues of the reduced matrix correspond to one-particle energy states of a quark. The gap of the eigenspectrum of the reduced matrix corresponds to the gap of the energy states, which causes the $μ$-independence of the fermion determinant for small $μ$ at T=0. Once $μ$ exceeds the gap, the quark number density becomes nonzero for each configuration, which causes the early onset of the quark number density.