Towards extremely dense matter on the lattice

Quantum chromodynamics (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 lattice QCD with the use of a dimensional reduction formula for the fermion determinant. We investigate the 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 N_t. The properties of the fermion determinant are examined using the reduction formula. We find that, as a consequence of the N_t-scaling law, the fermion determinant becomes insensitive to μ as T decreases, and is μ-independent at T=0 for μπ/2. The N_t-scaling law provides two types of low-temperature limit for the fermion determinant: (1) one for low density and (2) one for high density. The fermion determinant becomes real and the theory is free from the sign problem in both cases. In the case of (2), QCD approaches a theory in which quarks only interact 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 the Lee–Yang zero theorem. We find characteristic temperature dependences for the canonical distribution and the Lee–Yang zero trajectory. Using an assumption on the canonical partition function, we discuss the physical meaning of these temperature dependences and show that the changes in the canonical distribution and Lee–Yang zero trajectory are related to the existence/absence of μ-induced phase transitions. (author)