Analytical relation between the Polyakov loop and Dirac eigenvalues in temporally odd-number lattice QCD

Abstract

We derive an analytical gauge-invariant relation between the Polyakov loop $⟨L_P⟩$ and the Dirac eigenvalues ${\lambda }_n$ in QCD, i.e., $⟨L_P⟩\propto {\sum }_n{\lambda }_n^{N_t-1}⟨n|{\hat{U}}_4|n⟩$, on a temporally odd-number lattice, where the temporal lattice size $N_t$ is odd. Here, we use an ordinary square lattice with the normal (nontwisted) periodic boundary condition for link-variables in the temporal direction. This relation is a Dirac spectral representation of the Polyakov loop in terms of Dirac eigenmodes $|n⟩$. Because of the factor ${\lambda }_n^{N_t-1}$ in the Dirac spectral sum, this analytical relation indicates negligibly small contribution of low-lying Dirac modes to the Polyakov loop in both confined and deconfined phases, while the low-lying Dirac modes are essential for chiral symmetry breaking. Also, we numerically confirm the analytical relation, non-zero finiteness of $⟨n|{\hat{U}}_4|n⟩$, and tiny contribution of low-lying Dirac modes to the Polyakov loop in lattice QCD simulations. Thus, we conclude that low-lying Dirac modes are not essential modes for confinement, and there is no direct one-to-one correspondence between confinement and chiral symmetry breaking in QCD.

Comments: Proc. of LATTICE2013