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

We derive an analytical gauge-invariant relation between the Polyakov loop $\langle L_P \rangle$ and the Dirac eigenvalues $\lambda_n$ in QCD, i.e., $\langle L_P \rangle \propto \sum_n \lambda_n^{N_t -1} \langle n|\hat U_4|n \rangle$, 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 $|n\rangle$. 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 $\langle n|\hat U_4|n \rangle$, 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.