Polyakov loop fluctuations in the Dirac eigenmode expansion

We investigate correlations of the Polyakov loop fluctuations with eigenmodes of the lattice Dirac operator. Their analytic relations are derived on the temporally odd-number size lattice with the normal nontwisted periodic boundary condition for the link variables. We find that the low-lying Dirac modes yield negligible contributions to the Polyakov loop fluctuations. This property is confirmed to be valid in confined and deconfined phases by numerical simulations in SU(3) quenched QCD. These results indicate that there is no direct, one-to-one correspondence between confinement and chiral symmetry breaking in QCD in the context of different properties of the Polyakov loop fluctuation ratios.

Figure 1

The temperature dependence of the Polyakov loop susceptibilities ratio $R_A={\chi }_A/{\chi }_T$ from Eq. (6) and the chiral condensate of light quarks $⟨\overline{\psi }\psi {⟩}_l$, normalized to its zero-temperature value. The lattice QCD Monte Carlo results are from Refs. [13] and [31], respectively. The horizontal dashed lines are the limiting values of $R_A$ in a pure SU(3) lattice gauge theory [12]. The vertical dashed lines indicate the chiral crossover temperature and the range of its errors [31, 32, 33].

Figure 2

The link path structure on a temporally odd-number lattice with $N_{\tau }=5$ and with the periodic boundary condition. The left configuration is gauge variant, whereas the middle is gauge invariant. The right configuration represents a closed path with $N_{\tau }$ (link variables) because of the periodicity in a temporal direction, and thus this path corresponds to a gauge-invariant Polyakov loop.

Figure 3

The lattice QCD result of the Dirac eigenvalue distribution $\rho (\lambda )$ in the lattice unit: (a) the confinement phase at $\beta =5.6$ (i.e., $a\simeq 0.25\mathrm{fm}$) on ${10}^3\times 5$; (b) the deconfinement phase at $\beta =6.0$ (i.e., $a\simeq 0.10\mathrm{fm}$) on ${10}^3\times 5$ [26].

Figure 4

The absolute value of the bare chiral condensate $|⟨\overline{\psi }\psi ⟩|$ per a flavor in the confinement phase plotted against the quark mass $m$ in the lattice unit. The lattice QCD calculation is done at $\beta =5.6$ (i.e., $a\simeq 0.25\mathrm{fm}$) on ${10}^3\times 5$.

Figure 5

The numerical results for the $R_{\text{chiral}}$ and $R_{\mathrm{conf}}$ ratio from Eqs. (38) and (39), respectively, as a function of an infrared cutoff $\mathrm{\Lambda }$ introduced on Dirac eigenvalues, expressed in lattice units. The Monte Carlo calculations have been performed on the ${10}^3\times 5$ lattice at $\beta =5.6$ and for the quark mass of $m=5\mathrm{MeV}$.