Clear correlation between monopoles and the chiral condensate in SU(3) QCD

We study spontaneous chiral-symmetry breaking in SU(3) QCD in terms of the dual superconductor picture for quark confinement in the maximally Abelian (MA) gauge, using lattice QCD Monte Carlo simulations with four different lattices of ${16}^4$, ${24}^4$, ${24}^3\times 6$ at $\beta =6.0$ (i.e., the spacing $a\simeq 0.1\mathrm{fm}$), and ${32}^4$ at $\beta =6.2$ (i.e., $a\simeq 0.075\mathrm{fm}$), at the quenched level. First, in the confinement phase, we find Abelian dominance and monopole dominance in the MA gauge for the chiral condensate in the chiral limit, using the two different methods of (i) the Banks-Casher relation with the Dirac eigenvalue density and (ii) finite quark-mass calculations with the quark propagator and its chiral extrapolation. In the high-temperature deconfined phase, the chiral restoration is observed also for the Abelian and the monopole sectors. Second, we investigate local correlation between the chiral condensate and monopoles, which topologically appear in the MA gauge. We find that the chiral condensate locally takes a quite large value near monopoles. As an interesting possibility, the strong magnetic field around monopoles is responsible to chiral symmetry breaking in QCD, similarly to the magnetic catalysis.

Figure 1

The Dirac eigenvalue densities $\rho (\lambda )$ for SU(3) QCD, Abelian-projected QCD, and monopole and photon sectors, as functions of the Dirac eigenvalue $\lambda$, for the four types of different lattices: (a) $16^3\times 16$ at $\beta =6.0$, (b) $24^3\times 24$ at $\beta =6.0$, (c) $32^3\times 32$ at $\beta =6.2$, and (d) $24^3\times 6$ at $\beta =6.0$. The line is the best fit with a constant for the low-lying Dirac eigenvalue density.

Figure 2

The chiral condensates for SU(3) QCD, Abelian-projected QCD, and monopole and photon sectors, as functions of the bare quark mass $m$ in the lattice unit, for the four types of different lattices: (a) $16^3\times 16$ at $\beta =6.0$, (b) $24^3\times 24$ at $\beta =6.0$, (c) $32^3\times 32$ at $\beta =6.2$, and (d) $24^3\times 6$ at $\beta =6.0$. The solid line is the best fit with a linear function.

Figure 3

The local chiral condensate at $t=12$ and monopoles at $t=11.5$, 12.5 for a typical Abelian configuration of the ${24}^3\times 24$ lattice at $\beta =6.0$. The bare quark mass is taken as $m=0.02$. The value of the local chiral condensate $|⟨\overline{\chi }\chi (x){⟩}_u|$ is visualized with the color graduation. Monopoles at $t=11.5$ and 12.5 are plotted with upper and lower triangles, respectively.

Figure 4

The correlation function $C(x-y)$ between the local chiral condensate $⟨\overline{\chi }\chi (x){⟩}_u$ and the local monopole density ${\rho }_{\mathrm{L}}(y)$ plotted against $|x-y|$ for (left) ${24}^3\times 24$ at $\beta =6.0$ and (right) ${32}^3\times 32$ at $\beta =6.2$. The bare quark masses of $m=0.02$, 0.015, and 0.01 are used in the lattice unit.