Embedded monopoles in quark eigenmodes in SU(2) Yang-Mills theory

We study the embedded QCD monopoles (“quark monopoles”) using low-lying eigenmodes of the overlap Dirac operator in zero- and finite-temperature SU(2) Yang-Mills theory on the lattice. These monopoles correspond to the gauge-invariant hedgehogs in the quark-antiquark condensates. The monopoles were suggested to be agents of the chiral symmetry restoration since their cores should suppress the chiral condensate. We study numerically the scalar, axial, and chirally invariant definitions of the embedded monopoles and show that the monopole densities are in fact globally anticorrelated with the density of the Dirac eigenmodes. We observe that the embedded monopoles corresponding to low-lying Dirac eigenvalues are dense in the chirally invariant (high temperature) phase and dilute in the chirally broken (low-temperature) phase. We find that the scaling of the scalar and axial monopole densities towards the continuum limit is similar to the scaling of the stringlike objects while the chirally invariant monopoles scale as membranes. The excess of gluon energy at monopole positions reveals that the embedded QCD monopole possesses a gluonic core which is, however, empty at the very center of the monopole.

Figure 1

The densities of the (a) scalar, (b) axial, and (c) chirally invariant embedded monopoles in confinement phase for $\beta =2.3493$, 2.418, and 2.5 on vs the Dirac eigenvalue $\lambda$. (d) The same but for the deconfinement phase at $T=1.15T_c$. The densities are given in the units of the lattice spacing $a$.

Figure 2

The extrapolation of the scaling coefficients for the densities of the (a) scalar and axial, and (b) chirally invariant quark monopoles using various fits for $\lambda =235\mathrm{MeV}$.

Figure 3

The same as in Fig. 2 but for the zero mode, $\lambda =0$.

Figure 4

The scaling coefficients for the densities of the (a) scalar and axial, and (b) chirally invariant quark monopoles. The fits by functions (35, 36) are shown by solid and dashed lines.

Figure 5

The total, infrared and ultraviolet densities of the (a) scalar, (b) axial, and (c) chirally invariant embedded monopoles vs the Dirac eigenvalue $\lambda$.

Figure 6

The same as in Fig. 5 but for the deconfinement phase at $T=1.15T_c$.

Figure 7

(a) Example of the extrapolation of the UV fraction of the scalar and axial monopole densities at the eigenvalue $\lambda =235\mathrm{MeV}$. The fit is done by the linear formula (42). (b) The scaling coefficient $v^{\mathrm{UV}}$ vs $\lambda$ for the UV part of the scalar and axial monopole densities. The quantity $v^{\mathrm{UV}}$ is extrapolated the continuum limit.

Figure 8

(a) Example of extrapolation of the infrared-to-total ratios (44) corresponding to the scalar, axial, and chirally invariant embedded monopoles for the eigenvalue $\lambda =235\mathrm{MeV}$. The fit is done by the linear formula (45). (b) The infrared-to-total ratios of the scalar, axial and invariant monopole densities extrapolated by Eq. (45) to the continuum limit vs $\lambda$.

Figure 9

Spectral fermion density in the confinement ($T=0$, $\beta =2.3493$) and in the deconfinement ($T=1.15T_c$, $\beta =2.35$) phases.

Figure 10

The excess of the chromomagnetic action (50) at (a) the scalar and (b) the chirally invariant monopoles at $\lambda =0$ and $\lambda =235\mathrm{MeV}$. The solid and the dotted lines refer to the fits Eq. (60, 62), respectively.

Figure 11

The best fit parameters (a) $h$ and (b) $\delta$ of the fit function (62) vs the Dirac eigenvalue $\lambda$ for the scalar, axial, and chirally invariant embedded monopoles.

Figure 12

The local excess (50) of the Yang-Mills action at the position of the (a) scalar and (b) invariant monopoles, scaled by the power function, $f_S(a)a^{-(4+\delta )}$, with the anomalous scaling exponent $\delta =1/2$. The three values of the lattice coupling $\beta$ are shown.