Instantons and monopoles

This study is part of a research program aimed to investigate the relations between instantons, monopoles, and chiral symmetry breaking. Monopoles are important three-dimensional topological configurations existing in QCD, which are believed to produce color confinement. Instantons are four-dimensional topological configurations and are known to be related to chiral symmetry breaking. To study the relation between monopoles and instantons we generate configurations which add static monopole-antimonopole pairs of opposite charges to the vacuum state by use of a monopole creation operator. We observe that the monopole creation operator only adds long monopole loops to the configurations. We then count the number of fermion zero modes using overlap fermions as a tool. As a result we find that each monopole-antimonopole pair of magnetic charge one adds one zero mode of chirality $\pm 1$, i.e. one instanton of topological charge $\pm 1$.

Figure 1

An example of the distribution of eigenvalues $\lambda$, and improved eigenvalues ${\lambda }_{imp}$. In this configuration there is one zero mode. The right figure is an enlargement of the region $\lambda \approx 0$.

Figure 2

Spectral density of nonzero modes [$\rho (\lambda )$].

Figure 3

The number of observed zero modes $N_{\text{Zero}}$ versus the physical volume $V/r_0^4$. The continuous curve is $N_{\text{Zero}}=\sqrt{A·V/r_0^4}+B$, $A=4.9(3)$, $B=-0.18(5)$.

Figure 4

The average square of the topological charges $⟨Q^2⟩$ versus the physical volume $V/r_0^4$. The linear line is $⟨Q^2⟩=A·V/r_0^4+B$, $A=6.8(2)\times {10}^{-2}$, $B=-0.20(13)$.

Figure 5

A distribution of topological charges $Q$. The lattice is $V={16}^4$, $\beta =6.00$.

Figure 6

The topological susceptibility $⟨Q^2⟩r_0^4/V$ at $\beta =6.00$ compared to results of other groups [26, 27].

Figure 7

The topological susceptibilities $⟨Q^2⟩r_0^4/V$ in our simulations as shown in Table 2 versus the physical volume $V/r_0^4$. A dotted line marks the physical volume $V/r^4=49.96$.

Figure 8

The monopole density ${\rho }_m{r}_0^3$ versus the distance between the monopole and antimonopole. The lattice is $V={14}^4,\beta =6.00$.

Figure 9

The monopole density ${\rho }_m{r}_0^3$ versus the magnetic charge $m_c$, with $\text{distance}=8$. The data points of the normal configurations are slightly shifted horizontally from $m_c=0$ to distinguish them.

Figure 10

The histogram of the length of the monopole loops $L/a$. The normal configurations with two flavors of dynamical Wilson fermions, $V/r_0^4=1.565(13)\times {10}^3,r_0/a=6.05(5)$, are used.

Figure 11

The histogram of the length of the monopole loops $L/a$. Monopoles with charges $m_c$ ranging from 0 to 4 are added to the normal configurations. The lattice is $V={14}^4$, $\beta =6.00$

Figure 12

The average of the long and short monopole loops divided by the number of configurations. The lattice is $V={14}^4$, $\beta =6.00$. The data points of the normal configurations are slightly shifted from $m_c=0$ for graphical reasons.

Figure 13

The number of zero modes $N_{\text{Zero}}$ versus monopole charges $m_c$. The data point computed from the normal configurations is slightly shifted from $m_c=0$.

Figure 14

The average square of topological charges $⟨Q^2⟩$ versus monopole charges $m_c$. The data point computed from the normal configurations is slightly shifted from $m_c=0$.

Figure 15

The comparisons of the distributions of the topological charges computed by simulations ($\text{distance}=8$) and the prediction. The monopole charges $m_c$ are from 1 to 4.