Fractional topological charges and the lowest Dirac modes

We introduce vortex configurations with fractional topological charges where one unicolor or colorful intersection of two perpendicular vortex pairs contributes to the topological charge of the configurations. Using both the overlap and asqtad staggered fermion formulations, the lowest modes of the Dirac operator on the noninteger $Q$ configurations are studied in the fundamental and adjoint representations. We analyze the behavior of the fundamental and adjoint fermions in the background of the topological charge contributions of $|Q|=0.5$.

Figure 1

The total topological charge $Q$ of the vortex configuration corresponding to Fig. 2. The gluonic topological charge approaches $Q=-1.5$ by increasing the smoothing region $\mathrm{\Delta }t$ of the colorful vortex and the radius $R$ of the colorful region through increasing the lattice size.

Figure 2

The geometry and the topological charge contribution of the intersection points of two antiparallel vortex pairs. ${\sigma }_i$ on the vortices (dashed lines) means the links of these vortices are distributed over the ${\sigma }_i$-generated U(1) subgroup of SU(2). In diagram (b), the bold black line at the intersection point ($x_1$, $z_1$) indicates that the unicolor vortex is substituted in this region by a colorful region. $\overrightarrow{n}·\overrightarrow{\sigma }$ on the colorful region (bold black line) means the links belong to full SU(2) gauge group. In both diagrams, the intersection point around ($x_1$, $z_1$) contributes to the topological charge while the other intersection points have $Q=0$.

Figure 3

(a) The topological charge density for the configuration in the xz-plane depicted in Fig. 2. Two antiparallel $xy$- and $zt$-vortex pairs are combined with vortex centers ($z_1=6$, $z_2=14$) and ($x_1=6$, $x_2=14$) at $t_{\perp }=y_{\perp }=6$ with thickness $d=2$ on a $16^4$-lattice. Intersecting vortex sheets related to the same color direction contribute to the topological charge. Therefore, only the intersection point around ($x_1$, $z_1$), where the vortex sheets are related to ${\sigma }_3$, contributes to the topological charge. The intersection point around ($x_1=6$, $z_1=6$) gives rise to a lump of topological charge $Q=0.5$. (b) the configuration, depicted in Fig. 2, is the same as the left diagram but the unicolor region for the $xy$-vortices around the point ($x_1$, $z_1$) is substituted by a colorful region, negatively charged. The smoothing region is $\mathrm{\Delta }t=t_2-t_1=9-3=R=6$. The intersection point around ($x_1=6$, $z_1=6$) gives rise to a lump of topological charge $Q=-1.5$.

Figure 4

(a) The lowest eigenvalues of the overlap Dirac operator on a $16^4$ lattice in the fundamental representation for the vortex configurations with fractional topological charges schematically displayed in Fig. 2 with $Q=0.5$ [Fig. 2] and $Q=-1.5$ [Fig. 2], using periodic bcs in spatial directions and antiperiodic bcs in temporal direction for fermionic fields. (b) the same as (a) but for periodic bcs in temporal direction. There are two zero modes for fundamental overlap operator on the sharp $Q=-1.5$ configuration which persist through changing the bcs, but just one of them remains zero after increasing the smoothing region $\mathrm{\Delta }t$ of the colorful vortex. In the fundamental representation, we also get some low-lying eigenmodes for both configurations with smaller eigenvalues than the ones of the lowest eigenvectors for the trivial gauge field. These low lying modes could not be removed by changing the bcs.

Figure 5

The chiral densities of a, b) two fundamental zero modes for antiperiodic bcs c) the zero mode for periodic bcs d) six adjoint zero modes for the $Q=-1.5$ configuration in the xz-plane on a $16^4$ lattice. Therefore there is one exact topological zero mode for fundamental overlap operator on the $Q=-1.5$ configuration. The plot titles indicate the plane positions, the chirality ($\mathrm{chi}=0,\pm 1$), the number $n$ of plotted mode and the maximal density in the plotted area, “$\max =\dots$”.

Figure 6

The lowest eigenvalues of the overlap Dirac operator on a $16^4$ lattice in the adjoint representation for the vortex configurations with fractional topological charges schematically displayed in Fig. 2. We find six adjoint zero modes of positive chirality for the $Q=-1.5$ configuration and two adjoint zero-modes of negative chirality for the $Q=0.5$ configuration which persist regardless of the bcs.

Figure 7

$\lambda L$ of low-lying modes of the overlap Dirac operator in the fundamental representation for some values of $\zeta \equiv d/L$ for the $Q=-1.5$ configuration (a) through increasing both length scales $L$ and $d$ at a fixed $\zeta$ for analyzing the lattice spacing effect (b) for analyzing the finite volume effect, the largest $L$ of a) is fixed while $d$ and therefore $\zeta$ are decreased. There is a very slight difference between $L=12$ and $L=16$ data at fixed $\zeta$. Also, by varying the values of $\zeta$ at fixed $L$, the data stay without changes.

Figure 8

The same as Fig. 7 but for the $Q=0.5$ configuration. The behavior of $\lambda L$ of low-lying modes are the same as those of the $Q=-1.5$ configuration, plotted in Fig. 7.

Figure 9

$\lambda L$ of low-lying (nonzero) modes of the overlap Dirac operator in the fundamental representation versus $1/L$ as the inverse volume scaling of the eigenvalues (a) for the $Q=-1.5$ configuration (b) for the $Q=0.5$ configuration. For both configurations, there is no, or a very slight, difference between the $L=12$ and $L=16$ data. In fact, by increasing $L$, the gap between some eigenvalues becomes a little more. It seems that the lowest modes for the noninteger $Q$ configurations could not be identified as the near-zero modes.