Probing for instanton constituents with $\epsilon$-cooling

We use $\epsilon$-cooling, adjusting at will the order $a^2$ corrections to the lattice action, to study the parameter space of instantons in the background of nontrivial holonomy and to determine the presence and nature of constituents with fractional topological charge at finite and zero temperature for SU(2). As an additional tool, zero-temperature configurations were generated from those at finite temperature with well-separated constituents. This is achieved by “adiabatically” adjusting the anisotropic coupling used to implement finite temperature on a symmetric lattice. The action and topological charge density, as well as the Polyakov loop and chiral zero-modes are used to analyze these configurations. We also show how cooling histories themselves can reveal the presence of constituents with fractional topological charge. We comment on the interpretation of recent fermion zero-mode studies for thermalized ensembles at small temperatures.

Figure 1

A charge 1 configuration on a ${16}^4$ lattice with periodic boundary conditions, generated from a Monte Carlo configuration in the confined phase, first being cooled with $\epsilon =1$ to just above the one-instanton action, after which 500 sweeps of $\epsilon =0$ (full curves), $-1$ (dashed curves) and $-10$ (dotted curves) cooling were applied. After interpolation of the lattice data we plot the action density (top) and Polyakov loop (bottom, in one of the directions only) along the line connecting its extrema. From the behavior of the Polyakov loop we deduce that decreasing $\epsilon$ pushes the constituents further apart.

Figure 2

A charge 1 configuration on a ${16}^4$ lattice with the holonomies fixed to be trivial in one direction and maximally nontrivial in the other three directions, generated from a random start first being cooled with $\epsilon =1$ to just above the one-instanton action, after which 80 sweeps of $\epsilon =-1$ cooling were applied. We plot the Polyakov loop for two relevant directions, in a plane through the center of the instanton. In this plane the action density is shown in the middle.

Figure 3

Starting from a continuum caloron solution with well-separated lumps, discretized on the anisotropic lattice and adjusted by 100 $\epsilon =-10$ cooling sweeps, we performed the adiabatic cooling by reducing $\xi$ from 4 to 1, through $\xi =2\sqrt{2}$, 2 and $\sqrt{2}$, applying between each of the four steps 100 $\epsilon =-10$ cooling sweeps. Shown is on the top the action density and on the bottom the Polyakov loop in the time direction along a line through the constituent locations. The dotted, dashed and full curves are for $\xi =4$, 2 and 1, respectively.

Figure 4

Example of a charge 3 caloron solution with $\frac{1}{2}\mathrm{T}\mathrm{r}{\mathcal{P}}_{\infty }=-0.126$ on a $4\times {16}^3$ lattice obtained from $\epsilon =-1$ cooling. Shown are the surfaces where half the trace of the Polyakov loop takes on the values 0.5 (light, red) and $-0.65$ (dark, blue), corresponding, respectively, to the constituent monopoles with positive and negative magnetic charge.

Figure 5

A charge 2 caloron with $\mathrm{T}\mathrm{r}{\mathcal{P}}_{\infty }=0$ on a $4\times {16}^3$ lattice obtained from a Monte Carlo generated configuration at $4/g^2=2.2$. We first went down to slightly above the two instanton action with $\epsilon =1$. After that many thousands of $\epsilon =-2$ cooling sweeps (followed by 500 with $\epsilon =0$) were performed. This gives the finite volume modification of the so-called “rectangular” solution constructed in Ref. 15. Shown is a suitable surface of constant action density for the double doughnut structure, as well as (clockwise) the action density, the periodic zero-mode density and the Polyakov loop. The latter three are shown on a plane through the doughnut which supports the periodic zero-mode. The other doughnut seen by the action density and the antiperiodic zero-mode has the sign of the Polyakov loop inverted, but is not seen by the periodic zero-mode.

Figure 6

Sum of the two exact zero-mode densities for a charge 2 constant curvature configuration. The flux has nonzero components in the 1–2 and 0–3 planes. The result is plotted as a function of $x_0$ and $x_3$ for $x_1=x_2=z_1=z_2=0$, on the left at $z_0=z_3=0$ (the two zero-mode densities fall on top of each other) and on the right at $z_0=z_3=0.25$ (the two zero-modes densities are shifted by half a period in the $x_0$ and $x_3$ directions relative to each other).

Figure 7

The result of adiabatic cooling, starting at finite temperature with two charge 1 calorons in the crossed configuration (see the text) on a ${16}^4$ lattice with $\xi =4$ after 1000 cooling sweeps with $\epsilon =0$. The finite temperature solution is presented in the top two figures. The left plot shows the action density integrated over time, the right plot the Polyakov loop in the time direction, both in the $y$-$z$ plane at $x=8$ (where all constituents lie by construction). We changed $\xi$ through $2\sqrt{2}$, 2, $\sqrt{2}$ to reach 1, at each of these applying 1000 cooling sweeps with $\epsilon =0$. The result at $\xi =1$ is given in the bottom two figures (showing the same quantities as above).

Figure 8

A charge 1 configuration on a $4\times {16}^3$ lattice with twisted boundary conditions in the time direction, $\overrightarrow{k}=(1,0,0)$. It was obtained from a configuration, first cooled down with $\epsilon =1$ to slightly above the one-instanton action, applying ${10}^4$ cooling sweeps with $\epsilon =-10$ (500 cooling sweeps with $\epsilon =0$ were finally applied to bring it close to the continuum). The constituents had been pushed so far apart that one of them effectively changed its electric and magnetic charge. The doughnut characteristic for two coinciding magnetic monopoles, with its symmetry axis along $\overrightarrow{k}$, gives the ultimate fixed-point under over-improved cooling. The top plot shows an action density contour plot of the doughnut, at the bottom the action density is plotted over the $y$-$z$ plane slicing the doughnut in two. Gluing two of these boxes along the $\overrightarrow{k}$ direction can be compared to Fig. 5.

Figure 9

Results obtained with cooling on a ${12}^4$ lattice with twist in the time direction given by $\overrightarrow{k}=(1,1,1)$, starting from a random configuration. We first applied 1000 $\epsilon =1$ cooling sweeps to go down to slightly above the one-instanton action. We plot the action density (top) and the square of half the trace of the Polyakov loop (bottom) along the line connecting the two constituents, every time adjusting to the continuum by 500 $\epsilon =0$ additional cooling sweeps. Each curve, with constituents pushed further apart, is obtained after (1000, 2000, 2000, 44 000) additional $\epsilon =-10$ cooling sweeps.

Figure 10

Example for annihilation of constituents with opposite fractional topological charge on a $4\times {16}^3$ lattice generated from a configuration just below $T_c$ with ordinary cooling. Shown are, for two consecutive plateaux, the topological charge density (left) and the Polyakov loop (right) in the $x$-$y$ plane, averaged over $z$ (and $t$, though the configurations are nearly static). The annihilation is between constituents with opposite magnetic, but equal electric charge and equal Polyakov loop.

Figure 11

Example of a plateau configuration on a $4\times {16}^3$ lattice (with $S=1.92$ units and $\frac{1}{2}\mathrm{T}\mathrm{r}{\mathcal{P}}_{\infty }=-0.22$) before annihilation of the two bottom constituents in the inset [light (red) and dark (blue) shading distinguishes positive from negative topological charge]. The Polyakov loop at the center of each of the constituents is indicated by $P=\pm 1$. The crosses give the low-lying eigenvalues $\lambda$ of the clover-improved Wilson-Dirac operator with periodic boundary conditions in time. The two curves trace the two near zero-modes from their value with antiperiodic (left) boundary conditions to the periodic (right) case (the squares correspond from left to right with $n=5$, 4, 3, 2, 1, 0, defining the phase $\exp (2\pi in/10)$ for the fermion boundary conditions).

Figure 12

Sample of eight cooling histories on a $4\times {16}^3$ lattice at $4/g^2=2.2$ ($T\approx 0.8T_c$). Pluses give the Wilson action and crosses the absolute value of the (order $a^2$ improved clover averaged) topological charge. The curve “s” is an example discussed in the text.

Figure 13

Sample of eight cooling histories on a ${16}^4$ lattice at $4/g^2=2.3$ ($T\approx 0.25T_c$). The symbols are as discussed in the caption of Fig. 12.

Figure 14

Scatter plots of $|\Delta Q|$ (horizontally) versus $\Delta S$ (vertically) for (i) a $4\times {16}^3$ lattice at $4/g^2=2.4$ ($T\approx 1.2T_c$), (ii) a $4\times {16}^3$ lattice at $4/g^2=2.2$ ($T\approx 0.8T_c$), note the significant clustering around $\Delta S=1$, $|\Delta Q|=0$ characteristic of constituent annihilations, and (iii) a ${16}^4$ lattice at $4/g^2=2.3$ ($T\approx 0.25T_c$). Each case is based on 50 configurations.