Localization: Quantitative results

To get a more precise picture of the localization properties of the eigenvectors, we again use the inverse participation ratio (IPR) I of Eq. (2.14) and the pseu-doscalar inverse participation ratio (PIPR) I5 defined in Eq. (2.18). These give

0 0.1 0.2 0.3 0.4

Figure 2.11: The inverse participation ratio as a function of |Im λ|.

global measures of the localization of an eigenvector. Larger IPR means larger localization of the eigenvector. From the PIPR one gets information about the chiral properties of the localized objects. For a classical instanton I and I5 are equal. The smaller I5 is as compared to I, the less the localization is correlated with chirality. But this is not very conclusive. Therefore, we will subsequently use the local chirality variableX(x) defined in Eq. (2.23) to study this correlation in a more precise way.

In Fig. 2.11 the inverse participation ratio is plotted as a function of the imaginary part of the eigenvalue |Imλ|. In the upper plot the result below the phase transition is shown. We get similar results as for the zero temperature ensemble, see Fig. 2.3. The most localized states are near|Imλ|= 0. As has been discussed previously in Section 2.2.3, this can be understood as a consequence of a fluid of topological excitations. An isolated excitation generates a zero mode of the Dirac operator. This is shifted to finite imaginary part as several of them start to interact.

In the lower plots the function is shown in the chirally symmetric phase atβ1 = 8.45. For the real sector on the left there are no eigenvalues up to |Imλ| ≈0.14.

Thus, we are above the phase transition where the chiral condensate vanishes.

But despite this fact the picture is similar to the chirally broken phase at β1 = 8.10. Above the gap we see, again, that the most localized states are at small values of |Imλ|. The localization decreases for larger imaginary parts of the eigenvalue. On the other hand, this behavior is much less visible in the complex sector. This has been expected. The classical solutions show a much weaker

1 2 3 4 5 6 7

Figure 2.12: Distribution of the inverse participation ratio below (top plot) and above (bottom plots) the phase transition. In the high temperature phase, the picture differs significantly for the complex and the real sector.

localization in this sector, too, see the appendix of [GGR⁺01a].

The probability distribution for the IPR is plotted in Fig. 2.12 below the phase transition (top) and in the two sectors of the deconfined phase (bottom). In each of these cases we see a tendency towards stronger localization for larger volumes of the lattice. This is consistent with predictions made in models of topological excitations like the instanton model. As in the previous discussion of the IPR as a function of |Imλ|, the result for the chirally broken phase and the real sector of the chirally symmetric phase hardly differ. However, there is a large difference to the complex sector. There, the distribution is peaked at small values of the inverse participation ratio. This is consistent with the picture of weakly localized objects in this region.

To further study the dependence of topological excitations on the temperature

— that is onβ₁for fixed lattice geometry — we look at the number of eigenvectors which have an inverse participation ratio larger than some constant c. This number will be called N(c). Analogously we define N5(c) for the pseudoscalar inverse participation ratio. The zero modes are excluded from this analysis.

We have already discussed that the chiral condensate is built up by topological excitations which slightly overlap. By doing so, they create small imaginary eigenvalues of the Dirac operator. Via the Banks-Casher relation Eq. (2.1) the density of these eigenvalues is responsible for the chiral condensate. As the chiral condensate vanishes at the phase transition, we expect the number of localized

8.0 8.1 8.2 8.3 8.4 8.5 β₁

0 1 2 3

< N5(c) >

0 1 2 3

< N(c) >

< L > real

c = 2.5 c = 5 c = 10

8.0 8.1 8.2 8.3 8.4 8.5 8.6 β₁

< L > complex

Figure 2.13: Average of N(c) and N5(c) the numbers of eigenvector with I > c and I₅ > c respectively, as a function of β₁. Zero modes are omitted in the evaluation of N(c) and N5(c). We display our results from 6×16³ lattices for three values of the cutc.

modes to drop as β1 is increased.

The result for three different cuts c= 2.5, 5, and 10 is shown in Fig. 2.13 on the 6×16³ lattice for the four values ofβ₁. We observe a decline in the real as well in the complex sector with growing β1. The functional form of N and N5

versusβ1 is similar. This supports the picture of topological objects that decrease in density but keep their chirality.

2.3.6 Local chirality

To verify the last findings, we now have a look at the local chirality variable X(x) of Eq. (2.23), which we already used at zero temperature in Section 2.2.4.

It maps the ratio of the density of positive chirality to the density of negative chirality on to the interval [−1,1]. A double peak structure in the distribution ofX is the signal for the chirality of the eigenvectors. This variable is evaluated on the sites with a high scalar density. In this way the correlation between high density and chirality is probed. The cuts on the density are chosen such that on average 1%, 6.25% and 12.5% of all sites are used in the analysis.

To start, the distribution of the local chirality variable for the zero modes is presented in Fig. 2.14. It is computed on 55 gauge configurations at β1 = 8.10 on 6×20³ lattices. Theβ1 = 8.45 sample consists of 55 configurations, too. This

X

Figure 2.14: Local chirality for the zero modes. We present data for both sides of the phase transition. For the chirally symmetric phase, we distinguish between the real and the complex sector of the Polyakov loop. We use different values for the cut-off on the percentage of supporting lattice points: 1%, represented by filled circles, 6.25%, open squares and 12.5%, filled triangles. The data are computed on the 6×20³ lattice.

statistics has to be split into the complex and the real sector with an approximate 2 : 1 ratio. The curves are normalized such that the integral is equal to 1. The apparent asymmetry in the complex sector can be attributed to the low statistics.

As we can observe a clear double peak structure, the zero modes are pre-dominantly chiral. The curves do not depend much on the percentage of lattice points included in the analysis. The local chirality in the chirally broken phase seems to be weaker than above the phase transition. This can be explained by the stronger ultraviolet fluctuations for lower values ofβ1.

Let us now turn to the near-zero modes. The dependence of the distribution of the local chirality on the magnitude of the imaginary part as well on the percentage of lattice points included is shown in Fig. 2.15 on the left. The data is from the 6× 20³ lattice with β1 = 8.10. The top plot shows the modes with |Imλ| < c1, where c1 = 0.018 is chosen such that 20% of our eigenvectors contribute to this bin. In the middle plot we show the distribution in the next bin c1 ≤ |Imλ| < c2, which contains the result for 40% of the modes for c2 = 0.057.

The last 40% of the eigenvectors lead to the lower plot. Again, in each of the plots there are three curves that originate from three different cuts on the scalar

β = 8 . 10

Figure 2.15: Local chirality for non-zero modes in the chirally symmetric phase on the left and in the broken phase on the right. The lattice size is 6×20³ in both cases. Forβ1 = 8.45, we divide our data with respect to the sector of the Polyakov loop. The left column contains the results for real hLi, the right column shows data for complex hLi. We bin the eigenvectors with respect to the imaginary parts of the corresponding eigenvalues such that the first bin (|Im λ| < c1, top plot) shows the very near-zero modes (closest to the edge forβ₁ = 8.45), while the other two bins (c1 ≤ |Im λ|< c2, middle plot andc2 ≤ |Im λ|, bottom plot) show the local chirality for eigenvectors with eigenvalues higher up in the spectrum.

The choice of the thresholdsc_i is discussed in the text and different for the three columns.

density p(x). We see that even if the signal for the local chirality is strongest for the near-zero modes and becomes weaker as |Imλ| grows, the local chirality stays intact for the eigenvectors with larger |Imλ|. Furthermore, the result for the near-zero modes closest to the origin is almost indistinguishable from the one for the zero modes. The curves hardly depend on the p(x) cut for the modes closest to the origin, whereas the dependence is larger for the eigenvectors higher up in the spectrum. All this supports the assumption of a smooth transition from the zero mode physics to the near-zero modes. Hence, the picture of a fluid of topological objects — calorons and anti-calorons — that lose their identity the more they overlap and simultaneously generate a higher|Imλ|, is made plausible by these observations. For |Imλ| > c2 the highest peaks still carry chirality but as the density gets smaller, quantum fluctuations get more and more important.

In Fig. 2.15 on the right the same plots are shown for the chirally symmetric phase atβ1 = 8.45, while the sector with real Polyakov loop can be found on the left. The partitioning of the eigenvectors in three bins of 20%, 40%, and 40%

with respect to|Imλ| yields c1 = 0.207 andc2 = 0.230. These values differ from the chirally broken phase due to the creation of the gap around|Imλ|= 0 which makes the chiral condensate vanish. In the complex sector this gap is significantly smaller and, thus, c1 = 0.081 and c2 = 0.112.

In the real sector the local chirality has vanished entirely, regardless of the eigenvector’s imaginary part. The topological excitations do not exist in this phase, as the chiral condensate has vanished, too.

In the complex sector only 1% of the lattice points show a local chirality. There is no dip visible in the graphs with a higher percentage of the sites encountered.

This means that there, too, are no extended objects with a chiral character.