2.2 A lattice study of chiral symmetry breaking
2.2.3 Localization
Now we look into the localization properties of the near-zero modes. For an instanton configuration in the continuum the Dirac operator has an exact zero mode ψ, which is localized in space and time around the center of the instanton.
In order to quantify the localization, consider the gauge invariant scalar density p(x) and the pseudoscalar density p5(x)
p(x) = X
c,d
ψ(x, d, c)∗ψ(x, d, c)≡ψ+(x)ψ(x) , (2.10) p5(x) = X
d,d0,c
ψ(x, d0, c)∗(γ5)d0,dψ(x, d, c)≡ψ+(x)γ5ψ(x) . (2.11) where ψ(x, d, c) is the eigenvector of the lattice Dirac operator. We take the eigenvectors as normalized such that
X
x
p(x) = 1 . (2.12)
For an eigenvectorψ of D,γ5ψ is also an eigenvector with γ5ψ =±ψ, see discus-sion after Eq. (2.8). The sign depends on the configuration being an instanton or an anti-instanton. Thus, scalar and the pseudoscalar density are equal for a single anti-instanton configuration, whereas they have opposite sign for an instanton.
p5(x) =
(−p(x) for instantons
+p(x) for anti-instantons (2.13) These densities show a clear localization at the point of an isolated smooth instan-ton put on the lattice by hand [FLS+85, GGL+01]. As an alternative approach one can use cooling techniques to identify the instantons independently. It was found that the localization of the eigenvectors is concentrated at the same region where the cooling procedure finds an instanton, see, e.g., [CGHN94].
To quantify the localization further the inverse participation ratio I has been introduced
I =V X
x
p(x)2 . (2.14)
It is widely used in condensed matter physics. Let us consider some examples to understand the behavior of this observable. (Keep in mind that p(x)≥0!)
• The density has support only on one lattice pointx0. Using the normaliza-tion condinormaliza-tion Eq. (2.12) this states
p(x) = δxx0 . (2.15)
So,p(x)2 =δxx0 and if we sum this over allxwe get 1. By definition (2.14), the inverse participation ratio is I =V in this case.
• If the density is maximally spread on all sites, the density is p(x) = 1
V . (2.16)
Herep(x)2 = 1/V2 for all x and the inverse participation ratio gives 1.
• ForN non-overlapping objects with a volume V0 each and p(x) =cwithin the volume andp(x) = 0 outside we get c= 1/N V0. Thus
I =V XN
i=1
1
N2V02 = V
N V02 = 1 ρV0
, (2.17)
whereρ=N V0/V is the density of the objects. Thus for a constant density, the inverse participation ratio should be independent of the volume V. Again, the smaller the objects, the higher is the localization.
To summarize: I is large if the scalar density is localized. It decreases to 1 the more the density is spread out. It is thus an appropriate measure for the localization of an eigenmode.
From the instanton picture we expect that near-zero modes are dominated by weakly interacting instantons. They have a high localization and the inverse participation ratio should be large. Modes further away from the origin are dominated by configurations where the instantons and anti-instantons have a larger overlap. In other words they start to annihilate and lose their identity.
Thus the localization is expected to be weaker for modes with larger imaginary part than for those with a smaller one.
In Figure 2.3 we show the result of the simulation. The inverse participation ratio is plotted versus the imaginary part of the corresponding eigenvalue Imλ.
The real modes have are left out. As the distribution is symmetric with respect to Imλ = 0, we show the curve only for positive imaginary part. The data is from a 164 lattice at three values of β1.
The three curves have their maximum near the origin and decrease for larger values of Imλ as expected. Moreover, we can observe that the localization near the origin is largest for the smallest value of β1, i.e. β1 = 8.10. In Figure 2.2 we have seen that for this β1 the chiral condensate is larger than for the larger values ofβ1. It remains to be understood whether this is simply due to a larger
0 0.05 0.1 0.15
Im λ
0.0 2.0 4.0 6.0 8.0 10.0
<I> β1 = 8.10
β1 = 8.30 β1 = 8.45
Figure 2.3: The average inverse participation ratiohIi of the near-zero modes as a function of λ. The lattice size is 164. The real modes are left out.
number of localized states or the states themselves are more localized. Therefore, the probability distribution ofI is shown in Fig. 2.4 for those eigenvectors with 0 < |Imλ| ≤ 0.05. In this |Imλ| range the curves in Fig. 2.3 depend on β1 significantly. The curves are normalized such that their integral over I is 1. The distribution for β1 = 8.10 exhibits a maximum which is shifted to larger values ofI compared to the curves for largerβ1. This shows that the modes themselves are more localized for smaller values of β1.
A second signature deduced from the instanton model is the chirality of the low lying eigenmodes. To quantify this, we consider the pseudoscalar density p5(x) = ψ+(x)γ5ψ(x) which is defined in Eq. (2.11). This density should have a negative sign near an instanton peak and a positive sign near an anti-instanton peak.
Analogously to the inverse participation ratio of Eq. (2.14), the pseudoscalar inverse participation ratio is defined by
I5 =V X
x
p5(x)2. (2.18)
This is a measure for the localization of the pseudoscalar density p5(x) in the same way as I is for p(x). For a single (anti)-instanton, I5 is equal to the in-verse participation ratio I, because p(x) = |p5(x)| for these configurations, see Eq. (2.13). According to the definitions Eqs. (2.10) and (2.11)
|p5(x)| ≤p(x) (2.19)
where we have used the definition ofγ5 from Eq. (A.4). For the inverse partici-pation ratios this translates to
I5 ≤I . (2.20)
1.0 2.0 3.0 4.0 5.0 6.0 7.0
I
0.00 0.25 0.50 0.75 1.00
P(I) β1 = 8.10
β1 = 8.30 β1 = 8.45
Figure 2.4: The distribution of the inverse participation ratio for modes with 0<|Imλ| ≤0.05. The lattice size is 164.
0.00 0.50 1.00 1.50
Im λ [fm-1]
0.0 0.2 0.4 0.6 0.8
< I5 I-1 > β1 = 8.10
β1 = 8.30 β1 = 8.45
Figure 2.5: The ratio of the pseudoscalar density to the scalar densityI5/I as a function of Imλ. The data is shown in physical units on the 164 lattices.
If we consider an ensemble of instantons and anti-instantons which are well separated, p(x) and |p5(x)| are much the same over the whole volume and thus I ≈I5. As these (anti)-instantons start to overlap, p5 undergoes several changes of sign in places wherep(x) is far from zero; I5 will be significantly smaller than I. For modes close to the origin, where we expect well defined (anti)-instantons, the ratio I5/I should be close to 1 and it should decrease for larger imaginary parts of the eigenvalue.
The result for this observable is shown in Fig. 2.5 as a function of Imλ.
The expected decrease of the ratio hI5/Ii as |Imλ| grows is observed. This supports the instanton model as it combines the two properties of instantons:
the localization and the chirality of the excitations. The dependence on|Imλ| is rather similar for all three values of the gauge couplingβ1.