Exploring the structure of the quenched QCD vacuum with overlap fermions

E.-M. Ilgenfritz, K. Koller, Y. Koma, G. Schierholz, T. Streuer, and V. Weinberg (QCDSF Collaboration)

Phys. Rev. D 76, 034506 – Published 29 August 2007

Abstract

Overlap fermions have an exact chiral symmetry on the lattice and are thus an appropriate tool for investigating the chiral and topological structure of the QCD vacuum. We study various chiral and topological aspects of quenched gauge field configurations. This includes the localization and chiral properties of the eigenmodes, the local structure of the ultraviolet-filtered field strength tensor, as well as the structure of topological charge fluctuations. We conclude that the vacuum has a multifractal structure.

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Received 16 May 2007

DOI:https://doi.org/10.1103/PhysRevD.76.034506

Authors & Affiliations

E.-M. Ilgenfritz¹, K. Koller², Y. Koma³, G. Schierholz^4,5, T. Streuer⁶, and V. Weinberg^7,8 (QCDSF Collaboration)

¹Institut für Physik, Humboldt Universität zu Berlin, 12489 Berlin, Germany
²Sektion Physik, Universität München, 80333 München, Germany
³Institut für Kernphysik, Johannes-Gutenberg Universität Mainz, 55099 Mainz, Germany
⁴Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germany
⁵John von Neumann-Institut für Computing NIC, 15738 Zeuthen, Germany
⁶Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055, USA
⁷Deutsches Elektronen-Synchrotron DESY, 15738 Zeuthen, Germany
⁸Institut für Theoretische Physik, Freie Universität Berlin, 14196 Berlin, Germany

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Vol. 76, Iss. 3 — 1 August 2007

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Images

Figure 1
The full pion propagator compared with the truncated pion propagator constructed as an all-to-all correlator summing over the 40 lowest modes only with a quark mass $\frac{a m}{2 ρ} = 0.01$ [see Eq. (12)]. This calculation uses 250 configurations on the $16^{3} \times 32$ lattice generated at $β = 8.45$ .
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Figure 2
The effective range function $F (r)$ as a function of $r / a$ on the $16^{3} \times 32$ lattice at $β = 8.45$ for $ρ = 1.4$ , together with an exponential fit.
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Figure 3
The Ginsparg-Wilson circle and the analyzed part of the unimproved and improved spectra for a $Q = 3$ configuration on the $16^{3} \times 32$ lattice generated at $β = 8.45$ . The inset shows the lowest part of the spectrum magnified. The zero eigenvalue is threefold degenerate.
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Figure 4
The normalized distributions of topological charge $Q$ , in subfigures ordered according to descending $β$ , for (a) $12^{3} \times 24$ at $β = 8.45$ , (b) $16^{3} \times 32$ at $β = 8.45$ , (c) $24^{3} \times 48$ at $β = 8.45$ , (d) $12^{3} \times 24$ at $β = 8.10$ , and (e) $16^{3} \times 32$ at $β = 8.00$ . Note that the physical volumes corresponding to (b) and (d) are roughly equal and similarly for (c) and (e).
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Figure 5
The spectral densities $a^{3} ρ (λ, V)$ at $β = 8.45$ for the three lattice sizes under study ( $12^{3} \times 24$ , $16^{3} \times 32$ , and $24^{3} \times 48$ ) and the spectral densities at $β = 8.1$ and $β = 8, 0$ for lattice sizes ( $12^{3} \times 24$ and $16^{3} \times 32$ ), together with fits using random matrix theory predictions.
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Figure 6
The scalar density of two typical nonzero eigenmodes of the overlap operator, shown in two-dimensional profile for a series of subsequent time slices of a $Q = 0$ configuration on the $16^{3} \times 32$ lattice generated at $β = 8.45$ . (a) shows the first nonzero mode and (b) the highest (144th) analyzed mode.
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Figure 7
Normalized histograms of IPRs as functions of $λ$ , in subfigures ordered according to descending $β$ : (a) $12^{3} \times 24$ at $β = 8.45$ , (b) $16^{3} \times 32$ at $β = 8.45$ , (c) $24^{3} \times 48$ at $β = 8.45$ , (d) $12^{3} \times 24$ at $β = 8.10$ , and (e) $16^{3} \times 32$ at $β = 8.00$ , for zero modes and nonzero modes in $λ$ bins with width 100 MeV.
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Figure 8
The average IPR for zero modes and for nonzero modes in $λ$ bins of width 50 MeV for the five ensembles.
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Figure 9
The fractal dimension $d^{*} (p)$ obtained from fits of the volume dependence of the averages of the generalized $I_{p}$ , presented for zero modes and for nonzero modes in $λ$ bins of width 50 MeV for the three ensembles with different volume and common $β = 8.45$ .
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Figure 10
Cluster analysis of some individual eigenmodes as listed in the legend box, averaged over 170 configurations on $16^{3} \times 32$ generated at $β = 8.45$ . The $p_{cut} / p_{\max }$ dependence is shown of (a) the total number of separate clusters at the height $p_{cut}$ , (b) the connectivity as a signal for the percolation of the mode, (c) the missing norm of the mode, and (d) the effective dimension of the mode at height $p_{cut}$ determined by the return probability (22) of random walkers.
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Figure 11
Normalized histograms of the local chirality in the $Q = 0$ subsample (consisting of 37 configurations) of the $16^{3} \times 32$ lattices generated at $β = 8.45$ . Left: For the lowest ten pairs (with positive and negative $λ$ ) averaged over the subsample. Right: For all modes from the subsample, averaged over $λ$ bins of width 100 MeV. Upper row: With a cut for 1% of the sites with biggest scalar density $p (x)$ . Lower row: With a cut for 6.25% of the sites with biggest $p (x)$ .
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Figure 12
The same as Fig. 11 but with less exclusive cuts with respect to the scalar density. Upper row: with a cut for 12.5. Lower row: With a cut for 50% of the sites with biggest $p (x)$ .
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Figure 13
The topological charge density correlator for the $12^{3} \times 24$ lattice configurations at $β = 8.10$ . (a) The correlator of the mode-truncated density $q_{λ_{cut}} (x)$ for various $λ_{cut}$ compared with the correlator of the all-scale density $q (x)$ , measured only on 53 configurations. (b) magnifies the region in $r$ where the mode-truncated correlators become negative for sufficiently large $λ_{cut}$ .
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Figure 14
Comparison of the topological charge density correlator based on the all-scale density $q (x)$ for the $12^{3} \times 24$ lattice at $β = 8.10$ (53 configurations) with the same correlator for the $16^{3} \times 32$ lattice at $β = 8.45$ (only 5 configurations) and $β = 8.60$ (2 configurations). (b) magnifies the region in $r$ where the correlator becomes stronger negative when one gets closer to the continuum limit.
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Figure 15
Two-dimensional profile through the typical $Q = 0$ configuration generated at $β = 8.45$ on a $16^{3} \times 32$ lattice. The profile is periodically doubled in both directions for better visibility of extended structures. The cuts show, from (a) to (c), the mode-truncated density for descending levels of truncation $λ_{cut} = 200 MeV$ , $λ_{cut} = 400 MeV$ , and $λ_{cut} = 634 MeV$ , respectively. The all-scale topological density is shown in (d). Note the twentyfold larger vertical scale in (d).
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Figure 16
Cluster analysis of the all-scale topological density: The $q_{cut}$ dependence is shown of (a) the total number of separate clusters, (b) the size of the largest cluster relative to all clusters, (c) the distance between the two largest clusters in lattice units, and (d) the connectivity (see Sec. 2c). The data are plotted for the $12^{3} \times 24$ lattice configurations at $β = 8.10$ (53 configurations, in red), for the $16^{3} \times 32$ lattice configurations at $β = 8.45$ (only 5 configurations, in green), and for the $16^{3} \times 32$ lattice configurations at $β = 8.60$ (only 2 configurations, in blue) being averaged over.
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Figure 17
Cluster analysis of the all-scale topological density: The fractional volume [ $(a) + (b)$ ] and the total charge [ $(c) + (d)$ ] of the corresponding clusters, ranked according to their occupied volume, is shown for two values of $q_{cut} / q_{\max }$ : in the percolated regime (left) for $q_{cut} / q_{\max } = 0.10$ and in the cluster-separated regime (right) for $q_{cut} / q_{\max } = 0.30$ . The plot represents one single configuration of the $16^{3} \times 32$ lattice ensemble generated at $β = 8.45$ .
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Figure 18
The probability of return to the cluster center $x_{c}$ for random walks, restricted to topological clusters of the all-scale topological charge density, shown as a function of the step number. The clusters are distinguished by different cuts with respect to the density that they enclose. The cases $q_{cut} / q_{\max } = 0.4$ , 0.3, 0.2, 0.1, and 0.0 are considered on the coarser lattice $12^{2} \times 24$ at $β = 8.10$ . The results of the power fits (see the curves) are labeled by the effective dimensions $d^{*}$ .
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Figure 19
The covering sphere method (see text): (a) for the coarse lattice $12^{3} \times 24$ at $β = 8.10$ and (b) for the fine lattice $16^{3} \times 32$ at $β = 8.45$ . The fitted power behavior of the accumulated charge jumps from one-dimensional to three-dimensional at some percolation threshold $q_{cut} / q_{\max }$ (see text).
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Figure 20
Comparison of the cluster structure of the mode-truncated density with three (four) cutoffs $λ_{cut}$ with the all-scale topological charge density. The effect of $β$ at the same physical volume can be seen by comparing the left side ( $12^{3} \times 24$ lattices at $β = 8.10$ ) with the right side ( $16^{3} \times 32$ lattices at $β = 8.45$ ). Top row: The $q_{cut}$ dependence of the total number of clusters [(a),(b)]. Note the different multiplicity scale on the right for clusters of the all-scale topological charge density. Middle row: The packing fraction of all clusters together [(c),(d)]. Bottom row: The connectivity [(e),(f)].
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Figure 21
Comparison of the cluster structure of the all-scale topological charge density with the mode-truncated density for three values of the cutoffs $λ_{cut}$ : (a) the fractional size of the largest cluster and (b) the distance between the two largest clusters in lattice units, both as a function of $q_{cut} / q_{\max }$ . The data refer to the $16^{3} \times 32$ lattice at $β = 8.45$ only.
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Figure 22
The correlation function between the mode-truncated topological charge density for $λ_{cut} = 100 MeV$ and the local chirality of selected nonzero modes from the lowest one to the 120th mode. The correlation function is an average over the 37 configurations with $Q = 0$ on the $16^{3} \times 32$ lattice generated at $β = 8.45$ .
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Figure 23
Normalized histograms with respect to the local (anti-)self-duality of the field strength tensor in the $Q = 0$ subsample consisting of 37 configurations generated on the $16^{3} \times 32$ lattice at $β = 8.45$ , (a) taken over 100% of the lattice sites, but depending on the number of nonzero modes (2–20) included in the filter and (b) applying various cuts specifying the infrared action density for the case of 20 eigenmodes included in the filter.
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Figure 24
Isosurface plots of $R (x)$ in the order of decreasing tolerance with respect to deviations from perfect (anti-)self-duality: (a) $R_{cut} = 0.97$ , (b) $R_{cut} = 0.98$ , (c) $R_{cut} = 0.99$ , and (d) $R_{cut} = 0.999$ . The figure shows the time slice $t = 6$ of the typical $Q = 0$ configuration on the $16^{3} \times 32$ lattice generated at $β = 8.45$ . Dark grey (red in color online) and light grey (green in color online) surfaces enclose regions where anti-self-duality or self-duality is better fulfilled than in the rest of the volume. The filter is using the 5 lowest pairs of nonzero eigenmodes.
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Figure 25
$R$ -cluster analysis using a running cutoff $R_{cut}$ on the basis of the filtered action density $s_{IR}$ and topological charge density $q_{IR}$ used to define $R (x)$ . The $R_{cut}$ dependence is shown of (a) the total number of separate clusters, (b) the size of the largest cluster relative to all clusters, (c) the distance between the two largest clusters, and (d) the connectivity (see text). The data are averaged over the $16^{3} \times 32$ lattice ensemble generated at $β = 8.45$ .
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Figure 26
Results of the $R$ -cluster analysis at $R_{cut} = 0.96$ (left) and $R_{cut} = 0.99$ (right) concerning the relative volume of all emerging clusters [(a),(b)] and concerning the charge inside the clusters [(c),(d)]. The cluster charge is defined by the all-scale topological charge density (18). The distributions describe the same $Q = 0$ configuration from the $16^{3} \times 32$ lattice ensemble generated at $β = 8.45$ already considered in Figs. 6 and 24.
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Figure 27
Direct comparison between the isosurface plots with respect to $R$ [left: with $R_{cut} = 0.99$ in (a) and $R_{cut} = 0.98$ in (c)] and the isosurface plots with respect to the truncated topological density $q_{λ_{cut}}$ with $λ_{cut} = 200 MeV$ (9 pairs of nonzero modes) [right, with $q_{cut} / q_{\max } = 0.2$ in (b) and $q_{cut} / q_{\max } = 0.1$ in (d)]. All subfigures show the same time slice $t = 6$ of the typical $Q = 0$ configuration of the $16^{3} \times 32$ ensemble at $β = 8.45$ . The similarity is found in all time slices.
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Figure 28
Two-dimensional section through the typical $Q = 0$ configuration generated at $β = 8.45$ on a $16^{3} \times 32$ lattice. Direct comparison of (a) ${\tilde{q}}_{IR} (x)$ based on the infrared field strength tensor based on 10 nonzero modes (5 pairs), (b) the mode-truncated density $q_{λ_{cut}} (x)$ with $λ_{cut} = 200 MeV$ (for this configuration it amounts to 9 pairs of nonzero modes), and (c) the all-scale charge density $q (x)$ . (d),(e) show the positive and negative parts, respectively, of the local (anti-)self-duality variable $R (x)$ . Note the twentyfold larger vertical scale in (c) compared to (b). The scale of (a) is freely adapted to that of (b).
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Figure 29
The full pion propagator compared with the contribution of the zero modes alone and the cumulative approximation by various numbers of nonzero modes (see legend box) is shown in analogy to Fig. 1. For this calculation 250 configurations have been used from the $16^{3} \times 32$ lattice ensemble generated at $β = 8.45$ .
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Figure 30
The dependence of the pion propagator for various $t$ (see legend box) on the number of accumulated nonzero modes is presented for the $16^{3} \times 32$ lattice ensemble generated at $β = 8.45$ .
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Figure 31
The effective-mass plot vs. $t$ for the $16^{3} \times 32$ lattice ensemble generated at $β = 8.45$ . This is shown for the full pion propagator and for various numbers (see legend box) of nonzero modes included in the approximation.
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