Study of Vacuum Structure by Low-Lying Eigenmodes of Overlap-Dirac Operator

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1 Commun. Theor. Phys. (Beijing, China) 46 (2006) pp c International Academic Publishers Vol. 46, No. 3, September 15, 2006 Study of Vacuum Structure by Low-Lying Eigenmodes of Overlap-Dirac Operator WANG Zi-Qing, 1 LÜ Xiao-Fu,1,3,4 and WANG Fan 2 1 Department of Physics, Sichuan University, Chengdu , China 2 Department of Physics, Nanjing University, Nanjing , China 3 Institute of Theoretical Physics, the Chinese Academy of Sciences, Beijing , China 4 CCAST (World Laboratory), P.O. Box 8730, Beijing , China (Received November 23, 2005) Abstract APE smearing and overlap-dirac operator are combined to filter vacuum configurations. The structures of vacuum are studied by low-lying eigenmodes of the overlap-dirac operator, which exhibits that instanton liquid model can be used. PACS numbers: p, Lg, Pg Key words: APE smearing, overlap-dirac operator, structures of vacuum, instanton liquid model 1 Introduction The notion of the gauge field topology is important for understanding QCD vacuum. The instanton liquid model (ILM), [1] where correlation among instantons is introduced, was used as a basis for developing a rather successful semiclassically-motivated phenomenology. Even though termed as a liquid, it is quite dilute with (anti-) instantons of radius ρ 1/3 fm and density n = 1 fm 4. This setup allows for an interesting mechanism of spontaneous chiral symmetry breaking. Because of the important role of ILM we want to investigate this picture by lattice QCD. It is quite reasonable to expect that the lattice QCD will eventually provide us with detailed answers about the structure of QCD vacuum. However, finding a clean and satisfactory way to infer this information from lattice QCD has proven to be nontrivial since the gauge fields are rough. The necessity to eliminate the short-distance fluctuations resulted in manipulating the gauge fields in various cooling [2] or smoothing [3] procedures. The cooling is a local minimization procedure for the gauge action with initial state being the Monte Carlo generated QCD configuration. However, after a few cooling sweeps the gauge field undergoes large changes and become smooth. Further cooling possibly leads to smoother configuration, and eventually into the trivial configuration with nonperturbative effects removed. The cooling or smoothing has the subjective nature since we stop at sweeps here the gauge configuration shows the ILM structure. Horváth et al. [4] used lattice fermions to study the vacuum structure, because the space-time structure of the V (n) µ (x) = (1 α)v (n 1) µ (x) + α 6 ν µ low-lying is naturally smoother than that of the gauge fields themselves. They identified the possible vacuum structures by finding the local maxima of density d(n) = ψ nψ n. In their criterion, the local maxima is retained only if the average of d(n) decays monotonically from origin over the distance 3a for all directions. The details were given in Ref. [4]. Their results exhibited that the densities of structures are divergent with the lattice spacing decreasing. So they concluded that vacuum fluctuations of topological charge are not effectively dominated by instantons. We think that the lattice artifacts were not excluded in their paper. With the lattice spacing decreasing, the tiny fluctuations become more and more. But they still used 3a in their criterion. So almost all fluctuations were included in their results. In order to solve this problem, we take a few APE smearing [3,5,6] for the original lattice gauge fields and use a new criterion for structures. 2 Setup The method of inverse blocking [7] is one of the theoretically best supported smoothing algorithms and can keep the topological structure. But it is too costly to use it to measure the instanton distribution. The method of RG cycling [8] combines the inverse blocking with an additional blocking step that makes it possible to smooth the gauge configurations. But because RG cycling still contains an inverse blocking step, it cannot be used on large lattices. DeGrand et al. [6] used a simple transformation to fit the RG cycling, where each link is replaced by an APE-smeared link, [ V (n 1) ν (x)v µ (n 1) (x + ˆν)V ν (n 1) (x + ˆµ) + V ν (n 1) (x ˆν) V µ (n 1) (x ˆν)V ν (n 1) (x ˆν + ˆµ) ], (1) The project supported in part by the Key Research Plan of Theoretical Physics and Cross Science under Grant No

2 522 WANG Zi-Qing, LÜ Xiao-Fu, and WANG Fan Vol. 46 with V µ (n) (x) projected back onto SU(3) after each step, and V µ (0) (x) = U µ (x) the original link variable. V µ (n) (x) can be decomposed as [9] 1 V = V (V V ) (V V ) 1/2 O(V V ) 1/2 OH, (2) 1/2 where O is a U(3) matrix and H is a hermitean matrix. The matrix O can be reduced to an SU(3) matrix by multiplying it with a matrix I(ξ) proportional to the unity matrix with det [I(ξ)] = det [O ]. Therefore, V = ÕI(ξ) H, (3) where Õ = OI(ξ) is now an SU(3) matrix. The APE smearing reduces the short range fluctuations effectively and reproduces the result of RG cycling at least for instantons that are larger than about 1.5 lattice spacing. But some miss-identified vacuum fluctuations cannot be eliminated with APE smearing. [5] Here we take a few APE smearings to reduce the short range fluctuations, then use the overlap-dirac operator to perform a secondary filtering. We take N = 10 levels of APE smearing with α = 0.45 APE parameter for every configuration. Hasenfratz [5] took an estimate that the product Nα can be considered as characteristic number of the amount of smearing. So the smoothing effects here are just as much as 2 cooling steps (for the cooling α = 1), and it wouldn t destroy the topological structures. In Fig. 1, we show the distributions of action density in a z-t slice for a lattice before and after APE smearing, the left one is the distribution before APE smearing, the right one is the distribution after APE smearing. It exhibits that short range fluctuations are greatly reduced by APE smearing, but structures are still not clear, so it needs a secondary filtering by overlap-dirac operator. Fig. 1 The distributions of action density g 2 a 4 Tr[F µν(x)f µν(x)] in a z-t slice for a lattice before and after APE smearing, the left one is the distribution before APE smearing, the right one is the distribution after APE smearing. The massless overlap-dirac operator is [10] D = 1 2 [1 + γ 5ǫ(H w )], ǫ(h w ) = H w H w, (4) where ( ) B m C H w = C, (5) B + m C iα,jβ (n,n ) = 1 σ αβ [ µ U ij µ (n)δ n 2,n+ˆµ µ (U µ) ij (n )δ n,n +ˆµ], (6) B iα,jβ (n,n ) = 1 2 δ [ α,β 2δij δ nn Uµ ij (n)δ n,n+ˆµ µ (U µ) ij (n )δ n,n +ˆµ]. (7) where m refers to the overlap mass which is a parameter that has to be in the range [0,2]. We choose m = H 2 = D D ( D = γ 5 Dγ 5 ) commutes with γ5 and can be diagonalized in separate chiral in difference runs. [11] The eigenmodes of D can be obtained by combining the common eigenmodes of γ 5 and H 2, ψ = 1 2 (ψ R ± ıψ L ), (8) where ψ R and ψ L are the eigenmodes of H 2 with the same eigenvalue, and γ 5 ψ R = ψ R, γ 5 ψ L = ψ L. (9) One can define the chirality c(n) as follows: c(n) = ψ nγ 5 ψ n. (10) The gauge configurations are generated with the Iwasaki gauge action. [12] The parameters of the Iwasaki gauge ensembles are listed in Table 1. Configurations are separated by sweeps.

3 No. 3 Study of Vacuum Structure by Low-Lying Eigenmodes of Overlap-Dirac Operator 523 Table 1 Ensembles of Iwasaki gauge configurations β a [fm] V lat configs Results At first we study the distribution of the local chiral orientation X(n), which is defined by [13] ( π ) tan 4 (1 + X) = ψ L ψ R. (11) Our results for the X-distributions from the lowest two pairs of nonzero modes at the 2.5% of the lattice sites with the largest d(n) are shown in Fig. 2. In this figure, the histograms are normalized so that the sum of the values in all bins adds up to unity. In their peaks, the lowlying nonzero eigenmodes are fairly chiral, which suggests that the local chiral structure of fermionic eigenmodes is consistent with ILM. Fig. 2 X-distributions for the lowest two pairs of nonzero modes of the overlap-diract operator at 2.5% sites with the largest d(n) on the three ensembles with Iwasaki gauge action. Furthermore we use the chirality to explore vacuum structures. We found that the density d(n) and the chirality c(n) have the same space-time structures. The ILM also suggests this. But it is more convenient and clearer to use the chirality than the density, because using of chirality can distinguish the structures with different chirality. In our criterion, we select out some points which have the biggest chirality and the smallest chirality separately and examine whether they condensate as lumps. According to ILM, the contribution of one (anti-) instanton is definite, so the product of its volume and its mean density must be definite. Therefore the lumps with low densities and small volumes could not be regarded as (anti-) instantons. We analyzed the zero modes and the lowest non-zero modes of the overlap-dirac operator. As showed in Table 2, we selected different numbers of points to explore the details of structures. The numbers of points for every lump and their centers are listed in Table 2. The values of chirality of the isosurfaces, i.e., the surfaces of the lumps are listed in Table 2 also. We can find that the zero mode has one big lump and that the lowest non-zero mode has two big lumps with positive chirality and one big lump with negative chirality, while a big lump (marked by *) not only appears in the zero mode but also appears in the lowest nonzero mode. We regard these two lumps as one because the lump marked by * in the lowest nonzero mode is made up of two small lumps marked by 3 and 4 separately, moreover the lump marked by 4 is the same lump which appears in the zero mode and marked by * (The reasons will be further discussed below). So there are 3 big lumps totally, i.e. one instanton and two anti-instantons. There are still some small lumps, but they are so small that they cannot be regarded as anti-instantons. Furthermore some small lumps are close to each other and can form some big lumps, while

4 524 WANG Zi-Qing, LÜ Xiao-Fu, and WANG Fan Vol. 46 the big lumps have relatively big distances (about 1 fm) from each other. In Fig. 3, we show the chirality distribution of the lowest nonzero mode in a given time slice for the same lattice of Table 2 listing. We can find two lumps in left and down, i.e. the lumps marked by 1 and 2 in Table 2. They are close and connected each other. Moreover the left one has small volume and low chirality, so it just can be regarded as the tail of the bigger one. In like manner, we can draw a conclusion that there is a big lump in the right and up. So there are two structures in this slice, and they are separated very well. For these reasons, we can regard the small lumps as quantum fluctuations arising from as-yet-unknown mechanisms. In the same way, we explore the other configurations. The result is shown in Fig. 4, which is consistent with ILM. Fig. 3 Distribution of chirality c(n) of the lowest nonzero mode in a given time (t = 10) slice of a lattice at β = Periodic boundary conditions are imposed. The dark denotes high chirality. Fig. 4 Density of structures (in fm 4 ) as a function of the lattice spacing. Table 2 The chirality structures of the zero mode and the lowest nonzero mode of a lattice at β = The first subtable lists the structures of the zero mode which is positive chirality. The second one lists the structures of the lowest nonzero mode with positive chirality. The last one lists the structures of the lowest nonzero mode with negative chirality. In every subtable, the first line lists the numbers of points selected and the value of chirality c(n) on the isosurfaces (listed in brackets); the lower lines list the coordinates of the centers of every lump and the numbers of points (listed in brackets) for every lump. The structures of the zero mode (positive chirality): 64 ( ) 128 ( ) 512 ( ) 1024 ( ) 2048 ( ) (64) (128) (512) (1018) (2018) The structures of the lowest nonzero mode with positive chirality: (6) (30) 64 ( ) 128 ( ) 512 ( ) 1024 ( ) 2048 ( ) (64) (128) (304) (496) (4) (1517) (7) (101) (283) (9) (2) (57) (139) 3 The structures of the lowest nonzero mode with negative chirality: (37) (106) (531) 64 ( ) 128 ( ) 512 ( ) 1024 ( ) 2048 ( ) (64) (128) (512) (1024) (2048) There are still some properties of the Iwasaki vacuum which are different from the properties of the ideal ILM

5 No. 3 Study of Vacuum Structure by Low-Lying Eigenmodes of Overlap-Dirac Operator 525 vacuum. In appendix, we give out some properties of smooth SU(2) instanton backgrounds. In smooth SU(2) instanton backgrounds, only n (n is the smaller one of the numbers of instantons or anti-instantons) pairs of lowest nonzero eigenmodes exhibit double picked X-distributions, but in Iwasaki configurations, the higher eigenmodes still exhibit double peaked X-distribution. Maybe it is the complicity of vacuum that results in such phenomena. 4 Conclusions In this paper, we coupled the overlap fermions to the APE smeared gauge fields. The chiral orientation of the lowlying modes was studied, which exhibits good chirality and is consistent with ILM. Then by studying the structures of low-lying eigenmodes of overlap-dirac operator, we found that there are just a few space-time structures for every configuration. Further studying exhibits that some small structures are just the quantum fluctuations. The results are in agreement with ILM. But there are still some properties of the vacuum disagreeing with the ideal instanton model, which probably results from the complicity of vacuum. Appendix: Some Properties of Smooth SU(2) Instanton Backgrounds In this appendix we provide some properties of smooth SU(2) instanton backgrounds. [14] We first consider the low-lying eigenmodes of D in a configuration with one instanton and two anti-instantons. The result is shown in Fig. 5. The left one is the distribution of the chirality c(n) of the zero mode and the right one is that of the lowest nonzero mode. Then we consider the low-lying eigenmodes of D in a configuration with 2 instantons and 2 anti-instantons. The result is shown in Fig. 6. The left one is the distribution of the chirality c(n) of the lowest nonzero mode and the right one is that of the next lowest nonzero mode. These results suggest that low-lying eigenmodes of D reflect the structures of gauge background very well. Then we give out the X-distributions of the lowest 3 pairs of nonzero eigenmodes for both backgrounds. The results are showed in Fig. 7. Fig. 5 Distributions of c(n) of the zero mode (the left one) and the lowest nonzero mode (the right one) in a z-t slice for a smooth SU(2) one-instanton-and-two-anti-instanton background. The instanton and anti-instantons have a size of ρ = 1.5 and the distance between instanton and every anti-instanton is four lattice spacings. Fig. 6 Distributions of c(n) of the lowest nonzero mode (the left one) and the next lowest nonzero mode (the right one) in a z-t slice for a smooth SU(2) two-instanton-and-two-anti-instanton background. The instantons and anti-instantons have a size of ρ = 1.5 and the distance between every instanton and every anti-instanton is four lattice spacings.

6 526 WANG Zi-Qing, LÜ Xiao-Fu, and WANG Fan Vol. 46 Fig. 7 The X-distributions of the lowest 3 pairs of nonzero eigenmodes at the 10% sites with the largest d(n) for both backgrounds. The upper line is the distributions for the one-instanton-and-two-anti-instanton background. The lower line is the distributions for the two-instanton-and-two-anti-instanton background. References [1] E.V. Shuryak, Nucl. Phys. B 198 (1982) 83; D.I. Diakonov and V.Y. Detrov, Nucl. Phys. B 245 (1984) 259; T. Schäfer and E. Shuryak, Rev. Mod. Phys. 70 (1998) 323. [2] M.C. Chu, J.M. Grandy, S. Huang, and J.W. Negele, Phys. Rev. D 49 (1994) [3] M. Falcioni, M. Paciello, G. Parisi, and B. Taglienti, Nucl. Phys. B 251 (1985) 624; M. Albanese, et al., Phys. Lett. B 192 (1987) 163. [4] I. Horváth, S.J. Dong, et al., Phys. Rev. D 66 (2002) [5] A. Hasenfratz and C. Nieter, Phys. Lett. B 439 (1998) 366. [6] T. DeGrand, A. Hasenfratz, and T. Kovács, Nucl. Phys. B 520 (1998) 301. [7] M. Blatter, R. Burkhalter, P. Hasenfratz, and F. Niedermayer, Nucl. Phys. B (Proc. Suppl.) 42 (1995) 799; M. Blatter, R. Burkhalter, P. Hasenfratz, and F. Niedermayer, Phys. Rev. D 53 (1996) 923; R. Burkhalter, Phys. Rev. D 54 (1996) [8] T. DeGrand, A. Hasenfratz, and T. Kovács, Nucl. Phys. B 505 (1997) 417. [9] R. Petronzio and E. Vicari, Phys. Lett. B 248 (1990) 159. [10] H. Neuberger, Phys. Lett. B 417 (1998) 141; Phys. Lett. B 427 (1998) 353. [11] R.G. Edwards, U.M. Heller, and R. Narayanan, Nucl. Phys. B 540 (1999) 457. [12] Y. Iwasaki, Nucl. Phys. B 258 (1985) 141; M. Okamoto, et al., Phys. Rev. D 60 (1999) [13] I. Horváth, N. Isgur, J. McCune, and H.B. Thacker, Phys. Rev. D 65 (2002) [14] R.G. Edwards, U.M. Heller, and R. Narayanan, Nucl. Phys. B 522 (1998) 285.