Mobility edge and locality of the overlap-dirac operator with and without dynamical overlap fermions

Transcription

1 Mobility edge and locality of the overlap-dirac operator with and without dynamical overlap fermions JLQCD Collaboration: a,b, S. Aoki c,d, H. Fukaya e, S. Hashimoto a,b, K-I. Ishikawa f, K. Kanaya c, T. Kaneko a,b, H. Matsufuru a, M. Okamoto a, T. Onogi g a High Energy Accelerator Research Organization (KEK, Tsukuba 35-8,Japan b School of High Energy Accelerator Science, The Graduate University for Advanced Studies (Sokendai, Tsukuba 35-8, Japan c Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba , Japan d Riken BNL Research Center, Brookhaven National Laboratory, Upton, New York 973, USA e Theoretical Physics Laboratory, RIKEN, Wako 35-98, Japan f Department of Physics, Hiroshima University, Higashi-Hiroshima , Japan g Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 66-85, Japan We perform a systematic study of low-lying eigenmodes of H w with various gauge actions to find the optimal choice for dynamical overlap fermion simulations, with which one may achieve lower numerical cost for HMC and better locality property of the overlap kernel. For this purpose, our study is made with emphasis on the distribution of low-lying eigenvalues and the mobility edge with and without dynamical overlap fermions. PoS(LAT66 XXIV International Symposium on Lattice Field Theory July Tucson Arizona, US Speaker. norikazu.yamada@kek.jp c Copyright owned by the author(s under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.

2 . Introduction The JLQCD collaboration has started a large scale simulation of dynamical overlap fermions [], aiming at studying the QCD dynamics in the presence of very light quarks. For the overlap fermion formulation, large computational costs and non-locality of the overlap-dirac operator have been practical and theoretical concerns. The computational cost is closely related to the near-zero mode density of the Hermitian Wilson-Dirac operator, H w ( m = γ 5 D w ( m = γ 5 ( m + D w ( (m = + s, (. where D w ( is the standard Wilson-Dirac operator for the massless Wilson fermion. According to Ref. [], locality of the overlap-dirac operator can be studied by eamining the locality of eigenvectors of H w, defined by H w φ i ( = λ w,i φ i (. Since both the near-zero mode density and the corresponding eigenvectors are known to depend on the gauge action and β, in this work we eplore these two properties for several gauge actions. We consider three gauge actions, the standard plaquette (Plq, Iwasaki (RG, and a modified gauge action motivated by the admissibility condition (Adm. The Adm action [3, 4, 5] is given by ReTrP µ,ν (/3 β S Adm =,µ,ν ( ReTrP µ,ν /3/ε, when ReTrP µ,ν/3 < ε, (. otherwise where ε is a parameter to control the possible maimum value of ReTrP µ,ν (/3. With ε < /(6( +, the locality of the overlap-dirac operator is guaranteed [6]. This action reduces to the standard plaquette action when /ε. In addition, for each of these gauge actions we introduce two-flavors of etra-wilson fermions ψ and ghosts χ [7] as S et = ψ(d w ( m ψ( + χ ( ( D w ( m + iµγ 5 τ 3 χ(, (.3 where τ 3 acts on the flavor inde. Since integrating out these etra fields results in deth w( m /(H w( m + µ, the appearance of the modes with λ w,i < µ are suppressed with this weight. µ = corresponds to the standard quenched approimation. The same value of m is taken for both the overlap kernel and etra fields. The lattice size is fied to For the Adm action, /ε = /3 is used throughout this study. Three values of µ (.,., and.4 are eamined, and more than, trajectories are accumulated for each gauge action as shown in Tab.. The lattice spacing is set by r =.49 fm, and is about.5 fm for all lattices, unless otherwise stated. PoS(LAT66. Spectral density Three panels in Fig. show the distributions of the near-zero modes for three gauge actions, where all eigenvalues are plotted in the lattice unit. Different symbols represent different values of µ. We find that the etra fermions and ghosts suppress the low-lying modes as epected. The suppression is most effective with µ =.. Figure shows a comparison of the spectral density ρ for the three gauge action with µ =.. We observe that the RG action yields the smallest near-zero mode density. From these observations made in the quenched approimation, we decide to employ the Iwasaki RG gauge action with µ =. in the dynamical overlap simulations.

3 λ w λ w λ w action β µ /ε # of trj. Plq 5.83., 5.7., ,6 RG.43.,.37.,6.7.4, Adm.33. /3,.3. /3,.6.4 /3 4,8... Table : Simulation parameters. Plq, β=5.83 Plq, β=5.7, µ=. Plq, β=5.45, µ= conf id... RG, β=.43 RG, β=.37, µ=. RG, β=.7, µ= conf id... Adm, β=.33, /ε=/3 Adm, β=.3, /ε=/3, µ=. Adm, β=.6, /ε=/3, µ= conf id Figure : Distribution of near-zero modes for three different gauge actions, Plq, RG and Adm from top to bottom, and with three values of µ=.,.,.4 from left to right. ρ 8.e-3 6.e-3 4.e-3.e-3 Plq, β=5.7, µ=. Adm, β=.3, ε=/3, µ=. RG, β=.37, µ=..e+ -.. λ w Figure : Spectral densities for three quenched gauge actions, Plq (blue, Adm (black and RG (red with µ =.. ρ 8.e-3 6.e-3 4.e-3.e-3 Dynamical, β=.3, m ud =.5 Dynamical, β=.35, m ud =. RG, β=.37, µ=..e+ -.. λ w Figure 3: Comparison of the near-zero mode densities on the quenched (black, slightly fine (blue and coarse (red dynamical configurations. We made the similar study on dynamical configurations, and compare the results with that on the quenched results with µ=. in Fig. 3. The results from two dynamical ensembles with β =.3 and.35 are shown, their lattice spacings are a=. fm and. fm, respectively, and the sea quark mass is about m s /4 in both cases. While the density increases for the dynamical configurations, it is clear that the nearzero mode suppression works well even after incorporating the overlap sea quarks. More details on our dynamical configurations are found in Ref. []. 3. Mobility edge The overlap-dirac operator is proved to be eponentially local, if there is no low-lying mode PoS(LAT66 3

4 below some threshold [8]. In practice, the density of the near-zero mode is non-zero, unless we introduce the determinant factor such as dethw. Even in this case, the overlap-dirac operator is eponentially local if the near-zero modes themselves are eponentially localized []. Golterman, Shamir, and Svetitsky argued that the magnitude of the overlap operator behaves as ( D ov (,y λρ( λep y + O( ep( λ c y, (3. l l ( λ where λ denotes a near-maimum eigenvalue of low-lying localized modes (somewhat ambiguous, l l ( λ a localization length at λ. The parameter λ c stands for the mobility edge, which separates localized and etended modes. The first term is derived from the dependence of ρ(λ and l l (λ on λ, while the second term is a conjecture motivated by numerical eperiences []. In most cases, it is known that the second term dominates the first. Mobility edge is determined from the eigenvectors as follows. We first define ρ i ( and f i (r as ρ i ( = φ i (φ i(, ρ i ( = ma{ρ i (}, (3. f i (r = { ρ i ( r = }, (3.3 where ρ i ( is an average of ρ i ( over the lattice points which have the same distance r from. The eigenvalues λ w,i are binned with a certain bin size, and f i (r is averaged over the modes within a bin. The localization length at each bin, l l,i, is then obtained at large r by fitting to ( f i (r = ep r l l,i. (3.4 Mobility edge, λ c, is defined by λ w,i at which l l,i diverges. Figure 4 shows an eample of f i (r obtained on a single configuration. We can clearly see that the decay rate becomes smaller for larger eigenvalues. The determination of λ c is performed with a fied value of m =.6. In addition to the ensembles used in the study of spectral density, the determination is performed on three coarse lattices, which are generated with the Plq action with β =5.7 and 5.4 and the RG action with β=.43, to see the dependence on lattice spacing. l l,i etracted from the fit using eq. (3.4 is plotted as a PoS(LAT66.. e-6 Dynamical, β=.3, m ud = f i (r e-8 e- e- e-4 e r Figure 4: Eample of f i (r. Different symbols represent different bin in λ w, as denoted in the plots. Data for the dynamical configuration at β=.3, m sea =.5. 4

5 .5 Plq, β=5.45, µ=.4 Plq, β=5.7, µ=. Plq, β=5.83, µ=..5 RG, β=.7, µ=.4 RG, β=.37, µ=. RG, β=.43, µ=. /l l.5 /l l /l l /l l λ w λ w Plq, β=5.83, µ=. Plq, β=5.7, µ=. Plq, β=5.4, µ=. RG, β=.43, µ=. Plq, β=5.83, µ=. /l l /l l λ w Rg, β=.43, µ=. RG, β=., µ= λ w RG, β=.37, µ=. Dynamical, β=.3, m ud =.5 Dynamical, β=.3, m ud =.5 PoS(LAT λ w λ w Figure 5: Inverse localization length /l l as a function of λ w. function of λ w,i in Fig. 5 for various ensembles. In the two plots at the top of Fig. 5, the results at the same lattice spacing, a=.5 fm, but with different µ are plotted for the Plq (left and the RG (right. λ c is found to be.4-.7 for the plaquette action and for the RG action in the lattice unit. Dependence of λ c on µ turns out to be weak in the range of. < µ <.4. From the two plots in the middle showing the results with µ = for two or three different β, we find that λ c in the lattice unit decreases as β decreases. The data at β = 5.4 for the plaquette action (circle in the left-middle panel shows /l l (. Ref. [] suggests to use /l l ( = as the definition of the Aoki phase. In the left-bottom, the results with the plaquette and RG actions are compared at the same lattice spacing. Clearly the RG action gives lager λ c than that of the Plq action. Finally, the right-bottom figure shows a comparison of the results from quenched and dynamical runs with two different sea quark masses. Here we do not see any significant difference between the quenched 5

6 e+ e- Dynamical, β=.3, m ud =.5 RG, β=.43, µ=. RG, β=.37, µ=. RG, β=., µ=. h(r e-4 e-6 e-8 e r Figure 6: r dependence of h(r for four different ensembles. and dynamical runs nor clear dependence on sea quark mass within dynamical runs. λ c turns out to be 5 6 MeV in our dynamical lattices. We also measure /l ov from the r dependence of h(r defined by h(r = ma { } D ov (,yδ(y r = µ µ. (3.5 µ h(r is measured on the same ensembles, and some of the results obtained with the RG action are plotted in Fig. 6, where numerical data are in the lattice unit. We see that all results ecept for the one with β=. coincide with each other. β=. corresponds to a. fm, while others are a=.-.5 fm. By fitting this to the same form as eq. (3.4, we obtain /l ov. For our dynamical lattice at β=.3, /l ov is estimated to be about 8 MeV. In Fig. 7, λ c and /l ov obtained with the RG action are compared. While both /l ov and λ c show the similar dependence on lattice spacing, the difference is sizable. Within the range of the lattice spacing we have studied it turns out that /l ov > λ c. Although λ c 5-6 MeV in our dynamical lattice is not much larger than Λ QCD, /l ov 8 MeV is probably acceptable for simulations of QCD. In Ref. [9], /l ov has been studied at several different lattice spacings in the range of a.3. fm within the quenched approimation. They reported acceptably large values for /l ov at all lattices, which appears to be consistent with our observation. PoS(LAT66 /l ov, λ c [lattice unit] /l ov, µ=. /l ov, µ=. /l ov, Dynamical RG, m =.6 λ c, µ=. λ c, µ=. λ c, Dynamical.. a Figure 7: a dependence of /l ov and λ c. 6

7 4. Summary In this work, near-zero mode density for various gauge actions are eamined from the viewpoint of cost reduction in dynamical overlap simulation, and the RG Iwasaki action with etra fermions and ghosts ehibits smallest near-zero mode density among the actions studied. We also determine the mobility edge and the localization range of the overlap-dirac operator. While λ c and /l ov seem to agree with each other qualitatively, the difference is sizable. /l ov tells us the locality of an given overlap-dirac operator and is presumably all we need to know from the practical point of view, but the precise relationship between /l ov and λ c should be understood. Acknowledgment Numerical simulations are performed on Hitachi SR and IBM System Blue Gene Solution at High Energy Accelerator Research Organization (KEK under a support of its Large Scale Simulation Program (No This work is supported in part by the Grant-in-Aid of the Ministry of Education (Nos. 3354, 3353, 5545, 67447, 67456, 73466, 7747, 834, 83475, References [] H. Fukaya et al. [JLQCD collaboration], PoS LAT6, 5 (6; S. Hashimoto et al. [JLQCD collaboration], PoS LAT6, 5 (6; T. Kaneko et al. [JLQCD collaboration], PoS LAT6, 54 (6; H. Matsufuru et. al., [JLQCD collaboration], PoS LAT6, 3 (6. [] M. Golterman and Y. Shamir, Phys. Rev. D 68, 745 (3 [arxiv:hep-lat/36]; M. Golterman, Y. Shamir and B. Svetitsky, Phys. Rev. D 7, 75 (5 [arxiv:hep-lat/47]; Phys. Rev. D 7, 345 (5 [arxiv:hep-lat/5337]. [3] M. Luscher, Nucl. Phys. B 549, 95 (999 [arxiv:hep-lat/983]. [4] H. Fukaya, S. Hashimoto, T. Hirohashi, K. Ogawa and T. Onogi, Phys. Rev. D 73, 453 (6 [arxiv:hep-lat/56]. [5] W. Bietenholz, K. Jansen, K. I. Nagai, S. Necco, L. Scorzato and S. Shcheredin, JHEP 63, 7 (6 [arxiv:hep-lat/56]. [6] H. Neuberger, Phys. Rev. D 6, 855 ( [arxiv:hep-lat/994]. [7] T. Izubuchi and C. Dawson [RBC Collaboration], Nucl. Phys. Proc. Suppl. 6, 748 (; M. Luscher, private communication; H. Fukaya, arxiv:hep-lat/638; P. M. Vranas, Phys. Rev. D 74, 345 (6 [arxiv:hep-lat/664]; in these proceedings; H. Fukaya, S. Hashimoto, K. I. Ishikawa, T. Kaneko, H. Matsufuru, T. Onogi and [JLQCD Collaboration], arxiv:hep-lat/67. [8] P. Hernandez, K. Jansen and M. Lüscher, Nucl. Phys. B 55, 363 (999 [arxiv:hep-lat/988]. [9] T. Draper et al., PoS LAT5, (6 [arxiv:hep-lat/575]; T. Draper et al., arxiv:hep-lat/6934. PoS(LAT66 7