Chiral Ward identities for Dirac eigenmodes with staggered fermions
Hwancheol Jeong, Chulwoo Jung, Sunghee Kim, Weonjong Lee, Jeonghwan Pak
CChiral Ward identities for Dirac eigenmodes withstaggered fermions
Hwancheol Jeong ∗ , Sunghee Kim, Weonjong Lee, and Jeonghwan Pak Lattice Gauge Theory Research Center, CTP, and FPRD,Department of Physics and Astronomy,Seoul National University, Seoul 08826, South KoreaE-mail: [email protected] , [email protected] Chulwoo Jung
Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USAE-mail: [email protected]
SWME Collaboration
We study chiral properties of eigenvalue spectrum for staggered quarks. We present a new methodto identify would-be zero modes and nonzero modes using their symmetry and chiral properties.Here, we review the traditional method with HYP improved staggered quarks, and extend it to acompletely new method which uses the chiral Ward identities and leakage patterns to achieve thegoal. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] J a n hiral Ward identities for staggered fermions Hwancheol Jeongparameter valuegluon action tree level Symanzik [6–8]tadpole improvement yes β a N f Table 1: Input parameters for the numerical study. For more details, refer to Ref. [12].
1. Introduction
Here, we present recent progress in understanding chiral properties of staggered Dirac eigen-modes based on our previous works in Refs. [1, 2].
2. Eigenvalues of Dirac operators with staggered fermions
Let us consider Dirac operator D s for staggered fermions. Since D s is anti-Hermitian, eigen-values of D s are purely imaginary or zero: D s | f λ (cid:105) = i λ | f λ (cid:105) , (2.1)where λ is real, and | f λ (cid:105) is an eigenvector with its eigenvalue i λ . D s also anti-commutes with the operator Γ ε = [ γ ⊗ ξ ] which is the generator for U ( ) A symmetry. Here, we adopt the same notation as in Ref. [1]. Since Γ ε anti-commutes with D s , onecan show that an eigenstate | f λ (cid:105) with λ (cid:54) = | f − λ (cid:105) with eigenvalue − i λ [1].The partner eigenvector | f − λ (cid:105) can be obtained by applying Γ ε to the eigenvector | f + λ (cid:105) with phasedifference: Γ ε | f + λ (cid:105) = e + i θ | f − λ (cid:105) , Γ ε | f − λ (cid:105) = e − i θ | f + λ (cid:105) . (2.2)Here, the phase θ is real, and also turns out to be arbitrary [1].In practice, we do not calculate eigenvalues of D s directly. Instead, we use a Hermitian andpositive semi-definite operator D † s D s , which satisfies D † s D s | g λ (cid:105) = λ | g λ (cid:105) . (2.3)The corresponding eigenvectors | f ± λ (cid:105) are obtained by decomposing the eigenvector | g λ (cid:105) usingthe projection operators as in Ref. [1]. Since D † s D s is Hermitian, one can make use of Lanczosalgorithm [3] to calculate its eigenvalues and eigenvectors. Here, we use the implicitly restartedLanczos [4] with acceleration by Chebyshev polynomial [5].All the numerical calculations are performed on the gauge ensemble described in Table 1.We use HYP staggered fermions as valence quarks which reduce the taste-breaking for staggeredfermions, and thus show improved chiral behaviors [12–15].1 hiral Ward identities for staggered fermions Hwancheol Jeong -0.050.000.05 4 8 12 16 20 24 28 32 λ i i (a) Q t = -0.050.000.05 5 10 15 20 25 30 35 λ i i (b) Q t = − Figure 1: Eigenvalue spectra of staggered Dirac operator on gauge configurations with Q t = Q t = −
1. Here, i represents an index of eigenvalue λ i .Meanwhile, the index theorem [16] states that Q t = n − − n + , Q t = π (cid:90) E d x ε αβ µν Tr ( F αβ F µν ) (2.4)where Q t is the topological charge [17], the subscript E represents the Euclidean space, and n + ( n − )is the number of zero modes with right-handed (left-handed) helicity. In the continuum, Eq. (2.4)indeed comes from the axial Ward identity [2]. For staggered fermions, a similar relation holdsbut four-fold degeneracy which comes from the approximate SU ( ) taste symmetry should becounted [1]: Q t = ( n s − − n s + ) , (2.5)where n s ± represent the number of zero modes with right-handed ( + ) and left-handed ( − ) helicitiesfor staggered quarks. Here, n s ± must be multiples of four due to the taste symmetry.
3. Eigenvalue spectrum
In Fig. 1, we present tens of low-lying eigenvalues of the Dirac operator with HYP staggeredquarks on gauge configurations with topological charges Q t = , −
1. Here, we measure Q t usingthe Q ( ) operator defined in Ref. [18, 19] after 10 ∼
30 iterations of the APE smearing with α = .
45 [20–22]. In the plot, eigenvalues are sorted in ascending order of their absolute values | λ i | . Here, we assign the index (2 n ) of the eigenvalue such that it satisfies λ n = − λ n − . Even thewould-be zero modes have tiny but nonzero values of λ i at finite lattice spacing a (cid:54) =
0. Hence, eacheigenvalue has its parity partner with opposite sign even though it belongs to the would-be zeromodes. The solid green line for Q t = − ( ) taste symmetry is exactly conserved in the continuum. In Fig. 1b,one can see the would-be zero modes appear with four-fold degeneracy. For nonzero modes, oneeigenvalue must have four-fold degeneracy due to the SU ( ) taste symmetry in the continuum,and its U ( ) A parity partner should have the same four-fold degeneracy. Hence, for each nonzero2 hiral Ward identities for staggered fermions Hwancheol Jeong (a) Q t = (b) Q t = − (c) Q t = − (d) Q t = − Figure 2: Γ ( λ i ) for various topological charges. Here, i represents an index of eigenvalue λ i .eigenvalue, it has a set of eight-fold degeneracy due to the exact U ( ) A symmetry on the latticeand the SU ( ) taste symmetry in the continuum. In Fig. 1, one can see the nonzero modes show upwith eight-fold degeneracy.
4. Chirality for staggered fermions
Let us consider three chirality operators: Γ ε , Γ , and Ξ defined as Γ ε ≡ [ γ ⊗ ξ ] , Γ ≡ [ γ ⊗ ] , Ξ ≡ [ ⊗ ξ ] . (4.1) Γ ε represents a chirality of the conserved U ( ) A symmetry for staggered fermions. A taste singletoperator Γ corresponds to the generator for the anomalous U ( ) anomA symmetry in the continuum.Similarly, Ξ represents the parity partner for the chirality operator Γ .The Γ ε , Γ , and Ξ operators satisfy the same relations as the continuum chirality operator γ as follows, ( Γ ) n + = Γ , ( Γ ) n = , (4.2)where Γ ∈ { Γ ε , Γ , Ξ } . Furthermore, they are related to each other by Γ ε = Γ Ξ ; Γ = Ξ Γ ε ; Ξ = Γ Γ ε . (4.3)These properties insure that they are the best choice to examine the chiral symmetry for staggeredfermions.Let us define the chirality as Γ ( α , β ) ≡ (cid:104) f α | [ γ ⊗ ] | f β (cid:105) , Γ ( λ i ) ≡ Γ ( λ i , λ i ) . (4.4)In Fig. 2, we measure the chirality Γ ( λ i ) for topological charges Q t = , − , − , −
3, respectively.Comparing with Fig. 1, the would-be zero modes has a non-trivial chirality around 0.8 in magni-tude, while nonzero modes have values of Γ ( λ i ) close to zero. Consequently, would-be zero modesare manifestly distinguishable from nonzero modes by the Γ chirality as shown in Ref. [12–14].The magnitudes of the chirality for would-be zero modes are somewhat smaller than one, the con-tinuum expectation value. It is because the Γ operator is not conserved at a (cid:54) = hiral Ward identities for staggered fermions Hwancheol Jeongparameter value | Γ ( λ , λ ) | | Ξ ( λ , λ ) | | Ξ ( λ , λ ) | | Γ ( λ , λ ) | | Γ ( λ , λ ) | | Γ ( λ , λ ) | | Ξ ( λ , λ ) | | Ξ ( λ , λ ) | | Ξ ( λ , λ ) | | Ξ ( λ , λ ) | | Γ ( λ , λ ) | | Γ ( λ , λ ) | Table 2: Numerical demonstration of chiral Ward identity (WI) in Eq. (5.8). Here, λ = − λ , λ = − λ , λ = − λ .
5. Chiral Ward identity
Rewriting Eqs. (2.2) by implementing Eq. (4.2) and Eq. (4.3), we obtain the following chiralWard identities for staggered fermions: Γ | f + λ (cid:105) = e + i θ Ξ | f − λ (cid:105) , Γ | f − λ (cid:105) = e − i θ Ξ | f + λ (cid:105) . (5.1)Let us define the chirality matrix elements sandwiched between the two eigenvectors as Γ ε ( α , β ) ≡ (cid:104) f α | Γ ε | f β (cid:105) = (cid:104) f α | [ γ ⊗ ξ ] | f β (cid:105) , (5.2) Γ ( α , β ) ≡ (cid:104) f α | Γ | f β (cid:105) = (cid:104) f α | [ γ ⊗ ] | f β (cid:105) , (5.3) Ξ ( α , β ) ≡ (cid:104) f α | Ξ | f β (cid:105) = (cid:104) f α | [ ⊗ ξ ] | f β (cid:105) . (5.4)Using the Ward identity of Eqs. (5.1), we rewrite the chirality matrix elements as follows, Γ ( α , + β ) = e + i θ β Ξ ( α , − β ) , Γ ( α , − β ) = e − i θ β Ξ ( α , + β ) , (5.5) Γ (+ α , β ) = e − i θ α Ξ ( − α , β ) , Γ ( − α , β ) = e + i θ α Ξ (+ α , β ) . (5.6)If we take the norm of them, then | Γ ( α , β ) | = | Ξ ( α , − β ) | = | Ξ ( − α , β ) | = | Γ ( − α , − β ) | . (5.7)In addition, the Hermiticity insures interchanging α and β , which provides the final form of theWard identities: | Γ ( α , β ) | = | Ξ ( α , − β ) | = | Ξ ( − α , β ) | = | Γ ( − α , − β ) | = | Γ ( β , α ) | = | Ξ ( β , − α ) | = | Ξ ( − β , α ) | = | Γ ( − β , − α ) | . (5.8)Table 2 shows how the chiral Ward identities of Eq. (5.8) works in our numerical study. Here, itconfirms that they are valid within our numerical precision.
6. Leakage of chirality
Here, we focus on off-diagonal elements ( α (cid:54) = β ) of chirality Γ ( α , β ) and Ξ ( α , β ) . We areinterested in how much of the chirality of an eigenmode leaks into other eigenmodes. Traditionally,the diagonal elements of chirality Γ ( λ i ) were measured and studied as in Ref. [12–14]. Here,4 hiral Ward identities for staggered fermions Hwancheol Jeong | Γ ( λ i , λ ) | i (a) Q t = − | Γ ( λ i , λ ) | i (b) Q t = − Figure 3: Leakage patterns of would-be zero modes for the Γ operator.we study on the off-diagonal elements of chirality Γ ( α , β ) , and Ξ ( α , β ) with α (cid:54) = β . In thecontinuum, the SU ( ) taste symmetry is respected. The net consequence of the conserved tastesymmetry is that each nonzero eigenvalue has eight-fold degeneracy, and these eight degenerateeigenmodes will mix with one another within the eight-fold degenerate members. In other words,if | α | (cid:54) = | β | , then Γ ( α , β ) = Ξ ( α , β ) = a =
0) thanks to the SU ( ) tastesymmetry. However, at finite lattice ( a (cid:54) = ( ) taste symmetry is not exact, but mostlyrespected near the continuum. Hence, the leakage from one set of the eight-fold degeneracy toother set of eight-fold degeneracy will be very small near the continuum ( a ≈ Γ for would-be zero modes. Here, we observe that for Γ there is only one non-trivialsignal at the would-be zero mode itself and there is almost no leakage to nearest zero and nonzeroeigenmodes.For nonzero modes, leakage of the Γ operator for an eigenvalue λ i is supposed to go intothe four-fold parity partners with eigenvalue − λ i in the continuum. Near the continuum ( a ≈ ( ) taste symmetry is almost respected and the leakage to eigenmodes outside the eight-fold degeneracy is also almost prohibited. This kind of leakage patterns for nonzero eigenmodesare presented in Fig. 4. In Fig. 4a, the leakage of the Γ chirality for the eigenmode | f λ (cid:105) goesinto the eigenmodes with eigenvalue − λ : | f λ j (cid:105) with j = , , ,
12, as the theory predicts. InFig. 4b, the leakage of the Γ chirality for the eigenmode | f λ (cid:105) goes into the eigenmodes witheigenvalue − λ : | f λ j (cid:105) with j = , , , ( ) taste symmetry breaking in that a small amount of leakage of the Γ chirality for theeigenmode | f λ (cid:105) goes into eigenmodes outside of the eight-fold degeneracy such as | f λ j (cid:105) with j = , , , , , | Γ ( λ i , λ ) | i (a) Leakage for λ with Q t = − | Γ ( λ i , λ ) | i (b) Leakage for λ with Q t = − Figure 4: Leakage patterns of Γ for nonzero eigenmodes.5 hiral Ward identities for staggered fermions Hwancheol Jeong
7. Conclusion
We have studied Γ and Ξ chirality for eigenmodes of staggered fermions. Thanks to the Wardidentities, Γ chirality is completely correlated with Ξ chirality. We demonstrate how the leakagepatterns of Γ chirality can be used to distinguish zero eigenmodes and nonzero eigenmodes. Acknowledgments
We would like to express our sincere gratitude to Eduardo Follana for his kind help. The re-search of W. Lee is supported by the Mid-Career Research Program (Grant No. NRF-2019R1A2C2085685)of the NRF grant funded by the Korean government (MOE). This work was supported by SeoulNational University Research Grant in 2019. W. Lee would like to acknowledge the support fromthe KISTI supercomputing center through the strategic support program for the supercomputingapplication research (No. KSC-2017-G2-0009). Computations were carried out on the DAVIDcluster at Seoul National University.
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