Chiral symmetry and taste symmetry from the eigenvalue spectrum of staggered Dirac operators
Hwancheol Jeong, Chulwoo Jung, Seungyeob Jwa, Jangho Kim, Jeehun Kim, Nam Soo Kim, Sunghee Kim, Sunkyu Lee, Weonjong Lee, Youngjo Lee, Jeonghwan Pak
HHow to fish out chiral symmetry and taste symmetry embedded in the eigenvaluespectrum of staggered Dirac operators
Hwancheol Jeong, Chulwoo Jung, Seungyeob Jwa, Jangho Kim, Jeehun Kim, Nam SooKim, Sunghee Kim, Sunkyu Lee, Weonjong Lee, ∗ Youngjo Lee, and Jeonghwan Pak (SWME Collaboration) Lattice Gauge Theory Research Center, FPRD, and CTP,Department of Physics and Astronomy, Seoul National University, Seoul 08826, South Korea Physics Department, Brookhaven National Laboratory, Upton, NY11973, USA Institut f¨ur Theoretische Physik, Goethe University Frankfurt am Main,Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany Department of Electrical and Computer Engineering and the Institute of New Media and CommunicationsSeoul National University, Seoul 08826, South Korea Department of Statistics, Seoul National University, Seoul 08826, South Korea (Dated: May 22, 2020)We investigate general properties of the eigenvalue spectrum for improved staggered quarks. Weintroduce a new chirality operator [ γ ⊗
1] and a new shift operator [1 ⊗ ξ ], which respect the samerecursion relation as the γ operator in the continuum. Then we show that matrix elements of thechirality operator sandwiched between two eigenstates of the staggered Dirac operator are relatedto those of the shift operator by the Ward identity of the conserved U (1) A symmetry of staggeredfermion actions. We perform the numerical study in quenched QCD using HYP staggered quarksto demonstrate the Ward identity numerically. We introduce a new concept of leakage patternswhich collectively represent the matrix elements of the chirality operator and the shift operatorsandwiched between two eigenstates of the staggered Dirac operator. The leakage pattern providesa new method to identify zero modes and non-zero modes in the Dirac eigenvalue spectrum. Thisnew method of the leakage pattern is as robust as the spectral flow method but requires muchless computing power. Analysis using the machine learning technique confirms that the leakagepattern is universal, since the staggered Dirac eigenmodes on normal gauge configurations respectit. In addition, the leakage pattern can be used to determine a ratio of renormalization factors as abyproduct. We conclude that it might be possible and realistic to measure the topological charge Q using the Atiya-Singer index theorem and the leakage pattern of the chirality operator in thestaggered fermion formalism. PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.AwKeywords: lattice QCD, Lanczos algorithm, chiral symmetry, staggered fermion, taste symmetry
I. INTRODUCTION
It is important to understand the low-lying eigenvaluespectrum of the Dirac operator, which exhibits the topo-logical Ward identity of the Atiya-Singer index theorem[1], the Banks-Casher relationship [2], and the universal-ity of the distribution of the near-zero modes for fixedtopological charge sectors [3, 4]. Study on the eigenvaluespectrum of the Dirac operator is, by nature, highly non-perturbative. Hence, numerical tools available in latticegauge theory provide a perfect playground to study ondiverse properties of the Dirac eigenvalue spectrum.In lattice QCD, there are a number of popular meth-ods to implement a discrete version of the continuumDirac operator on the lattice. Among them, we are in-terested in one particular class of lattice fermions thatare widely used in lattice QCD community: improvedstaggered quarks [5–7]. Here, we study the eigenvaluespectrum of staggered Dirac operators in quenched QCD ∗ E-mail: [email protected] to show that the small eigenvalues near zero modes ofthe staggered Dirac operators reproduce the continuumproperties very closely, which was originally noticed inRefs. [8–10]. To reach this conclusion of Refs. [8, 9],they performed a number of tests including (1) the Atiya-Singer index theorem that describes the chiral Ward iden-tity relating the zero modes to the topological charge;(2) the Banks-Casher relationship that relates the chi-ral condensate to the density of eigenvalues at the zeromode; (3) the universality of the small eigenvalue spec-trum in the ε -regime that is predicted from the randommatrix theory. In addition, in Ref. [11, 12], they usedthe spectral flow method of Adams [13] to identify thezero modes from the mixture with non-zero modes. Thespectral flow method is robust but highly expensive in acomputational sense.Here, we introduce a new advanced chirality opera-tor [ γ ⊗ γ .Using this chirality operator, we show that its matrix el-ements between eigenstates are related to those of theshift operator [1 ⊗ ξ ] through the Ward identity of theconserved U (1) A symmetry of staggered fermions. In ad-dition, we introduce a new concept of leakage pattern to a r X i v : . [ h e p - l a t ] M a y distinguish zero modes from non-zero modes. Using theleakage pattern of the chirality and shift operators, weshow that it is possible to measure the zero modes asreliably as the spectral flow method. Hence, it would bepossible to determine the topological charge Q using theleakage pattern with much smaller computational costthan the spectral flow methods. We also show that it ispossible to determine the ratio of renormalization con-stants, Z P × S /Z P × P using the leakage pattern.In Section II, we briefly review the continuum theoryof the eigenvalue spectrum and its relation to the quarkcondensate (cid:104) ¯ ψψ (cid:105) . We also review the Atiya-Singer in-dex theorem in brief. In Section III, we briefly reviewthe eigenvalue spectrum of staggered Dirac operatorsobtained using the Lanczos algorithm. In Section IV,we briefly review the conserved U (1) A symmetry in thestaggered fermion formalism and explain its role in theeigenvalue spectrum of staggered Dirac operators. Wealso present numerical examples to help readers to un-derstand basic concepts and notations. In Section V, wedefine the chirality operator [ γ ⊗
1] and the shift oper-ator [1 ⊗ ξ ]. We show that they respect the continuumrecursion relation of γ . Then we derive the chiral Wardidentity of the U (1) A symmetry to show that the matrixelements of the chirality operator are related to those ofthe shift operator through the Ward identity. Then, wediscuss the eigenvalue spectrum in the continuum limitand introduce a new notation of quartet indices. Then,we introduce the concept of leakage patterns for the chi-rality operator and the shift operator. We also presentthe numerical examples to demonstrate that the leak-age patterns are completely different between zero modesand non-zero modes. In Section VI, we review the ma-chine learning technique, and describe how to apply it tothe task of digging out the quartet structure of non-zeromodes efficiently using leakage patterns. In Section VII,we explain how to the leakage pattern of the zero modescan be used to determine ratio of the renormalizationfactors non-perturbatively. In Section VIII, we conclude.The appendices contain technical details on Lanczos al-gorithms and mathematical proofs, and show more plotsof leakage patterns for diverse topological charge valuesof Q .Preliminary results of this paper are published inRef. [14–16]. II. QUARK CONDENSATE IN THECONTINUUM
In continuum the quark condensate is given by (cid:10) ¯ ψψ (cid:11) = 1 N f (cid:88) f (cid:10) | ¯ ψ f ψ f | (cid:11) (1)= − V N f (cid:90) d x Tr (cid:18) D + m (cid:19) , (2) where D is the Dirac operator, m is the quark mass, x is the space-time coordinate, V is the volume, and N f is the number of flavors with the same mass m . Thetrace is a sum over spin and color. Let us think of theeigenvalues of the Dirac operator. D is anti-Hermitian,so its eigenvalues are purely imaginary or zero. D † = − D (3) Du λ ( x ) = iλu λ ( x ) (4)where λ is a real eigenvalue, and u λ ( x ) is the correspond-ing eigenvector.By spectral decomposition [4], S f ( x, y ) = (cid:104) ψ f ( x ) ¯ ψ f ( y ) (cid:105) = (cid:88) λ iλ + m u λ ( x ) u † λ ( y ) (5) (cid:104) ¯ ψψ (cid:105) = − V (cid:88) λ iλ + m (cid:90) d x Tr( u λ ( x ) u † λ ( x )) (6)= − V (cid:88) λ iλ + m . (7)where we adopt a normalization convention: (cid:104) u a | u b (cid:105) = (cid:90) d x u † a ( x ) u b ( x ) = δ ab . (8)Thanks to the chiral symmetry, γ D = − Dγ (9) Dγ | u λ (cid:105) = − iλγ | u λ (cid:105) (10)Hence, let us define u − λ ≡ γ u λ , and then Du − λ = − iλu − λ . Hence, if there exists u λ , then its parity partnereigenstate u − λ with negative eigenvalue − iλ must existaccordingly as a pair except for zero modes with λ = 0.Now let us separate the zero mode contribution fromthe spectral decomposition. (cid:104) ¯ ψψ (cid:105) = − V (cid:88) λ> mλ + m − n + + n − mV . (11)Here, n + ( n − ) is the number of right-handed (left-handed) zero modes per flavor. Let us define the sub-tracted quark condensate (cid:104) ¯ ψψ (cid:105) sub : (cid:104) ¯ ψψ (cid:105) sub = (cid:10) ¯ ψψ (cid:11) + n + + n − mV = − V (cid:88) λ> mλ + m . (12)= − V (cid:88) n mλ n + m with λ n > − (cid:90) + ∞−∞ dλ mλ + m ρ s ( λ ) , (14)where the spectral density ρ s ( λ ) is ρ s ( λ ) = 1 V (cid:88) n δ ( λ − λ n ) . (15)Here, ρ s is a spectral density on a single gauge configu-ration with volume V . Now let us average over a full en-semble of gauge field configurations and take the limit ofinfinite volume ( V → ∞ ). Then, in that limit, the spec-tral density ρ ( λ ) = (cid:104) ρ s ( λ ) (cid:105) has a well defined (smoothand continuous) value as λ →
0. Then, we can define thechiral condensate asΣ = −(cid:104) | ¯ ψψ | (cid:105) sub ( m = 0)= lim m → (cid:90) + ∞−∞ dλ mλ + m ρ ( λ ) = πρ (0) , (16)which is the Banks-Casher relation. The subtractedquark condensate (cid:104) ¯ ψψ (cid:105) sub is expected to behave well inthe chiral limit, even though the contribution from thezero modes is divergent as a simple pole in the chirallimit. Hence, in the numerical study on the lattice, itis important to identify the would-be zero modes whichcorrespond to the zero modes in the continuum limit, andremove them in the calculation of quark condensate.Before proceeding, let us briefly go through the indextheorem. In the continuum, the axial Ward identity is ∂ µ A µ ( x ) = 2 mP ( x ) − N f q ( x ) (17)in the Euclidean space [17]. Here A µ ≡ ¯ ψγ µ γ ψ is theaxial vector current in the flavor singlet representation, P ≡ ¯ ψγ ψ is the corresponding pseudo-scalar operator,and q ≡ π Tr[ F µν (cid:101) F µν ] is the topological charge density(= winding number density). Now the topological charge Q is Q ≡ (cid:90) d x (cid:104) q ( x ) (cid:105) (18)= − N f (cid:90) d x (cid:104) ∂ µ A µ ( x ) − mP ( x ) (cid:105) (19)= mN f (cid:90) d x (cid:10) ¯ ψγ ψ (cid:11) (20)Using the spectral decomposition, we can rewrite Q asfollows, Q = − m (cid:88) λ iλ + m (cid:90) d x (cid:104) u † λ ( x ) γ u λ ( x ) (cid:105) . (21)By the way, γ u λ ( x ) = u − λ ( x ), and so for λ (cid:54) = 0, (cid:90) d x (cid:104) u † λ ( x ) γ u λ ( x ) (cid:105) = (cid:104) u λ | u − λ (cid:105) = 0 . (22)Hence, only zero modes with λ = 0 contribute to Q . Forthe zero modes, it is convenient to choose the helicityeigenstates as the basis vectors so that (cid:104) u L | γ | u L (cid:105) = − (cid:104) u R | γ | u R (cid:105) = +1, where the superscripts L, R repre-sent left-handed and right-handed helicity, respectively.Then, it is straight-forward to derive the index theorem[1]: Q = n − − n + , (23)where n + ( n − ) is the number of the right-handed (left-handed) zero modes. III. SPECTRAL DECOMPOSITION WITHSTAGGERED FERMIONS
In the staggered fermion formalism, there are a num-ber of improved versions such as HYP-smeared stag-gered fermions [5], asqtad imporved staggered fermions[18], and HISQ staggered fermions [7]. Here, we callall of them “staggered fermions” collectively. Staggeredfermions have four tastes per flavor by construction [19].Hence, quark condensate for staggered fermions is definedas (cid:104) ¯ χχ (cid:105) = − V N t (cid:28) Tr 1 D s + m (cid:29) U , (24)where χ represents staggered quark fields, D s is the stag-gered Dirac operator for a single valence flavor, V is thelattice volume, and N t is the number of tastes. We mea-sure the quark condensate using the stochastic method.( D s + m ) x,y χ ( y ) = ξ ( x ) (25) χ ( x ) = (cid:20) D s + m (cid:21) x,y ξ ( y ) (26)Tr 1 D s + m = lim N ξ →∞ N ξ (cid:88) ξ (cid:88) y ξ † ( y ) χ ( y ) , (27)where x, y are representative indices which represent thespace-time coordinate and taste, color indices collec-tively. Here, ξ ( x ) represents either Gaussian randomnumbers or U (1) noise random numbers which satisfya simple identity:lim N ξ →∞ N ξ (cid:88) ξ ξ † ( x ) ξ ( y ) = δ xy , where N ξ is the number of the random vector samples.Staggered fermions have a taste symmetry of SU (4) L ⊗ SU (4) R ⊗ U (1) V in the continuum limit at a = 0 [20].However, this symmetry breaks down to a subgroup of U (1) V ⊗ U (1) A on the lattice with a (cid:54) = 0 [19, 20]. Theremaining axial symmetry of U (1) A plays an importantrole in protecting the quark mass from receiving an ad-ditive renormalization. In addition, it does not have anyaxial anomaly.The Dirac operator ( D s ) of staggered fermions areanti-Hermitian: D † s = − D s . Hence, its eigenvalues arepurely imaginary: D s | f sλ (cid:105) = iλ | f sλ (cid:105) , (28)where λ is real. Here, the subscript s and superscript s represent staggered quarks.In practice, when we obtain eigenvalues of D s nu-merically, we use the following relationship instead ofEq. (28): D † s D s | g sλ (cid:105) = λ | g sλ (cid:105) . (29)where the | g sλ (cid:105) state is a mixture of two eigenvectors: | f s + λ (cid:105) and | f s − λ (cid:105) . In other words, | g sλ (cid:105) = c | f s + λ (cid:105) + c | f s − λ (cid:105) (30)where c i are complex numbers and they satisfy the nor-malization condition: | c | + | c | = 1 (31)The numerical algorithm is a variation of Lanczos algo-rithm adapted for lattice QCD [21]. Details on the nu-merical algorithms as well as comprehensive referencesare given in Appendix A.Why do we obtain λ instead of iλ ? The first rea-son is to use the even-odd preconditioning [22], whichmakes Lanczos run on only even or odd sites on the lat-tice. This leads to two benefits: one is that there is asubstantial gain in the speed of the code and the other isthat the code uses only half of the memory that is oth-erwise used. Details on the even-odd preconditioning aredescribed in Appendix B. The second reason is that itallows us to implement the polynomial acceleration al-gorithms [23] into Lanczos easier, since the eigenvaluesof D † s D s are positive definite, and have a lower boundof λ >
0. Here, note that staggered fermions can havewould-be zero modes whose eigenvalues are small andpositive ( λ >
0) in rough gauge configurations. In otherwords, there is no exact zero modes ( λ = 0) with stag-gered fermions on rough gauge configurations [24]. De-tails on our implementation of polynomial acceleration isdescribed in Appendix A.Hence, the Lanczos algorithm solves the eigenvalueequation Eq. (29) and obtain the solution | g sλ (cid:105) as well asthe corresponding eigenvalue λ . Then we use the projec-tion method to obtain | f s + λ (cid:105) and | f s − λ (cid:105) as follows. First,let us define projection operators as P + = ( D s + iλ ) (32) P − = ( D s − iλ ) (33)where P + is the projection operator to select onlythe | f s + λ (cid:105) component and remove the | f s − λ (cid:105) component.Then, we can use the projection operator P + to selectonly the | f s + λ (cid:105) component of | g sλ (cid:105) as follows, | χ + (cid:105) = P + | g sλ (cid:105) (34) | χ − (cid:105) = P − | g sλ (cid:105) (35)Then, we can find the orthonormal eigenvectors as fol-lows, | f s + λ (cid:105) = | χ + (cid:105) (cid:112) (cid:104) χ + | χ + (cid:105) (36) | f s − λ (cid:105) = | χ − (cid:105) (cid:112) (cid:104) χ − | χ − (cid:105) . (37) IV. CHIRAL SYMMETRY OF STAGGEREDFERMIONS
The two vectors | f s ± λ (cid:105) are related to each other throughthe chiral Ward identity in staggered fermion formalism.Let us address this issue of chiral symmetry of staggeredfermions and its consequences. Let us begin with nota-tions and definitions for later usage. We define staggeredbilinear operators as O S × T ( x ) ≡ ¯ χ ( x A )[ γ S ⊗ ξ T ] AB χ ( x B )= ¯ χ a ( x A )( γ S ⊗ ξ T ) AB U ( x A , x B ) ab χ b ( x B )(38)where χ b are staggered quark fields, and a, b are colorindices. Here, the coordinate is x A = 2 x + A and A, B are the hypercubic vectors with A µ , B µ ∈ { , } .( γ S ⊗ ξ T ) AB = 14 Tr( γ † A γ S γ B γ † T ) (39)where γ S represents the Dirac spin matrix, and ξ T rep-resents the 4 × U ( x A , x B ) ≡ P SU (3) (cid:20) (cid:88) p ∈C V ( x A , x p ) V ( x p , x p ) · · · V ( x p n , x B ) (cid:21) (40)where P SU (3) represents the SU (3) projection, and C rep-resents a complete set of the shortest paths from x A to x B . V ( x, y ) represents the HYP-smeared fat link [5, 6] forHYP staggered fermions, the Fat7 fat link [6, 25–27] forasqtad or HISQ staggered fermions, and the thin gaugelink for unimproved staggered fermions.The conserved U (1) A axial symmetry transformationis Γ (cid:15) ( A, B, a, b ) ≡ [ γ ⊗ ξ ] AB ; ab = ( γ ⊗ ξ ) AB · δ ab = (cid:15) ( A ) · δ AB · δ ab (41)where Γ (cid:15) is often called “distance parity”, and (cid:15) ( A ) ≡ ( − S A (42) S A ≡ (cid:88) µ =1 A µ (43)Under the U (1) A transformation, the staggered Dirac op-erator transforms as follows,Γ (cid:15) D s Γ (cid:15) = D † s = − D s (44)Γ (cid:15) D s = − D s Γ (cid:15) (45)Therefore, D s | f s + λ (cid:105) = + iλ | f s + λ (cid:105) D s Γ (cid:15) | f s + λ (cid:105) = − iλ Γ (cid:15) | f s + λ (cid:105) (46)Hence, f s − λ can be obtained from f s + λ through Γ (cid:15) trans-formation as follows.Γ (cid:15) | f s + λ (cid:105) = e + iθ | f s − λ (cid:105) Γ (cid:15) | f s − λ (cid:105) = e − iθ | f s + λ (cid:105) . (47)In general, there is no constraint for the real phase θ so that we expect that its probability distribution mustbe random. In practice, however, we make use of theeven-odd preconditioning, by which we obtain the oddsite fermion fields ( | g o (cid:105) ) from the even site fermion fields( | g e (cid:105) ) with the relation | g o (cid:105) = η D oe | g e (cid:105) where D oe isa portion of D s which connects even site fields to oddsite fields, and η is a random complex number. Hence,the distribution of θ depends on our choice of η . In ournumerical study, we set η to η = 1. Then, θ is given by θ = π + 2 β , β = arctan( λ ) . (48)Details on the even-odd preconditioning and the deriva-tion of Eq. (48) are explained in Appendix B.We expect that if there exists an eigenvector of | f s + λ (cid:105) ,there must be a corresponding parity partner of | f s − λ (cid:105) due to the exact chiral symmetry Γ (cid:15) . In other words,this Ward identity of Eq. (47) comes directly from theconserved U (1) A axial symmetry. A. Numerical Examples
TABLE I. Input parameters for numerical study in quenchedQCD. For more details, refer to Ref. [9].parameters valuesgluon action tree level Symanzik [28–30]tadpole improvement yes β a . /a . N f N f = 0 (quenched QCD) Now let us show numerical examples to demonstratehow the above theory works in quenched QCD. In TableI, details on gauge configurations are presented.We measure the topological charge Q using gauge links.We use the Q (5Li) operator defined in Ref. [34, 35] after10 ∼
30 iterations of the APE smearing with α = 0 . Q = 0 in Fig. 1. Since Q = 0, we do not expect to findany zero modes for this gauge configuration. In Fig. 1 (a),we show eigenvalues of λ for eigenvectors | g sλ (cid:105) defined in λ i i (a) λ i -0.06-0.030.000.030.06 1 5 9 13 17 21 25 29 λ i i (b) λ i FIG. 1. Eigenvalue spectrum of staggered Dirac operator ona Q = 0 gauge configuration. Eq. (29). Here, we observe eight-fold degeneracy for non-zero eigenmodes due to the conserved U (1) A axial sym-metry. Here, λ = − λ and, in general, λ n = − λ n − for n > n ∈ Z . In other words, λ n is the paritypartner of λ n − . For each λ i , there exists four-fold de-generacy due to approximate SU (4) taste symmetry. Foreach of these four-fold degenerate eigenvalues (for exam-ple λ , λ , λ , λ in Fig. 1 (a)), there exists a parity part-ner eigenvalue due to the U (1) A symmetry: λ = − λ , λ = − λ , λ = − λ , and λ = − λ (refer to Fig. 1 (b)).Let us turn to the Q = − Q = − Q = −
1. Asone can see in Fig. 2 (a) and 2 (b), we find four-fold de-generate would-be zero modes: λ , λ , λ , λ . Thanks tothe U (1) A chiral Ward identity in Eq. (47), we find that λ = − λ and λ = − λ . As in the case of Q = 0, we findthat the non-zero eigenmodes are eight-fold degenerate.This pattern of four-fold degeneracy for would-be zeromodes and eight-fold degeneracy for non-zero modes isalso observed in the case of Q = − Q = −
3, whichare presented in Appendix C.At this point, you might have already concluded thatwe can distinguish would-be zero modes of staggeredquarks from non-zero modes by counting the degeneracyof the eigenvalues [8, 9, 39]. This is true but has somepossibility to lead to a wrong answer in practice. Thereason is that, on large lattices, the eigenvalues are sodense that it is not easy to distinguish 4-fold and 8-fold λ i i (a) λ i -0.06-0.030.000.030.06 1 5 9 13 17 21 25 λ i i (b) λ i FIG. 2. The same as Fig. 1 except for Q = − π π
0 0.05 0.1 0.15 0.2 θ λθ = π + 2 arctan( λ ) FIG. 3. The phase θ as a function of λ . Red circle symbolsrepresent numerical results for θ . The blue line represents theprediction from the theory. Here, we use a gauge configurationwith Q = − degeneracies in our eyes. Hence, we need a significantlymore robust method to identify would-be zero modes andnon-zero modes in staggered fermion formalism. This isthe main subject of the next section: Sec. V.Using the chiral Ward identity of Eq. (47), we can mea-sure the phase θ numerically. In Fig. 3, we show numer-ical results (red circle symbols) for θ . Here, the blue linerepresents the prediction from the theory in Eq. (48).We find that results are consistent with the theoreticalprediction within numerical precision. V. CHIRALITY MEASUREMENT
In order to simplify the notation, let us introduce thefollowing convention for eigenvalue indices. D s | f j (cid:105) = iλ j | f j (cid:105) (49)where | f j (cid:105) = | f sλ j (cid:105) which is defined in Eq. (28). We definethe chirality operator as follows.Γ ( λ i , λ j ) ≡ (cid:104) f i | [ γ ⊗ | f j (cid:105)≡ (cid:90) d x [ f sλ i ( x A )] † [ γ ⊗ x ; AB f sλ j ( x B ) (50)where x A and [ γ ⊗
1] are defined in Eqs. (38)-(40), and λ i and λ j represent eigenvalues of D s . For notationalsimplification, let us define(Γ ) ij ≡ Γ ( λ i , λ j ) (51) | Γ | ij ≡ | Γ ( λ i , λ j ) | (52)The chirality operator [ γ ⊗
1] satisfies the same relation-ships as the continuum chirality operator γ .[ γ ⊗ n +1 = [ γ ⊗ , (53)[ γ ⊗ n = [1 ⊗ , (54)[ 12 (1 ± γ ) ⊗ n = [ 12 (1 ± γ ) ⊗ , (55)[ 12 (1 + γ ) ⊗ − γ ) ⊗
1] = 0 , (56)where n ≥ n ∈ Z . A rigourous proof of Eqs. (53)-(56) is given in Appendix D.Our definition of the chirality operator [ γ ⊗
1] is dif-ferent from that conventionally used in Refs. [8, 13, 24].The old chirality operator used in Refs. [8, 13, 24] doesnot satisfy the recursion relation of Eqs. (53)-(56). Inaddition, it does not satisfy the chiral Ward identity ofEqs. (62)-(64). This difference is addressed in AppendixD. The bottom line is that the conventional chiralityoperator does not satisfy the recursion relationships inEqs. (53)-(56), even though it is classified according tothe true irreducible representation (irrep) of the latticerotational symmetry group [40–42].For our further discussion, we need to define anotheroperator [1 ⊗ ξ ], which we call “(maximal) shift operator”as follows,Ξ ( λ i , λ j ) ≡ (cid:104) f i | [1 ⊗ ξ ] | f j (cid:105)≡ (cid:90) d x [ f sλ i ( x A )] † [1 ⊗ ξ ] x ; AB f sλ j ( x B ) (57)For notational convenience, let us define(Ξ ) ij ≡ Ξ ( λ i , λ j ) (58) | Ξ | ij ≡ | Ξ ( λ i , λ j ) | (59)This shift operator satisfies the following recursion rela-tions: [1 ⊗ ξ ] n +1 = [1 ⊗ ξ ] , (60)[1 ⊗ ξ ] n = [1 ⊗ , (61)where n ≥ n ∈ Z . The conserved U (1) A symmetrytransformation can be expressed in terms of the chiralityoperator and the shift operator as follows,Γ (cid:15) ≡ [ γ ⊗ ξ ]= [ γ ⊗ ⊗ ξ ]= [1 ⊗ ξ ][ γ ⊗ . (62)In addition, the chirality and shift operators satisfy thefollowing relations:Γ (cid:15) [ γ ⊗
1] = [ γ ⊗ (cid:15) = [1 ⊗ ξ ] (63)Γ (cid:15) [1 ⊗ ξ ] = [1 ⊗ ξ ]Γ (cid:15) = [ γ ⊗
1] (64)Therefore, we can obtain the following Ward identities: e + iθ [ γ ⊗ | f − i (cid:105) = [1 ⊗ ξ ] | f + i (cid:105) e − iθ [ γ ⊗ | f + i (cid:105) = [1 ⊗ ξ ] | f − i (cid:105) (65)where | f ± i (cid:105) ≡ | f s ± λ i (cid:105) (66)Hence, we can define the spectral decomposition as[ γ ⊗ | f j (cid:105) = (cid:88) i (Γ ) ij | f i (cid:105) (67)(Γ ) ij = (cid:104) f i | [ γ ⊗ | f j (cid:105) = Γ ( λ i , λ j ) (68)Similarly, [1 ⊗ ξ ] | f j (cid:105) = (cid:88) i (Ξ ) ij | f i (cid:105) (69) (Ξ ) ij = (cid:104) f i | [1 ⊗ ξ ] | f j (cid:105) = Ξ ( λ i , λ j ) (70)Thanks to the Ward identity of Eq. (65), we obtain e − iθ Γ ( λ i , + λ j ) = Ξ ( λ i , − λ j ) e − iθ (Γ ) i + j = (Ξ ) i − j | Γ | i + j = | Ξ | i − j . (71)Similarly, e + iθ Γ ( λ i , − λ j ) = Ξ ( λ i , + λ j ) e + iθ (Γ ) i − j = (Ξ ) i + j | Γ | i − j = | Ξ | i + j . (72)Let us apply Γ (cid:15) on both sides of Eq. (67), and then weobtain [1 ⊗ ξ ] | f j (cid:105) = (cid:88) (cid:96) (Γ ) (cid:96)j e iθ (cid:96) | f − (cid:96) (cid:105) (73)= (cid:88) i (Ξ ) ij | f i (cid:105) . (74)Hence, we obtain another Ward identity: | Γ | − ij = | Ξ | + ij (75)Similarly, we can obtain the Ward identity: | Γ | − i − j = | Ξ | + i − j (76) | Γ | + ij = | Ξ | − ij (77)We can summarize all the results of Eqs. (71)-(77) intothe following form: | Γ | ij = | Ξ | − ij = | Ξ | i − j = | Γ | − i − j , (78) ⇔ | Γ ( λ i , λ j ) | = | Ξ ( − λ i , λ j ) | = | Ξ ( λ i , − λ j ) | = | Γ ( − λ i , − λ j ) | (79)In addition, the Hermiticity insures interchanging λ i and λ j . This gives the final form of the chiral Ward identities. | Γ | ij = | Ξ | − ij = | Ξ | i − j = | Γ | − i − j = | Γ | ji = | Ξ | − ji = | Ξ | j − i = | Γ | − j − i (80) ⇔ | Γ ( λ i , λ j ) | = | Ξ ( − λ i , λ j ) | = | Ξ ( λ i , − λ j ) | = | Γ ( − λ i , − λ j ) | = | Γ ( λ j , λ i ) | = | Ξ ( − λ j , λ i ) | = | Ξ ( λ j , − λ i ) | = | Γ ( − λ j , − λ i ) | (81)The ( | Γ | ij ) represents the leakage probability of thechirality operator if i (cid:54) = j or ( λ i (cid:54) = λ j ). We call | Γ | ij theleakage parameter for the chirality operator. Similarly,the ( | Ξ | ij ) represents the leakage probability of the shiftoperator if i (cid:54) = j . We call | Ξ | ij the leakage parameter for the shift operator. By monitoring the leakage pattern, wecan distinguish zero modes and non-zero modes, which isthe main subject of the next subsections. A. Eigenvalue spectrum of D s in the continuum Here, we consider staggered quark actions in the con-tinuum at a = 0. Let us define a general form of the shiftoperator which corresponds to a generator of the SU (4)taste symmetry:Ξ F = [1 ⊗ ξ F ] (82) ξ F ∈ { ξ , ξ µ , ξ µ , ξ µν } for µ (cid:54) = ν (83)where ξ µ respects the Clifford algebra { ξ µ , ξ ν } = 2 δ µν inthe Euclidean space.Let us consider the following quantity W in the con-tinuum at a = 0: W ≡ (cid:104) f (cid:96) | Ξ F D s | f n (cid:105) (84) D s | f n (cid:105) = iλ n | f n (cid:105) (85)Since the SU (4) taste symmetry is exactly conserved inthe continuum, we know that[Ξ F , D s ] = 0 (86)Hence, we find the following Ward identity: W = (cid:104) f (cid:96) | Ξ F D s | f n (cid:105) = iλ n (cid:104) f (cid:96) | Ξ F | f n (cid:105) (87)= (cid:104) f (cid:96) | D s Ξ F | f n (cid:105) = iλ (cid:96) (cid:104) f (cid:96) | Ξ F | f n (cid:105) (88)The Ward identity leads to the following condition: i ( λ (cid:96) − λ n ) · (cid:104) f (cid:96) | Ξ F | f n (cid:105) = 0 (89)Hence, in the continuum ( a = 0), we find the followingproperties of the eigenvalue spectrum. • If λ (cid:96) (cid:54) = λ n , (Ξ F ) (cid:96)n = (cid:104) f (cid:96) | Ξ F | f n (cid:105) = 0. In otherwords, if the eigenvalues are different ( λ (cid:96) (cid:54) = λ n ),there is no leakage ((Ξ F ) (cid:96)n = 0) between the twoeigenmodes. • If λ j ≡ λ (cid:96) = λ n , (Ξ F ) (cid:96)n (cid:54) = 0 is possible. In otherwords, if the eigenvalues are degenerate ( λ j = λ (cid:96) = λ n ) and they belong to a quartet such that theysatisfy D s | f j,m (cid:105) = iλ j | f j,m (cid:105) (90) | f (cid:96) (cid:105) , | f n (cid:105) ∈ {| f j,m (cid:105) with m = 1 , , , } (91)Here, | f (cid:96) (cid:105) and | f n (cid:105) are linear combinations of thequartet {| f j,m (cid:105)} and they are orthogonal to eachother by construction due to Lanczos algorithm.Here, j is a quartet index and m is a taste indexwhich represents the four-fold degeneracy for theeigenvalue λ j . • We know that the staggered fermion field χ c ( x A ) ismapped into the continuum fermion field ψ cα ; t ( x ),where α represents a Dirac spinor index, c repre-sents a color index, t = 1 , , , λ j , there re-main four degrees of freedom which come from thetaste index. Accordingly for a given eigenvalue λ j ,there are four degenerate eigenstates | f j,m (cid:105) with m = 1 , , , • If we know all the four eigenstates {| f j,m (cid:105)} for acertain eigenvalue λ j , we find thatTr(Ξ F ) = (cid:88) m =1 (Ξ F ) j,mj,m = (cid:88) m =1 (cid:104) f j,m | Ξ F | f j,m (cid:105) = 0 (92)This is because the SU (4) group generators aretraceless in the fundamental representation.However, on the lattice at a (cid:54) = 0, the taste symmetryis broken by those terms of order a α ns with n ≥ a (cid:54) = 0, D s | f j,m (cid:105) = iλ j,m | f j,m (cid:105) (93)and λ j,m (cid:54) = λ j,m (cid:48) in general for m (cid:54) = m (cid:48) , which reflectsthe taste symmetry breaking effect at a (cid:54) = 0. We knowthat λ j,m = λ j,m (cid:48) for all m, m (cid:48) in the continuum ( a = 0)due to the exact taste symmetry. Hence, on the finitelattice, we expect a small deviation from the above con-tinuum properties. A good barometer to measure thiseffect is to monitor T T ≡
14 Tr(Ξ ) = 14 (cid:88) m (Ξ ) j,mj,m (94)and measure how much it deviates from zero (= the con-tinuum value). Another direct barometer to measure aneffect of the taste symmetry breaking is to monitor theleakage S from one quartet ( λ (cid:96) ) to another quartet ( λ j )with λ (cid:96) (cid:54) = λ j . S ≡ (cid:88) m,m (cid:48) | Ξ | (cid:96),mj,m (cid:48) = 116 (cid:88) m,m (cid:48) |(cid:104) f (cid:96),m | Ξ | f j,m (cid:48) (cid:105)| (95)The size of the leakage S indicates directly how muchthe taste symmetry is broken at a (cid:54) = 0, since S = 0 inthe continuum at a = 0. We present numerical resultsfor T and S in the next subsection. B. Numerical study on chirality and leakage
Here, we use dual notations for the eigenmodes: oneis the normal index i for λ i and the other is the quartetindex j with taste index m for λ j,m . The normal indexis convenient for the plots, tables, and leakage patternssuch as | Γ | ab , while the quartet index is convenient toexplain the eigenstates classified by the taste symmetrygroup. The one-to-one mapping from the normal indexsystem i to the quartet index system j, m is given inTable II for the quartet index j = 0 , ± Q = ± j = ± TABLE II. One to one mapping of a normal index i of the λ i eigenstate into a quartet index j and a taste index m forthe λ j,m . λ i = λ j,m . Here, λ n = − λ n − and λ − j,m = − λ + j,m . The zero represents would-be zero modes. The non-zero represents non-zero modes. Here, we assume that Q = ± λ i λ j,m i j m mode λ λ , λ λ , λ λ , λ λ , λ λ +1 , λ λ +1 , λ λ +1 , λ λ +1 ,
11 +1 4 non-zero λ λ − , − λ λ − , − λ λ − , − λ λ − , − In Fig. 4, we present the leakage pattern of the zeromode of λ and its parity partner λ = − λ . Since Q = − j = 0).lim a → λ i = 0 for i = 1 , , , . (96)In the continuum limit ( a = 0), the SU (4) taste sym-metry becomes exactly conserved and so would-be zeromodes become exact zero modes. However, at finite lat-tice spacing a (cid:54) = 0, the gauge configuration is so roughthat would-be zero modes have non-zero eigenvalues: λ = − λ , λ = − λ , and λ (cid:54) = λ for λ , λ > TABLE III. Numerical values for leakage patterns from the λ eigenstate to the λ i eigenstate in Fig. 4. Here, j representsa quartet index for the λ i eigenstate. The leakage representsleakage patterns of |O| i = |O ( λ i , λ ) | = |(cid:104) f i |O| f (cid:105)| for O =Γ , Ξ . j leakage value Ward id.0 | Γ | | Ξ | | Ξ | | Ξ | | Ξ | | Γ | | Γ | | Γ | | Γ | . × − | Γ | . × − | Γ | . × − +1 | Γ | . × − − | Γ | . × − +2 | Γ | . × − − | Γ | . × − j=0 j=±1 j=±2 | Γ | i i (a) | Γ | i = | Γ ( λ i , λ ) | j=0 j=±1 j=±2 | Ξ | i i (b) | Ξ | i = | Ξ ( λ i , λ ) | j=0 j=±1 j=±2 | Ξ | i i (c) | Ξ | i = | Ξ ( λ i , λ = − λ ) | j=0 j=±1 j=±2 | Γ | i i (d) | Γ | i = | Γ ( λ i , λ = − λ ) | FIG. 4. Leakage pattern for would-be zero modes at Q = − λ i =2 n − > i odd number, and the blue bar represents leakage to its paritypartner λ i =2 n = − λ n − with i even number. In Fig. 4 (a), we show the leakage pattern of | Γ | i = | Γ ( λ i , λ ) | = |(cid:104) f i | Γ | f (cid:105)| . We find that there is, inpractice, no leakage and so the only non-zero compo-nent is | Γ | = | Γ ( λ , λ ) | and the rest is practicallyzero. In Fig. 4 (b), 4 (c), and 4 (d), we find that theWard identity of Eqs. (80) and (81) is well respected inthe numerical results. In other words, the Ward identity | Γ | = | Ξ | = | Ξ | = | Γ | is satisfied within the nu-merical precision of the computer. Please refer to TableIII for numerical details. This confirms that the theoret-ical prediction from the Ward identity in Eqs. (80) and(81) is correct.In Fig. 4 (a), we find that there is a small leakage intoother quartets ( j = ± , ± j = 0 quartet ( e.g. | Γ | ) is of order 10 − . Wealso observe small leakage patterns of order 10 − ∼ − from the would-be zero modes, j = 0 quartet to the non-zero modes, j = ± , ± e.g. | Γ | ).Now let us switch the gear to non-zero modes in the j = +1 quartet. In Fig. 5, we present the leakage patternfor the non-zero modes of λ and its parity partner λ = − λ . Even in the continuum limit ( a = 0), λ (cid:54) = 0 and soit is a non-zero mode. Thanks to the approximate SU (4)taste symmetry and the exact U (1) A axial symmetry,there will be eight-fold degeneracy in the family of eighteigenstates composed of the j = +1 quartet to which λ belongs and j = − j = ± { λ i } with 5 ≤ i ≤
12 in Fig. 5.Let us scrutinize the leakage pattern of the non-zeromode λ = λ j =+1 ,m =1 . In Fig. 5 (a), first, note that thereis practically no leakage in the Γ chirality measurementfrom λ into λ n − with n > n ∈ Z . In otherwords, | Γ | n − = | Γ ( λ n − , λ ) | ∼ = 0. This implies thatthe measurement of the chirality operator on the non-zeromode with λ > λ <
0. In Fig. 5 (a), second, notethat the nontrivial leakage goes to those eigenstates in the j = − { λ , λ , λ , λ } = { λ j,m | j = − , m = 1 , , , } . In addition, we find that the Wardidentity of Eqs. (80) and (81) is well respected within thenumerical precision in Fig. 5 (a), 5 (b), 5 (c), and 5 (d). InTable IV, we present numerical values of | Γ | i in Fig. 5 (a). Let us examine the Γ = [ γ ⊗
1] leakage pattern of the j = +1 quartet of the non-zero modes { λ , λ , λ , λ } .In Fig. 6, we find that the chirality measurement van-ishes; (Γ ) ii = Γ ( λ i , λ i ) = 0 for λ i in the j = +1 quartetof the non-zero modes. We also find that the Γ leakageof λ +1 ,m > j = +1 quartet goes to the paritypartners of λ − ,m (cid:48) < j = − j = ± j = − | Γ | − ,m +1 ,m (cid:48) are summarized in TableV.Let us examine the Ξ = [1 ⊗ ξ ] leakage pattern of the j = +1 quartet of the non-zero modes: { λ , λ , λ , λ } .In Fig. 7, we find that the Ξ leakage from the j = +1quartet to the j = − is related tothe leakage pattern of Γ by the Ward identity | Ξ | j,mj (cid:48) ,m (cid:48) = | Γ | − j,mj (cid:48) ,m (cid:48) , (97)Fig. 7 can be obtained from Fig. 6 using the Ward iden-tity. We find that the Ξ leakage from the j = +1 quartetto other quartets such as j = ± j = +1 quartet).The leakage patterns of the Γ chirality and Ξ shiftoperators for diverse topological charges are given in Ap-pendix F. j=0 j=±1 j=±2 | Γ | i i (a) | Γ | i = | Γ ( λ i , λ ) | j=0 j=±1 j=±2 | Ξ | i i (b) | Ξ | i = | Ξ ( λ i , λ ) | j=0 j=±1 j=±2 | Ξ | i i (c) | Ξ | i = | Ξ ( λ i , λ = − λ ) | j=0 j=±1 j=±2 | Γ | i i (d) | Ξ | i = | Γ ( λ i , λ = − λ ) | FIG. 5. Leakage pattern for non-zero modes at Q = − Let us summarize the leakage pattern for would-be zeromodes and that for non-zero modes. Let us fist beginwith the leakage pattern for the zero modes.1. A zero mode in staggered fermions appears in aform of four-fold degeneracy (we call them a quar-tet). In other words, for the topological charge Q , the number of zero modes is 4 × ( n + + n − )and Q = n − − n + (Atiyah-Singer Index Theorem),where n + ( n − ) is the number of right-handed (left-handed) zero mode quartets.2. In the chirality Γ = [ γ ⊗
1] measurement, thezero mode has practically no leakage to other eigen-states.3. In the shift Ξ = [1 ⊗ ξ ] measurement, the zeromode with eigenvalue λ has a full (100%) leakageinto its parity partner mode with eigenvalue − λ ,and no leakage into any other eigenmodes.1 TABLE IV. Numerical values for data in Fig. 5. j leakage value Ward identities − | Γ | | Ξ | = | Ξ | = | Γ | − | Γ | | Ξ | = | Ξ | = | Γ | = | Γ | = | Ξ | = | Ξ | = | Γ | − | Γ | | Ξ | = | Ξ | = | Γ | = | Γ | = | Ξ | = | Ξ | = | Γ | − | Γ | | Ξ | = | Ξ | = | Γ | = | Γ | = | Ξ | = | Ξ | = | Γ | +1 | Γ | . × − = | Ξ | = | Ξ | = | Γ | +1 | Γ | . × − = | Ξ | = | Ξ | = | Γ | = | Γ | = | Ξ | = | Ξ | = | Γ | +1 | Γ | . × − = | Ξ | = | Ξ | = | Γ | = | Γ | = | Ξ | = | Ξ | = | Γ | +1 | Γ | . × − = | Ξ | = | Ξ | = | Γ | = | Γ | = | Ξ | = | Ξ | = | Γ | | Γ | . × − | Γ | . × − +2 | Γ | . × − − | Γ | . × − TABLE V. | Γ | − ,m +1 ,m (cid:48) values in Fig. 6. λ i λ j λ λ λ λ λ λ λ λ The leakage pattern for nonzero modes is1. A non-zero mode in staggered fermions appears ina form of eight-fold degeneracy composed of a quar-tet (+ j quartet) and its parity partner quartet ( − j quartet). In other words, non-zero eigenmodes canbe grouped into sets with eight elements in eachset. This is due to the approximate SU (4) tastesymmetry and the conserved U (1) A axial symme-try.2. In the chirality Γ = [ γ ⊗
1] measurement, the non-zero mode with eigenvalue λ j,m has no leakage toits own quartet ( j quartet), but has leakage onlyto the parity partner − j quartet with λ − j,m (cid:48) . Ithas no leakage to any eigenmode which belongs toother quartets such as (cid:96) (cid:54) = ± j quartets.3. In the shift Ξ = [1 ⊗ ξ ] measurement, the non-zero mode with λ j,m has no leakage to its paritypartner − j quartet at all. But it has leakage onlyto the eigenstates in its own j quartet. This comesdirectly from the Ward identity. In other words, theΞ leakage pattern is a mirror image reflecting Γ by the mirror of Ward identity. It has no leakageto any eigenmode which belongs to other quartetsuch as (cid:96) (cid:54) = ± j quartets.4. Thanks to the conserved U (1) A symmetry, the leakage pattern of | Γ | − j,m(cid:96),m (cid:48) is identical to that of | Ξ | + j,m(cid:96),m (cid:48) by the Ward identity.In Appendix E, we provide more examples to demon-strate that our claim on the leakage pattern for zeromodes holds valid in general. In Appendix F, we givemore examples to demonstrate that our claim on theleakage pattern for non-zero modes holds valid in gen-eral. We have repeated numerical tests over hundreds ofzero modes and tens of thousands of nonzero modes. Weperform the numerical study on hundreds of gauge con-figurations in order to check the above leakage pattern,and find that the above leakage pattern is valid for all ofthem except for those gauge configurations with unstabletopological charge.1. We find a number of gauge configurations whichdoes not have a stable topological charge.2. We have found about 10 gauge configurations withunstable topological charge among the 100 gaugeconfigurations with the 12 lattice geometry at β =4 . lattice geometry at β =5 . TABLE VI. Numerical results for T . To obtain the results,we use 292 gauge configurations with input parameters set toTable I. N q represents the number of quartets used to obtainthe statistical error. Here, j = 0 represents would-be zeromodes, and j > j | Re( T ) | | Im( T ) | N q j = 0 7 . × − . × − j > . × − . × − j=0 j=±1 j=±2 | Γ | i i (a) | Γ | i = | Γ ( λ i , λ ) | j=0 j=±1 j=±2 | Γ | i i (b) | Γ | i = | Γ ( λ i , λ ) | j=0 j=±1 j=±2 | Γ | i i (c) | Γ | i = | Γ ( λ i , λ ) | j=0 j=±1 j=±2 | Γ | i i (d) | Γ | i = | Γ ( λ i , λ ) | FIG. 6. [ γ ⊗
1] leakage pattern for non-zero modes at Q = − In Table VI, we present results for T defined inEq. (94), which is a direct barometer to estimate theeffect of taste symmetry breaking. If the taste symme-try is exactly conserved, then T must vanish. Hence,a non-trivial value of T indicates size of taste symme-try breaking. In Table VI, we find that | Re( T ) | is ofthe order of a sub-percent level 10 − , while | Im( T ) | = 0essentially. This indicates that the effect of taste symme-try breaking is very small (in the sub-percent level perquartet).In Fig. 8, we present S defined in Eq. (95) as a func-tion of | (cid:96) − j | with (cid:96), j ≥
0. Here, | (cid:96) − j | = 1 representsa pair of nearest neighbor quartets, | (cid:96) − j | = 2 representsa pair of next to the nearest neighbor quartets, and soon. The values of S are as big as their statistical error.This indicates that this taste symmetry breaking effectgives just a random noise to the physical signal ( S = 0).For | (cid:96) − j | = 1, it gives a random noise of ≈ j=0 j=±1 j=±2 | Ξ | i i (a) | Ξ | i = | Ξ ( λ i , λ ) | j=0 j=±1 j=±2 | Ξ | i i (b) | Ξ | i = | Ξ ( λ i , λ ) | j=0 j=±1 j=±2 | Ξ | i i (c) | Ξ | i = | Ξ ( λ i , λ ) | j=0 j=±1 j=±2 | Ξ | i i (d) | Ξ | i = | Ξ ( λ i , λ ) | FIG. 7. [1 ⊗ ξ ] leakage pattern for non-zero modes at Q = − | (cid:96) − j | = 2, it gives a random noise of ≈ | (cid:96) − j | increases. Thenumerical values of S in Fig. 8 are presented in TableVII. TABLE VII. Numerical results for S . Here, we measure S between two different quartets ( (cid:96) (cid:54) = j and (cid:96), j ≥ N p represents the number of ( (cid:96) , j ) pairs with (cid:96) (cid:54) = j . | (cid:96) − j | S N p . × − . × − . × − . × − . × − S [ × ] | l - j | FIG. 8. S as a function of | (cid:96) − j | . Numerical values are givenin Table VII. VI. MACHINE LEARNING
In previous sections, we have shown that staggeredfermions respect the U (1) A symmetry which induces thechiral Ward identities in Eq. (80), and also respect anapproximate SU (4) taste symmetry which brings in thequartet behavior of the eigenvalue spectrum. Further-more, a combined effect of those symmetries gives us dis-tinctive leakage patterns for the chirality operator Γ andthe shift operator Ξ . In this section, we apply the ma-chine learning technique to the following tasks.1. We want to know how much the non-zero modesrespect the quartet classification rules, which comefrom the SU (4) taste symmetry.2. We want to know how efficiently we can measurethe topological charge Q using the index theoremfrom the quartet structure of the non-zero modes.3. We want to find out any anomalous behavior ofthe eigenvalue spectrum, which does not follow thestandard leakage pattern of the non-zero modes.4. We want to figure out what causes the anomalousbehavior of the eigenvalue spectrum.Let us explain our sampling method for the machinelearning. In Fig. 9, we show matrix elements | Γ | ij on agauge configuration with Q = 2. Fig. 9(a) is for the 200lowest eigenmodes and Fig. 9(b) is a zoomed-in version ofFig. 9(a) for the 32 lowest eigenmodes. Here, the depth ofthe blue color represents the size of | Γ | ij matrix element,and i, j run over the range of [0 , Q = 2. Excluding the would-bezero modes, we randomly choose a 15 ×
15 sub-matrixof | Γ | ij along the diagonal line of | Γ | ij matrix elements.This 15 ×
15 sub-matrix is the largest square sub-matrixof | Γ | which contains all elements of only one quartet ofnon-zero modes and its parity partner quartet.In Fig. 10, we present 8 different classes for arbitrarysamples. Our purpose of the machine learning is to find (a)200 × × FIG. 9. Matrix elements of | Γ | for 200 and 32 of the low-est eigenmodes on a gauge configuration with Q = 2. Here,indices on both axes are the eigenvalue index. The color ofeach square represents the magnitude of corresponding ma-trix element. Black lines indicate borders of non-zero modequartets, and red lines are of zero mode quartets. borders (black lines) of the non-zero mode quartet (oroctet when the parity partners are included) in each sam-ple. We classify arbitrary samples into eight differentclasses according to the location of the border line. Eachclass is labeled as in Fig. 10.We use a deep learning model which combines themulti-layer perceptron (MLP) [43] and the convolutionalneural network (CNN) [43]. In Table VIII, we present4 (a)class 0 (b)class 1(c)class 2 (d)class 3(e)class 4 (f)class 5(g)class 6 (h)class 7 FIG. 10. Examples for our samples. Every sample containsonly one non-zero mode quartet. There are eight kinds ofclasses according to the location of the borders of the quartet. our basic setup for the machine learning. We use thegauge configuration ensemble described in Table I. Thedata measured over 292 gauge configurations are dis-tributed over training set, validation set, and test set asin Table VIII. For each gauge configuration, we generatearound tens of 15 ×
15 matrix samples from the 200 lowesteigenmodes without overlapping. We make popular andsuitable choices for loss function , optimization method ,and activation functions relevant to our purpose, which Popular and basic loss functions such as the mean squared error(MSE) and mean absolute error (MAE) are usually used for re-
TABLE VIII. Parameters for machine learning.parameters valuesnumber of training configurations 120number of training samples 1223number of validation configurations 30number of validation samples 308number of test configurations 142number of test samples 1448loss function categoricalcross-entropy [43, 44]optimization method Adam [45]activation function for hidden layers ReLU [43]activation function for output layer softmax [43]TABLE IX. Hyper-parameters for neural networks. Here, weshow one of the examples of best performance, in which weuse only MLP but not CNN.layer type number of units activationinput - 225 -hidden is summarized in Table VIII. The best hyper-parameterssuch as the number of layers and the number of units foreach layer are determined by Keras Tuner [44].The accuracy of classification per gauge configurationis obtained by averaging the accuracies of the machinelearning (ML) prediction for all the samples on a singlegauge configuration. Our best model achieves an aver-age accuracy of 96 . gression problems. On the contrary, the categorical cross-entropyloss is best applicable to the multi-class classification problems. Popular optimization methods available in the market arestochastic gradient descent, AdaGrad, RMSprop, and Adam.Adam [45] is one of the popular algorithms recently. Popular activation functions are tanh, sigmoid, and ReLU. Here,we make use of ReLU for the hidden layers since it is the simplestand fastest among them. Softmax function is essential for theoutput layer of the multi-class classification. . .
4% implies that one can find completelycorrect quartet groups for all the normal gauge configu-rations of the test set in the end. It also demonstratesthat our claim on the leakage pattern is universal overall the normal gauge configuration ensembles. Details onresults of this ML research will be reported separately inRef. [46].
VII. ZERO MODES AND RENORMALIZATION
As explained in Sec. V, we know that there is practi-cally no leakage for the zero modes in the chirality mea-surement. Hence, it is possible to determine the renor-malization factor κ P by imposing the index theorem asfollows. For Q (cid:54) = 0,4 × Q = − κ P × (cid:88) λ ∈ S (cid:104) f sλ | [ γ ⊗ | f sλ (cid:105) (98) κ P = − QC (99) C = (cid:88) λ ∈ S Γ ( λ, λ ) (100)where S is the set of the zero modes in staggered fermionformalism and κ P = Z P × S ( µ ) Z P × P ( µ ) (101)where O S = ¯ χ [ γ ⊗ χ (102) O P = ¯ χ [ γ ⊗ ξ ] χ (103)[ O S ] R ( µ ) = Z P × S ( µ )[ O S ] B (104)[ O P ] R ( µ ) = Z P × P ( µ )[ O P ] B (105)where the subscript [ · · · ] R ([ · · · ] B ) represents a renor-malized (bare) operator. The Z P × S and Z P × P are therenormalization factors for the bilinear operators O S and O P , respectively. One advantage of this scheme is that κ P is independent of valence quark masses, even thoughwe perform the measurement with arbitrary masses forvalence quarks. Numerical results for κ P are summarizedin Table X.There are a few key issues on the physical interpreta-tion of κ P . • Since the topological charge Q is independent ofrenormalization scale and the C is independent ofrenormalization scale, κ P must be independent ofthe renormalization scale µ . • This means that the scale dependence of Z P × S ( µ )must cancel off that of Z P × P ( µ ). • It would be nice to cross-check this property of κ P in the RI-MOM scheme [47], and in the RI-SMOMscheme [48]. TABLE X. Numerical results for κ P .topological charge number of samples κ P | Q | = 1 72 1.26(13) | Q | = 2 68 1.22(3) | Q | = 3 45 1.23(2)weighted average 241 1.23(2) VIII. CONCLUSION
We study the general property of the eigenvalue spec-trum of Dirac operators in staggered fermion formalism.As an example, we use the Dirac operator for HYP stag-gered quarks. In Section V, we introduce a new chiral-ity operator Γ and a new shift operator Ξ and provethat they respect the continuum recursion relationshipas explained in Eqs. (53)-(56) and Eqs. (60)-(61). Us-ing these advanced operators with nice chiral property,we find that the leakage pattern of | Γ | − j,m(cid:96),m (cid:48) is related tothat of | Ξ | j,m(cid:96),m (cid:48) through the Ward identity of the con-served U (1) A symmetry.We find that the leakage pattern of Γ and Ξ for thezero modes is quite different from that for the non-zeromodes. This difference in leakage pattern allows us todistinguish the zero modes from the non-zero modes eventhough we do not know a priori about the topologicalcharge. We find that using the leakage pattern of Γ andΞ , it is possible to determine the topological charge asreliably as typical field theoretical methods in the marketsuch as the cooling method.We use the machine learning (ML) technique to checkthe universality of this leakage pattern over the entireensemble of gauge configurations. Our best-trained deeplearning model identifies the quartet of non-zero modeswith 98.7(34)% accuracy per a single normal gauge con-figuration. We find that the ML can identify all quartetgroups on an eigenvalue spectrum correctly if we choosethe highest probable prediction by the ML and comparethe prediction with the correct answer later. In addi-tion, the ML technique finds out even the wrong answersby our input mistakes since the ML prediction does notagree with the wrong answer by giving the predictionwith low accuracy ( < κ P = Z P × S ( µ ) /Z P × P ( µ ) from the chirality measurement of Γ .6The leakage pattern is a completely new concept in-troduced in this paper. It can be used to study the lowlying eigenvalue spectrum of staggered Dirac operatorssystematically. It helps us to understand how to fish outthe taste symmetry and chiral symmetry embedded inthe staggered eigenvalue spectrum. It will help us to digout its related physics more efficiently. ACKNOWLEDGMENTS
We would like to express sincere gratitude to EduardoFollana for helpful discussion and providing his code tocross-check results of our code. The research of W. Lee issupported by the Mid-Career Research Program (GrantNo. NRF-2019R1A2C2085685) of the NRF grant fundedby the Korean government (MOE). This work was sup-ported by Seoul National University Research Grantin 2019. W. Lee would like to acknowledge the sup-port from the KISTI supercomputing center through thestrategic support program for the supercomputing ap-plication research [No. KSC-2016-C3-0072, KSC-2017-G2-0009, KSC-2017-G2-0014, KSC-2018-G2-0004, KSC-2018-CHA-0010, KSC-2018-CHA-0043]. Computationswere carried out in part on the DAVID supercomputerat Seoul National University.
Appendix A: Lanczos algorithm
Lanczos is a numerical algorithm to calculate eigen-values and eigenvectors of a Hermitian matrix [21]. Ittransforms an n × n Hermitian matrix H to tridiagonalmatrix T through a unitary transformation Q , which isrepresented by T = Q † HQ . (A1)Here, columns of Q are composed of basis vectors of n -thKrylov subspace K n ( H, b ) generated by H and a startingvector b of our choice. Each iteration of Lanczos com-putes a column of Q and T in sequence. At the end,diagonalizing the tridiagonal matrix T yields eigenvaluesand eigenvectors of H .In principle, Lanczos is a direct method that takes n iterations to construct the n × n tridiagonal matrix T .However, since these columns of T are computed in or-der, a sequence of m < n iterations also constructs an m × m tridiagonal matrix T (cid:48) which is a submatrix of T .In practice, the real benefit of Lanczos is that eigenval-ues of T (cid:48) approximate some eigenvalues of T . As itera-tion continues and the size of the submatrix T (cid:48) increases,eigenvalues of T (cid:48) converge to eigenvalues of T . Theirconvergence condition is somewhat complicated. Theyconverge to the largest, the smallest, or the most sparseeigenvalue first. The speed of convergence depends onthe density of eigenvalues. The less dense, the fasterthey converge. In this paper, we make use of two popular improve-ment techniques of Lanczos: (1) implicit restart [49], and(2) polynomial acceleration with Chebyshev polynomial[50]. The implicit restart method gets rid of convergedeigenvalues in the middle of the Lanczos iteration. Ittakes effect as if we restarted the Lanczos with a shiftedmatrix H (cid:48) given by H (cid:48) ≡ H − (cid:88) i λ i I , (A2)where λ i are eigenvalues we want to remove. Then H (cid:48) is still Hermitian but does not have such eigenvalues λ i .Hence, Lanczos with H (cid:48) converges to remaining eigenval-ues faster. Besides, the implicitly restarting proceduregives us a new submatrix, which has a reduced dimen-sion (( m − r ) × ( m − r )) by the number of eigenvalues wehave removed ( r ). Then we iterate Lanczos r times torefill the submatrix to restore the structure of m × m ma-trix. Then we repeat the implicit restart to obtain a newsubmatrix of ( m − r ) × ( m − r ), and so on. It allows usto control the size of submatrix, the computational costand the memory usage while the submatrix T (cid:48) contains( m − r ) eigenmodes more precise (much closer to the trueeigenmodes of the full matrix H ) for each iteration.A polynomial operation on a matrix changes the eigen-value spectrum accordingly while retaining the eigen-vectors. Since the polynomial of a Hermitian matrixis also Hermitian, Lanczos is still available to calculateits eigenvalues and eigenvectors. By choosing a properpolynomial, one can manipulate density of the eigen-value spectrum so that the convergences to the desiredeigenvalues are accelerated. Chebyshev polynomial is apopular choice for this purpose. Using the Chebyshevpolynomial, we want to map the first region of eigen-modes of no interest to [ − , −∞ , − − ,
1] wherethe eigenvalues are enough dense to make the Lanczosnot converge. In addition, Chebyshev polynomial rapidlychanges in the second region such that it makes the den-sity of eigenmodes enough low to accelerate the conver-gence of Lanczos faster. Here, we apply Chebyshev poly-nomial for D † s D s whose eigenvalues are λ ≥
0. We setthe lower bound of the first region to a value somewhatgreater than the largest eigenvalue that we want to get.This strategy will not only suppress high unwanted eigen-modes but also accelerate the speed of Lanczos for thelow eigenmodes of our interest.Numerical stability is essential for Lanczos algorithm.Each Lanczos iteration generates Lanczos vectors, whichare column vectors of the unitary matrix Q in Eq. (A1).After several iterations, however, Lanczos vectors losetheir orthogonality due to gradual loss of numerical pre-cision. This loss would induce spurious ghost eigenvalues[51]. A straightforward prescription to the problem isperforming a reorthogonalization for every calculation ofLanczos vectors. There are also alternative approaches toeliminate those ghost eigenvalues without the reorthogo-7nalization, such as Cullum-Willoughby method [52, 53].Here, we take the first solution to perform the full re-orthogonalization for each Lanczos iteration.For a large scale simulation using Lanczos, Multi-GridLanczos [54] and Block Lanczos [55] are available in themarket. Multi-Grid Lanczos is also based on the implicitrestart and Chebyshev acceleration. Along with that, itreduces the memory requirement significantly by com-pressing the eigenvectors using their local coherence [56].It constructs a spatially-blocked deflation subspace fromsome of the lowest eigenvectors of Dirac operator. Thenthe coherence of eigenvectors allows us to represent othereigenvectors on this subspace and run Lanczos with muchless memory size. Meanwhile, Block Lanczos utilizes theSplit Grid method [55]. This algorithm deals with mul-tiple starting vectors for Lanczos, where the Split Gridmethod divides the domain of the Dirac operator appli-cation into multiple smaller domains so that each partialdomain runs in parallel on a partial grid (lattice) with alower surface to volume ratio compared to that of the fullgrid. Hence, one can optimize the off-node communica-tion by adjusting the block (grid) size. It would give asignificant speed-up compared with our method. We planto implement Multi-Grid Lanczos and Block Lanczos innear future. Appendix B: Even-odd preconditioning and phaseambiguity
Even-odd preconditioning reorders a fermion field χ ( x )so that even site fermion fields are obtained first, and oddsite fermion fields are obtained from them: χ ( x ) = (cid:18) χ e χ o (cid:19) , (B1)where χ e ( χ o ) is the fermion field collection on even (odd)sites. On this basis, the massless staggered Dirac opera-tor D s can be represented as a block matrix: D s = (cid:18) D eo D oe (cid:19) , (B2)where D oe ( D eo ) relates even (odd) site fermion fields toodd (even) site fermion fields. Since D † s = − D s , we alsofind that D † oe = − D eo and D † eo = − D oe .On this basis, D † s D s is expressed as D † s D s = (cid:18) − D eo − D oe (cid:19) (cid:18) D eo D oe (cid:19) (B3)= (cid:18) − D eo D oe − D oe D eo (cid:19) . (B4)Hence, the eigenvalue equation of D † s D s (Eq. (29)) canbe divided into two eigenvalue equations as follows, − D eo D oe | g e (cid:105) = λ | g e (cid:105) , (B5) − D oe D eo | g o (cid:105) = λ | g o (cid:105) , (B6) where | g e ( o ) (cid:105) is the collection of even (odd) site compo-nents of | g sλ (cid:105) . Here, we omit the superscript s and thesubscript λ for notational simplicity. Now, let us mul-tiply D oe from the left on both sides of Eq. (B5). Thenwe find that − D oe D eo ( D oe | g e (cid:105) ) = λ ( D oe | g e (cid:105) ) , (B7)which is identical to Eq. (B6). Hence, we find that | g o (cid:105) = η D oe | g e (cid:105) where η = re iα is an arbitrary com-plex number with r > ≤ α < π . Here, r represents the scaling behavior and α represents a ran-dom phase. Since − D eo D oe (= D † oe D oe ) is Hermitian andpositive semi-definite, one can solve Eq. (B5) using theLanczos algorithm introduced in Appendix A. From theresult of | g e (cid:105) , it is straightforward to obtain the eigenvec-tor | g sλ (cid:105) of Eq. (29) as follows, | g sλ (cid:105) = (cid:18) | g e (cid:105) η D oe | g e (cid:105) (cid:19) . (B8)where η is a random complex number in general.Now, we apply the projection operator P + , defined inEq. (32), to | g sλ (cid:105) . Using Eq. (B5), we find that | χ + (cid:105) = P + | g sλ (cid:105) = (cid:18) iλ D eo D oe iλ (cid:19) (cid:18) | g e (cid:105) η D oe | g e (cid:105) (cid:19) = (1 + iηλ ) (cid:18) iλ | g e (cid:105) D oe | g e (cid:105) (cid:19) . (B9)Similarly, for the projection operator P − defined inEq. (33), we find that | χ − (cid:105) = P − | g sλ (cid:105) = (1 − iηλ ) (cid:18) − iλ | g e (cid:105) D oe | g e (cid:105) (cid:19) . (B10)Since η only appears in the overall factor for both cases,it gives only the relative phase difference between thenormalized eigenvectors | f s ± λ (cid:105) defined in Eqs. (36) and(37).We can proceed further to obtain the eigenvectors | f s ± λ (cid:105) . The norm of | χ + (cid:105) is given by (cid:104) χ + | χ + (cid:105) = [(1 − iη ∗ λ )(1 + iηλ )] · λ (cid:104) g e | g e (cid:105) . (B11)Hence, | f s + λ (cid:105) is | f s + λ (cid:105) = 1 N (cid:115) iηλ − iη ∗ λ (cid:18) iλ | g e (cid:105) D oe | g e (cid:105) (cid:19) , (B12)where N ≡ (cid:112) λ (cid:104) g e | g e (cid:105) . (B13)Similarly, | f s − λ (cid:105) = 1 N (cid:115) − iηλ iη ∗ λ (cid:18) − iλ | g e (cid:105) D oe | g e (cid:105) (cid:19) . (B14)8These results for | f s ± λ (cid:105) indicate that the phase difference θ for Γ (cid:15) transformation defined in Eq. (47) depends onthe value of η .In our numerical study in this paper, we set η to η = re iα = 1: r = 1 and α = 0. Hence, the relativerandom phase between | f s ± λ (cid:105) states is removed by hand.Therefore, our value of θ defined in Eq. (47) includes abias from our choice of η = 1.For η = 1 (our choice), Γ (cid:15) | f s + λ (cid:105) isΓ (cid:15) | f s + λ (cid:105) = 1 N (cid:114) iλ − iλ (cid:18) iλ | g e (cid:105)− D oe | g e (cid:105) (cid:19) , (B15)while | f s − λ (cid:105) is | f s − λ (cid:105) = 1 N (cid:114) − iλ iλ (cid:18) − iλ | g e (cid:105) D oe | g e (cid:105) (cid:19) . (B16)Then the following contraction gives e iθ as follows, (cid:104) f s − λ | Γ (cid:15) | f s + λ (cid:105) = 1 N (cid:115)(cid:18) − iλ iλ (cid:19) ∗ iλ − iλ · ( − N )= − iλ − iλ = e i ( π +2 β ) = e iθ , (B17)where β ≡ arctan( λ ). From Eq. (47), we find that θ = π + 2 β . (B18)In Fig. 3, we measure the phase θ for hundreds of eigen-vectors on a gauge configuration with Q = −
1. Theresults for θ is consistent with our theoretical predictionof Eq. (B18) within numerical precision. Appendix C: Eigenvalue spectrum for Q = − and Q = − In Figs. 11 and 12, we present examples of the eigen-value spectrum for Q = − Q = −
3, respectively.Figs. 11(a) and 12(a) show eigenvalues λ for eigenvec-tors | g sλ (cid:105) defined in Eq. (29). In Fig. 11, we find twosets of four-fold degenerate eigenstates: { λ , λ , λ , λ } and { λ , λ , λ , λ } . Each of them indicates a quartetof would-be zero modes. The number of quartets corre-sponds to the topological charge Q = − n − = 0 and n + = 2). Apartfrom the would-be zero modes, we observe that non-zeromodes are eight-fold degenerate as in the cases of Q = 0(Fig. 1) and Q = − n − = 0 and n + = 3 ( Q = − { λ , λ , λ , λ } , { λ , λ , λ , λ } , and { λ , λ , λ , λ } .Because the number of quartets equals the absolute valueof the topological charge | Q | = 3, it is possible to de-duce that all the would-be zero modes have the same λ i i (a) λ i -0.06-0.030.000.030.06 1 5 9 13 17 21 25 29 λ i i (b) λ i FIG. 11. The same as Fig. 1 except for Q = − λ i i (a) λ i -0.06-0.030.000.030.06 1 5 9 13 17 21 25 λ i i (b) λ i FIG. 12. The same as Fig. 1 except for Q = − sign of chirality in accordance with the index theorem ofEq. (23). For non-zero modes, we observe the pattern ofeight-fold degeneracy as in other examples for Q = 0 inFig. 1, Q = − Q = − Appendix D: Recursion relationships for chiralityoperators
We define the chirality operator as (cid:104) f sα | [ γ ⊗ | f sβ (cid:105) ≡ (cid:90) d x [ f sα ( x A )] † ( γ ⊗ AB U ( x A , x B ) f sβ ( x B ) (D1)( γ S ⊗ ξ T ) AB = 14 Tr( γ † A γ S γ B γ † T ) (D2) U ( x A , x B ) = P SU (3) (cid:20) (cid:88) p ∈C V ( x A , x p ) V ( x p , x p ) V ( x p , x p ) V ( x p , x B ) (cid:21) (D3)First, let us prove the following theorem. Theorem D.1. [ γ ⊗ γ ⊗
1] = [1 ⊗
1] (D4)
Proof.
Let us first rewrite [ γ ⊗ as follows,[ γ ⊗ AC = (cid:88) B ( γ ⊗ AB U ( x A , x B ) · ( γ ⊗ BC U ( x B , x C )= (cid:88) B [( γ ⊗ AB ( γ ⊗ BC ] · [ U ( x A , x B ) U ( x B , x C )] (D5)By the way we know that( γ ⊗ AB = 14 Tr( γ † A γ γ B δ B ¯ A [ η ( A ) η ( A ) η ( A ) η ( A )]= δ B ¯ A η ( A ) (D6)where ¯ A µ = ( A µ + 1) mod 2, and η µ ( A ) = ( − X µ , for µ = 1 , , , , (D7) X µ = (cid:88) ν<µ A ν , (D8) η ( A ) = η ( A ) η ( A ) η ( A ) η ( A ) = ( − A + A (D9)Similarly, we find that( γ ⊗ BC = δ CB η ( B ) (D10)Hence, we can rewrite Eq. (D5) as follows[ γ ⊗ AC = (cid:88) B [ δ B ¯ A η ( A ) δ C ¯ B η ( B )] · [ U ( x A , x B ) U ( x B , x C )]= δ AC [ U ( x A , x ¯ A ) U ( x ¯ A , x A )] (D11)where we use the helpful identity: η ( ¯ A ) = η ( A ).By the way, thanks to the SU (3) projection in Eq. (D3), U ( x ¯ A , x A ) = [ U ( x A , x ¯ A )] † ∈ SU (3). Hence,[ U ( x A , x ¯ A ) U ( x ¯ A , x A )] = 1. Therefore, we can rewriteEq. (D11) as follows,[ γ ⊗ AC = δ AC = [1 ⊗ AC (D12)Hence, we just prove that [ γ ⊗ = [1 ⊗ γ ⊗ n +1 = (cid:0) [ γ ⊗ (cid:1) n · [ γ ⊗
1] (D13)= ([1 ⊗ n · [ γ ⊗
1] (D14)= [1 ⊗ · [ γ ⊗
1] (D15)= [ γ ⊗ . (D16)Using the results of Eq. (D4), we can prove another re-cursion relationship as follows,[ γ ⊗ n = (cid:0) [ γ ⊗ (cid:1) n (D17)= ([1 ⊗ n (D18)= [1 ⊗
1] (D19)Finally, we can prove the following theorem.
Theorem D.2. [ 1 + γ ⊗ γ ⊗
1] = [ 1 + γ ⊗
1] (D20)
Proof. [ 1 + γ ⊗ = 14 ([1 ⊗
1] + [ γ ⊗ = 14 (cid:0) [1 ⊗
1] + 2[ γ ⊗
1] + [ γ ⊗ (cid:1) = 12 ([1 ⊗
1] + [ γ ⊗ γ ⊗
1] (D21)(Q.E.D.)Using Eq. (D20), we can prove that for n > n ∈ Z , [ 1 + γ ⊗ n = [ 1 + γ ⊗
1] (D22)by induction.At this stage, it will be trivial to prove that[ 1 + γ ⊗ − γ ⊗
1] = 0 (D23)
Appendix E: Examples for the leakage pattern forzero modes
Let us begin with the case of Q = −
2. In Fig. 13,we show leakage patterns of the chirality operator for0 j=0-1 R j=0-2 R j=±1 j=±2 | Γ | i i (a) | Γ | i = | Γ ( λ i , λ ) | j=0-1 R j=0-2 R j=±1 j=±2 | Γ | i i (b) | Γ | i = | Γ ( λ i , λ ) | FIG. 13. [ γ ⊗
1] leakage pattern for the first quartet of would-be zero modes at Q = − j=0-1 R j=0-2 R j=±1 j=±2 | Ξ | i i (a) | Ξ | i = | Ξ ( λ i , λ ) | j=0-1 R j=0-2 R j=±1 j=±2 | Ξ | i i (b) | Ξ | i = | Ξ ( λ i , λ ) | FIG. 14. [1 ⊗ ξ ] leakage pattern for the first quartet of would-be zero modes at Q = − the first set of the zero modes at Q = −
2. In Fig. 14,we present the leakage patterns of the shift operator forthe first set of the zero modes at Q = −
2. By comparingFig. 13 with Fig. 14, we find that the chiral Ward identityof Eqs. (80) and (81) is well respected.In Fig. 15, we show leakage patterns of the chiralityoperator for the second set of the zero modes at Q = − Q = − Q = − Q = −
2. Hence, we j=0-1 R j=0-2 R j=±1 j=±2 | Γ | i i (a) | Γ | i = | Γ ( λ i , λ ) | j=0-1 R j=0-2 R j=±1 j=±2 | Γ | i i (b) | Γ | i = | Γ ( λ i , λ ) | FIG. 15. [ γ ⊗
1] leakage pattern for the second quartet ofwould-be zero modes at Q = − j=0-1 R j=0-2 R j=±1 j=±2 | Ξ | i i (a) | Ξ | i = | Ξ ( λ i , λ ) | j=0-1 R j=0-2 R j=±1 j=±2 | Ξ | i i (b) | Ξ | i = | Ξ ( λ i , λ ) | FIG. 16. [1 ⊗ ξ ] leakage pattern for the second quartet ofwould-be zero modes at Q = − choose the third set of the zero modes as our example. InFig. 17, we show leakage patterns of the chirality operatorfor the third set of the zero modes at Q = −
3. In Fig. 18,we present the leakage pattern of the shift operator forthe third set of the zero modes at Q = −
3. By comparingFig. 17 with Fig. 18, we find that the chiral Ward identityof Eqs. (80) and (81) is well preserved.
Appendix F: Examples for the leakage pattern fornon-zero modes
Let us begin with an example with Q = 0. Sincethe gauge configuration with Q = 0 usually has no zero1 j=0-1 R j=0-2 R j=0-3 R j=±1 | Γ | i i (a) | Γ | i = | Γ ( λ i , λ ) | j=0-1 R j=0-2 R j=0-3 R j=±1 | Γ | i i (b) | Γ | i = | Γ ( λ i , λ ) | FIG. 17. [ γ ⊗
1] leakage pattern for the third quartet ofwould-be zero modes at Q = − j=0-1 R j=0-2 R j=0-3 R j=±1 | Ξ | i i (a) | Ξ | i = | Ξ ( λ i , λ ) | j=0-1 R j=0-2 R j=0-3 R j=±1 | Ξ | i i (b) | Ξ | i = | Ξ ( λ i , λ ) | FIG. 18. [1 ⊗ ξ ] leakage pattern for the third quartet ofwould-be zero modes at Q = − mode ( n − = n + = 0), it is relatively easy to study non-zero modes. In Fig. 19, we present leakage patterns ofthe chirality operator Γ = [ γ ⊗
1] for non-zero modes { λ , λ , λ , λ } = { λ j,m | j = +1 , m = 1 , , , } in the j = +1 quartet when Q = 0. The results show that theΓ leakages for non-zero modes λ +1 ,m mostly go into theirparity partners of { λ , λ , λ , λ } = { λ j,m | j = − , m =1 , , , } in the j = − j = ± , ± j = − Q = − = [1 ⊗ ξ ] for the non-zero modes { λ , λ , λ , λ } of λ +1 ,m in the j = +1 quartet when Q = 0. For the Ξ j=±1 j=±2 j=±3 | Γ | i i (a) | Γ | i = | Γ ( λ i , λ ) | j=±1 j=±2 j=±3 | Γ | i i (b) | Γ | i = | Γ ( λ i , λ ) | j=±1 j=±2 j=±3 | Γ | i i (c) | Γ | i = | Γ ( λ i , λ ) | j=±1 j=±2 j=±3 | Γ | i i (d) | Γ | i = | Γ ( λ i , λ ) | FIG. 19. [ γ ⊗
1] leakage pattern for the first quartet of non-zero modes at Q = 0. operator, we find a great part of leakages from non-zeromodes λ +1 ,m within their quartet members of j = +1.Meanwhile, there are only negligible amounts of leakagesto its parity partner quartet elements of j = − j = ± , ±
3. This observation cor-responds to the case of Q = − in Fig. 19 and Ξ in Fig. 20 arerelated to each other by the Ward identity of Eq. (97).In Figs. 21 and 22, we present leakage patterns ofΓ and Ξ operators, respectively, for non-zero modes { λ , λ , λ , λ } = { λ j,m | j = +2 , m = 1 , , , } in the j = +2 quartet when Q = 0. Similar to the above casesfor j = +1, Γ leakages for non-zero modes of j = +2mostly go to their parity partner quartet elements of j = − { λ , λ , λ , λ } = { λ j,m | j = − , m = 1 , , , } ,and Ξ leakages for them mostly go to within their quar-tet members of j = +2: { λ , λ , λ , λ } . There are2 j=±1 j=±2 j=±3 | Ξ | i i (a) | Ξ | i = | Ξ ( λ i , λ ) | j=±1 j=±2 j=±3 | Ξ | i i (b) | Ξ | i = | Ξ ( λ i , λ ) | j=±1 j=±2 j=±3 | Ξ | i i (c) | Ξ | i = | Ξ ( λ i , λ ) | j=±1 j=±2 j=±3 | Ξ | i i (d) | Ξ | i = | Ξ ( λ i , λ ) | FIG. 20. [1 ⊗ ξ ] leakage pattern for the first quartet of non-zero modes at Q = 0. only negligible amount of leakages to other quartets forboth operators.Now let us examine the leakage patterns when would-be zero modes exist ( Q (cid:54) = 0). In Figs. 23 and 24, wepresent leakage patterns of Γ and Ξ operators, re-spectively, for non-zero modes { λ , λ , λ , λ } in the j = +1 quartet when Q = −
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