Domain-Wall Fermion with $ R_5 $ Symmetry

We present the domain-wall fermion operator which is reflection symmetric in the fifth dimension, with the approximate sign function $ S(H) $ of the effective 4-dimensional Dirac operator satisfying the bound $ |1-S(\lambda)| \le 2 d_Z $ for $ \lambda^2 \in [\lambda_{min}^2, \lambda_{max}^2] $, where $ d_Z $ is the maximum deviation $ | 1- \sqrt{x} R_Z(x) |_{\rm max} $ of the Zolotarev optimal rational polynomial $ R_Z(x) $ of $ 1/\sqrt{x} $ for $ x \in [1, \lambda_{max}^2/\lambda_{min}^2] $, with degrees $ (n-1,n) $ for $ N_s = 2n $, and $ (n,n) $ for $ N_s = 2 n + 1 $.


I. INTRODUCTION
In lattice QCD with exact chiral symmetry [1][2][3], the overlap Dirac operator with bare quark mass m q in general can be written as where H = cH w (1 + dγ 5 H w ) −1 , and c and d are constants. Here H w = γ 5 D w , and D w is the standard Wilson-Dirac operator minus the parameter m 0 ∈ (0, 2).
The eigenmodes of (1) consist of complex conjugate pairs, and (for topologically nontrivial gauge field) real eigenmodes with definite chiralities at m q and 1/r satisfying the chirality sum rule [4], n + − n − + N + − N − = 0, where n ± (N ± ) denote the number of eigenmodes at m q (1/r) with ± chirality. Empirically, the real eigenmodes always satisfy either (n − = N + = 0, n + = N − ) or (n + = N − = 0, n − = N + ). Thus, we have where H 2 ± = P ± (D † D), and P ± = (1 ± γ 5 )/2. It follows that the pseudofermion action for any number of flavors of overlap fermion can be expressed in terms of n ± and H 2 ± (Hermitian and positive-definite), thus is amenable to the hybrid Monte Carlo simulation (HMC) [5]. However, this approach requires the computation of the change of n ± at each step of the molecular dynamics in HMC, which is prohibitively expensive for large lattices [6,7]. Moreover, the discontinuity of the fermion determinant at the topological boundary highly suppresses the crossing rate between different topological sectors, thus renders HMC failing to sample all topological sectors ergodically.
These difficulties can be circumvented by using domain-wall fermion (DWF) with finite N s in the fifth dimension. Then HMC of lattice QCD with DWF on the 5-dimensional lattice can sample all topological sectors ergodically and also keep the chiral symmetry at a good precision. This has been demonstrated for 2-flavors QCD [8,9], and (1+1)-flavors QCD [10]. For DWF with finite N s , it is vital to preserve the chiral symmetry maximally, or equivalently, the approximate sign function S(H) of the effective 4-dimensional Dirac operator satisfies the bound, where sn(v s ; κ ′ ) is the Jacobian elliptic function with argument v s (see Eq. (13) of Ref. [11]) and modulus κ ′ = 1 − 1/b, b = λ 2 max /λ 2 min . Here λ 2 max and λ 2 min are the upper-bound and lower-bound for the eigenvalues of H 2 . It should be emphasized that λ min and λ max have to be fixed properly for each set of simulations, depending on the parameters β = 6/g 2 , quark masses, and lattice size, such that the desired precision of chiral symmetry can be attained with the minimal cost of the simulation.
Nevertheless, (2) breaks the R 5 (reflection) symmetry in the fifth dimension, which is essential for obtaining the exact pseudofermion action for hybrid Monte Carlo simulation of one-flavor DWF [10], as well as other applications. In this paper, we obtain the weights satisfying the R 5 symmetry (ω s = ω Ns−s+1 , s = 1, · · · , N s ) such that the approximate sign function S(H) of the effective 4-dimensional Dirac operator satisfies the bound, |1 − S(λ)| ≤ 2d Z , which is twice of that of the optimal DWF without R 5 symmetry [11]. In other words, if one imposes the R 5 symmetry, this is the optimal chiral symmety one can have.

II. OPTIMAL DOMAIN-WALL FERMION
In general, the 5-dimensional lattice operator of DWF can be written as [12] [D(m)] xx ′ ;ss ′ = (ρ s D w + 1) where N s is the number of sites in the fifth dimension, m ≡ rm q , m q is the bare quark mass, and r = 1/[2m 0 (1 − dm 0 )]. Including the action of the Pauli-Villars fields (with bare mass m P V = 1/r), the partition function of DWF in a gauge background can be integrated successively to obtain the fermion determinant of the effective 4-dimensional Dirac operator, For the optimal DWF without R 5 symmetry [11], the weights {ω s } are fixed according to the formula (2), then S(H) is equal to the Zolotarev optimal rational approximation of , with degrees (n, n) for N s = 2n + 1, and (n − 1, n) for N s = 2n. However, it breaks the R 5 symmetry, i.e., In the following, we construct the optimal DWF satisfying the R 5 symmetry.

III. OPTIMAL DOMAIN-WALL FERMION WITH R 5 SYMMETRY
First we recall the basic features of the optimal rational approximation of the ratio of two positive and continuous According to de la Vallée-Poussin's theorem and Chebycheff's theorem, the necessary and sufficient condition for R (m,n) (x) to be the optimal rational approximation of f (x)/g(x) for has (m + n + 2) alternate change of sign in the interval [1, b], and attains its maxima and minima (all with equal magnitude), say, at consecutive points (x i , i = 1, · · · , m + n + 2) For f (x)/g(x) = 1/ √ x, Zolotarev obtained two optimal rational polynomials in 1877, [13], in terms of the Jacobian elliptic functions. The basic formulas of Zolotarev optimal rational polynomial are collected in the Appendix. (See also Refs. [14,15].) For the optimal DWF without R 5 symmetry [11], the approximate sign function S(H) of the effective 4-dimensional lattice Dirac operator is exactly equal to HR , as pointed out in Ref. [15].
A. N s = 2n (even) For the case N s = 2n (even), the requirement of R 5 symmetry implies that ω s = ω Ns+1−s , and unlike the δ (n−1,n) Z (x) (shown in Fig. 1(a)) which varies alternatively between +d Z and −d Z . Nevertheless, one can shift δ Z (x) by a constant +d Z , simply by changing the overall Then the maximal deviation becomes 2d Z , which is twice of that of the Zoloterev optimal rational polynomial. Obviously, this is the optimal chiral symmetry one can have for DWF with R 5 symmetry. This can be attained by requiring δ(λ) to have 2n+1 alternative maxima and minima for λ ∈ [λ min , λ max ], at the consecutive points (λ min = λ 1 < λ 2 < · · · < λ 2n+1 = λ max ), where d Z is the maximum . This immediately implies that the weights in (3) can be fixed by the positions of the minima of δ(x), where sn(u; κ ′ ) is the Jacobian elliptic function with modulus κ ′ = 1 − 1/b, and K ′ is the complete elliptic function of the first kind with modulus κ ′ . Then the approximate sign

function S(H) of the effective 4-dimensional Dirac operator (4) becomes
which is exactly the same as the Zolotarev optimal rational approximation of the sign function (see Eq. (18)) except replacing D 0 by D ′ 0 . Note that if we replace D 0 by D ′′ 0 , then δ(x) is shifted by a constant −d Z , thus its maxima become 0, and minima −2d Z .
However, this cannot be realized by DWF with even N s and R 5 symmetry, due to the constraint (6), δ(x) ≥ 0.
For the case N s = 2n + 1, the requirement of R 5 symmetry implies that ω s = ω Ns+1−s , and (1 + ω s λ) 2 , which can be positive, zero, or negative. Thus, we have two options to attain the optimal chiral symmetry with R 5 symmetry, i.e., to shift δ(λ) by a constant +d z , or −d z .
The first option is to require δ(λ) to have 2n + 2 alternative maxima and minima in the [1, b]. Thus the weights in (3) can be fixed by the positions of the minima of δ(x), Note that the smallest weight ω n+1 is chosen to be the unpaired one at the center of the fifth dimension, such that δ(λ) does not change sign for all λ ∈ [λ min , λ max ]. Then the approximate sign function S(H) of the effective 4-dimensional Dirac operator (4) becomes which is exactly the same as the Zolotarev optimal rational approximation of the sign function (see Eq. (19)) except for the overall coefficent d ′ 0 , The second option is to require δ(λ) to have 2n + 2 alternative maxima and minima in the interval [λ min , λ max ], and λ max = 6.20. In (a), the weights are computed according to (12), while in (b), according to (15).

IV. NUMERICAL TEST
Theoretically, the sign function error |1 − S(H)| of the optimal DWF with R 5 symmetry is twice of that without R 5 symmetry. Nevertheless, it is interesting to check their difference in large-scale simulations of lattice QCD, by computing the residual mass [12] (9). The strange quark is simulated with the exact pseudofermion action for one-flavor DWF [10], and the up and down quarks are simulated with the 2-flavors algorithm as outlined in [9]. We generate the initial 400 trajectories with 2 Nvidia GTX-TITAN GPUs working together. After discarding the initial 240 trajectories for thermalization, we sample one configuration every 5 trajectories, resulting 32 "seed" configurations. Then we use these 32 seed configurations as the initial configurations for 32 independent simulations on 32 Nvidia GTX-TITAN GPUs. Each GPU generates 100-110 trajectories, and we accumulate a total of 3300 trajectories. From the saturation of the binning error of the plaquette, as well as the evolution of the topological charge, we estimate the autocorrelation time to be around 10 trajectories. Thus we sample one configuration every 10 trajectories, and obtain 330 configurations for this ensemble.
Then we compute two sets of quark propagators with point source at the origin and m q a = 0.005, one set with the {ω s } exactly the same as the sea-quarks (with R 5 symmetry), and the other set with the {ω s } (2) without R 5 symmetry. The residual mass of the set with R 5 symmetry is (m res a) R 5 = 0.00024 (1), while that of the set without R 5 symmetry is m res a = 0.00014 (1). Thus the ratio (m res ) R 5 /m res ≃ 1.71 (14), consistent with the theoretical expectation.

V. CONCLUDING REMARKS
With the weights (9), (12) and (15) for optimal DWF with R 5 symmetry, together with those (2) for optimal DWF without R 5 symmetry [11], this completes the study of DWF with the approximate sign function S(H) in the effective 4-dimensional lattice Dirac operator where d Z is the maximum deviation |1 − √ xR Z (x)| max of the Zolotarev optimal rational polynomial R Z (x) of 1/ √ x for x ∈ [1, λ 2 max /λ 2 min ], with degrees (n − 1, n) for N s = 2n, and (n, n) for N s = 2n + 1.
The correspondence between the approximate sign function S(H) of the optimal DWF and the Zolotarev optimal rational polynomial gives the optimal rational approximation a more general viewpoint. In general, the deviation δ(x) = 1 − √ xR Z (x) can be shifted by a constant ǫ, simply by adjusting the overall coefficient D 0 in (18) or d 0 in (19). In particular, for R (n−1,n) Z , if D 0 is replaced by D ′ 0 (7), then ǫ = +d Z , δ max = 2d Z and δ min = 0; while by D ′′ 0 (11), ǫ = −d Z , δ max = 0 and δ min = −2d Z . Similarly, for R (n,n) Z , if d 0 is replaced by d ′ 0 (14), ǫ = +d z , δ max = 2d z and δ min = 0; while by d ′′ 0 (14), ǫ = −d z , δ max = 0 and δ min = −2d z . Even though this does not satisfy the criterion that the maxima and the minima of δ(x) all have the same magnitude and δ min = −δ max , the most salient featues of the optimal rational approximation are preserved, namely, the number of alternate maxima and minima is (m + n + 2), with (n + 1) maxima and (m + 1) minima, as well as all maxima (minma) are equal. We can regard this as the generalized optimal rational approximation (with a constant shift ǫ). For ǫ > 0, this can be realized by DWF with/without R 5 symmetry.