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The Möbius Domain Wall Fermion Algorithm

Richard C. Brower, Harmut Neff, Kostas Orginos

arXiv:1206.5214v2hep-lat

TL;DR

The paper reviews generalized Möbius domain wall fermions to improve finite-L_s chiral symmetry while preserving the same overlap action as L_s approaches infinity. Its tuned scaling and preconditioning reduce chiral violations, with residual mass changing from O(1/L_s) for Shamir to O(1/L_s^2) for Möbius at large L_s.

  • Problem

    The paper addresses how to improve the algorithmic efficiency and finite-L_s chiral properties of domain wall fermions while relating them to the overlap formulation.

  • Method

    It reviews Möbius domain wall fermions, derives their currents and Ward-Takahashi identities, maps the five-dimensional action to a four-dimensional overlap operator, and uses 4d red-black preconditioning.

  • Results

    At large L_s, residual mass scales as O(1/L_s^2) for appropriately scaled Möbius fermions versus O(1/L_s) for Shamir, while Möbius at L_s = 32 may roughly correspond to Shamir with L_s = O(10^3).

  • Takeaways & Limitations

    Möbius rescaling can substantially reduce the required fifth-dimensional extent, with simulations potentially using L_s = 4 instead of Shamir’s L_s = 16 at a tolerable compromise in chirality.

  • Takeaways & Limitations

    The mapped currents are non-local in four dimensions, and the Pauli–Villars contribution is required for some observables such as conserved current-current correlators.

Abstract

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We present a review of the properties of generalized domain wall Fermions, based on a (real) Möbius transformation on the Wilson overlap kernel, discussing their algorithmic efficiency, the degree of explicit chiral violations measured by the residual mass ($m_{res}$) and the Ward-Takahashi identities. The Möbius class interpolates between Shamir's domain wall operator and Boriçi's domain wall implementation of Neuberger's overlap operator without increasing the number of Dirac applications per conjugate gradient iteration. A new scaling parameter ($α$) reduces chiral violations at finite fifth dimension ($L_s$) but yields exactly the same overlap action in the limit $L_s \rightarrow \infty$. Through the use of 4d Red/Black preconditioning and optimal tuning for the scaling $α(L_s)$, we show that chiral symmetry violations are typically reduced by an order of magnitude at fixed $L_s$. At large $L_s$ we argue that the observed scaling for $m_{res} = O(1/L_s)$ for Shamir is replaced by $m_{res} = O(1/L_s^2)$ for the properly tuned Möbius algorithm with $α= O(L_s)$

1 Introduction

The paper reviews domain wall and overlap fermions as approximate realizations of chiral symmetry, then extends their correspondence and Ward–Takahashi analysis to Möbius fermions.

  • Exact lattice chiral symmetry separates chiral-symmetry breaking from Lorentz breaking at finite lattice spacing.
  • Finite fifth dimension or finite sign-function approximation produces small chiral violations measured through the Ginsparg–Wilson defect.
  • Möbius fermions introduce a scaling parameter that improves the finite-Ls approximation while leaving the infinite-Ls overlap action unchanged.
  • The review extends domain wall–overlap correspondence to vector and axial currents, Ward–Takahashi identities, and residual-mass analysis.
  • The paper organizes general chiral-action analysis, Möbius construction, preconditioning, numerical tests, overlap mapping, and current identities.

2 M¨obius Domain Wall Fermions

Möbius domain wall fermions generalize the kernel through a real Möbius transformation and rescale its spectrum to improve finite-Ls chiral approximations while preserving the effective four-dimensional action.

  • Finite-Ls chiral violation is represented by the Ginsparg–Wilson defect ΔLs[H5], equivalently measuring imperfect chiral projectors.
  • The residual mass is obtained from the axial Ward identity and measures finite-Ls chiral-symmetry violation through the operator ψγ5ΔLψ.
  • The generalized kernel contains Shamir and Boriçi formulations as special cases without additional Wilson-kernel applications per domain wall iteration.
  • Möbius rescaling preserves the operator’s eigenvectors and therefore implements essentially the same lattice Dirac action with an improved chiral approximation.
  • At Ls = 12 and α = 4, Möbius and Shamir at Ls = 48 are indistinguishable on a loglog plot, with sign-function differences below 10^-3 for all H5 eigenvectors.
  • The five-dimensional action maps explicitly to a four-dimensional overlap operator, enabling reconstruction, current construction, and small-Ls preconditioning.
  • Removing redundant bulk pseudo-fermions can reduce stochastic-estimator noise, but the force term requires two inversions instead of one.

3 Red Black Preconditioning

Efficient Möbius inversion requires four-dimensional red–black preconditioning, which avoids an expensive Wilson-operator inner inversion and substantially reduces iteration cost.

  • Standard five-dimensional even–odd preconditioning is inefficient for Möbius fermions because the Wilson operator connects both space-time and fifth-dimensional neighbors.
  • The proposed partition uses a four-dimensional checkerboard with the same color across all fifth-axis sites, yielding a tridiagonal red-red and black-black structure.
  • The resulting chiral-sector solves use O(Ls) multiply-add steps and negligible cost relative to a single Dirac application.
  • The new Möbius red–black preconditioning provides about a factor 2.5 speedup on the tested lattice set.
  • For Shamir fermions, the new four-dimensional preconditioning performs comparably to the original five-dimensional scheme in iteration count.
  • With fixed Shamir parameters M5 = 1.8 and a5 = 1 at m = 0.06, residual-mass behavior is evaluated against Wilson Dirac applications as Ls varies.

4 Numerical Results

The numerical study evaluates how Möbius parameters tune residual chiral-symmetry breaking and computational cost relative to Shamir domain wall fermions. Optimal scaling α substantially improves residual-mass behavior at fixed L_s, while parameter choices such as a_5 and M_5 provide secondary optimization effects.

  • Retuning α(L_s) makes residual mass continue to fall exponentially over the studied range, before the tuned Möbius asymptotic behavior crosses to O(1/L_s^2).Standard Shamir crosses earlier from exponential decrease to O(1/L_s) behavior.
  • m_res = 6 × 10^-4 at L_s = 24 and 3 × 10^-5 at L_s = 32, whereas Shamir apparently requires L_s = 10^2 and 10^3, respectively.The comparison supports the estimated O(1/L_s) Shamir versus O(1/L_s^2) Möbius asymptotic behavior.
  • 4.2 Dependence on a5 and M5 at fixed Ls: At L_s = 8 and M_5 = 1.5, a_5 = 1.5 achieves roughly the Shamir residual mass at L_s = 16 with roughly half the Dirac applications.The study identifies a_5 = 1.5 as optimal in this scan, while α changes both residual mass and application count.
  • 4.2 Dependence on a5 and M5 at fixed Ls: With a_5 = 1.5 and L_s = 8, M_5 = 1.5 and M_5 = 1.4 perform equally well, with M_5 = 1.4 giving slightly lower residual mass at modestly higher cost.The overall tuning selects M_5 = 1.5, because α is the most important optimization parameter.
  • 4 Numerical Results: The optimization choices made at bare quark mass m = 0.06 remain optimal when repeated at m = 0.02.
  • 4.3 Zolotarev Polynomials for the quenched fields: Zolotarev approximation uses fewer fifth-dimensional sites but requires more Wilson Dirac applications and has not been made competitive with polar-form Möbius.The standard Zolotarev approximation also loses reflection symmetry and residual-mass positivity, so m_res is not an appropriate chiral-violation measure there.

5 Domain Wall/Overlap Operator Correspondence

The paper establishes a complete finite-Ls correspondence between overlap and domain wall fermion correlators, extending it to vector and axial currents and their Ward identities. It shows how current insertions and propagator identities map between the four-dimensional overlap and five-dimensional domain wall formulations, including contact terms and Pauli–Villars contributions.

  • 5 Domain Wall/Overlap Operator Correspondence: The correspondence maps overlap and domain wall fermion path integrals and correlators, with five-dimensional Dirac and Pauli–Villars fields underlying the domain wall formulation.The mapping is formulated through field redefinitions and a central preconditioned domain wall operator.
  • 5 Domain Wall/Overlap Operator Correspondence: Wick’s theorem and source-based generating functions extend the basic propagator identity to the full set of fermionic correlator identities.The propagator relation is presented as an example of the broader correspondence.
  • 5 Domain Wall/Overlap Operator Correspondence: The boundary-field definition smears the boundary by one lattice unit, simplifying overlap–domain wall correlation identities and making them insensitive to normalization changes in the domain wall operator.The paper presents this definition as more uniform across Shamir, Möbius, and future variants.
  • 5.1 Domain Wall/Overlap correspondence for currents: Vector-current identities equate overlap current variations with domain wall current insertions, including a five-dimensional bulk sum over the fifth coordinate.The bulk representation provides the domain wall realization of the overlap current kernel.
  • 5.1 Domain Wall/Overlap correspondence for currents: The current correspondence includes a gauge-field-dependent contact term at coincident points, while Pauli–Villars contributions can be required for conserved current–current correlators.The contact term arises because the chosen quark field depends on the gauge field.
  • 5.1 Domain Wall/Overlap correspondence for currents: The overlap axial current is defined by imposing a condition because the global lattice axial symmetry is non-local and gauge-background dependent.For non-local overlap actions, gauging the action is described as the preferable current-construction approach.

6 Ward-Takahashi Identities

The paper derives conserved and partially conserved vector and axial currents for Möbius domain wall fermions and relates their Ward–Takahashi identities to finite-L_s chiral violations. It identifies the residual mass as a measure of imperfect Ginsparg–Wilson symmetry and analyzes its scaling.

  • Vector current: The Möbius vector current is obtained by gauging the five-dimensional action, with the four-dimensional current formed by summing over the fifth dimension.The summed current is conserved because no current leaks through the fifth-dimensional boundaries.
  • Axial current: The axial Ward–Takahashi identity follows from five-dimensional flux conservation, with the mid-plane providing the natural axial-current definition.Changing the mid-plane location only redefines the current by a term with zero total divergence.
  • Residual chiral violations: Finite L_s produces a breaking term Δ_Ls that measures chiral-symmetry violation and corresponds to the defect in the Ginsparg–Wilson relation.The same finite-L_s approximation controls violations in both domain wall and overlap formulations.
  • Singlet anomaly: The singlet axial current retains the anomaly, whose L_s→∞ limit reproduces the lattice Atiyah–Singer index and continuum topological charge density.Pauli–Villars fields cancel the heavy cutoff-mode contributions.
  • Residual chiral violations: For Shamir fermions m_res scales as O(1/L_s), whereas appropriately scaled Möbius fermions are expected to achieve O(1/L_s^2) asymptotically.The improved scaling is attributed to suppressing errors from small eigenvalues with α proportional to L_s.
  • Residual chiral violations: Tuning m_res to zero with oscillating sign-function approximations does not ensure exact chiral symmetry because higher-dimensional chiral-breaking operators may remain.The relevant concern is higher-order terms, especially dimension-5 operators.

7 Conclusions

The conclusions present Möbius fermions as an efficient domain wall formulation with improved finite-L_s chirality and a systematic domain wall–overlap correspondence. Numerical tests support quadratic residual-mass scaling and substantial reductions in the fifth-dimensional extent.

  • 7 Conclusions: Möbius fermions require no additional Wilson-kernel applications per domain-wall iteration, but efficient implementation requires four-dimensional checkerboard preconditioning.Conventional five-dimensional red-black preconditioning would introduce a Wilson-operator inverse as an inner loop.
  • 7 Conclusions: Conserved and partially conserved axial currents, their Ward–Takahashi identities, and the finite-L_s overlap correspondence are formulated for Möbius fermions.The formalism also gives a derivation of the overlap operator at finite chemical potential.
  • 7 Conclusions: m_res scales as O(1/L_s^2) for appropriately scaled Möbius fermions, compared with O(1/L_s) for Shamir at large L_s.Tests suggest Möbius at L_s=32 may correspond roughly to Shamir with L_s=O(10^3), though direct verification is impractical.
  • 7 Conclusions: Möbius can reduce the fifth-dimensional extent substantially; a Shamir simulation at L_s=16 may be run with Möbius at L_s=4 with tolerable chirality compromise.Figure 10 compares the corresponding sign-function approximations.
  • 7 Conclusions: Combining Möbius fermions with mass preconditioning or multigrid methods remains an open avenue for balancing chirality, performance, and topological-sector thermalization.The authors explicitly describe these interactions as requiring further exploration.

A M¨obius generalization of Domain Wall Operator

The generalized domain wall operator extends the fifth-dimensional action with next-to-nearest-neighbor couplings and supports alternative matrix forms for implementation. These forms preserve the cyclic fifth-dimensional structure while offering possible coding advantages.

  • A Möbius generalization of Domain Wall Operator: The Möbius domain wall operator includes next-to-nearest-neighbor interactions along the fifth dimension.The fifth-dimensional index is cyclic modulo L_s, with chiral projectors defining the couplings.
  • A Möbius generalization of Domain Wall Operator: The operator is represented in matrix form, with an equally valid left form introduced for potential implementation efficiency.The left representation differs by how the fifth-dimensional structure is arranged.

A.1 LDU decomposition

The appendix reduces the five-dimensional domain wall matrix to the four-dimensional overlap form through permutation, LDU decomposition, Gaussian elimination, and back substitution. The resulting factorization establishes determinant and operator correspondences between the formulations.

  • A.1 LDU decomposition: A standard LDU decomposition relates the five-dimensional domain wall matrix to a four-dimensional overlap operator.The reduction proceeds through pivoting, Gaussian elimination, and back substitution.
  • A.1 LDU decomposition: Reflecting the fifth axis exchanges the upper- and lower-triangular conventions without changing the underlying factorization.The transformation converts the U matrix into an L matrix and vice versa.
  • A.1 LDU decomposition: The transfer-matrix construction organizes propagation between neighboring fifth-dimensional slices and encodes the Möbius operator through s-dependent coefficients.The transfer matrices satisfy special diagonal and adjacent-slice relations.
  • A.1 LDU decomposition: The factorized five-dimensional operator contains D_ov(m) as its nontrivial four-dimensional diagonal block.The remaining diagonal blocks are identities in the displayed factorization.
  • A.1 LDU decomposition: The determinant identity shows equivalence between the overlap measure and the domain wall measure supplemented by Pauli–Villars pseudofermions.The auxiliary factors cancel in the determinant product.

A.2 Domain Wall Correlators

This section develops the propagators and correlators needed for domain-wall calculations, including bulk, boundary-to-bulk, and quark-sector quantities used in Ward–Takahashi identities and the chiral anomaly.

  • A.2 Domain Wall Correlators: Bulk-to-bulk propagators are introduced for general domain-wall propagation and for propagators between quark spinors.The general bulk-to-bulk propagator is identified with Eq. 2.28.
  • A.2 Domain Wall Correlators: Boundary-to-bulk propagators connect domain-wall boundaries to interior points and are required for the Ward–Takahashi identities.The passage identifies these propagators as a fundamental set.
  • A.2 Domain Wall Correlators: The construction also distinguishes two varieties of anti-quark spinors and includes a domain-wall inverse formula.The supplied passages mention both ingredients without providing their complete expressions.
  • A.2 Domain Wall Correlators: A factorization formula is given for the mass term: M(m) = ADW (m)M(1).The formula is presented as a convenient factorization for the relevant quantity.
  • A.2 Domain Wall Correlators: Two correlators are introduced for the flavor chiral Ward identity, while an additional correlator is needed for the axial identity.The passages distinguish the flavor chiral Ward identity from the axial case.
  • A.2 Domain Wall Correlators: The singlet-current chiral anomaly requires a dedicated correlator.The passage states that this correlator is needed for the singlet-current chiral anomaly.
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