Equivalence of Chiral Fermion Formulations

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Equivalence of Chiral Fermion Formulations. A D Kennedy School of Physics, The University of Edinburgh Robert Edwards, Blint Jo , Kostas Orginos ( JLab ) Urs Wenger (ETHZ). On-shell chiral symmetry Neubergers Operator Into Five Dimensions Kernel Schur Complement Constraint - PowerPoint PPT Presentation

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• Equivalence of Chiral Fermion Formulations A D KennedySchool of Physics, The University of Edinburgh

Robert Edwards, Blint Jo, Kostas Orginos (JLab) Urs Wenger (ETHZ)

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ContentsOn-shell chiral symmetryNeubergers OperatorInto Five DimensionsKernelSchur ComplementConstraintApproximationtanhRepresentationContinued Fraction Partial FractionCayley TransformChiral Symmetry BreakingNumerical StudiesConclusions

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Chiral Fermions

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On-shell chiral symmetry: IIt is possible to have chiral symmetry on the lattice without doublers if we only insist that the symmetry holds on shell

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On-shell chiral symmetry: II

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Neubergers Operator: IWe can find a solution of the Ginsparg-Wilson relation as follows

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Into Five DimensionsH Neuberger hep-lat/9806025A Borii hep-lat/9909057, hep-lat/9912040, hep-lat/0402035A Borii, A D Kennedy, B Pendleton, U Wenger hep-lat/0110070R Edwards & U Heller hep-lat/0005002 (T-W Chiu) hep-lat/0209153, hep-lat/0211032, hep-lat/0303008R C Brower, H Neff, K Orginos hep-lat/0409118 Hernandez, Jansen, Lscher hep-lat/9808010

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Neubergers Operator: IIIs DN local?It is not ultralocal (Hernandez, Jansen, Lscher)It is local iff DW has a gapDW has a gap if the gauge fields are smooth enoughq.v., Ben Svetitskys talk at this workshop (mobility edge, etc.)It seems reasonable that good approximations to DN will be local if DN is local and vice versaOtherwise DWF with n5 may not be local

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Neubergers Operator: IIIFour dimensional space of algorithmsRepresentation (CF, PF, CT=DWF)Constraint (5D, 4D)

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Kernel

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Schur ComplementIt may be block diagonalised by an LDU factorisation (Gaussian elimination)

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Constraint: ISo, what can we do with the Neuberger operator represented as a Schur complement?Consider the five-dimensional system of linear equations

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Constraint: IIand we are left with just det Dn,n = det DN

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Approximation: tanhPandey, Kenney, & Laub; Higham; NeubergerFor even n (analogous formul for odd n)j

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Approximation: j

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Approximation: ErrorsThe fermion sgn problemApproximation over 10-2 < |x| < 1Rational functions of degree (7,8)

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Representation: Continued Fraction I

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Representation: Continued Fraction II

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Representation: Partial Fraction IConsider a five-dimensional matrix of the form (Neuberger & Narayanan)

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Representation: Partial Fraction IICompute its LDU decomposition

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Representation: Partial Fraction III

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Representation: Cayley Transform ICompute its LDU decompositionNeither L nor U depend on C

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Representation: Cayley Transform IIIn Minkowski space a Cayley transform maps between Hermitian (Hamiltonian) and unitary (transfer) matrices

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Representation: Cayley Transform IIIP-P-P+P+

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Representation: Cayley Transform IV

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Representation: Cayley Transform VWith some simple rescaling

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Representation: Cayley Transform VIIt therefore appears to have exact off-shell chiral symmetryBut this violates the Nielsen-Ninomiya theoremq.v., Pelissetto for non-local versionRenormalisation induces unwanted ghost doublers, so we cannot use DDW for dynamical (internal) propagatorsWe must use DN in the quantum action insteadWe can us DDW for valence (external) propagators, and thus use off-shell (continuum) chiral symmetry to manipulate matrix elements

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Chiral Symmetry Breakingmres is just one moment of L G is the quark propagator

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Numerical StudiesMatched mass for Wilson and Mbius kernelsAll operators are even-odd preconditionedDid not project eigenvectors of HW

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Comparison of RepresentationConfiguration #806, single precision

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Matching m between HS and HW

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Computing mres using L

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mres per Configuration

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Cost versus mres

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ConclusionsRelatively goodZolotarev Continued FractionRescaled Shamir DWF via Mbius (tanh)Relatively poor (so far)Standard Shamir DWFZolotarev DWF ( )Can its condition number be improved?Still to doProjection of small eigenvaluesHMC5 dimensional versus 4 dimensional dynamicsHasenbusch acceleration5 dimensional multishift?Possible advantage of 4 dimensional nested Krylov solversTunnelling between different topological sectorsAlgorithmic or physical problem (at =0)Reflection/refractionAssassination of Peter of Lusignan (1369) (for use of wrong chiral formalism?)

Fix the bottom equation!!!