1. Tuesday, 20 October 2015Tuesday, 20 October 2015Tuesday, 20 October 2015Tuesday, 20 October 2015 Workshop on Computational Hadron Physics Hadron Physics I3HP Topical Workshop Equivalence of Chiral Fermion Formulations A D Kennedy School of Physics, The University of Edinburgh Robert Edwards, Bálint Joó, Kostas Orginos (JLab) Urs Wenger (ETHZ)

2. 2 Tuesday, 20 October 2015A D Kennedy Contents On-shell chiral symmetry Neuberger’s Operator Into Five Dimensions Kernel Schur Complement Constraint Approximation tanh Золотарев Representation Continued Fraction Partial Fraction Cayley Transform Chiral Symmetry Breaking Numerical Studies Conclusions

3. 3 Tuesday, 20 October 2015A D Kennedy Chiral Fermions Conventions We work in Euclidean space γ matrices are Hermitian We write We assume all Dirac operators are γ 5 Hermitian

4. 4 Tuesday, 20 October 2015A D Kennedy It is possible to have chiral symmetry on the lattice without doublers if we only insist that the symmetry holds on shell On-shell chiral symmetry: I Such a transformation should be of the form (Lüscher) is an independent field from has the same Spin(4) transformation properties as does not have the same chiral transformation properties as in Euclidean space (even in the continuum)

5. 5 Tuesday, 20 October 2015A D Kennedy On-shell chiral symmetry: II For it to be a symmetry the Dirac operator must be invariant For an infinitesimal transformation this implies that Which is the Ginsparg-Wilson relation

6. 6 Tuesday, 20 October 2015A D Kennedy Both of these conditions are satisfied if(f?) we define (Neuberger) Neuberger’s Operator: I We can find a solution of the Ginsparg-Wilson relation as follows Let the lattice Dirac operator to be of the form This satisfies the GW relation iff It must also have the correct continuum limit Where we have defined where

7. 7 Tuesday, 20 October 2015A D Kennedy Into Five Dimensions H Neuberger hep-lat/9806025hep-lat/9806025 A Boriçi hep-lat/9909057, hep-lat/9912040, hep-lat/0402035hep-lat/9909057hep-lat/9912040hep-lat/0402035 A Boriçi, A D Kennedy, B Pendleton, U Wenger hep-lat/0110070hep-lat/0110070 R Edwards & U Heller hep-lat/0005002hep-lat/0005002 趙 挺 偉 (T-W Chiu) hep-lat/0209153, hep-lat/0211032, hep-lat/0303008hep-lat/0209153hep-lat/0211032hep-lat/0303008 R C Brower, H Neff, K Orginos hep-lat/0409118hep-lat/0409118 Hernandez, Jansen, Lüscher hep-lat/9808010hep-lat/9808010

8. 8 Tuesday, 20 October 2015A D Kennedy Is D N local? It is not ultralocal (Hernandez, Jansen, Lüscher) It is local iff D W has a gap D W has a gap if the gauge fields are smooth enough q.v., Ben Svetitsky’s talk at this workshop (mobility edge, etc.) It seems reasonable that good approximations to D N will be local if D N is local and vice versa Otherwise DWF with n 5 → ∞ may not be local Neuberger’s Operator: II 10μ

9. 9 Tuesday, 20 October 2015A D Kennedy Four dimensional space of algorithms Neuberger’s Operator: III RepresentationRepresentation (CF, PF, CT=DWF) ConstraintConstraint (5D, 4D) Approximation Kernel

10. 10 Tuesday, 20 October 2015A D Kennedy Kernel Shamir kernel Möbius kernel Wilson (Boriçi) kernel

11. 11 Tuesday, 20 October 2015A D Kennedy Schur Complement It may be block diagonalised by an LDU factorisation (Gaussian elimination) In particular Consider the block matrix Equivalently a matrix over a skew field = division ring The bottom right block is the Schur complement

12. 12 Tuesday, 20 October 2015A D Kennedy Constraint: I So, what can we do with the Neuberger operator represented as a Schur complement? Consider the five-dimensional system of linear equations The bottom four-dimensional component is

13. 13 Tuesday, 20 October 2015A D Kennedy Constraint: II Alternatively, introduce a five-dimensional pseudofermion field Then the pseudofermion functional integral is So we also introduce n-1 Pauli-Villars fields and we are left with just det D n,n = det D N

14. 14 Tuesday, 20 October 2015A D Kennedy Approximation: tanh Pandey, Kenney, & Laub; Higham; Neuberger For even n (analogous formulæ for odd n) ωjωj

15. 15 Tuesday, 20 October 2015A D Kennedy Approximation: ЗолотаревЗолотарев sn(z/M,λ) sn(z,k) ωjωj

16. 16 Tuesday, 20 October 2015A D Kennedy Approximation: Errors The fermion sgn problem Approximation over 10 -2 < |x| < 1 Rational functions of degree (7,8) ε(x) – sgn(x) log 10 x 0.01 0.005 -0.01 -0.005 -21.5-0.50.50 Золотарев tanh(8 tanh -1 x)

17. 17 Tuesday, 20 October 2015A D Kennedy Representation: Continued Fraction I Consider a five-dimensional matrix of the form Compute its LDU decomposition where then the Schur complement of the matrix is the continued fraction

18. 18 Tuesday, 20 October 2015A D Kennedy Representation: Continued Fraction II We may use this representation to linearise our rational approximations to the sgn function as the Schur complement of the five- dimensional matrix

19. 19 Tuesday, 20 October 2015A D Kennedy Representation: Partial Fraction I Consider a five-dimensional matrix of the form (Neuberger & Narayanan)

20. 20 Tuesday, 20 October 2015A D Kennedy Compute its LDU decomposition So its Schur complement is Representation: Partial Fraction II

21. 21 Tuesday, 20 October 2015A D Kennedy Representation: Partial Fraction III This allows us to represent the partial fraction expansion of our rational function as the Schur complement of a five-dimensional linear system

22. 22 Tuesday, 20 October 2015A D Kennedy Representation: Cayley Transform I Consider a five-dimensional matrix of the form Compute its LDU decomposition So its Schur complement is Neither L nor U depend on C If where, and, then

23. 23 Tuesday, 20 October 2015A D Kennedy Representation: Cayley Transform II The Neuberger operator is T(x) is the Euclidean Cayley transform of For an odd function we have In Minkowski space a Cayley transform maps between Hermitian (Hamiltonian) and unitary (transfer) matrices

24. 24 Tuesday, 20 October 2015A D Kennedy The Neuberger operator with a general Möbius kernel is related to the Schur complement of D 5 (μ) Representation: Cayley Transform III with and P-P- μP-μP- P+P+ μP+μP+

25. 25 Tuesday, 20 October 2015A D Kennedy Cyclically shift the columns of the right-handed part where Representation: Cayley Transform IV P+P+ P-P- μP+μP+ μP-μP-

26. 26 Tuesday, 20 October 2015A D Kennedy Representation: Cayley Transform V With some simple rescaling The domain wall operator reduces to the form introduced before

27. 27 Tuesday, 20 October 2015A D Kennedy Representation: Cayley Transform VI It therefore appears to have exact off-shell chiral symmetry But this violates the Nielsen-Ninomiya theorem q.v., Pelissetto for non-local version Renormalisation induces unwanted ghost doublers, so we cannot use D DW for dynamical (“internal”) propagators We must use D N in the quantum action instead We can us D DW for valence (“external”) propagators, and thus use off-shell (continuum) chiral symmetry to manipulate matrix elements We solve the equation Note that satisfies

28. 28 Tuesday, 20 October 2015A D Kennedy Chiral Symmetry Breaking Ginsparg-Wilson defect Using the approximate Neuberger operator  L measures chiral symmetry breaking The quantity is essentially the usual domain wall residual mass (Brower et al.) m res is just one moment of  L G is the quark propagator

29. 29 Tuesday, 20 October 2015A D Kennedy Numerical Studies Used 15 configurations from the RBRC dynamical DWF dataset Matched π mass for Wilson and Möbius kernels All operators are even-odd preconditioned Did not project eigenvectors of H W

30. 30 Tuesday, 20 October 2015A D Kennedy Comparison of Representation Configuration #806, single precision

31. 31 Tuesday, 20 October 2015A D Kennedy Matching m π between H S and H W

32. 32 Tuesday, 20 October 2015A D Kennedy Computing m res using Δ L

33. 33 Tuesday, 20 October 2015A D Kennedy m res per Configuration m res is not sensitive to this small eigenvalueBut m res is sensitive to this one ε

34. 34 Tuesday, 20 October 2015A D Kennedy Cost versus m res

35. 35 Tuesday, 20 October 2015A D Kennedy Conclusions Relatively good Zolotarev Continued Fraction Rescaled Shamir DWF via Möbius (tanh) Relatively poor (so far…) Standard Shamir DWF Zolotarev DWF ( 趙 挺 偉 ) Can its condition number be improved? Still to do Projection of small eigenvalues HMC 5 dimensional versus 4 dimensional dynamics Hasenbusch acceleration 5 dimensional multishift? Possible advantage of 4 dimensional nested Krylov solvers Tunnelling between different topological sectors Algorithmic or physical problem (at μ=0) Reflection/refraction Assassination of Peter of Lusignan (1369) (for use of wrong chiral formalism?)