Equivalence of Chiral Fermion Formulations

1. 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) Workshop on Computational Hadron Physics Hadron Physics I3HP Topical Workshop

2. 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 Contents A D Kennedy

3. Conventions • We work in Euclidean space • γmatrices are Hermitian • We write • We assume all Dirac operators are γ5Hermitian Chiral Fermions A D Kennedy

4. 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) On-shell chiral symmetry: I It is possible to have chiral symmetry on the lattice without doublers if we only insist that the symmetry holds on shell A D Kennedy

5. For it to be a symmetry the Dirac operator must be invariant For an infinitesimal transformation this implies that Which is the Ginsparg-Wilsonrelation On-shell chiral symmetry: II A D Kennedy

6. 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 • 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 A D Kennedy

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

8. 0 μ 1 Neuberger’s Operator: II • Is DN local? • It is not ultralocal (Hernandez, Jansen, Lüscher) • It is local iff DW has a gap • DW 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 DN will be local if DNis local and vice versa • Otherwise DWF with n5→ ∞ may not be local A D Kennedy

9. Kernel • Approximation Neuberger’s Operator: III Four dimensional space of algorithms • Constraint (5D, 4D) • Representation (CF, PF, CT=DWF) A D Kennedy

10. Wilson (Boriçi) kernel Shamir kernel Möbius kernel Kernel A D Kennedy

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

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

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

14. Approximation: tanh • Pandey, Kenney, & Laub; Higham; Neuberger • For even n (analogous formulæ for odd n) ωj A D Kennedy

15. sn(z/M,λ) sn(z,k) Approximation: Золотарев ωj A D Kennedy

16. 0.01 ε(x) – sgn(x) 0.005 log10 x -2 1.5 -1 -0.5 0 0.5 -0.005 Золотарев tanh(8 tanh-1x) -0.01 Approximation: Errors • The fermion sgn problem • Approximation over 10-2 < |x| < 1 • Rational functions of degree (7,8) A D Kennedy

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

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

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

20. So its Schur complement is Representation: Partial Fraction II • Compute its LDU decomposition A D Kennedy

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

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

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

24. The Neuberger operator with a general Möbius kernel is related to the Schur complement of D5 (μ) • with and Representation: Cayley Transform III μP+ μP- P- P+ A D Kennedy

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

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

27. We solve the equation Note that satisfies 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 DDWfor dynamical (“internal”) propagators • We must use DNin the quantum action instead • We can us DDWfor valence (“external”) propagators, and thus use off-shell (continuum) chiral symmetry to manipulate matrix elements A D Kennedy

28. 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.) Chiral Symmetry Breaking • G is the quark propagator • mres is just one moment of L A D Kennedy

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

30. mres is not sensitive to this small eigenvalue But mres is sensitive to this one mres per Configuration ε A D Kennedy

31. Cost versus mres A D Kennedy

32. 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?) A D Kennedy