Domain Wall Fermions and other 5D Algorithms

1. Domain Wall Fermionsand other 5D Algorithms A D Kennedy University of Edinburgh DWF@X BNL

2. 0 μ 1 Neuberger’s Operator • All the methods that are used to put chiral fermions on the lattice are rational approximations to Neuberger’s operator • They are not just analogous, there is a well-defined mapping between them A D Kennedy

3. 5D History • First 5D algorithm in LGT was Lüscher’s multiboson algorithm • His multiple pseudofermions can be viewed as one 5D pseudofermion field • This led to PHMC and RHMC • These are the analogous methods with 4D pseudofermions A D Kennedy

4. Kernel • Approximation Unified View of Algorithms • Today I will concentrate on the choice of constraint • But first I want to point out that we can draw some conclusions just from the 4D-5D equivalence per se • And I am not able to resist one brief rant about the choice of kernel • Constraint (5D, 4D) • Representation (CF, PF, CT=DWF) A D Kennedy

5. Significance of 5D Locality • We can choose an awful approximation for which the Neuberger operator fails to be local even on smooth configurations • Nevertheless, the corresponding DWF operator is manifestly local in 5D • We may thus conclude that “5D locality” does not eo ipso imply physical (4D) locality A D Kennedy

6. Reflection/Refraction • Do we need “reflection/refraction” to resolve the discontinuity in the derivative of the Neuberger operator? • This is more-or-less the same for 4D or 5D formulations • MD will “see” the barrier if • The rational approximation is poor enough • The MD integration step-size is small enough • Perhaps a good compromise is to use as poor an approximation for the MD as we can get away with, while using a good (Zolotarev?) approximation for the MC acceptance step • Does the problem manifest itself for large volumes anyway? A D Kennedy

7. Wilson (Boriçi) kernel Shamir kernel Möbius kernel Kernels A D Kennedy

8. Choice of Kernel • We leave it to the following talks to discuss the merits of various choice of kernel • We just observe that the choice of a kernel with a non-trivial denominator inhibits many useful algorithmic techniques • E.g., the use user of a nested inner-outer CG solver with an inner multishift (to implement the rational approximation in partial fraction form) and an outer multishift (to implement multiple partially quenched valence masses • With the Shamir kernel this requires three levels of nesting A D Kennedy

9. Schur Complement • The Neuberger operator is intrinsically a non-linear function of its Dirac operator kernel • It is a function of two non-commuting 4D operators • We may write it as a linear operator by introducing an extra dimension • The size of the fifth dimension is just the degree of the rational approximation used • To be precise, Ls = # poles • The basic idea is that the 4D operator is the Schur complement of the 5D operator A D Kennedy

10. The bottom right block is the Schur complement Schur Complement Consider the block matrix • It may be block diagonalised by an LDU factorisation (Gaussian elimination) A D Kennedy

11. The bottom four-dimensional component is 4D Pseudofermions 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

12. Alternatively, introduce a five-dimensional pseudofermion field • Then the pseudofermion functional integral is • So we also introduce n-1Pauli-Villars fields 5D Pseudofermions • and we are left with just det Dn,n = det DN A D Kennedy

13. Disadvantages of 5D pseudofermions • Introduce extra noise into the 4D world • Letting the 5D pseudofermions cancel stochastically with their Pauli-Villars partners (“pseudo-pseudo-fermions”) is a very bad idea • Cancelling them explicitly is better, but one is still has Ls-1 unnecessary noisy estimators of 1 • These increase the maximum force on the 4D gauge fields and force the MD integration step-size to be smaller A D Kennedy

14. Disadvantages of 5D pseudofermions • Extent of the fifth dimension is fixed • At least over an entire HMC trajectory • With 4D pseudofermions one can adjust the degree of the rational approximation at each MD step to cover the spectrum of the kernel with fixed maximum error A D Kennedy

15. Disadvantages of 4D Pseudofermions • Cannot evaluate roots of Neuberger operator • We cannot use the multishift solver techniques used with 5D pseudofermions because we would need constant shifts of the 4D operator and not the 5D one • Not obvious how to implement odd number of flavours efficiently with 4D pseudofermions • Not obvious how to use nth root RHMC acceleration trick • But we can still use Hasenbusch’s technique • These techniques can be implemented using nested 4D CG solver with multishift on both inner and outer solvers A D Kennedy

16. Desiderata & Conclusions • It would be nice to make progress in the following areas • Express nth roots of Neuberger operator as a 5D system • Work out how to systematically improve the condition number of 5D systems • The Schur complement of a 5D system is uniquely defined, but there are many 5D matrices with the same Schur complement • With each class of such 5D matrices it is often possible to greatly change the condition number by fairly simple transformations • It would be nice to know how to do this systematically • The evidence at present indicates that better approximations (e.g., Zolotarev rather than tanh) are not intrinsically worse conditioned A D Kennedy

17. Questions? A D Kennedy