Exact Pseudofermion Action for Hybrid Monte Carlo Simulation of One-Flavor Domain-Wall Fermion

1. Exact Pseudofermion Action for Hybrid Monte Carlo Simulation of One-Flavor Domain-Wall Fermion Yu-Chih Chen (for the TWQCD Collaboration) Physics Department National Taiwan University Collaborators: Ting-Wai Chiu

2. Content • Lattice Dirac Operator for Domain-Wall Fermion (DWF) • Two-Flavor Algorithm (TFA) • TWQCD’s One-Flavor Algorithm (TWOFA) • Rational Hybrid Monte Carlo (RHMC) Algorithm • TWOFA vs. RHMC with Domain-Wall Fermion • Concluding Remarks

3. Lattice Dirac Operator for Domain-Wall Fermion (DWF) • For domain-wall fermion, in general, the lattice Dirac operator reads • where , and is the standard Wilson-Dirac operator with , • are the matrices in the fifth dimension and depend on the quark mass.

4. Lattice Dirac Operator for Domain-Wall Fermion (DWF) • For , the form of are • Using, the can be written as Matrix in 4D Matrices in 5-th dimension

5. Lattice Dirac Operator for Domain-Wall Fermion (DWF) 1. If are the optimal weights given in Ref. [1], it gives Optimal Domain-Wall Fermion where , and [1] T. W. Chiu, Phys. Rev. Lett. 90, 071601 (2003)

6. Lattice Dirac Operator for Domain-Wall Fermion (DWF) 2. If , , it gives Domain-Wall Fermion with Shamir Kernel 3. If , , it gives Domain-Wall Fermion with Scaled () Shamir Kernel where , and

7. Lattice Dirac Operator for Domain-Wall Fermion (DWF) For a physical observable If , where the matrix K is independent of the gauge field

8. Lattice Dirac Operator for Domain-Wall Fermion (DWF) Using the redefined operator where satisfies: H is Hermitian H is positive-definite

9. Lattice Dirac Operator for Domain-Wall Fermion (DWF) • For DWF, since and are independent of the gauge field,

10. Two-Flavor Algorithm (TFA) [2] • For the DWF Dirac operator • we can apply the Schur decompositionwith the even-odd preconditioning • where • We then have [2] T. W. Chiu, et al. [TWQCD Collaboration],PoS LAT 2009, 034 (2009); Phys. Lett. B 717, 420 (2012) .

11. Two-Flavor Algorithm (TFA) • The pseudofermion action for HMC simulation of 2-flavor QCD with DWF is Two flavor • The field can be generated by the Gaussian noise field generated from Gaussian noise

12. TWQCD’s One Flavor Algorithm (TWOFA) For one-flavor of domain-wall fermion in QCD, we have devised an exact pseudofermion action for the HMC simulation, without taking square root. where In Dirac space

13. TWQCD’s One Flavor Algorithm (TWOFA) Use type I Schur decomposition to we then have where ,

14. TWQCD’s One Flavor Algorithm (TWOFA) Use type II Schur decomposition to we then have where ,

15. TWQCD’s One Flavor Algorithm (TWOFA) • By using the Sherman-Morrison formula, we have found that the fifth dimensional matrices can be rewritten as • where is a scalar function of , and , are the vector functions of , and , and here we have defined • With these form of , can be rewritten as

16. TWQCD’s One Flavor Algorithm (TWOFA) • Next we use , one then has Positive definite and Hermitian

17. TWQCD’s One Flavor Algorithm (TWOFA) • Again, with , one also has Positive definite and Hermitian

18. TWQCD’s One Flavor Algorithm (TWOFA) One flavor determinant

19. TWQCD’s One Flavor Algorithm(TWOFA) Use , and some algebra, the pseudofermion action of one-flavor domain-wall fermion can be written as where ,

20. TWQCD’s One Flavor Algorithm(TWOFA) The initial pseudofermion fields of each HMC trajectory are generated by Gaussian noises as follows. Gaussian noise

21. TWQCD’s One Flavor Algorithm (TWOFA) • The invesre square root can be approximated by the Zolotarev optimal rational approximation Gaussian noise

22. Rational Hybrid Monte Carlo(RHMC) Algorithm A widely used algorithm to do the one-flavor HMC simulation is the rational hybrid Monte Carlo (RHMC)[3], which can be used for any lattice fermion. • 1. 2. Positive definite and Hermitian The fields are generated by the Gaussian noise fields Gaussian noise [3] M. A. Clark and A. D. Kennedy, Phys. Rev. Lett. 98, 051601 (2007)

23. Rational Hybrid Monte Carlo(RHMC) Algorithm rational approximation where is the number of poles for rational approximation. The numbers of inverse matrices can be obtained from the multiple mass shift conjugate gradient.

24. TWOFA vs. RHMC with DWF I. Memory Usage : We list memory requirement (in unit of bytes) for links, momentum and 5D vectors as follows, , link variables , momentum , 5D vector Then the ratio of the memory usage for RHMC and TWOFA is where is the number of poles for MMCG in RHMC algorithm

25. TWOFA vs. RHMC with DWF

26. TWOFA vs. RHMC with DWF II. Efficiency: The lattice setup is HMC Steps: (Gauge, Heavy, Light) = (1, 1, 10),for RHMC We compare RHMC and TWOFA for the following cases: DWF with and (Optimal DWF) DWF with and (Shamir) DWF with () and (Scaled Shamir)

27. TWOFA vs. RHMC with DWF on the Lattice 1. Optimal Domain-Wall Fermion : Maximum Forces Maximum Forces

28. TWOFA vs. RHMC with DWF on the Lattice 1. Optimal Domain-Wall Fermion: )

29. TWOFA vs. RHMC with DWF on the Lattice 1. Optimal Domain-Wall Fermion: • HMC Steps: (Gauge, Heavy, Light) = (1, 1, 10),for RHMC ODWF (kernel ) with and

30. TWOFA vs. RHMC with DWF on the Lattice 2. Shamir Kernel () : Maximum Forces Maximum Forces

31. TWOFA vs. RHMC with DWF on the Lattice 2. Shamir Kernel () :

32. TWOFA vs. RHMC with DWF on the Lattice 2. Shamir Kernel () : for RHMC DWF (Shamir kernel) with and

33. TWOFA vs. RHMC with DWF on the Lattice 3. Scaled Shamir Kernel ( and ) : Maximum Forces Maximum Forces

34. TWOFA vs. RHMC with DWF on the Lattice 3. Scaled Shamir Kernel ( and ) :

35. TWOFA vs. RHMC with DWF on the Lattice 3. Scaled Shamir Kernel ( and ) : for RHMC DWF (scaled Shamir kernel) with and

36. Concluding Remarks We have derived a novel pseudofermion action for HMC simulation of one-flavor DWF, which is exact, without taking square root. It can be used for any kinds of DWF with any kernels, and for any approximations (polar or Zolotarev) of the sign function. 3. The memory consumption of TWOFA is much smaller than that of RHMC. This feature is crucial for using GPUsto simulate QCD. 4. The efficiency of TWOFA of is compatible with that of RHMC. For the cases we have studied, TWOFA outperforms RHMC. 5. TWQCD is now using TWOFA to simulate (2+1)-flavors QCD, and (2+1+1)-flavors QCD, on lattices.

37. Thank You