Report

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 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 Lattice Dirac Operator for Domain-Wall Fermion (DWF) For domain-wall fermion, in general, the lattice Dirac operator reads = + + − where = + , = − and = diag 1 , 2 , ⋯ , , and is the standard Wilson-Dirac operator with −0 0 < 0 < 2 , = + + + − − = + ,′ + 0 0 − ′ ,−1 , 1 < ≤ = −′ , , = 1, with = , = 1/ 20 1 − 0 − = + ± () are the matrices in the fifth dimension and depend on the quark mass. Lattice Dirac Operator for Domain-Wall Fermion (DWF) For = 4, the form of ± are 0 1 L 0 0 0 0 0 0 1 0 0 1 m 0 0 0 0 0 L 0 m 1 0 0 1 0 0 0 0 0 0 1 0 Using = + and = − , the () can be written as () = + () + − () Matrix in 4D Matrices in 5-th dimension + − + − () Lattice Dirac Operator for Domain-Wall Fermion (DWF) () = + () + − () + − () 1. If are the optimal weights given in Ref. [1], it gives Optimal Domain-Wall Fermion 2 ≈ () = ∶ Zolotarev Optimal Rational Approximation 2 + () = 1− 1+ =1 =1 → 2 , as → ∞ where = (1 − ) (1 + ), = and = 1 + 5 [1] T. W. Chiu, Phys. Rev. Lett. 90, 071601 (2003) −1 = 5 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 2 ≈ () = () = ′ ∶ Polar Approximation 2 + ′ 1− 1+ =1 =1 → 2 , as → ∞ where = (1 − ) (1 + ), and = 1 + 5 −1 Lattice Dirac Operator for Domain-Wall Fermion (DWF) For a physical observable () 1 () = 1 = ()exp − − ()det[ ()]exp − If () = ∗ (), where the matrix K is independent of the gauge field ()det[ ()]exp − det[ ()]exp − = ()det[()]exp − det[()]exp − Lattice Dirac Operator for Domain-Wall Fermion (DWF) Using the redefined operator () 1 () = 1 = ()det[()]exp − † () exp − † −1 − where satisfies: 1) det = det 2) H is Hermitian 3) H is positive-definite Lattice Dirac Operator for Domain-Wall Fermion (DWF) For DWF, since and are independent of the gauge field, + +−() () + + −−() () = + () → = + = + + ± = −1/2 ± + −1 −1 −1/2 ± = 1 + ± () 1 − ± () −1 − −1 Two-Flavor Algorithm (TFA) [2] For the DWF Dirac operator = + () = + + + + − − () we can apply the Schur decomposition with the even-odd preconditioning 4 − 0 + () = () where = 5 () 0 () 5 ()−1 ≡ 4 − 0 + () 5 ()−1 0 0 5 ()−1 () 5 ()−1 0 5 () = I − 5 () 5 () We then have det = det[5 ]−2 × det[()] [2] T. W. Chiu, et al. [TWQCD Collaboration],PoS LAT 2009, 034 (2009); Phys. Lett. B 717, 420 (2012) . Two-Flavor Algorithm (TFA) The pseudofermion action for HMC simulation of 2-flavor QCD with DWF is 1 † † = (1) (1) † () det (1) 2 Two flavor The field can be generated by the Gaussian noise field = † † 1 1 1 † 1 = † 1 1 = 1 ⟺ = () (1) generated from Gaussian noise 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. det[()] det[()] det[ (, 1)] = × det[(1)] det[(1)] det[ (, 1)] where = + () = + + + + − − () In Dirac space − 0 + + () = − ⋅ † ⋅ − 0 + + () − 0 + + () , 1 = − ⋅ † ⋅ − 0 + + (1) TWQCD’s One Flavor Algorithm (TWOFA) Use type I Schur decomposition to , 1 , and () = −1 0 0 0 − −1 −1 0 we then have det , 1 det det[ + Δ− ()] 1 = = det + Δ− () det[ ] where = 5 − 0 + − + ⋅ † 1 ⋅ − 0 + + () Δ− = 5 − 1 − − () , 5 = +′, TWQCD’s One Flavor Algorithm (TWOFA) Use type II Schur decomposition to 1 , and (, 1) −1 = 0 − −1 0 0 −1 0 we then have det 1 det , 1 det[(1)] 1 = = det + Δ+ () det[ 1 − Δ+ ()] 1 − Δ+ () where 1 1 = 5 − 0 + + 1 + ⋅ + ⋅ − 0 + − (1) Δ+ = 5 + 1 − + () , 5 = +′, † 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 ± = ± 0 + −1 With these form of ± (), Δ± () can be rewritten as Δ± = 5 ± 1 − ± () = −1/2 ± ± −1/2 1− = 1 − 1 + (1 − 2) TWQCD’s One Flavor Algorithm (TWOFA) Next we use det + = det[ + ], one then has det , 1 det 1 = det + Δ− () =det + −1/2 − − −1/2 =det + − −1/2 1 1 −1/2 − Positive definite and Hermitian TWQCD’s One Flavor Algorithm (TWOFA) Again, with det + = det[ + ], one also has det 1 det , 1 1 = det + Δ+ () 1 − Δ+ () =det + −1/2 + + −1/2 =det + + −1/2 1 1 − Δ+ () 1 −1/2 + 1 − Δ+ () Positive definite and Hermitian TWQCD’s One Flavor Algorithm (TWOFA) 1 = 1 † + − −1/2 2 = 2 † + + −1/2 × 1 −1/2 − 1 1 −1/2 + 2 1 − Δ+ () det , 1 det 1 det det , 1 det[()] = det[(1)] One flavor determinant TWQCD’s One Flavor Algorithm(TWOFA) Use 1 , 2 and some algebra, the pseudofermion action of one-flavor domain-wall fermion can be written as 1 0 † −1/2 −1/2 = 0 1 − − − 1 () + 2 † 0 + + −1/2 1 −1/2 + 1 − Δ+ ()+ where = 5 5 () , 5 = +′, Δ± = −1/2 ± ± −1/2 1− = 1 − 1 + (1 − 2) 2 0 TWQCD’s One Flavor Algorithm(TWOFA) The initial pseudofermion fields of each HMC trajectory are generated by Gaussian noises as follows. 1 = 1 † + − −1/2 1 −1/2 − 1 = 1 † 1 1 −1/2 ⇒ 1 = + − −1/2 − 2 = 2 † + + −1/2 −1/2 1 1 −1/2 + 2 1 − Δ+ () 1 −1/2 ⇒ 2 = + + −1/2 + 1 − Δ+ () Gaussian noise −1/2 2 TWQCD’s One Flavor Algorithm (TWOFA) The invesre square root can be approximated by the Zolotarev optimal rational approximation 1 1 = =1 1 + 2 2 = =1 Gaussian noise 1 + 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. 1/4 1/4 1 † † † = (1) 1 (1) 1 1/2 () † 1. det (1) = det (1) † 1 1/4 1 () † 1/2 (1) † 1 1/4 2. Positive definite and Hermitian The fields are generated by the Gaussian noise fields = 1 (1) † 1 1/4 () † 1/4 Gaussian noise [3] M. A. Clark and A. D. Kennedy, Phys. Rev. Lett. 98, 051601 (2007) Rational Hybrid Monte Carlo(RHMC) Algorithm = † † (1) 1 1/ = 0 + =1 + 1/4 1 () † 1/2 † (1) 1 1/4 rational approximation where is the number of poles for rational approximation. The numbers of inverse matrices 1 ( + ) can be obtained from the multiple mass shift conjugate gradient. TWOFA vs. RHMC with DWF I. Memory Usage : We list memory requirement (in unit of bytes) for links, momentum and 5D vectors as follows, 1) ≡ 8 ∗ 3 ∗ 2) = 48 ∗ , link variables 3) = 32 ∗ , momentum 4) = 24 ∗ ∗ , 5D vector Then the ratio of the memory usage for RHMC and TWOFA is 20 + 3 3 + 2 = 32 + 10.5 where is the number of poles for MMCG in RHMC algorithm TWOFA vs. RHMC with DWF 20 + 3 3 + 2 = 32 + 10.5 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: 1) DWF with = . , = . and = . . (Optimal DWF) 2) DWF with = . , = . and = (Shamir) 3) DWF with = . , = . ( = ) and = (Scaled Shamir) TWOFA vs. RHMC with DWF on the × × Lattice Maximum Forces 1. Optimal Domain-Wall Fermion : Maximum Forces = . = . = . = . TWOFA vs. RHMC with DWF on the × × Lattice 1. Optimal Domain-Wall Fermion: ∆ ∆ −∆ = . () erfc = . () Accept = . () −∆ = . () erfc = . () Accept = . () 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 = . . Algorithm Plaquette Force (Gauge) Force (heavy) Force (light) TWOFA 1.3 0.4 0.58051(09) 5.15555(34) 0.18971(13) 0.01401(29) RHMC 1.3 0.4 0.58100(10) 5.15762(36) 0.35334(11) 0.06946(44) Algorithm Accept erfc( ∆ ) −∆ Memory . . (sec.) TWOFA 0.980(8) 0.981(11) 0.9992(16) 1.00 1.00 0() RHMC 0.987(7) 0.994(18) 1.0003(16) 6.58 1.21 () TWOFA vs. RHMC with DWF on the × × Lattice Maximum Forces 2. Shamir Kernel ( = ) : Maximum Forces = . = . = . = . TWOFA vs. RHMC with DWF on the × × Lattice 2. Shamir Kernel ( = ) : ∆ ∆ −∆ = . erfc = . () Accept = . () −∆ = . erfc − = . () Accept = . () TWOFA vs. RHMC with DWF on the × × Lattice 2. Shamir Kernel ( = ) : = . , = . , = , = , = , HMC Steps: (Gauge, Heavy, Light) = (1, 1, 10), = for RHMC DWF (Shamir kernel) with = . , = . and = Algorithm Plaquette Force (Gauge) Force (heavy) Force (light) TWOFA 1.8 0.1 0.59061(09) 5.17686(34) 0.14663(45) 0.03578(18) RHMC 1.8 0.1 0.59094(14) 5.17866(35) 0.28522(68) 0.10757(06) Algorithm Accept erfc( ∆ ) −∆ Memory . . (sec.) TWOFA 0.987(7) 0.987(13) 0.9999(16) 1.00 1.00 () RHMC 0.997(3) 0.953(06) 1.0074(18) 6.58 1.00 () TWOFA vs. RHMC with DWF on the × × Lattice Maximum Forces 3. Scaled Shamir Kernel ( = and = ) : Maximum Forces = . = . = . = . TWOFA vs. RHMC with DWF on the × × Lattice 3. Scaled Shamir Kernel ( = and = ) : ∆ ∆ −∆ = . erfc = . () Accept = . () −∆ = . erfc − = . () Accept = . () TWOFA vs. RHMC with DWF on the × × Lattice 3. Scaled Shamir Kernel ( = and = ) : = . , = . , = , = , = , HMC Steps: (Gauge, Heavy, Light) = (1, 1, 10), = for RHMC DWF (scaled Shamir kernel) with = . , = . ( = )and = Algorithm Plaquette Force (Gauge) Force (heavy) Force (light) TWOFA 1.8 0.1 0.59061(09) 5.17854(34) 0.14646(13) 0.03359(13) RHMC 1.8 0.1 0.59032(09) 5.17670(32) 0.28559(39) 0.10775(06) Algorithm Accept erfc( ∆ ) −∆ Memory . . (sec.) TWOFA 0.983(7) 0.990(14) 1.0000(15) 1.00 1.00 () RHMC 0.997(3) 0.967(07) 1.00038(18) 6.58 1.17 () Concluding Remarks 1. We have derived a novel pseudofermion action for HMC simulation of one-flavor DWF, which is exact, without taking square root. 2. 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 GPUs to 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 323 × 64 × 16, and 243 × 48 × 16 lattices. Thank You