1. Lattice QCD simulation with exact chiral symmetry
--- JLQCD's dynamical overlap project ---
24 March 2008, JSP Annual meeting, Kinki Univ.
Hideo Matsufuru for the JLQCD Collaboration
High Energy Accerelator Research Organization (KEK)

2. Collaboration JLQCD Collaboration
S.Hashimoto, T.Kaneko, J.Noaki, E.Shintani, N.Yamada, K.Takeda, H.Ikeda, H.M. (KEK)
S.Aoki, K.Kanaya, A.Ukawa, T.Yoshie (Tsukuba Univ)
H.Fukaya (Niels Bohr Institute)
T.Onogi, H.Ohki (YITP, Kyoto Univ)
K-I.Ishikawa, M.Okawa (Hiroshima Univ)
Also collaborating with
TWQCD Collaboration
T-W.Chiu, K.Ogawa (Natl.Taiwan Univ)
T-H.Hsieh (RCAS, Academia Sinica)

3. Project goals Nf =2, 2+1 dynamical overlap fermions Goals:
Exact chiral symmetry
Small enough quark masses, e-regime
Since on new KEK system (x50 upgraded)
Let us try new formalism!
Goals:
Exploring the chiral regime
Confirming the chiral symmetry breaking scenario
Testing the effective chiral Lagrangian predictions
Matrix elements with controlled chiral extrapolation
Without artificial operator mixing
Precision computation for flavor physics

4. Machines Main machine: IBM Blue Gene/L at KEK
57.6 Tflops peak (10 racks)
0.5TB memory/rack
8x8x8(16) torus network
~30% performance for Wilson kernel (Doi and Samukawa, IBM Japan)
Overlap HMC: 10~15% on one rack
Also using
Hitachi SR11000 (KEK)
2.15TFlops/0.5TB memory
NEC SX8 (YITP, Kyoto)
0.77TFlops/0.77TB memory

5. Contents Introduction Taming overlap fermion Overlap operator
Suppressing near zero modes of HW
Fixed topology simulation
Algorithms
Simulation
Results
Topological susceptibility
Meson spectrum
Matrix elements
Summary/outlook
Comment on data sharing

6. Introduction

7. Lattice QCD Lattice QCD: gauge theory on 4D Euclidean lattice
Regularized by lattice
gluon: SU(3) link variables
quark: Grassmann field on sites
Continuum limit (a→0) → QCD
Numerical simulation by Monte Carlo
Nonperturbative calculation
Quantitative calculation has become possible
Matrix elements for flavor physics
Phase structure/plasma properties
Chiral extrapolation is still a source of largest systematic errors!

8. Lattice fermion Fermion doubling Wilson fermion Staggered fermion
naïve discretization causes 16-fold doubling
Nielsen-Ninomiya's No-go theorem
Doublers appear unless chiral symmetry is broken
Wilson fermion
adds Wilson term to kill 15 doublers
breaks chiral symmetry explicitly
Improved versions, twisted mass versions are widely used
Staggered fermion
16=4 spinors x 4 flavors ('tastes')
Remnant U(1) symmetry
Fourth root trick

9. Chiral fermions Domain-wall fermion (Kaplan 1992, Shamir 1993)
5D formulation, light modes appear at the edges
Symmetry breaking effect mres  0 as N5  
Costs O(20) times than Wilson fermions
Ginsparg-Wilson relation (1982)
Exact chiral symmetry on the lattice (Luscher 1998)
Satisfied by
Overlap fermion (Neuberger, 1998)
Fixed point fermion (Hasenfratz-Niedermayer, 199x)

10. Overlap fermion (Neuberger, 1998) Theoretically elegant
HW : hermitian Wilson-Dirac operator
(Neuberger, 1998)
Theoretically elegant
Satisfies Ginsparg-Wilson relation
Infinite Ns limit of Domain-wall fermion (No mres )
Numerically cost is high
Calculation of sign function
Discontinuity at zero eigenvalue of HW
Has become feasible with
Improvement of algorithms
Large computational resources

11. Costs
Large-scale dynamical simulation projects

12. Why overlap fermion? Exact chiral symmetry
Exploring the chiral dynamics
Confirming the chiral symmetry breaking scenario
Phase transition at T>0
Matrix elements with controlled chiral extrapolation
no unwanted operator mixing
Testing the effective chiral Lagrangian predictions
with exact chiral symmetry, continuum ChPT applies
(Wilson or staggered require extension)
Let us get rid of large cost by large computer power and
improved algorithms !

13. Project Nf =2, 2+1 dynamical overlap fermions
Small enough quark masses: ms/6--ms, e-regime
Iwasaki gauge action
Extra Wilson fermion/ghost to suppress near-zero modes of HW
Fixed Q at 0, 2, 4
Present status
Nf=2: 163x32, a=0.12fm
--- Production done, physics measurements in progress
Nf=2+1: 163x48, a=0.11fm
--- Production almost finished
Nf=2+1, 243x just started

14. Taming
overlap fermion

15. Overlap fermion Sign function means:
HW : hermitian Wilson-Dirac operator
Sign function means:
In practice, all eigenmodes cannot be determined
Reasonable solution:
Eigenmodes determined at low frequency part
Approximation formula for high mode part
ex. Chebychev polynomial, partially fractional, etc.
(, ): eigenvalue/vector of HW

16. Sign function Zolotarev's Rational approximation
: calculated by multishift CG simultaneously
Valid for |l| (eigenmode of HW) [lthrs ,lmax]
Smaller lthrs, larger N is needed for accuracy: ~exp(-lthrs N)
Projecting out low-modes of HW below lthrs
explicitly determined
--- Cost depends on the low-mode density
(lthrs=0.045, N=10 in this work)

17. Sign function discontinuity
Overlap operator is discontinuous at l=0
--- Needs care in HMC when l changes sign, thus changing topological charge Q
Reflection/refraction
(Fodor, Katz and Szabo , 2004)
Change momentum at l=0
Additional inversions at l=0
Topology fixing term
(Vranas, 2000, Fukaya, 2006)
l~0 modes never appear
Tunneling HMC
(Golterman and Shamir, 2007)
Project out low modes in MD steps
Needs practical feasibility test
topology fixing
term
reflection
refraction
THMC
l=0
momentum

18. Suppressing near-zero modes of HW
Topology fixing term: extra Wilson fermion/ghost
(Vranas, 2000, Fukaya, 2006, JLQCD, 2006)
--- avoids l~0 during MD evolution
No need of reflection/refraction
Cheeper sign function
Nf = 2, a~0.125fm, msea ~ ms , m =0.2
with SE
without SE

19. Suppressing near-zero modes of HW
HMC/MD history of lowest eigenvalue
Nf = 2, a~0.125fm, msea ~ ms , m =0.2
with SE
without SE
With extra Wilson fermion/ghost, l=0 does not occur

20. Simulation at fixed topology
(Bowler et al., 2003, Aoki, Fukaya, Hashimoto, & Onogi, 2007)
Is fixing topology a problem ?
In the infinite V limit,
Fixing topology is irrelevant
Local fluctuation of topology is active
In practice, V is finite
Topology fixing finite V effect
q=0 physics can be reconstructed
Finite size correction to fixed Q result (with help of ChPT)
Must check local topological fluctuation
topological susceptibility, h' mass
Remaining question: Ergodicity ?

21. Algorithm

22. Overlap solver Solver algorithm for overlap operator
Most time-consuming part of HMC
Two algorithms were used: Nested (4D) CG and 5D CG
Nested CG (Fromer et al., 1995, Cundy et al., 2004)
Outer CG for D(m), inner CG for (multishift)
Relaxed CG: ein is relaxed as outer loop iteration proceeds
Cost mildly depends on N CG
Relaxed CG
Nf=2, a=0.12fm, on single config.
Without low-mode projection 5D

23. 5D solver 5-dimensional CG (Borici, 2004, Edwards et al., 2006)
One can solve by solving (example: N=2 case) ,
: overlap operator (rational approx.)
Even-odd preconditioning
Projection of low-mode of HW
O(3-4) times faster than 4DCG
Nf=2+1 HMC on BG/L 1024-node
(Nsbt=8)

24. Comparison with Domain-wall
Rough comparison of cost of CG solver
Overlap 5DCG (N=2 case)
Domain-wall (Ns=4 case)
CG iteration/#DW -mult for D-1
At m=ms/2, a−1=0.17GeV, 163x32/48, up to residual 10-10:
Overlap(N=10): O(1200) / O(50,000) (mres = 0 MeV)
DW(Ns=12): O(800) / O(20,000) (mres =2.3 MeV)
(Y.Aoki et al., Phys.Rev.D72,2005)
Currently factor ~2.5 difference

25. Overlap HMC Hybrid Monte Carlo (HMC)
Standard algorithm for dynamical simulation
Introduce momenta conjugate to U (link var.)
Fermion determinant: pseudo-fermion ---> external field
Iwasaki gauge action (rectangular improved)
Extra Wilson fermion/ghost
Concerning overlap fermion,
Determine low eigenmodes of HW
Inversion of linear equation
If topology is not fixed (not our case!),
monitoring lowest eigenmode
reflection/refraction at =0 --- additional inversion

26. HMC for Nf=2 Improvement of HMC
Mass preconditioning (Hasenbusch, 2001)
Multi-time step (Sexton-Weingarten, 1992)
different time steps for overlap, preconditioner, gauge/exWg
Nf=2 (JLQCD, in preparation)
5D solver without projection of low-modes of HW
Noisy Metropolis (Kennedy and Kuti, 1985)
--> correct error from low modes of HW
At early stage, 4D solver w/o noisy Metropolis (twice slower)

27. HMC for Nf=2+1 Nf=2+1 Nf=1 part: one chirality sector
(Bode et al., 1999, DeGrand and Schaefer, 2006)
--> use one of chirality
5D solver with low-mode projection (no noisy Matropolis)
(S.Hashimoto et al., Lattice 2007)
Further improvements are called for!
Faster inversion algorithms
Acceleration of HMC (chronological estimator, better integration scheme, etc.)

28. Simulation

29. Runs Nf=2: 163x32, a=0.12fm (production run finished)
6 quark masses covering (1/6~1) ms
10,000 trajectories with length 0.5
20-60 min/traj on BG/L 1024 nodes
Q=0, Q=−2,−4 (msea ~ ms/2)
e-regime (msea ~ 3MeV)
Nf=2+1 : 163x48, a=0.11fm (almost finished)
2 strange quark masses around physical ms
5 ud quark masses covering (1/6~1)ms
2500 trajectories with length 1
About 2 hours/traj on BG/L 1024 nodes
Nf=2+1 : 243x48 (just started!)

30. Lattice scale Scale: set by r0 = 0.49fm Static quark potential
Milder b-shift than
Wilson-type fermion

31. Results

32. Results Physics measurements (mainly on Nf=2)
e-regime talk by H. Fukaya (25th)
Topological susceptibility JLQCD-TWQCD, 2007
Meson spectroscopy talk by J. Noaki (next)
p+-p0 mass difference Shintani et al., Lattice 2007
Coupling const
BK JLQCD, 2008
Pion form factors talk by T. Kaneko
Nucleon sigma term talk by T. Onogi (previous)
Ps-NG boson mass/S-parameter --- talk by N.Yamada
More measurements are in progress and planned
What is advantage/specific feature of overlap simulation ?

33. Physics at fixed topology
One can reconstruct fixed q physics from fixed Q physics
(Bowler et al., 2003, Aoki, Fukaya, Hashimoto, & Onogi, 2007)
Partition function at fixed topology
For , Q distribution is Gaussian
Physical observables
Saddle point analysis
for
Example: pion mass

34. Topological susceptibility
Is local topological fluctuation sufficient ?
(JLQCD-TWQCD, 2007)
Topological susceptibility ct can be extracted from correlation functions (Aoki et al., 2007)
where

35. Meson spectrum: finite volume effect
Talk by J.Noaki
Finite volume correction
R: ordinary finite size effect
Estimated using two-loop ChPT (Colangelo et al, 2005)
T: Fixed topology effect (Aoki et al, 2007)
At most 5% effect --- largely cancel between R and T
Nf=2, a=0.12fm, Q=0

36. Meson spectrum: Q-dependnece
Nf=2, msea = 0.050
Q = 0, -2, -4
- No large Q-dependence
(consistent with expectation)
afp

37. Meson spectrum: two-loop ChPT test
Chiral extrapolation
Continuum ChPT formula applicable (NNLO)
Fit parameter: (fp is mass dependent)
NLO vs NNLO of ChPT
--- NLO tends to fail; NNLO is successful
Talk by J.Noaki
Nf=2, a=0.12fm, Q=0

38. Meson spectrum: low energy consts
Talk by J.Noaki (next)
Low energy constants
--- Quantitative test is possible
Nf=2, a=0.12fm, Q=0

39. Kaon bag parameter BK Kaon bag parameter
JLQCD, 2008
Kaon bag parameter
Nonperturbative renorm. with RI-MOM scheme
NLO of PQChPT
Nf=2, a=0.12fm result:
Exact chiral symmetry simplifies
operator mixing structure.
Wilson fermion has O(1/a)
power divergence in chiral limit.

40. p+-p0 mass difference plays essential role !
E.Shintani et al., Lattice 2007
Das-Guralnik-Mathur-Low-Young sum rule (1967)
Vacuum polarization
Low Q2: pole dominance
High Q2: OPE
Nf=2, a=0.12fm
m2 =975(30)(67)(166)MeV2
Cf. exp MeV2
Exact chiral symmetry
plays essential role !

41. Strong coupling const. Determination of s from vacuum polarization
For details, please contact E.Shintani
Determination of s from vacuum polarization
(Shifman, Vainshtein, Zakharov, 1979)
3 fit parameters: Λ, <GG>, constant (scheme dep.)
αs(Q2,Λ)
input known value
With Wilson fermion, condensate is difficult to determine due to
O(a) chiral symmetry breaking

42. strong coupling const. Using 3-loop Π, 4-loop αs : good agreement with
cf. 245(16)(16) MeV
by ALPHA collaboration
Gluon condensate
depends on loop order
indicates the truncation
(renormalon) ambiguity in QCD.

43. Summary/Outlook
We are performing dynamical overlap project at fixed topological charge
Nf=2 on 163x32, a~0.12fm: producing rich physics results
Nf=2+1 on 163x48, a~0.11fm: almost finishing
Understanding chiral dynamics with exact chiral symmetry
Application to matrix elements in progress
Configurations will be supplied to ILDG
(After first publication of spectrum paper)
Outlook
More measurements are planned
Larger lattices 243x48: just started, but need further improved algorithms
Finite temperature ?