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Computational Methods for Chiral Fermions

This book explores computational methods for chiral fermions and covers topics such as domain wall actions, overlap operators, topological indices, and rational polynomial approximations. It provides insights into the study of chiral symmetry and its breaking in various systems.

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Computational Methods for Chiral Fermions

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  1. Computational Methods for Chiral Fermions Robert Edwards June 30, 2003

  2. 5D Domain Wall • Domain wall action: • 5D Domain wall kernel: with quark mass m, and • Integrate out Ls-1 extra fields to obtain • HereP is such that (P-1Y)1 = q is the light fermion

  3. Induced 4D action – truncated overlap • Core piece of induced kernel: Two variants: • Domain wall: H = HT = g5 Dw /(2 + a5 Dw) • Overlap:lim a5! 0 : H = Hw = g5 Dw • Algorithmically: solve Dtov(m)y = b via D(5)DWf = (b,0,…,0)T and y = (P-1f)1

  4. Ginsparg-Wilson Relation • Overlap operators defined with hermitian H(-m) • Propagator has a subtraction (contact term) • Ginsparg-Wilson relation. Chiral symmetry broken at a single point. • Contact terms: • Automatic operator improvement! Use directly in Ward identities. • Choice of H -- often use hermitian Wilson-Dirac operator with a large negative massHw(-m) = g5 Dw(-m). Massive version is

  5. Massless fermions • Massless Overlap-Dirac operator has exact chiral symmetry. • Topological index = number of zero eigenvalues. Eigenvs have definite chirality. • Non-topological eigenvalues come in complex conjugate pairs. • How do you get an odd number of zero modes with a system with an even number of degrees of freedom?? • Pair zero modes with modes at the cutoff. • Eigenvalues of g5e(g5 Dw(-m;U)) and Dw(-m;U) for two m.

  6. Chiral fermions on the cheap • Massless overlap Dirac operator with index Q • Eigenvalue problem: • Compute index by counting the number deficit of positive eigenvalues of H(m) (for SU(N)). • Easier way: follow spectral flow of H(m) for m > 0. Track level crossings and direction of crossings up to some m. • Get topological index as a function of m

  7. Wilson spectral flow for smooth SU(2) • Spectral flow of H(m) . Single instanton, 84, Dirichlet BC, r=2.0, cm=4.5

  8. Zero mode for smooth SU(2) • The modes associated with the crossings. • The continuum solution

  9. Overlap spectral flow for smooth SU(2) • Spectral flow of overlap Ho(m) = g5 Do(m). • Single instanton, 84, Dirichlet BC, r=1.5, cm = 4.5. • The zero modes after the crossing, m=0.6, 0.7, and 0.8. • The continuum solution

  10. A sample Wilson spectral flow on the lattice • SU(3) pure gauge configuration • b=5.85, 6312 lattice

  11. Spectral flow in SU(3): typical case • Spectral flow of H(m) for quenched Wilson b=5.85 and 6.0 • 50 configs, 10 eigenvalues overlayed • Observe significant fill-in

  12. Zero mode size distribution • Size of zero modes at each crossing. • Modes become small • Upshot: large contamination from small quantum fluctuations

  13. Main problem for chiral fermions • The density of zero eigenvalues r(0;m) is computed by fitting the integrated density

  14. Topology and small zero modes • As a function of lattice spacing we find very roughly r(0;m)/s3/2s exp(-exp(b)) • These small modes enable topology to change! • Berruto, Narayanan, Neuberger proved a class of gauge fields exist nontrivial only in a 2-unit hypercube that have Hw(U;-m) = 0. • Can superimpose them. Suggests, r(0;m)>0for all couplings • Might be zero density, but finite number! Andersen localization - recent work: Golterman, Shamir

  15. Numerical implementations of e(Hw) In practice, we only need the application of D(m) on a vector • Chebyshev approximation of e(x)= x /p x2 over some interval [d,1]. For small dtoo many terms needed. • A fractional inverse method using Gegenbauer polynomials for 1 /p x2. Poor convergence since not optimal polynomials. • Use a Lanczos based method to compute 1/p x2 based on the sequence generated for the computation of 1/x • Since no 5D gauge fields, can try 1D geometric. • Can solve multi-mass systems for fixed c. MR, CG, BiCG,…

  16. Projection • Can enforce accuracy of e(H) by projecting out lowest few Heigenvectors and adding their correct contribution exactly. • Eigenvector projection both increases accuracy of approximation of e(H) and decreases condition number, e.g. of inner CG. • Caveat: Projection complicated if H complicated. • Straightforward for Hw

  17. Rational Approximation • Approximate e(Hw) by a rational polynomial approximation. Can be rewritten as a sum over poles: • The application of c(e(Hw)y can be done by the simultaneous solution of the shifted linear systems: • We refer to this as the inner CG, since it is usually accompanied by an outer CG for computing overlap fermion propagators or eigenvalues. • A good approximation is hard to achieve for small x.

  18. Rational Polynomials • How do we determine P(x2) and Q(x2) ? • Polar decomposition denoted by eN(x) since (1+x)2Ns e2Nx. • Form induced by domain wall • Referred to as truncated overlap. • Has property eN(x) = eN(1/x) • Have sufficient accuracy in interval 0 < xmin <= x <= xmax, with xmin and xmax depending on the order and version of the approx and accuracy • Can rescale e(sx) = e(x)

  19. More rational approximations • Optimal rational approximation: • Smallest and largest ev’s of H determine fit interval • Fit P(x2)/Q(x2) to 1/p x2 over xmin to 1. Use Remez

  20. Zolotarev • Analytic solution exists!!! • Zolotarev approximation: fit x/p x2 over xmin to xmax • Solution for bk & ck in terms of elliptic integrals • Plot of error vs. x • High accuracy achievable. • Caveat: range of coefficients exceeds precision for small.

  21. Why consider 5D methods? 4D-inner+4D-outer joint Krylov?? Standard DWF DWF+O(a25) Polar, Zolotarev 4D+Cheb. poly Exact 4D Overlap 1 + g5e(H) 4D+Rat. Poly+2-pass • For 5D actions, have a 5D Krylov method - optimal search directions. • For overlap, outer CG method (a 4D Krylov space) and inner search method -- maybe CG. Inefficient since inner space not used to help outer search. • Joint Krylov search (e.g., Partlett)?? • Real measure of success: condition number and time!! 4D+Gegen poly 4D+Lanczos 5D Partial fracs Approx 4D 4D+Rat. Poly+1-pass

  22. Partial Fractions • Rewrite rational approximation of e(Hw) as a continued fraction. • Rewrite solution of Dy=(1+g5 H P(H)/Q(H))y=b as tridiagonal operator in 5D. A single (5D) Krylov space for CG. Many variants! • The four dimensional operator written as a 5D chain • Care needed: e.g., sometimes huge condition numbers

  23. Guidance for 5D methods • What we are missing is some theoretical guidance: • N&N : inspired 5D action from study of convergence bounds • Using polar decomp. coefficients (arguments apply for Zolotarev) • For each s, use combination gaussian integration/partial fractions via new fields Y = (y, c1,f1,...,cn,fn)T : • Bounds: cond. number for • Similar bound for DWF. Argue no benefit from DWF. • In practice new 5D operator still more expensive – quark masses.

  24. Two-pass method • Lower memory at expense of flops in inner CG - improve performance • Observation: in multishift only need total solution • Soln. depends on Lanczos coeffs, a(k), b(k), gs(k) • Scheme: first pass: compute coeffs., second pass: update solution • Only 5 large vectors used • 1-pass cost (flops) grows with N • 2-pass cost (roughly) independent of N • Large enough N, 2-pass wins! • N ~ 50in theory, N ~ 16 in practice! • Memory is bottleneck • Sample avg. times / float-op mem. saturated on large lattice • Caveat: will not hold when in cache

  25. Algorithm improvements Comments for 4D and 5D approaches: • Generically one has to live with small eigenvalues of Hw(-M) or its variants. • Improve gauge action -- lowers density of small eigenvalues. • Improve fermion operator kernel (something other thanDw). Usually more expensive without enough gain. • Smear gauge fields -- radically lowers small mode density! Subtlety is two length scales present -- correlation length and the smearing distance.

  26. Further ruminations • Choose Ls large enough and ignore chiral symmetry violations. Rely on density of zero modes decreasing faster than lattice spacing. • Extrapolate 5D extent Ls!1. Problematic. Difficulty distinguishing power versus exponential subleading corrections. • Project out a few small eigenvectors and treat them exactly. For standard DWF, straightforward to do! Can always eliminate Lsdependence. • Goal is to approx. a discontinuous function. Inefficient with finite number (Ls) terms. Projection by-passes this problem.

  27. Improving the gauge action • Gauge action improvements – reduce fluctuations – lowers • Comparison of density of small (zero) evs. r(0) from Hw for various gauge actions • DBW2 (renorm. group) smallest • Surprise! dyn. fermions induce fluctuations! • Fermions screen b-func., hence gauge coupling runs more slowly to short distance • Accuracy problem worse for dynamical chiral fermions!

  28. Projection for domain wall • Projection is possible also for domain wall fermions: • Induces H = HT = g5 Dw /(2 + a5 Dw) , D+ = a5Dw(-M) + 1 • Need evs. of HTvi = livi , Use generalized ev. solver Two variants: • Boundary corrections: • Bulk corrections: D+’( D+ s.t. l(Ht) shifted from 0

  29. Preconditioning (even/odd) • Generically can always even-odd precondition: • Write matrix Das a two by two block matrix • Transform: • Suitable if A-1EE easy to apply • Classic even/odd precond. not suitable for overlap • Is suitable for DWF, even with projection (boundary version)! Factor of 3 improvement in speed! • Have inner (4-volume) CG with A’ee – well conditioned • Even-odd not particularly suitable with projection on bulk terms

  30. Effect on Gellmann-Oakes-Renner Relation • Exact chiral symmetry implies identity • Stochastic estimate for • Finite Ls and no projection lead to strong violations:

  31. Induced quark mass dependence • From 5D axial Ward identity, define an induced quark mass mres • mres for different quenched gauge actions, a ~ 0.1fm • Improving gauge action lowers mres • Projection: slight improvement at Ns=16, big improvement at Ns=32 • Consistent with at small N, bulk modes contribute – unaffected by projection • If r(0) > 0, have mressr(0) / Ns

  32. Effect on Spectroscopy • Use pseudoscalar and vector channels to set the quark mass scale. • Also compare extrapolations - exact results have mp2(m=0) consistent with zero. Result for Ls=10 shows clear chiral symmetry breaking.

  33. Cost • Cost in number of Dirac operator apps - spectroscopy calculation. • At fixed scale (stange mass), cost of EO preconditioned & projected DWF (for Ls=30) about a third of 4-D overlap using Hw. However, have multi-mass shifting for 4-D method. • Preconditioning essential for DWF - cuts cost by three. • Projection overhead is small for DWF. In fact, can speed up inversion! • Preconditioned Clover at same scale about 800 ops.

  34. Dynamical chiral fermions • First ignore projection: 5D forms straightforward to implement • Action in Hybrid Monte Carlo (HMC): where f are pseudofermion fields • Key step: straightforward • 4D Overlap: • Use some smooth rational approx. to e(Hw) for guiding Hamiltonian. Accept/reject off exact Hamiltonian. Derivative doable. Simplest! • Projection: 4D & 5D, use 1st order pert. theory to evolve evs. • Use previous evs for init. guess in inversions

  35. Dynamical Overlap • Exploit [H2o(m),g5]=0 property for Nf>0HMC • Extract zero-mode contribution • A pseudofermion action in chiral sector opposite to zero mode where • Must reweight. For general Nf, use Nfpseudofermions: • Suppression of exact zeros moving into simulated chiral sector. Topology can change in opposite chiral sector. • Can work at m=0 ! Works too well, not enough Q=0 samples

  36. Conclusions • These things are not cheap! • Given sufficient mods (projection, etc.) have chiral fermions with the same chiral symmetries as continuum fermions. • No fermion doubling and have correct topological index. • No free lunch theorem still holds. Chiral fermions more expensive than traditional methods considering only inversion cost (hidden costs in traditional approaches?). • To avoid finite volume errors, still need large box sizes to hold a light pion. • What is killer application of chiral fermions? Relationship of topology and chiral symmetry, thermodynamics, and electroweak. Also possibly operator improvement (structure functions)?

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