Computational Methods for Computational Methods for Chiral FermionsChiral Fermions

Robert Edwards

June 30, 2003

5D Domain Wall5D Domain Wall Domain wall action:

5D Domain wall kernel:

with quark mass , and Integrate out Ls-1 extra fields to obtain

Here P is such that (P-1 )1 = q is the light fermion

Induced 4D action – truncated overlapInduced 4D action – truncated overlap

Core piece of induced kernel:

Two variants: Domain wall: H = HT = 5 Dw /(2 + a5 Dw) Overlap: lim a5! 0 : H = Hw = 5 Dw

Algorithmically: solve Dtov() = b via

D(5)DW = (b,0,…,0)T and = (P-1 )1

Ginsparg-Wilson RelationGinsparg-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 mass Hw(-m) = g5 Dw(-m). Massive version is

Massless fermionsMassless 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 5(5 Dw(-m;U)) and Dw(-m;U) for two m.

Chiral fermions on the cheapChiral 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

Wilson spectral flow for smooth SU(2)Wilson spectral flow for smooth SU(2)

Spectral flow of H(m) . Single instanton, 84, Dirichlet BC, =2.0, c=4.5

Zero mode for smooth SU(2)Zero mode for smooth SU(2) The modes associated with the crossings.

The continuum solution

Overlap spectral flow for smooth SU(2)Overlap spectral flow for smooth SU(2)

Spectral flow of overlap Ho(m) = 5 Do(m).

Single instanton, 84, Dirichlet BC, =1.5, c = 4.5.

The zero modes after the crossing, m=0.6, 0.7, and 0.8.

The continuum solution

A sample Wilson spectral flow on the latticeA sample Wilson spectral flow on the lattice

SU(3) pure gauge configuration =5.85, 6312 lattice

Spectral flow in SU(3): typical caseSpectral flow in SU(3): typical case

Spectral flow of H(m) for quenched Wilson =5.85 and 6.0

50 configs, 10 eigenvalues overlayed

Observe significant fill-in

Zero mode size distributionZero mode size distribution

Size of zero modes at each crossing.

Modes become small

Upshot: large contamination from small quantum fluctuations

Main problem for chiral fermionsMain problem for chiral fermions

The density of zero eigenvalues r(0;m) is computed by fitting the integrated density

Topology and small zero modesTopology and small zero modes

As a function of lattice spacing we find very roughly (0;m)/3/2s exp(-exp()) 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, (0;m)>0 for all couplings

Might be zero density, but finite number! Andersen localization - recent work: Golterman, Shamir

Numerical implementations of Numerical implementations of (H(Hww))

In practice, we only need the application of D() on a vector

Chebyshev approximation of (x)= x /p x2 over some interval [,1]. For small too 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 MR, CG, BiCG,…

ProjectionProjection

Can enforce accuracy of (H) by projecting out lowest few H eigenvectors and adding their correct contribution exactly.

Eigenvector projection both increases accuracy of approximation of (H) and decreases condition number, e.g. of inner CG.

Caveat: Projection complicated if H complicated.

Straightforward for Hw

Rational ApproximationRational Approximation

Approximate (Hw) by a rational polynomial approximation. Can be rewritten as a sum over poles:

The application of ( (Hw) 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.

Rational PolynomialsRational Polynomials How do we determine P(x2) and Q(x2) ? Polar decomposition denoted by N(x)

since (1+x)2N s e2Nx. Form induced by domain wall Referred to as truncated overlap. Has property N(x) = N(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 (sx) = (x)

More rational approximationsMore 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

ZolotarevZolotarev

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.

Why consider 5D methods?Why consider 5D methods?

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 !!

Standard DWF DWF+O(a25)

Polar, Zolotarev

5D Partial fracs

Exact 4D Overlap1 + 5(H)

4D+Cheb. poly

4D+Gegen poly 4D+Lanczos

Approx 4D4D+Rat. Poly+1-pass

4D+Rat. Poly+2-pass

4D-inner+4D-outer joint Krylov??

Partial FractionsPartial Fractions

Rewrite rational approximation of (Hw) as a continued fraction.

Rewrite solution of D=(1+5 H P(H)/Q(H))=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

Guidance for 5D methodsGuidance 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 = (, 1,1,...,n,n)T :

Bounds: cond. number for

Similar bound for DWF. Argue no benefit from DWF.

In practice new 5D operator still more expensive – quark masses.

Two-pass methodTwo-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, (k), (k), s(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 ~ 50 in theory, N ~ 16 in practice!

Ops Flop/site Time (ns) Time(ns)

SSE2 no SSE2

c*A+c*B 72 2.97 2.98

A+c*B 48 4.33 4.34

h A,Ai 36 3.33 3.34

Hw*B 1644 0.69 1.46

Memory is bottleneck Sample avg. times / float-op mem. saturated on large lattice

Caveat: will not hold when in cache

Algorithm improvementsAlgorithm 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 than Dw). 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.

Further ruminationsFurther 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 Ls dependence.

Goal is to approx. a discontinuous function. Inefficient with finite number (Ls) terms. Projection by-passes this problem.

Improving the gauge actionImproving the gauge action

Gauge action improvements – reduce fluctuations – lowers Comparison of density of small (zero) evs. (0) from Hw for

various gauge actions

DBW2 (renorm. group) smallest

Surprise! dyn. fermions induce fluctuations!

Fermions screen -func., hence gauge coupling runs more slowly to short distance

Accuracy problem worse for dynamical chiral fermions!

Projection for domain wallProjection for domain wall

Projection is possible also for domain wall fermions:

Induces H = HT = 5 Dw /(2 + a5 Dw) , D+ = a5Dw(-M) + 1

Need evs. of HT vi = i vi , Use generalized ev. solver

Two variants: Boundary corrections: Bulk corrections: D+

’ ( D+ s.t. (Ht) shifted from 0

Preconditioning (even/odd)Preconditioning (even/odd) Generically can always even-odd precondition: Write matrix D as 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

Effect on Gellmann-Oakes-Renner RelationEffect on Gellmann-Oakes-Renner Relation

Exact chiral symmetry implies identity

Stochastic estimate for

Finite Ls and no projection lead to strong violations:

Induced quark mass dependenceInduced 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 (0) > 0, have mres s (0) / Ns

Effect on SpectroscopyEffect on Spectroscopy Use pseudoscalar and vector channels to set the quark mass

scale. Also compare extrapolations - exact results have m

2(=0) consistent with zero. Result for Ls=10 shows clear chiral symmetry breaking.

CostCost 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.

Dynamical chiral fermionsDynamical chiral fermions

First ignore projection: 5D forms straightforward to implement Action in Hybrid Monte Carlo (HMC):

where are pseudofermion fields Key step: straightforward

4D Overlap: Use some smooth rational approx. to (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

Dynamical OverlapDynamical Overlap

Exploit [H2o(),5]=0 property for Nf>0 HMC

Extract zero-mode contribution

A pseudofermion action in chiral sector opposite to zero mode

where

Must reweight. For general Nf, use Nf pseudofermions:

Suppression of exact zeros moving into simulated chiral sector. Topology can change in opposite chiral sector.

Can work at =0 ! Works too well, not enough Q=0 samples

ConclusionsConclusions

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)?