Domain Wall Fermions and other 5D Algorithms

Wednesday, April 19, 2023 DWF@X BNL

Domain Wall FermionsDomain Wall Fermionsand other 5D Algorithmsand other 5D Algorithms

A D KennedyUniversity of Edinburgh

Wednesday, April 19, 2023

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All the methods that are used to put chiral fermions on the lattice are rational approximations to Neuberger’s operator

They are not just analogous, there is a well-defined mapping between them

Neuberger’s Operator

1, 1 1 sgn

52D H H

N 10 μ


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5D History

First 5D algorithm in LGT was Lüscher’s multiboson algorithm

His multiple pseudofermions can be viewed as one 5D pseudofermion field

This led to PHMC and RHMCThese are the analogous methods with 4D pseudofermions


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Unified View of Algorithms

Representation (CF, PF, CT=DWF)

Constraint (5D, 4D)

,

( )sgn( ) ( )

( )n

n mm

P HH H

Q HApproximation

5 WH D MKernel

Today I will concentrate on the choice of constraintBut first I want to point out that we can draw some conclusions just from the 4D-5D equivalence per seAnd I am not able to resist one brief rant about the choice of kernel


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Significance of 5D Locality

We can choose an awful approximation for which the Neuberger operator fails to be local even on smooth configurationsNevertheless, the corresponding DWF operator is manifestly local in 5DWe may thus conclude that “5D locality” does not eo ipso imply physical (4D) locality


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Reflection/Refraction

Do we need “reflection/refraction” to resolve the discontinuity in the derivative of the Neuberger operator?

This is more-or-less the same for 4D or 5D formulationsMD will “see” the barrier if

The rational approximation is poor enoughThe MD integration step-size is small enough

Perhaps a good compromise is to use as poor an approximation for the MD as we can get away with, while using a good (Zolotarev?) approximation for the MC acceptance stepDoes the problem manifest itself for large volumes anyway?


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Kernels

Shamir kernel

55

5

;2

WT T T

W

a D MH D aD

a D M

Möbius kernel

5 5

55 5

;2

WM M M

W

b c D MH D aD

b c D M

5W WH D M

Wilson (Boriçi) kernel


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Choice of Kernel

We leave it to the following talks to discuss the merits of various choice of kernelWe just observe that the choice of a kernel with a non-trivial denominator inhibits many useful algorithmic techniques

E.g., the use user of a nested inner-outer CG solver with an inner multishift (to implement the rational approximation in partial fraction form) and an outer multishift (to implement multiple partially quenched valence massesWith the Shamir kernel this requires three levels of nesting


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Schur Complement

The Neuberger operator is intrinsically a non-linear function of its Dirac operator kernelIt is a function of two non-commuting 4D operatorsWe may write it as a linear operator by introducing an extra dimensionThe size of the fifth dimension is just the degree of the rational approximation used

To be precise, Ls = # poles

The basic idea is that the 4D operator is the Schur complement of the 5D operator


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1 0 0 111 11 0 0 1

A A B

CA D CA B

1 0 0 1

1 0 0 1

1 0

1 11 0

A B

CA D CA B

Schur Complement

It may be block diagonalised by an LDU factorisation (Gaussian elimination)

A B

C D

Consider the block matrix

The bottom right block is the Schur complement


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1

2

2

1

n

n

D

00

00

1

2

52

1

n

n

D

4D Pseudofermions

1

2

2

1

n

n

DU

So, what can we do with the Neuberger operator represented as a Schur complement?Consider the five-dimensional system of linear

equations00

00

1L 1LL

The bottom four-dimensional component is, Nn n DD


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5D Pseudofermions

Alternatively, introduce a five-dimensional pseudofermion field 1 12 n

Then the pseudofermion functional integral is

† 15†

5 ,1

det det det detn

Dj j

j

d d e D LDU D D

So we also introduce n-1 Pauli-Villars fields

†,

11 1

†,

1 1

detj j jj

n nD

j j j jj j

d d e D

and we are left with just det Dn,n = det DN


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Disadvantages of 5D pseudofermions

Introduce extra noise into the 4D world

Letting the 5D pseudofermions cancel stochastically with their Pauli-Villars partners (“pseudo-pseudo-fermions”) is a very bad ideaCancelling them explicitly is better, but one is still has Ls-1 unnecessary noisy estimators of 1

These increase the maximum force on the 4D gauge fields and force the MD integration step-size to be smaller


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Disadvantages of 5D pseudofermions

Extent of the fifth dimension is fixed

At least over an entire HMC trajectoryWith 4D pseudofermions one can adjust the degree of the rational approximation at each MD step to cover the spectrum of the kernel with fixed maximum error


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Disadvantages of 4D Pseudofermions

Cannot evaluate roots of Neuberger operator

We cannot use the multishift solver techniques used with 5D pseudofermions because we would need constant shifts of the 4D operator and not the 5D oneNot obvious how to implement odd number of flavours efficiently with 4D pseudofermionsNot obvious how to use nth root RHMC acceleration trick

But we can still use Hasenbusch’s technique

These techniques can be implemented using nested 4D CG solver with multishift on both inner and outer solvers


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Desiderata & Conclusions

It would be nice to make progress in the following areas

Express nth roots of Neuberger operator as a 5D systemWork out how to systematically improve the condition number of 5D systems

The Schur complement of a 5D system is uniquely defined, but there are many 5D matrices with the same Schur complementWith each class of such 5D matrices it is often possible to greatly change the condition number by fairly simple transformationsIt would be nice to know how to do this systematicallyThe evidence at present indicates that better approximations (e.g., Zolotarev rather than tanh) are not intrinsically worse conditioned


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Questions?

Documents

Domain Wall Fermions and other 5D Algorithms