Chiral Enough? A D Kennedy University of Edinburgh Wednesday, 22 - - PowerPoint PPT Presentation

Wednesday, 22 February 2012 New-Type Fermions on the Lattice

Chiral Enough?

A D Kennedy

University of Edinburgh

2

Outline

Wednesday, 22 February 2012 A D Kennedy

We shall review algorithms for on-shell chiral lattice (approximate overlap) fermions In QCD chiral symmetry is explicitly broken by quark masses So there is no reason to require “too much” chirality, we just need



res q

m m

We shall show results from a preliminary dynamical study to compare the cost of using different approximations to achieve this

3 Wednesday, 22 February 2012 A D Kennedy

Chiral Fermions

Conventions

We work in Euclidean space

γ matrices are Hermitean

We write We assume all Dirac operators are γ5 Hermitean

γ γ =

† 5 5

D D

µ µ

γ = ⋅ D D

4

Such a transformation should be of the form (Lüscher)

is an independent field from has the same Spin(4) transformation properties as does not have the same chiral transformation properties as in Euclidean space (even in the continuum)

ζ is a free parameter

Wednesday, 22 February 2012 A D Kennedy

It is possible to have chiral symmetry on the lattice without doublers if we only insist that the symmetry holds on shell

On-shell chiral symmetry: I

y

†

y y

( )

α ζ γ αγ ζ

ψ ψ ψ ψ

    − − −    

→ →

5 5

1 2 1 1 2

;

i aD i aD

e e

y

†

y

†

y

5 Wednesday, 22 February 2012 A D Kennedy

On-shell chiral symmetry: II

For it to be a symmetry the Dirac operator must be invariant

( )

ζ ζ

α γ αγ

  − − −      

→ =

5 5 1 2 1 1 2 aD aD

i i

D e De D

For an infinitesimal transformation this implies that

( )

ζ γ γ ζ   − − + − =      

5 5

1 2 1 1 2 aD D D aD

γ γ γ + =

5 5 5

2 D D aD D

Which is the Ginsparg-Wilson relation

6 Wednesday, 22 February 2012 A D Kennedy

( ) ( ) ( )

γ γ γ −   = = −   − −

5 5 5 †

sgn ˆ

W w W W

D M D M D M D M Both of these conditions are satisfied if we define (Neuberger)

Neuberger’s Operator: I

We can find a solution of the Ginsparg-Wilson relation as follows

[ ]

γ γ γ γ γ γ = + = ⇒ =

† † 1 5 5 5 5 5 5 2 1

; ˆ ˆ ˆ aD aD aD Let the lattice Dirac operator to be of the form This satisfies the GW relation iff γ

=

2 5

1 ˆ It must also have the correct continuum limit

Where we have defined where

( )

W W

D Z O a

→ ∂ / +

2

W

Z M aZ

=

( )

( ) ( )

γ γ γ   → ∂ / ⇒ = ∂ / − + = − +    

2 2 5 5 5

2 1 1 ˆ

W

D D Z aZ O a O a M

7 Wednesday, 22 February 2012 A D Kennedy

Into Five Dimensions

H Neuberger hep-lat/9806025 A Boriçi hep-lat/9909057, hep-lat/9912040, hep-lat/0402035 A Boriçi, A D Kennedy, B Pendleton, U Wenger hep-lat/0110070 R Edwards & U Heller hep-lat/0005002

趙挺偉 (T-W Chiu) hep-lat/0209153,

hep-lat/0211032, hep-lat/0303008 R C Brower, H Neff, K Orginos hep-lat/0409118 Hernandez, Jansen, Lüscher hep-lat/9808010

菊川芳夫 & 野口達也 (Y Kikukawa &

T Noguchi) hep-lat/9902022

8 Wednesday, 22 February 2012 A D Kennedy

Is DN local?

It is not ultralocal (Hernandez, Jansen, Lüscher)

It is local iff DW has a gap DW has a gap if the gauge fields are smooth enough Reflection positivity?

It seems reasonable that good approximations to DN will be local if DN is local and vice versa

Otherwise DWF with n5 → ∞ may not be local

Neuberger’s Operator: II

( ) ( ) ( )

µ µ µ γ   = + + −  

1 , 1 1 sgn 5 2 D H H N

1 0 μ

9 Wednesday, 22 February 2012 A D Kennedy

Four dimensional space of algorithms

Neuberger’s Operator: III

Representation (CF , PF , CT= DWF) Constraint (5D, 4D)

ε ≈ =

,

( ) sgn( ) ( ) ( )

n n m m

P H H H Q H Approximation

( )

γ = −

5 W

H D M Kernel

10 Wednesday, 22 February 2012 A D Kennedy

Kernel

Shamir kernel

( ) ( )

5 5 5 5

; 2

W T T T W

a D M H D a D a D M

γ − = = + −

Möbius kernel

( ) ( ) ( ) ( )

5 5 5 5 5 5 5 5

; 2

W M M M W

a b c D M H D a D a b c D M

γ + − = = + − −

( )

γ = −

5 W W

H D M Boriçi (Wilson/Whatever) kernel

11 Wednesday, 22 February 2012 A D Kennedy

−       = ⋅ ⋅       − −       −      

1 1 1 1 1 1 1 A A B CA D CA B

      = ⋅ ⋅            

1 1 1 1

    = ⋅     − −     −    

1 1 1 1 A B CA D CA B

Schur Complement

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

( )

−

  = −    

1

det det A B AD ACA B C D In particular

     

A B C D

Consider the block matrix

Equivalently a matrix over a skew field = division ring The bottom right block is the Schur complement

12 Wednesday, 22 February 2012 A D Kennedy 1 2 2 1

ψ φ φ φ φ

− −

              = 

n n

D χ               

1 2 5 2 1

φ φ φ φ ψ

− −

      =               

n n

D

Constraint: I

1 2 2 1

φ φ φ φ ψ

− −

      =               

n n

DU

So, what can we do with the Neuberger

• perator represented as a Schur complement?

Consider the five-dimensional system of linear equations

χ       =         

1 −

L

1 −

L L

The bottom four-dimensional component is

, N n n

D D

ψ ψ χ = =

13 Wednesday, 22 February 2012 A D Kennedy

Constraint: II

Alternatively, introduce a five-dimensional pseudofermion field

( )

1 1 2 n

φ φ φ ψ

−

Φ = 

Then the pseudofermion functional integral is

† 1 5

† 5 , 1

det det det det

n D j j j

d d e D LDU D D

−

−Φ Φ =

Φ Φ ∝ = = =∏

∫

So we also introduce n-1 Pauli-Villars fields

† ,

1 1 1 † , 1 1

det

j j j j

n n D j j j j j j

d d e D

ξ ξ

ξ ξ

− − − − = =

  ∝    

∏ ∏ ∫

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

14 Wednesday, 22 February 2012 A D Kennedy

Approximation: tanh

Pandey, Kenney, & Laub; Higham; Neuberger (KLN) For even n (analogous formulæ for odd n)

( )

( )

ε

  −   − +   −   −   +  

− = = +

1 1 1 1, 1 1

1 tanh tanh 1

n x x n n n x x

x n x

( )

( )

π π − + = + + =

=      

∏ ∏

2 2 2 2 2 2

1 tan 1 1 2 tan 1

n n

k x n k k x n k

xn

( ) ( )

π π     − + − =        

=

∑

2 2 2 2 1 1 2 2

cos sin 1

2 1

n

x k k k n n

x n

ωj

15 Wednesday, 22 February 2012 A D Kennedy

Approximation: Золотарев

( ) ( ) ( ) ( ) ( ) ( )

( )

λ

    =

− ′ = − ′ −

∏

2 2 / 2 2 1 2 1 2

sn ; 1 sn / ; sn 2 / ; 1 sn ; sn ; 1 sn 2 / ;

n m

z k z M iK m n k z k M z k iK m n k

sn(z/M,λ) sn(z,k)

ωj

18 Wednesday, 22 February 2012 A D Kennedy

Approximation: Errors

The fermion sgn problem

Approximation over 10-2 < | x| < 1 Rational functions of degree (7,8)

ε(x) – sgn(x) log10 x

0.01 0.005

• 0.01
• 0.005
• 2

1.5

• 1
• 0.5

0.5

Золотарев tanh(8 tanh-1x)

19

Approximation: Errors II

Wednesday, 22 February 2012 A D Kennedy

Dependence on degree

20

Approximation: Errors III

Wednesday, 22 February 2012 A D Kennedy

Error over approximation interval as function of degree and interval

21

Approximation: Slope at x=0

Wednesday, 22 February 2012 A D Kennedy

Derivative of Золотарев approximation at origin (maximum MD force)

   

, .

. . ;

Z N KLN

N N

e e e     

0 87

0 49 0 025

22

Approximation: ΔH

Wednesday, 22 February 2012 A D Kennedy

HMC energy change (323 × 64 lattice)

Nigel Cundy, A.D. Kennedy, Andreas Schäfer, Nucl.Phys. B845, 30—47 (2011) [arXiv:1010.5629]

23

Approximation: mRes

Wednesday, 22 February 2012 A D Kennedy

N dependence of Residual mass (323 × 64 lattice)

24

Approximation: Low Mode mRes

Wednesday, 22 February 2012 A D Kennedy

Ratio of “bulk” to “low mode” contributions to residual mass (323 × 64 lattice)

25 Wednesday, 22 February 2012 A D Kennedy

Representation: Continued Fraction I

1 2 3

1 1 1 1 1 1 A A A A

             

Consider a five-dimensional matrix of the form

1 1 1 1 1 1 1 2 1 2 1 3 2

1 1 1 1 1 1 1 1 S S S S S S S S S S

− − − − − −

                      = ⋅ ⋅                          

Compute its LDU decomposition

1

1 ;

n n n

S A S A S −

= + =

where

3 3 3 3 2 2 2 1 1

1 1 1 1 1 1 S A A A S A A S A A

= − = − = − − − −

then the Schur complement of the matrix is the continued fraction

26 Wednesday, 22 February 2012 A D Kennedy

Representation: Continued Fraction II

We may use this representation to linearise our rational approximations to the sgn function

1 1 2 1 1 1 1 2 2 , (

)

n n n n n n

c c H H H H H c c c c c c ε β β β β

− −

= + + + + 

1 2 1 2 1 2 1 1 2 2 1 2 2 1 2 1 1 1 c c c n n n c c c n n n c c c c c c c H n H n H H H c β β β β β       −   −             −         − − −    

as the Schur complement of the five- dimensional matrix

27 Wednesday, 22 February 2012 A D Kennedy

Representation: Partial Fraction I

2 2 2 1 1 1 2 2

1 1 1 1 1 1 1 1

x p p x x p q p x q

R −                       −  

Consider a five-dimensional matrix of the form (Neuberger & Narayanan)

28 Wednesday, 22 February 2012 A D Kennedy

Compute its LDU decomposition

So its Schur complement is

1 2 2 1 2 2 2 2

p x x q p x R x q + − + −

1 1 1 1 1

p x p q x q x x q x q p x p q x q x x q x q − − − − − + + −

                   

( ( ) ) x p p x q xq p x x q x p p x q R xq p x x q − − + − − +

                         

1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1

p p x x p x q x q x q x q p x q x q x q x q − − + − + + − +

                     

1 1 1 1 1 1 1 1 1 2 2 1 2 1 x p p x q x R x p p q             =               − −

Representation: Partial Fraction II

29 Wednesday, 22 February 2012 A D Kennedy

Representation: Partial Fraction III

This allows us to represent the partial fraction expansion of our rational function as the Schur complement of a five-dimensional linear system

ε

− =

= −

∑

1, 2 2 1

( )

n j n n j j

p H H H q

30 Wednesday, 22 February 2012 A D Kennedy

1 2 1 3 2 1 4 3 2 1

1 1 1 1 1 1 1 A C A A C A A A C C A A A A C

+ + + − +

    −     −         −         −    

1 2 1 3 2 1 4 3 2 1

1 1 1 A C A A C A A A C C A A A A C

+ + + − +

  −   −     −     −  

2 3 4

1 1 1 1 A A A

    −     −     −  

=

Representation: Cayley Transform I

1 2 3 4

1 1 1 A C A A A C

+ −

  −   −     −     −  

Consider a five-dimensional matrix of the form

Compute its LDU decomposition

CT 1 2 1 n n

S C A A A AC

− − +

= − 

So its Schur complement is

L does not depend on C

( )

( )

( ) ( )

CT 5 2 5 1 1

1 1 1 1 1 T S T T

µ µ γ µ γ

−

−   = +   − + + −   +    

If where , and , then C P P

µ

± ±

= −



1 n

T T T

= 

1 s s

A T

−

=

31 Wednesday, 22 February 2012 A D Kennedy

Representation: Cayley Transform II

The Neuberger operator is

( ) ( ) ( )

CT N CT

, 1 S D H S

µ µ ≈

In Minkowski space a Cayley transform maps between Hermitean and unitary matrices

For an odd function we have

( ) ( ) ( ) ( )

1 x x T x T x

ε ε − = − ⇔ − =

( ) ( )

1 T

ε = ⇔ =

( )

j j j

x T x x

ω ω − = +

T(x) is the Euclidean Cayley transform of

( ) ( ) ( ) ( ) ( )

1 1 ( ) ; 1 1 T x x x T x T x x

ε ε ε − − = = + +

, (

) sgn( )

n m x

x

ε ≈

32

D P P P D P P D P P P D

µ µ

+ + − − + + − + + + − +

−     − −     − −   −  

The Neuberger operator with the Shamir kernel and the KLN approximation is related to the Schur complement of the Domain Wall operator

1 1 1 1 1 1 D D D D P P D D D D

µ µ

+ + + + − + + + + +

                           

− − − = + − − −

Wednesday, 22 February 2012 A D Kennedy

Representation: Cayley Transform III

with and

( )

+ =

− +

5 5

1

W

a D a D M

( )

γ

± =

±

1 5 2 1

P

P+

μP+

P-

μP-

33 Wednesday, 22 February 2012 A D Kennedy

(1) (1) (1) (2) (2) (2) (3) (3) (3) ( 4) ( 4) ( 4)

D D P D P D P D D P D P D D P D P D P D

µ µ

+ − + − − − − + − + − − + − + + + − − +

  −         −  

The Neuberger operator with a Möbius kernel and any approximation is related to the Schur complement of D5 (μ)

(1) (1) (1) (1) (2) (2) (2) (2) (3) (3) (3) (3) ( 4) ( 4) ( 4) ( 4)

D D D D D D D D P P D D D D D D D D

µ µ

+ − + − − + + − − + − + + − − + − +

    −         = +         −    

Representation: Cayley Transform IV

with and

( ) ( ) 5 ( ) ( ) 5

( ) 1 ( ) 1

s s s W s s s W

D b D M D c D M

α α

+ −

  = − +     = − −  

( )

γ

± =

±

1 5 2 1

P

( ) ( ) ( ) ( ) 5 5 5 5 5 5 5 5

;

s s s s s

b c b c b c b c

ω + + = − = −

P+

μP+

P-

μP-

34 Wednesday, 22 February 2012 A D Kennedy

(1) (1) (1) (1) (2) (2) (2) (2) 5 (3) (3) (3) (3) ( 4) ( 4) ( 4) ( 4)

( ) D D D D D D D D D P P D D D D D D D D

µ µ µ

        + − − +             − + + −     − +     − + + −             − + + −    

− = + − 

1 1 1 1 1 1 1 1 1

P P

−

                −            

≡ + 

Cyclically shift the columns of the right-handed part where

5 5

D D

→ 

Representation: Cayley Transform V

P+ P-

μP+ μP-

35 Wednesday, 22 February 2012 A D Kennedy

( ) ( ) ( ) s s s

Q D P D P

± + ± −

≡ +



Representation: Cayley Transform VI

1 ( ) ( ) s s s M s s M

H T Q Q H

ω ω

− + −

− ≡ − = +

With some simple rescaling

1 1 1 2 1 3 1 4

1 1 1 5 1

( )

T C T T T C

D

µ

− + − − − −

− − − − −

    =        ฀

The domain wall operator reduces to the form introduced before

( ) ( ) ( ) ( )

1 2 3 4

Q Q Q Q

− − − − −

    =       ฀

36 Wednesday, 22 February 2012 A D Kennedy

Representation: Cayley Transform VII

It therefore appears to have exact off-shell chiral symmetry But this would violate the Nielsen-Ninomiya theorem! However, DDW is a very non-local operator

We can only use DDW for valence (“external”) propagators, and thus use off-shell (continuum) chiral symmetry to manipulate matrix elements

We solve the equation

{ }

5

,

DW

D

γ =

Hence Note that satisfies

1 1 DW N

D D a

− −

= −

{ } { }

{ }

1 1 1 1 5 5 5 5 5

, , 2 , 2

DW N N N N

D D a D D D a

γ γ γ γ γ

− − − −

= − = − =

5 5

( ) (1) D D

µ φ χ =

37 Wednesday, 22 February 2012 A D Kennedy

Chiral Symmetry Breaking

Ginsparg-Wilson defect

5 5 5 5

2

L

D D aD D

γ γ γ γ + − = ∆

Using the approximate Neuberger operator

( )

1 5 2 1

aD H

γ ε   = +   ∆L measures chiral symmetry breaking

( )

2 1 2 1 L

a H

ε   ∆ = −  

The quantity is essentially the usual domain wall residual mass (Brower et al.)

† res †

tr tr

L

G G m G G

∆ =

mres is just one moment of ∆L G is the quark propagator

38 Wednesday, 22 February 2012 A D Kennedy

Conclusions: Kernel

For state-of-the-art dynamical QCD computations with on-shell chiral fermions Boriçi (Wilson or Whatever) kernel seems

nicest, except for historical compatibility

Möbius kernel can be used to shift the spectrum to make better use of tanh approximation

39 Wednesday, 22 February 2012 A D Kennedy

Conclusions: Constraint

4D pseudofermions seem simpler than 5D ones 5D Domain Wall requires pseudofermions = Pauli- Villars fields (pseudo-pseudofermions) to avoid being killed by noise

This seems unnecessary work

Hasenbusch acceleration

5 dimensional multishift? Possible advantage of 4 dimensional nested Krylov solvers

Quantitative comparison still required (only old results

• n small lattices with heavy quarks)

40 Wednesday, 22 February 2012 A D Kennedy

Conclusions: Approximation

Zolotarev approximation with moderate ε seems to give much smaller (x 10-4) residual mass than tanh for a small (≈20%) extra cost DWF using Möbius kernel may also be competitive

But its benefits seem to follow from shifting the spectrum to make better use of the tanh approximation So why not just use shifted tanh approximation? And if so, why not use “truncated” Zolotarev approximation?

41 Wednesday, 22 February 2012 A D Kennedy

Conclusions: Representation

There appear to be opportunities to tune five dimensional representations Zolotarev DWF (Chiu)

Old data indicated it was much slower than alternatives Should be compared with DWF at same small residual mass Condition number improved significantly by naïve “tuning” of

the parameter αs (caveat: unpublished result)

Can its condition number be further improved?

Quantitative comparison of CF, PF, and CT (DWF) for realistic systems would be nice

But expensive…

42 Wednesday, 22 February 2012 A D Kennedy

Conclusions: Topology

Slow tunnelling between topological sectors

Algorithmic or physical problem (at μ=0)?

Include extra determinant terms in the action to suppress defects without suppressing topological modes? Does global topology change matter?

Finite volume effect? Cluster decomposition Local topology change suffices? (Whatever this means)

Lots of interesting things still to study!