Criterion for the index theorem on the lattice
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Criterion for the Index Theorem on the lattice. Pedro Bicudo Dep Física IST Lisboa. Coimbra, 25-29 September 2002. 0. Introduction. Ginsparg-Wilson relation.

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Criterion for the Index Theorem on the lattice

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Criterion for the index theorem on the lattice

Criterion for the Index Theorem on the lattice

Pedro Bicudo

Dep Física IST Lisboa

Coimbra, 25-29 September 2002

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

0. Introduction

Ginsparg-Wilson relation

From the onset of Lattice theory it was realized that the finite difference Dirac action would suffer from doubling of the fermion spectrum and from the cancellation of the axial anomaly. LaterGinspargand Wilson derived a relation to recover the axial anomaly,

D g5 + g5 D =D g5 RD

where R is proportional to the lattice spacing a. This is not strictly chiral invariant. Recently

Lusherproved that chiral invariance can be recovered in an extended form,

dY= g5 (1- 1 R D) Y , dY= g5 (1- 1 D R) Y

and Hasenfratz Laliena and Niedermayer showed that the Atiyah-Singer Index Theorem is also recovered on the lattice,

n--n+=q , r= 1 tr{g5 RD}

where n-is the number of zero modes of with negative chirality. The Index Teorem on the lattice is confirmed by the Fixed point action and by the overlap action of Neuberger .

2

2

2

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Charge

Indeed1 tr{g5 RD} is a charge density , and it is local if the product RD is local.

Nevertheless the topological charge on the lattice is neither unique nor conserved.

2

Another possible choice for the topological density is the simplest simple gauge definition. In d=2 QED ( the discrete version of the Schwinger model) r is defined with the plaquette,

r1= 1 Arg ( P ) F12

continuum

P(n) = U1(n) U2(n+1) U*1(n+2) U*2(n)

Moreover gauge configurations with

topological charge can be explicitely

constructed with a topological parameter

Q which coincides with the charge Q1

when Q is integer

2 p

0

0

0

0

Ex. Of a 3x3 sub-square

with chargeQ:

2pQ3/9

2pQ6/9

0

-2pQ2/9

-2pQ2/9

-2pQ2/9

0

-2pQ1/9

-2pQ1/9

0

0

0

0

U2

U1

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Question ?

The topological charge on the lattice is neither unique nor conserved, and the discrete version of theDirac action is not unique. In this talk we address the extremum question,

‘’ What is the lattice Dirac action with the best index ? ‘’

Putting it in other words, which is the one with the best topological charge?

Because this is a vast problem, we specialize to Dirac operators constructed with the Wilson action, and to local lattice operators. We will also perform our tests in d=2.

The summary of the study presented here is,

1. Topological properties of the Wilson action

2. The index of the Neuberger overlap action

3. A criterion

4. Conclusion

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

1. Topological properties of the Wilson action

Wilson included in the first and simplest lattice version of the Dirac action,

wD(n,n’)=Sigi(Ui(n) dn+i,n’- Ui(n)+dn,n’+i ) + r a Si( 2 dn,n’ - Ui(n) dn+i,n’- Ui(n)+dn,n’+i ) + m I

a 2 a2

a Gauge invariant first derivative , and a Gauge invariant Laplacian

which fixes the doubling problem of the finite difference first derivative.

In the free limit the first derivative has the Fourier Transform,

(dn+i,n’- dn,n’+i ) sin(k)

a

p

0

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

A short footage 15 s film of the topology of the Wilson action

showing the

eigenvalues of

the 4x4 Wilson

action as a

function of the

arbitrary

topological

parameter of

the gauge

configuration,

Q= 0 2.

Im l

Re l

UNITS:

a =1

r =1

R=1

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

End of film

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

End of film

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

End of film

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

End of film

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l

Re l

End of film

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Re l

Q

This shows the real eigenvalues of the 4x4 Wilson action as a function of the topological parameter Q.

Quadruplets of eigenvalues appear from the complex plane at discrete values of Q.

When the density r is small the at all points of the lattice, the real eigenvalues are close to

1, 2 or 4

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Re c

Q

This shows the chirality c of the real eigenvalues of the 4x4 Wilson action as a function of the topological parameter Q.

v+ g5 v

c =

v+v

When the density r is small at all points of the lattice, the real eigenvalues have a chirality close to

1 or -1

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

# l

*signc

Q1

Q

This shows the number of real eigenvalues with

l <1.

of the Wilson action in a 4x4 lattice, compared with the topological charge Q1

When the density r is small at all points of the lattice, # l and Q1 are equal.

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

General NXN result

Back to the general NXN case we find that for small topological charge densities, that

the Wilson action is close to comply with the Index Theorem. The difference is proportional

to a small number ,

e = | q2 | / N2

Where is the q2 number of small real eigenvalues.

Actually the real eigenvalues and the respective chiralities are,

l = 0 + 3.0 e +o(e 2) , c = [-1 +o(e 2) ] sign (q2)

l = 2 +o(e 2) , c = [+1 -2 .0 e +o(e 2) ] sign (q2)

l = 4 - 3.0 e +o(e 2) , c = [-1 +o(e 2) ] sign (q2)

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

2. The index of the Neuberger overlap action

The Neuberger overlap action,

nD = 1 1 + X , X = nD -m0 I , X+ = g5 X g5

R X+X

is a solution of the Ginsparg-Wilson relation. Moreover it produces the trace,

tr { nD} = 1 tr g5 X

R |g5 X |

= 1S sign( l’i )

R i

where l’iare the eigenvalues of g5 X , which is hermitean .

Now let us start at a low parameter m0 and increase it continuously. Whenever g5 X passes by a root, l’i changes sign, and the charge R tr { nD}/2 has a discrete step of +1.

These steps occur at the zero modes of X, and this coincides with m0=li, an eigenvalue of the Wilson action.

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

q3

m0

This shows the index of the Neuberger Action when we run the parameter

m0= -1 5

This checks

that the index jumps precisely

at the location

of the real eigenvalues

of the Wilson

action.

...

Position of

Wilson real

eigenvalues

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Im l,l’

Re l,l ’

This figure

shows an

arbitrary

interpolation

between the

eigenvalues of

the Neuberger

action and the

eigenvalues

of the Wilson

action in a 4x4

lattice for Q=1.

The Neuberger

Action projects

the eigenvalues

of the Wilson

action on the

trigonometric circle.

In particular the real eingenvalues are projected on +1.

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

3. A criterion

For low density the real eigenvalues of the Wilson action wD have chirality ~ +1. This implies that the corresponding eigenvectors of wD are very close to eigenvectors of wD+ .

It is then possible to derive the criterion,

‘’ A hermitean solution of the Ginsparg-Wilson relation wgrD with constant R and constructed with the Wilson action wD , and wD+, complies maximally with the Index Theorem if and only if

wgrD2q( l- l0 ) (q is the Heaviside step function)

R

when we replace wD,wD+l where l is a real number 0< l < 4 and where l0 belongs to a narrow subinterval of ]0,2[ ‘’.

In particular for the d=2 lattice Schwinger model 1.0< l0 < 1.2

The Neuberger action is the simplest one that produces the desired step function, therefore it is the solution that maximises our criterion.

2

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

4. Conclusion

.The Wilson action approximately complies with the Index Theorem.

.The Neuberger overlap action with parameter m0=1.0 to 1.2 is the action with the best index, among the actions constructed with the Wilson action.

.This should not depend on the dimension, nevertheless I ought to extend that to d=4, with a better computer.

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

References

Atiyah Singer AnMa87, 485;546;531 (1968)

Wilson PRD10, 2445 (1974)

Nielsen Ninomiya NPB185, 20 (1981); NPB193, 173 (1981)

Ginsparg Wilson PRD25, 2649 (1982)

Neuberger PLB417, 141 (1998); PLB427, 353 (1998)

Hasenfratz laliena Niedermayer PLB427, 125 (1998)

Luscher NPB538, 515 (1999)

Luscher NPB428, 342 (1998)

Chiu PRD58,074511 (1998)

The end

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


Criterion for the index theorem on the lattice

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)


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