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

Coimbra, 25-29 September 2002

Pedro Bicudo

Dep Física IST Lisboa


0. Introduction

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. Later Ginsparg and Wilson derived a relation to recover the axial anomaly,

D + D =D RD

where R is proportional to the lattice spacing a. This is not strictly chiral invariant. RecentlyLusher proved that chiral invariance can be recovered in an extended form,

= (1- 1 R D) = (1- 1 D R)

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

n--n+=q , = 1 tr{ 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

Ginsparg-Wilson relation


Indeed 1 tr{ 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

Charge

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) is defined with the plaquette,

1= 1 Arg ( P ) F12 continuum

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

Moreover gauge configurations withMoreover gauge configurations withtopological charge can be explicitely topological charge can be explicitely constructed with a topological parameterconstructed with a topological parameterQ which coincides with the charge which coincides with the charge Q1

when when Q is integer is integer

-2Q1/9

-2Q2/9-2Q2/9

2Q6/9Ex. Of a 3x3 sub-squarewith charge Q:

0

0

0

0 0 0

0 0

-2Q2/9

2Q3/9

-2Q1/9 0

0

U1

U2


The topological charge on the lattice is neither unique nor conserved, and the discrete version of the Dirac 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 action2. The index of the Neuberger overlap action3. A criterion4. Conclusion

Question ?


1. Topological properties of the Wilson action

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

wD(n,n’)=ii(Ui(n) n+i,n’- Ui(n)+ n,n’+i ) + r a

i( 2 n,n’ - Ui(n) n+i,n’- Ui(n)+ n,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,

(n+i,n’- n,n’+i ) sin(k) a 0


Im

Re

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 thearbitrary topologicalparameter of the gauge configuration,

Q= 0 2.

UNITS:

a =1 r =1 R=1


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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 is small the at all points of the lattice, the real eigenvalues are close to 1, 2 or 4


Re

Q

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

v+vv+v

When the density is small at all points of the lattice, the real eigenvalues have a chirality close to 1 or -1


# sign

Q1

Q

This shows the number of real eigenvalues with

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

When the density is small at all points

of the lattice, #

and Q1 are equal.


General NXN result

Back to the general NXN case we find that for small topological charge densities, thatthe Wilson action is close to comply with the Index Theorem. The difference is proportionalto a small number ,

= | q2 | / N2

Where is the q2 number of small real eigenvalues.

Actually the real eigenvalues and the respective chiralities are,

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

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

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


2. The index of the Neuberger overlap action

The Neuberger overlap action,

nD = 1 1 + X , X = nD -m0 I , X+ = 5 X 5

R X+X

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

tr { nD} = 1 tr 5 X R | 5 X |

= 1 sign( ’i ) R i

where ’i are the eigenvalues of 5 X , which is hermitean .

Now let us start at a low parameter m0 and increase it continuously. Whenever 5 X passes by a root, ’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 = i, an eigenvalue of the Wilson action.


q3

m0

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

m0= -1 5

This checksthat the index jumps preciselyat the locationof the real eigenvaluesof the Wilsonaction.

Position of Wilson real eigenvalues

...


Im ’

Re ’

This figure shows an arbitrary interpolation between the eigenvalues of the Neuberger action and the eigenvaluesof 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.


3. A criterionFor 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

wgrD 2 ) ( is the Heaviside step function) R

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

In particular for the d=2 lattice Schwinger model

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

2


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.


The end

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)











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