Criterion for the Index Theorem on the lattice

1
Criterion for the Index Theorem on the lattice
Pedro Bicudo Dep Física IST Lisboa

• Coimbra, 25-29 September 2002

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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. Later
Ginsparg and 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 Lusher
proved 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--nq , r 1 trg5
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 .
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2
2
3
Charge
Indeed 1 trg5 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 d2 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(n1) U1(n2) U2(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
2pQ6/9
Ex. Of a 3x3 sub-square with charge Q
2pQ3/9
0
-2pQ2/9
-2pQ2/9
-2pQ2/9
0
-2pQ1/9
-2pQ1/9
0
0
0
0
U2
U1
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Question ?
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 d2. 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
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1. Topological properties of the Wilson action
Wilson included in the first and simplest lattice
version of the Dirac action, wD(n,n) Si gi
(Ui(n) dni,n - Ui(n) dn,ni ) r a Si ( 2
dn,n - Ui(n) dni,n - Ui(n) dn,ni ) 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, (dni,n - dn,ni )
sin(k) a
p
0
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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 R1
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Im l
Re l
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Im l
Re l
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Im l
Re l
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Re l
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End of film
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Im l
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Im l
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End of film
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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
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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

113
l signc Q1

Q
This shows the number of real eigenvalues with
l lt1. 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.
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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)
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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 XX is a
solution of the Ginsparg-Wilson relation.
Moreover it produces the trace, tr nD 1 tr
g5 X R
g5 X 1 S sign( li
) R i where li are
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, li 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.
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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
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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 Q1. The Neuberger
Action projects the eigenvalues of the Wilson
action on the trigonometric circle. In
particular the real eingenvalues are projected on
1.
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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
wgrD 2 q( l- l0 )
(q is the Heaviside step function)
R when we replace wD, wD l
where l is a real number 0lt l lt 4 and where l0
belongs to a narrow subinterval of 0,2 .
In particular for the d2 lattice Schwinger
model 1.0lt l0 lt 1.2 The Neuberger action is the
simplest one that produces the desired step
function, therefore it is the solution that
maximises our criterion.
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4. Conclusion
.The Wilson action approximately complies with
the Index Theorem. .The Neuberger overlap
action with parameter m01.0 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 d4, with a better computer.
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References Atiyah Singer AnMa87, 485546531
(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
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