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

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

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998) 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. Recently Lusher 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 22 Ginsparg-Wilson relation

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998) 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 ) F 12 continuum P(n) = U 1 (n) U 2 (n+1) U* 1 (n+2) U* 2 (n) Moreover gauge configurations with topological charge can be explicitely constructed with a topological parameter which coincides with the charge Q which coincides with the charge Q 1 when is integer when Q is integer -2  Q1/9 -2  Q2/9 2  Q6/9 Ex. Of a 3x3 sub- square with charge Q: 0 0 0 0 00 00 -2  Q2/9 2  Q3/9 -2  Q1/9 0 0 U1U1 U2U2

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998) 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 action 2. The index of the Neuberger overlap action 3. A criterion 4. Conclusion Question ?

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998) 1. Topological properties of the Wilson action Wilson included in the first and simplest lattice version of the Dirac action, wD(n,n’)=   i   i  (U i (n)  n+i,n’  - U i (n) +  n,n’+i ) + r a  i  ( 2  n,n’ - U i (n)  n+i,n’  - U i (n) +  n,n’+i ) + m I a 2 a 2 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

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998) 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 the arbitrary topological parameter of the gauge configuration, Q= 0 2. 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) Im  Re

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998) End of film Im  Re

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

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998) Re   Q This shows the chirality  of the real eigenvalues of the 4x4 Wilson action as a function of the topological parameter Q.  v +  v   v +  v When the density  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) #  sign  Q 1  Q This shows the number of real eigenvalues with  of the Wilson action in a 4x4 lattice, compared with the topological charge Q 1 When the density  is small at all points of the lattice, # and Q 1 are equal.

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998) 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,  = | q 2 | / N 2 Where is the q 2 number of small real eigenvalues. Actually the real eigenvalues and the respective chiralities are, = 0 + 3.0  +o(  2 ),  = [-1 +o(  2 ) ] sign (q 2 ) = 2 +o(  2 ),  = [+1 -2.0  +o(  2 ) ] sign (q 2 ) = 4 - 3.0  +o(  2 ),  = [-1 +o(  2 ) ] sign (q 2 )

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998) 2. The index of the Neuberger overlap action The Neuberger overlap action, n D = 1 1 + X, X = n D -m 0 I, X + =  5 X  5 R X + X is a solution of the Ginsparg-Wilson relation. Moreover it produces the trace, tr { n D} = 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 m 0 and increase it continuously. Whenever  5 X passes by a root, ’ i changes sign, and the charge R tr { n D}/2 has a discrete step of + 1. These steps occur at the zero modes of X, and this coincides with m 0 = i, 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) q 3 m 0  This shows the index of the Neuberger Action when we run the parameter m 0 = -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) Im  ’ Re  ’  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) 3. A criterion For low density the real eigenvalues of the Wilson action w D have chirality ~ + 1. This implies that the corresponding eigenvectors of w D are very close to eigenvectors of w D +. It is then possible to derive the criterion, ‘’ A hermitean solution of the Ginsparg-Wilson relation wgr D with constant R and constructed with the Wilson action w D, and w D +, complies maximally with the Index Theorem if and only if wgr D 2   ) (  is the Heaviside step function) R when we replace w D, w D +  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

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998) 4. Conclusion.The Wilson action approximately complies with the Index Theorem..The Neuberger overlap action with parameter m 0 =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) 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)

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

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

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Presentation on theme: "Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998) Criterion."— Presentation transcript:

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

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Presentation on theme: "Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998) Criterion."— Presentation transcript:

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