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value and chirality +1, which corresponds to the isolated ground state of D† D
D |ψ+ , 0 = 0 ” D |ψ+ = 0 , σ3 |ψ+ , 0 = |ψ+ , 0
with
ψ+ (x) = e’S(x) .
On the other hand, the formal solution of D† |ψ’ = 0,
ψ’ (x) = eS(x) ,
is not normalizable and, therefore, no eigenvector with negative chirality is found.
We conclude that the operator D has only one eigenvector with zero eigen-
value corresponding to positive chirality: the index of D is one. Note that ex-
pressions for the index of the Dirac operator in a general background have been
derived. In the present example, they yield
[sgn A(+∞) ’ sgn A(’∞)]
1
Index = 2

in agreement with the explicit calculation.

The Resolvent. For later purpose it is useful to exhibit some properties of the
resolvent
’1
G = (D ’ k) ,
for real values of the parameter k. Parametrizing G as a 2 — 2 matrix:
G11 G12
G= ,
G21 G22
one obtains
’1
G11 = ’k D† D + k 2
’1
G21 = ’D D† D + k 2
’1
G12 = D† DD† + k 2
’1
G22 = ’k DD† + k 2 .
Chiral Anomalies and Topology 225

For k 2 real one veri¬es G21 = ’G† .
12
A number of properties then follow directly from the analysis presented in
Appendix B.
When k ’ 0 only G11 has a pole, G11 = O(1/k), G22 vanishes as k and
G12 (x, y) = ’G21 (y, x) have ¬nite limits:

’ k ψ+ (x)ψ+ (y)/ ψ+ ’G21 (y, x)
1 2 1 ψ+ (x)ψ+ (y)
G(x, y) ∼ ∼’ (1+σ3 ).
G12 (x, y) 0 ψ+ 2
2k
k’0


Another limit of interest is the limit y ’ x. The non-diagonal elements are
discontinuous but the limit of interest for domain wall fermions is the average
of the two limits

G(x, x) = 1 (1 + σ3 )G11 (x, x) + 1 (1 ’ σ3 )G22 (x, x) + iσ2 G12 (x, x) .
2 2

When the function A(x) is odd, A(’x) = ’A(x), in the limit x = 0 the matrix
G(x, x) reduces to

G(0, 0) = 1 (1 + σ3 )G11 (0, 0) + 1 (1 ’ σ3 )G22 (0, 0).
2 2



Examples.
12
(i) In the example of the function S(x) = 2x , the two components of the
hamiltonian H become

DD† = ’d2 + x2 + 1 , D† D = ’d2 + x2 ’ 1 .
x x


We recognize two shifted harmonic oscillators and the spectrum of D contains

one eigenvalue zero, and a spectrum of opposite eigenvalues ±i 2n, n ≥ 1.
(ii) Another example useful for later purpose is S(x) = |x|. Then A(x) =
sgn(x) and A (x) = 2δ(x). The two components of the hamiltonian H become

DD† = ’d2 + 1 + 2δ(x), D† D = ’d2 + 1 ’ 2δ(x). (84)
x x


Here one ¬nds one isolated eigenvalue zero, and a continuous spectrum µ ≥ 1.
(iii) A less singular but similar example that can be solved analytically cor-
responds to A(x) = µ tanh(x), where µ is for instance a positive constant. It
leads to the potentials

µ(µ “ 1)
V (x) = A2 (x) ± A (x) = µ2 ’ .
cosh2 (x)

The two operators have a continuous spectrum starting at µ2 and a discrete
spectrum

µ2 ’ (µ ’ n)2 , n ∈ N ¤ µ, µ2 ’ (µ ’ n ’ 1)2 , n ∈ N ¤ µ ’ 1.
226 J. Zinn-Justin

8.2 Field Theory in Two Dimensions
A natural realization in quantum ¬eld theory of such a situation corresponds to
a two-dimensional model of a Dirac fermion in the background of a static soliton
(¬nite energy solution of the ¬eld equations).
¯ ¯
We consider the action S(ψ, ψ, •), ψ, ψ being Dirac fermions, and • a scalar
boson:
2
¯ ¯
S(ψ, ψ, •) = dx dt ’ψ (‚ + m + M •) ψ + 1
(‚µ •) + V (•) .
2


We assume that V (•) has degenerate minima, like (•2 ’ 1)2 or cos •, and ¬eld
equations thus admit soliton solutions •(x), static solitons being the instantons
of the one-dimensional quantum • model.
Let us now study the spectrum of the corresponding Dirac operator
D = σ1 ‚x + σ2 ‚t + m + M •(x).
We assume for de¬niteness that •(x) goes from ’1 for x = ’∞ to +1 for
x = +∞, a typical example being
•(x) = tanh(x).
Since time translation symmetry remains, we can introduce the (euclidean) time
Fourier components and study
D = σ1 dx + iωσ2 + m + M •(x).
The zero eigenmodes of D are also the solutions of the eigenvalue equation
D |ψ = ω |ψ , D = ω + iσ2 D = σ3 dx + iσ2 m + M •(x) ,
which di¬ers from (81) by an exchange between the matrices σ3 and σ1 . The
possible zero eigenmodes of D (ω = 0) thus satisfy
σ1 |ψ = |ψ , = ±1
and, therefore, are proportional to ψ (x), which is a solution of
ψ + m + M •(x) ψ = 0 .
This equation has a normalizable solution only if |m| < |M | and = +1. Then
we ¬nd one fermion zero-mode.
A soliton solution breaks space translation symmetry and thus generates a
zero-mode (similar to Goldstone modes). Straightforward perturbation expan-
sion around a soliton then would lead to IR divergences. Instead, the correct
method is to remove the zero-mode by taking the position of the soliton as a col-
lective coordinate. The integration over the position of the soliton then restores
translation symmetry.
The implications of the fermion zero-mode require further analysis. It is found
that it is associated with a double degeneracy of the soliton state, which carries
1/2 fermion number.
Chiral Anomalies and Topology 227

8.3 Domain Wall Fermions

Continuum Formulation. One now considers four-dimensional space (but the
strategy applies to all even dimensional spaces) as a surface embedded in ¬ve-
dimensional space. We denote by xµ the usual four coordinates, and by t the
coordinate in the ¬fth dimension. Physical space corresponds to t = 0. We then
study the ¬ve-dimensional Dirac operator D in the background of a classical
scalar ¬eld •(t) that depends only on t. The fermion action reads

¯ ¯
S(ψ, ψ) = ’ dt d4 x ψ(t, x)Dψ(t, x)

with
D = ‚ + γ5 dt + M •(M t),
where the parameter M is a mass large with respect to the masses of all physical
particles.
Since translation symmetry in four-space is not broken, we introduce the
corresponding Fourier representation, and D then reads

D = ipµ γµ + γ5 dt + M •(M t).

To ¬nd the mass spectrum corresponding to D, it is convenient to write it as

D = γp [i|p| + γp γ5 dt + γp M •(M t)] ,
2
where γp = pµ γµ /|p| and thus γp = 1. The eigenvectors with vanishing eigen-
value of D are also those of the operator

D = iγp D + |p| = iγp γ5 dt + iγp M •(M t),

with eigenvalue |p|.
We then note that iγp γ5 , γp , and ’γ5 are hermitian matrices that form a
representation of the algebra of Pauli matrices. The operator D can then be
compared with the operator (81), and M •(M t) corresponds to A(x). Under
the same conditions, D has an eigenvector with an isolated vanishing eigenvalue
corresponding to an eigenvector with positive chirality. All other eigenvalues, for
dimensional reasons are proportional to M and thus correspond to fermions of
large masses. Moreover, the eigenfunction with eigenvalue zero decays on a scale
t = O(1/M ). Therefore, for M large one is left with a fermion that has a single
chiral component, con¬ned on the t = 0 surface.
One can imagine for the function •(t) some physical interpretation: • may
be an additional scalar ¬eld and •(t) may be a solution of the corresponding
¬eld equations that connects two minima • = ±1 of the • potential. In the
limit of very sharp transition, one is led to the hamiltonian (84). Note that such
an interpretation is possible only for even dimensions d ≥ 4; in dimension 2,
zero-modes related to breaking of translation symmetry due to the presence of
the wall, would lead to IR divergences. These potential divergences thus forbid
228 J. Zinn-Justin

a static wall, a property analogous to the one encountered in the quantization
of solitons in Sect. 8.2.
More precise results follow from the study of Sect. 8.1. We have noticed that
G(t1 , t2 ; p), the inverse of the Dirac operator in Fourier representation, has a
short distance singularity for t2 ’ t1 in the form of a discontinuity. Here, this
is an artifact of treating the ¬fth dimension di¬erently from the four others. In
real space for the function G(t1 , t2 ; x1 ’ x2 ) with separate points on the surface,
x1 = x2 , the limit t1 = t2 corresponds to points in ¬ve dimensions that do not
coincide and this singularity is absent. A short analysis shows that this amounts
in Fourier representation to take the average of the limiting values (a property
that can easily be veri¬ed for the free propagator). Then, if •(t) is an odd
function, for t1 = t2 = 0 one ¬nds

i
D’1 (p) = d1 (p2 )(1 + γ5 ) + (1 ’ γ5 )p2 d2 (p2 ) ,
2p

where d1 , d2 are regular functions of p2 . Therefore, D’1 anticommutes with γ5
and chiral symmetry is realized in the usual way. However, if •(t) is of more
general type, one ¬nds

i
D’1 = d1 (p2 )(1 + γ5 ) + (1 ’ γ5 )p2 d2 (p2 ) + d3 (p2 ),
2p

where d3 is regular. As a consequence,

γ5 D’1 + D’1 γ5 = 2d3 (p2 )γ5 ,

which is a form of Ginsparg“Wilson™s relation because the r.h.s. is local.


Domain Wall Fermions: Lattice. We now replace four-dimensional contin-
uum space by a lattice but keep the ¬fth dimension continuous. We replace the
Dirac operator by the Wilson“Dirac operator (80) to avoid doublers. In Fourier
representation, we ¬nd

D = ±(p) + iβµ (p)γµ + γ5 dt + M •(M t).

This has the e¬ect of replacing pµ by βµ (p) and shifting M •(M t) ’ M •(M t) +
±(p). To ensure the absence of doublers, we require that for the values for which
βµ (p) = 0 and p = 0 none of the solutions to the zero eigenvalue equation is
normalizable. This is realized if •(t) is bounded for |t| ’ ∞, for instance,

|•(t)| ¤ 1

and M < |±(p)|.
The inverse Dirac operator on the surface t = 0 takes the general form

D’1 = iβ δ1 (p2 )(1 + γ5 ) + (1 ’ γ5 )δ2 (p2 ) + δ3 (p2 ),
Chiral Anomalies and Topology 229

where δ1 is the only function that has a pole for p = 0, and where δ2 , δ3 are
regular. The function δ3 does not vanish even if •(t) is odd because the addition
of ±(p2 ) breaks the symmetry. We then always ¬nd Ginsparg“Wilson™s relation

γ5 D’1 + D’1 γ5 = 2δ3 (p2 )γ5

More explicit expressions can be obtained in the limit •(t) = sgn(t) (a situation
analogous to (84)), using the analysis of the Appendix B.
Of course, computer simulations of domain walls require also discretizing the
¬fth dimension.


Acknowledgments

Useful discussions with T.W. Chiu, P. Hasenfratz and H. Neuberger are grate-
fully acknowledged. The author thanks also B. Feuerbacher, K. Schwenzer and
F. Ste¬en for a very careful reading of the manuscript.


Appendix A. Trace Formula for Periodic Potentials

We consider a hamiltonian H corresponding to a real periodic potential V (x)
with period X:
V (x + X) = V (x).

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. 49
( 78 .)



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