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).