Chiral Fermions, Anomalies and

arXiv:hep-lat/9212030v1 21 Dec 1992UCSD/PTH 92-39November 1992Chiral Fermions, Anomalies andChern-Simons Currents on the Lattice 1Karl JansenUniversity of California at San DiegoDepartment of Physics-0319La Jolla, CA 92093-0319USAAbstractI discuss the zeromode spectrum of lattice chiral fermions in the domain wall modelsuggested recently. In particular I give the critical momenta where the fermions ceaseto be chiral and show that the spectrum is directly related to the behaviour of theChern-Simons current on the lattice.

First results for the domain wall model on thefinite lattice indicate that the relevant features of the model in the infinite systemsurvive for the finite lattice.1Talk presented at the International Conference on Lattice Field Theory, LATTICE 92, (1992) Amsterdam

1IntroductionIn this talk I want to report some first results for the domain wall model suggested byKaplan [1]. The basic idea of this model is to start with an odd dimensional –and thereforevectorlike– theory and to add a mass term for the fermions which depends on the extradimension and has the form of a soliton or kink, generating in this way a domain wall.

Itis well known that in a situation like this one finds zeromodes bound to the domain wall[2]. Kaplan was able to show that these zeromodes represent chiral fermions on the latticeand that the unwanted doubler modes can be removed by introducing the usual Wilson-term.

These results [1, 3], which hold for the free theory and for the infinite lattice, couldbe demonstrated to survive even on a finite lattice [4]. In addition the cancellation of thezeromode anomaly by the divergence of the Goldstone-Wilczek current as described in [1]could also been seen on the finite lattice [4].In this talk I will discuss the peculiar behaviour of the zeromode spectrum.

I will give thecritical momenta of these zeromodes where they lack to be chiral and relate the zeromodespectrum to the behaviour of the Chern-Simons current on the lattice.2The infinite latticeTo be specific I will discuss the spectrum and also the anomaly for the case of a 3-dimensionalmodel. The results generalized to arbitrary dimensions can be found in [3, 5].

I start withthe Dirac-Wilson operator on an infinite lattice with lattice spacing a = 1K3D =3Xµ=1σµ∂µ + mǫ(s) + r23Xµ=1∆µ(1)where ∂denotes the lattice derivative ∂µ = 12 [δz,z+µ −δz,z−µ], ∆the lattice Laplacian ∆µ =[δz,z+µ + δz,z−µ −2δz,z], the σµ are the usual Pauli matrices and r the Wilson coupling. Iwill denote by s the extra dimension along which the mass defect appears, while x, t are the2-dimensional coordinates.

The domain wall is taken to be a step function ǫ,ǫ(s) =−1s < 00s = 0+1s > 0(2)where the height of the domain wall is given by the mass parameter m which I will chooseto be positive throughout this paper.We are looking for solutions which are plane waves in the (x, t)-planeΨ± = ei(ptt+pxx)Φ(s)u±(3)1

where u± are the eigenspinors of σ3, σ3u± = ±u±. With this ansatz the Dirac operatorbecomesK3D =2Xi=1iσi sin(pi) + σ3∂s + mǫ(s) + r2Xi=1(cos(pi) −1) + r2∆s .

(4)The final goal is to diagonalize the 3 dimensional Dirac operator in such a way that it reducesto the 2 dimensional Dirac operator for free massless fermions,K3DΨ = K2DΨ where K2Dacting on Ψ is given byK2D =2Xµ=1σµ∂µ = i(σ1 sin(pt) + σ2 sin(px)) . (5)Hence the equation to solve isσ3∂s + mǫ(s) −rF + r2∆sΦu± = 0(6)where F =Pi=t,x(1 −cos(pi)).

Following [1] I choose an exponential ansatz for Φ away fromthe domain wall Φ(s + 1) = zΦ(s). Inserting this into (6) one finds four solutionsz =r −meff ±qmeff(meff −2r) + 1r ± 1(7)where meff = mǫ(s) −rF, the ± in the nominator stand for the two roots and the ± in thedenominator stand for the chirality.

Note that in the limit r = 1 eq. (7) can be reduced tothe corresponding expressions in [1].One has to impose the condition that the solutions are normalizable to obtain sensiblewavefunctions.

This means that |z| > 1 for s < 0 and |z| < 1 for s > 0. The boundaries ofthe regions where chiral solutions exist are obtained by setting |z| = 1.

Explicit matchingof the normalizable solutions for positive and negative s at s = 0 enables one to determinethe regions with chiral fermions. One finds that existence and chirality of the solutions isindependent of the sign of r and that a negative m leads to opposite chiralities.

Dependingon the values of m/r one gets m = rF and m = r(F + 2) as the boundaries for the criticalmomenta, where F is defined as above.The results can be summarized as follows. Starting with m/r = 0 one finds no chiralfermions.

For increasing 0 < m/r < 2 the region in momentum space around ⃗p = (0, 0)where chiral modes exist grows. This region is bounded by m = rF which gives the uppercritical momenta.

Increasing m/r above m/r = 2 opens up the two “doubler” modes at⃗p = (0, π) and ⃗p = (π, 0) which have flipped chirality, while the original mode at ⃗p = (0, 0)disappears. Here the boundaries of the regions in momentum space are given by m = rF forthe lower and m = r(F +2) for the upper critical momenta.

For m/r > 4 the two “doublers”disappear and one gets a zero mode at ⃗p = (π, π) with the same chirality as the mode at⃗p = (0, 0). The boundary for the lower critical momenta is given by m = r(F + 2).

Thismode is finally also lost as m/r is increased to m/r ≥6.It should be remarked that this spectrum stems from Ψ+ solutions only, and that thereare no Ψ−solutions for positive m. The change of the chirality is the usual reinterpretationof the chirality at different corners of the Brillouin zone.2

The zero mode spectrum as found here is directly related to the coefficient of the latticeChern-Simons current induced by heavy fermions. This current is responsible for the anomalycancellation.As there is a chiral zeromode bound to the domain wall, there exists thecorresponding anomaly.

However, we started with a vectorlike theory and consequently thetheory should be anomaly free. As is explained in [1] this contradiction is resolved by thefact that the domain wall induces a Goldstone-Wilczek current [6] far offthe domain wallthe divergence of which exactly cancels the zeromode anomaly on the domain wall [7].Recently, this current was also calculated on the lattice.

The radiatively induced Chern-Simons action in d=3 dimensions is given by [8]ΓCS = ǫµνρZd3xAµ∂νAρ . (8)The effective action is then given by cΓCS.

The coefficient c can be calculated in perturbationtheory [9, 5] and is given byc = iǫµνρ∂∂(q)νI|q=0(9)with I the following integralI =Zd3p(2π)3Tr [S(p)Λµ(p, p −q) S(p −q)Λρ(p + q, p)](10)where S is the fermion propagator and Λµ denotes the photon vertex. Imposing the WardidentityΛµ(p, p) = −i∂/∂pµS−1(p)(11)the relevant integral becomesZd3p(2π)3TrhS∂µS−1i hS∂νS−1i hS∂ρS−1i.

(12)The fermion propagator can be written in a generic form, S−1(p) = a(p)+i⃗b(p)⃗σ which hasthe structure of S−1(p) = N(p)V (p), where V (p) is a SU(2) matrix and N(p) a normalizationfactor. Provided that S−1 does not vanish for any p the integral eq.

(12) does not depend onN(p) and has a simple topological interpretation: It is the winding number of the map Vfrom the torus T 3 to SU(2) or the sphere S3. (In general this map is a map from the torusT d to the sphere Sd.

)Taking the usual Wilson fermion propagator the winding number can be calculated byusing the fact that the integral has to be computed only in infinitesimal regions near theBrillouin corners. One obtains (see [5] for more details)c =−i32π33Xk=0(−1)k m −2rk|m −2rk| .

(13)This expression should be compared with the (unregulated) continuum result [7]ccont =−i32π3m|m| . (14)3

Therefore one finds that for m/r < 0, c = 0. For m/r < 2 the lattice result is twice thecontinuum value.

For 2 < m/r < 4 it is −4ccont and for 4 < m/r < 6 it is again twice thecontinuum value.For the case of the domain wall model in which we are interested here the result has thefollowing implication: For the calculation of the Goldstone-Wilczek current one goes far offthe domain wall and assumes there the mass to be constant. Then one can calculate thecurrent with the method described above.

For m/r < 2 one finds therefore that the currentonly flows on one side of the domain wall and has twice the value of the continuum result. Ofcourse, the divergencies will come out the same.

The behaviour of the lattice Chern-Simonscoefficient finds its exact correspondence in the zeromode spectrum as discussed above wherewe found 1 lefthanded, 2 righthanded and 1 lefthanded chiral fermion for 0 < m/r < 2,2 < m/r < 4 and 4 < m/r < 6, respectively.3And the Finite LatticeAny numerical work on this system will necessarily involve finite lattices, and so I nowcompare the results obtained on the infinite system with the ones of a finite lattice. Onthe finite lattice one has to choose some boundary conditions.

Taking periodic boundaryconditions generates a second anti-domain wall. The mass term is therefore modified to bemǫL(s) withǫL(s) =−12 ≤s ≤Ls2+1Ls2 + 2 ≤s ≤Ls0s = 1, Ls2 + 1.

(15)The zeromodes on the finite lattice can be searched for by solving the Hamiltonianproblem numerically. If one again assumes plane waves in the x-direction the Hamiltonianis given by [4]H = −σ1 [iσ2 sin(px) + σ3∂s + mǫL(s) + r(cos(px) −1) + r2∆s).

(16)The eigenvalues and eigenfunctions of the Hamiltonian eq. (16) were calculated numeri-cally .

To find the critical momenta the ratioR =¯ΨΨ¯Ψσ1Ψ(17)was studied, which is a normalized measure for whether the fermions are chiral or not. It iszero if the fermions are chiral and R > 0 for non-chiral modes (see fig.2b in [4]).

To determinewhether one still has chiral fermions a threshold value for R was defined. If R < 0.01 thefermions were regarded to be chiral.Comparing the results from the infinite system with the finite lattice calculations withL = 100 [3], one finds that the two are practically indistinguishable.

For L = 20, a lattice4

size realistic for simulations, a small shift occurs. Fixing m and r we find for m/r < 2 asmaller value and for 2 < m/r < 4 a larger value of the critical momentum.It is also possible to extract the Chern-Simons current on the finite lattice in the presenceof a smooth external gauge field configuration.

The current is most easily computed bythe inverse fermion matrix of the model which can be obtained by standard methods likeconjugate gradient. Note that this is an exact solution of the numerical problem and thatno simulation is involved [4].I show in fig.

1 the Chern-Simons current on a 163 lattice with m = 0.81 and r = 0.9.Note that these values of m, r are quite large. The picture nicely demonstrates that theadvocated flow of the current as obtained above is reproduced on the finite lattice: It flowsonly on one side of the domain wall and is zero on the other.It is a simple task to get the divergence of the current and one can demonstrate [4] thatit obeys the continuum anomaly equation to a very good accuracy.

It is also possible tofind non-trivial anomaly cancellation like in the 3-4-5 model where the individual fermioncurrents are anomalous but the sum of them cancel.In summary the domain wall model shows a lot of promising features on the finite latticelike the chiral zeromode spectrum and the correct anomaly behaviour. Although these prop-erties were only found for the free theory or with weak external gauge fields they certainlypoint into the right direction and are encouraging.

The next step is to include dynamicalgauge fields and see whether one can find the desired behaviour.AcknowledgementsI want to thank M.F.L. Golterman, D. Kaplan and M. Schmaltz for giving me the opportunityto present our work at this conference.

I also want to thank them and J. Kuti for numeroushelpful and stimulating discussions.This work is supported by DOE grant DE-FG-03-90ER40546.5

References[1] D.B. Kaplan, Phys.Lett.B288 (1992) 342.

[2] R.Jackiw and C. Rebbi, Phys.Rev.D 13 (1976) 3398. [3] K. Jansen and M. Schmaltz, Critical Momenta of Lattice Chiral Fermions, UCSDpreprint, UCSD/PTH 92-29, to appear in Phys.Lett.B.

[4] K. Jansen, Phys.Lett.B288 (1992) 348. [5] M.F.L.

Golterman, K. Jansen and D. Kaplan, Chern-Simons Currents and ChiralFermions on the Lattice, UCSD preprint, UCSD/PTH 92-28. [6] J. Goldstone and F. Wilczek, Phys.Rev.Lett.

47 (1981) 986. [7] C.G.

Callan, Jr. and J.A. Harvey, Nucl.Phys.

B250 (1985) 427. [8] A.J.

Niemi and G.W. Semenoff,Phys.Rev.Lett51(1983) 2077;A.N.

Redlich,Phys.Rev.D29 (1984) 2366. [9] A. Coste and and M. L¨uscher, Nucl.Phys.

B323 (1989) 631.Figure CaptionFig.1 I show the Chern-Simons current in arbitrary units as obtained on a 163 lattice. Itshows the expected peculiar behaviour that it flows only on one side of the domain wall andis zero on the other.

The locations of the domain walls is at s=1 and at s=9.6

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