How much are 2d Yukawa models similar to the Gross-Neveu models?∗

arXiv:hep-lat/9212024v1 18 Dec 19921How much are 2d Yukawa models similar to the Gross-Neveu models?∗A.K. Dea, E. Fochtb,c, W. Franzkib,c and J. Jers´akb,caWashington University, Department of Physics, St. Louis MO 63130, USAbInstitute for Theoretical Physics E, RWTH Aachen, Sommerfeldstr., 5100 Aachen, GermanycHLRZ c/o KFA J¨ulich, P.O.

Box 1913, 5170 J¨ulich, GermanyWe present numerical evidence that the 2d Yukawa models with strong quartic selfcoupling of the scalar fieldhave the same phase structure and are asymptotically free in the Yukawa coupling like the Gross-Neveu models.1. Yukawa models in 2 dimensionsThe 2d Yukawa models (Y2 ) with chiral Z(2)or U(1) symmetries are natural extensions of theusual or chiral 2d Gross-Neveu (GN) models, re-spectively.

Starting from the auxiliary field repre-sentation of the 4-fermion coupling, one can addboth the kinetic term and a self-interaction of thisfield φ into the GN action. On the lattice the Z(2)symmetric Y2 action is then (we introduce NF /4“naive” Dirac fermion fields ψα)S=−2κXx,µφxφx+µ +Xxφ2x + λXx(φ2x −1)2+Xx,αψαx∂/ψαx + yXx,αψαxφxψαx .

(1)The φ4 scalar selfcoupling has been chosen, out ofmany possibilities in 2d, for the sake of analogywith the 4d theories. For κ = λ = 0 the Z(2)GN model is obtained if the Yukawa coupling y isrelated to the usual GN coupling g by y =√2g.By choosing the above hopping parameter κformulation of the scalar field sector the kineticterm can be turned on or offgradually, elucidatingthe smoothness of the transition from the auxil-iary to a dynamical field.

The relationshipy0 =ya√2κ(2)between the Yukawa coupling y0 used in contin-uum and y is singular at κ = 0, however. Another∗Talk presented by E. Focht at the ”Lattice ’92” confer-ence, Amsterdam.virtue of the hopping parameter formulation isthat spin models with the Z(2) or U(1) symme-try (the Ising or the XY models, respectively) areeasily recovered at y = 0 and κ finite by choosingλ = ∞.

The Y2 models thus interpolate betweenthe GN and spin models.2. Expected scaling propertiesMotivated by the recent discussion of a rela-tionship between the Nambu–Jona-Lasinio typefour-fermion theories and the Standard Model [1–3] we address here the question in which regionsof the parameter space the Y2 models still possessthe most cherished properties of the GN models,namely the asymptotic freedom at y →0, thefermion mass generation and, in the case of theZ(2) model, the symmetry breaking [4,5].

In theGN models these properties are derived by meansof the 1/NF expansion.For Y2 models this expansion is applicable forλ = O(1/NF) [1,2] and here the same results asfor the GN models are found. In particular, aslong as κ < κc(λ = 0) = 1/4, the fermion massamF is expected to scale asamF ∝exp−12β0a2m2Z1y2.

(3)For λ = 0 we havem2 = m20 =1 −κκc(0)1a2κ ,Z = 12κ ,(4)with m0 being the scalar field mass at λ = y = 0.

2For large λ the 1/NF expansion is `a priori notapplicable, not to speak of the perturbation the-ory. Nevertheless, M.A.

Stephanov suggested [6]that in this case the mean field (MF) theory couldbe a good guide. An observation of long range fer-romagnetic couplings in the effective scalar fieldtheory at y > 0 by E. Seiler [7] supports the ap-plicability of the MF theory.

The resulting ex-pectations are [8]:(i)The Y2 models at λ = ∞or large pos-sess at arbitrarily small y only the broken sym-metry (in the Z(2) case) or the spin-wave (in theU(1) case) phase, whereas the symmetric or vor-tex phase present in the y = 0 case in the purescalar theory is absent at y > 0. (ii)The fermion mass and the magnetizationy⟨φ⟩scale according to eq.

(3). The coefficientm2 is for κ > 0 the squared scalar mass at y = 0and Z its wave function renormalization constant.In general, Z/m2 could be replaced by the scalarfield propagator at zero momentum, i.e.

the sus-ceptibility. This should hold for any κ < κc(λ)where κc(λ) is the line of critical points in thepure scalar theory at y = 0.Thus according to the MF theory the asymp-totic freedom and the other mentioned propertiesof the GN models might occur also for large λ inthe Y2 models.3.

Methods of data analysisWe have tried three methods:(i)The asymptotic scaling law, which couldbe used to fit the data, is as (3) multiplied byy−β1/β20. Here β0 and β1 are the β-function coef-ficients which in the GN limit have the perturba-tive values [9,10]β0 = NF −12π,β1 = −NF −1(2π)2(5)in the model with Z(2) symmetry andβ0 = NF2π,β1 = −NF2π2(6)in the U(1) case.

This method does not take thefinite volume effects into account. Nevertheless,one can roughly determine the values of the co-efficient β0 and compare with the values (5) and(6).

The 2-loop contribution given by β1 is lessimportant than the finite size effects. (ii)A more appropriate way of analyzing thedata is provided by a modified gap equationon finite lattices,a2m2Z1y2 = πβ0VX{p}1Pµsin2 pµ + (amF /s)2(7)where the sum is performed over the fermion mo-menta with one periodic and one antiperiodicboundary condition on the L2 lattices.

Thanksto the IR divergence it leads for small amF in theinfinite volume limit to the scaling law (3).The coefficient β0 is considered as a free pa-rameter in order to allow for its possible devia-tion from the perturbative value. The parameters takes into account the fact that the gap equa-tion determines the value of the mass gap withinsufficient precision.

In order to fit the data it isnecessary to treat the mass gap as a free param-eter.It would be interesting to determine the massgap and compare it with the recently obtainedexact results for the GN models [11,12].Withour present accuracy it could be only estimatedto be roughly consistent with these results.We note that one could take the finite size ef-fects into account also beyond the leading 1/NForder [13,14]. As we do not expect this expansionto be applicable for large λ, we do not attemptsuch refinements.Once the parameters in the gap equation havebeen determined by a fit, the quantity h(κ, λ),h(κ, λ) =12β0a2m2Z⇒amF ∝exp−h(κ, λ)y2(8)has been extracted.

(iii)We have tried to fit the data also by analternative to the essential singularity at y = 0,namely by a hypothetical power law behaviormF = a(y −yc)b.(9)4. Results at λ = 0To gain experience we have first performed sim-ulations at λ = 0 in the interval −0.1 ≤κ <κc(λ = 0) = 1/4.

We have determined in both

3models the y-dependence of amF and of y⟨φ⟩inthe Z(2) model at fixed values of κ on latticesof size 162 - 642. The following observations areuseful for the study of the models at large λ:(i)The data analysis by means of the asymp-totic law (3) both without and with the 2–loopcorrection is possible if those points at small y,which show finite size effects, are excluded (fig.

1).However, the obtained values of β0 change withlattice size, when lower values of amF can betaken into account on larger lattices, so that oneprobably does not see the true asymptotic be-haviour. (ii)The gap equation (7) can describe thedata obtained on different lattices consistently bymeans of one set of parameter values, includingβ0 (fig.

1). The onset of finite size effects at smallamF, as well as the general tendency of the dataat large y are well reproduced.

This analysis issuperior to that by means of (3). (iii)The fermion masses in both models andy⟨φ⟩in the Z(2) model behave in a very analo-gous way.

The magnetization in the U(1) case ispresent on finite lattices, but it shows a signifi-cant size dependence consistent with its vanish-ing in the infinite volume. Thus we observe thedynamical fermion mass generation taking placein spite of the absence of spontaneous symmetrybreaking in 2d [5].

(iv)The κ-dependence is consistent with theexpected one, eq. (4).

This is demonstrated infig. 2, where h(κ, 0) is shown.

The values of β0obtained at different κ are for both models con-sistent with the values (5) and (6) within 10 %. (v)The power law formula (9) can fit thedata well for any given lattice size.

The parame-ters b and yc depend strongly on the lattice size,however. This makes an algebraic singularity lessprobable than the essential one, but a reliable ex-clusion of the former on the basis of data aloneseems very difficult.

So one should be cautiousat large λ, when the analytic information on thetype of singularity is less reliable than at λ = 0.5. Results at large λGoing to large λ we have simulated both mod-els at λ = 0.5 and the U(1) model at λ = ∞in theinterval −0.3 ≤κ < κc(λ).

The most importantresults are:(i)The y-dependence of the fermion mass,including the onset of the finite size effects, isdescribed by the gap equation (7) nearly as wellas in the λ = 0 cases. This is demonstrated infig.

3 for the U(1) model at λ = ∞and κ = 0 on162 - 642 lattices. (ii)The power law behaviour (9) is not ex-cluded but, similarly to λ = 0, disfavored by thestrong dependence of yc and b on the lattice size.However, the finite size scaling analysis based onthe Ansatz (9) has yet to be done.

(iii)These facts lead us to the tentative con-clusion that at large λ, including λ = ∞, onlythe broken symmetry (Z(2) model) or spin wave(U(1) model) phase is present at arbitrary smally. (iv)Except at κ = 0, we do not yet havean independent determination of m2/Z for largeλ.Therefore we cannot yet determine the val-ues of β0 and give the results in form of the val-ues of the coefficient h(κ, λ).

As an example weshow h(κ, ∞) for the U(1) model in fig. 4 (hereκc(∞) ≈0.56).

We hope to determine β0 at largeλ for various κ in the near future. At present ourestimates indicate that its values are roughly con-sistent with β0 in the GN cases, eqs.

(5) and (6).Thus we have found some evidence that theY2 models with Z(2) and U(1) chiral symmetrieswith a strong φ4 interaction behave for κ < κc(λ)as the GN models of the same symmetry, in par-ticular they are asymptotically free.The tran-sition from the GN to the spin model universal-ity classes takes place probably at small y andκ ≈κc(λ).6. AcknowledgementsWe thank E. Seiler and M.A.

StephanovforhelpfulsuggestionsandA.Hasenfratz,P. Hasenfratz, R. Lacaze, F. Niedermayer andM.M.

Tsypin for valuable discussions. The calcu-lations have been performed on the CRAY Y-MPof HLRZ J¨ulich.

This work has been supportedby Deutsches Bundesministerium f¨ur Forschungund Technologie and by Deutsche Forschungsge-meinschaft.

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M. Campostrini, G. Curci and P. Rossi, Nucl.Phys. B314 (1989) 467.Figure 1.The 1/y2 dependence of y⟨φ⟩in theZ(2) GN model (κ = λ = 0) with fits by eq.

(7).The straight line is a fit by eq. (3).Figure 2.The κ dependence of the coefficienth(κ, 0) in the Z(2) model at λ = 0 is consistentwith eq.

(4) (dashed line).Figure 3.The 1/y2 dependence of amF in theU(1) Y2 model with NF = 16 at λ = ∞with fitsby eq. (7).Figure 4.The κ dependence of the coefficienth(κ, ∞) in the U(1) model at λ = ∞.

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