We study the dynamical breaking of chiral symmetry in the 3-d Thirring model
for a small number of fermion species. The critical point is identified by
fitting lattice data to an equation of state. The spectrum of the theory is
studied to confirm the phase structure of the model.Comment: 3 pages, 3 figures. Talk presented at Lattice '9

Del Debbio, L.

English

arXiv

We study the dynamical breaking of chiral symmetry in the 3-d Thirring model for a small number of fermion species. The critical point is identified by fitting lattice data to an equation of state. The spectrum of the theory is studied to confirm the phase structure of the model

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     Edinburgh Research Explorer                                      Chiral symmetry breaking in the 3-d Thirring model for small$N_f$Citation for published version:Del Debbio, L 1997 'Chiral symmetry breaking in the 3-d Thirring model for small $N_f$' ArXiv.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Preprint (usually an early version)General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 20. Feb. 2015arXiv:hep-lat/9709034v1  12 Sep 19971Chiral Symmetry Breaking in the 3-d Thirring model for small Nf .L. Del Debbio a∗ (for the UKQCD Collaboration)aCentre de Physique The´orique, CNRS Luminy, F-13288 Marseille Cedex 9, FranceWe study the dynamical breaking of chiral symmetry in the 3-d Thirring model for a small number of fermionspecies. The critical point is identified by fitting lattice data to an equation of state. The spectrum of the theoryis studied to confirm the phase structure of the model.1. IntroductionThe Thirring model is a three dimensional QFTwith Nf fermion species and a current–currentfour fermion interaction. As such, it is obviouslynon–renormalisable in the usual perturbative ex-pansion. It has been pointed out several yearsago [1] that a continuum limit can be definedfor some theories in a non–perturbative (NP) ap-proach. For the Gross–Neveu model, a four–fermitheory with a (pseudo–)scalar interaction, the ex-istence of such an UV fixed point has been shownwithin the 1/Nf expansion and confirmed by nu-merical simulations. The critical point separatesa strong coupling phase with chiral symmetrybreaking from a symmetric weak–coupling one.For the Thirring model, solid conclusions cannot be drawn from analytic results as differentNP techniques yield different results. We aim toobtain a complete description of the phase struc-ture of this model by lattice simulation, usingstaggered fermions and a HMC algorithm. Themotivations, the lattice formulation of the the-ory and the technical details of our analysis werepresented in great details in earlier works [2], towhich we refer the interested reader. This con-tribution summarizes our most recent results forNf = 4, 6.The order parameter for detecting chiral sym-metry breaking is the chiral condensate 〈χ¯χ〉. Itsbehaviour around the critical point is describedby fitting to an equation of state (EOS) [3].Finite–size effects are included in the EOS andthe critical exponents are obtained for Nf = 4.∗This is work is supported by CNRS and the Angelo DellaRiccia foundation.The results are compared with those for Nf = 2.Once again, lattice data show a clear signal ofchiral symmetry breaking and fit rather well theRG predictions.Moreover the chiral symmetry breaking transi-tion should manifest itself in the behaviour of thesusceptibilities and in the spectrum of the theory.This issue was already addressed forNf = 2 in [2].Preliminary results for Nf = 4 are reported here.The general pattern, which had already emergedin our previous works, is confirmed by the newresults. We briefly outline the next developmentsin our program.2. Fitting to the Equation of StateThe EOS is a functional relation between theorder parameter, the coupling, g, and the baremass, m, in the regularised theory, which can bederived form the RG equations in the vicinity ofa fixed point (see e.g. [3]). It therefore provides arelation to which lattice data can be fitted. Theresults of this approach for Nf = 2 and fixedlattice size were presented in [2]. In order to takeinto account finite size effects, it is sufficient toconsider the inverse linear size of the lattice, 1/Las an additional scaling field of dimension 1. Theexact functional form of the EOS is unknown, butit can be expanded in Taylor series, yielding afitting function for the lattice data with only afew unknown parameters:m = B〈χ¯χ〉δ +A(t+ CL−1/ν)〈χ¯χ〉δ−1/β (1)where t = 1/g2 − 1/g2c .For Nf = 4, we fit to the EOS data from latticewhose sizes range from 84 to 164. The results are2Table 1Results from fits including finite size scaling.Parameter Fit Parameter FitNf = 2 1/g2c 1.92(2) Nf = 4 1/g2c 0.70(2)δ 2.75(9) δ 3.41(31)β 0.57(2) β 0.41(10)η 0.60(2) η 0.34(20)ν 0.71(4) ν 0.61(10)A 0.334(7) A 0.70(5)B 2.7(3) B 5.8(2.1)C 2.1(7) C 2.9(2.2)χ2/d.o.f 1.76 χ2/d.o.f 0.80summarized in Tab. 1.It is interesting to stress that the results fromthe fit do take into account the effects due to thefinite volume of the lattice and should thereforebe interpreted as values in the thermodymamicallimit. The values for the critical coupling and thecritical exponents are stable with respect to theprevious results obtained for Nf = 4 at fixed lat-tice size [2]. Such a stability in our fits can beseen as a sign that the lattice data really followthe behaviour expected in the vicinity of an UVfixed point. The critical exponents differ signifi-cantly between Nf = 2 and Nf = 4, confirminga result that had already emerged from the fixedsize fits. We should be able to reduce the er-rors on the fitted values as statistics increase overthe next months. It is worthwhile to stress onceagain that, in order to identify unambiguously afixed point, we should study the renormalised tra-jectories in coupling space, for which we have tomeasure 3- and 4-point correlation functions.3. Susceptibilities and spectrumIndependent information on the phase struc-ture of the theory can be obtained from the sus-ceptibilities and the spectrum of the theory. Onceagain precise definitions and the results for theNf = 2 case are given in previous works andwe want to concentrate on the new results forNf = 4, 6. We briefly recall the definitions andthe expected behaviour for some quantities of in-terest.The transverse and longitudinal susceptibilitiesare defined respectively as:χl =1V∑x〈χ¯χ(0) χ¯χ(x)〉c (2)χt =1V∑x〈χ¯εχ(0) χ¯εχ(x)〉c =1m〈χ¯χ〉 (3)They are related to the scalar and pion masses:χl,t = Zs,pi/M2s,pi (4)Another interesting quantity is their ratio R =χl/χt. If there is a critical point in the phase dia-gram where chiral symmetry is dynamically bro-ken, then:limm→0R ={0, g2 > g2c ;1δ− 1β, g2 < g2c .(5)The results for the susceptibilities are shown inFig. 1. Even though the results were producedat non-vanishing bare mass, i.e. in presence ofan explicit symmetry breaking term, the patternreflects the existence of a phase transition. Onceagain the near degeneracy of χl and χt in thesymmetric phase (t > 0) supports the hypothesisδ − 1/β = 1, used in the fits reported in Tab. 1.In the strong coupling regime (t < 0), thetransverse susceptibility is larger than the lon-gitudinal, reflecting the fact that the pion is al-most a Goldstone boson in that regime. The sametrend is confirmed by the masses extracted fromthe aymptotic behaviour of two–point functions.Preliminary results are reported in Fig. 2; the er-rors on the scalar mass is still very large and hasnot been included in the plot. A more detailed3-0.40 -0.20 0.00 0.20 0.400.00.51.01.52.0Figure 1. Transverse (circles) and longitudinal(diamonds) susceptibilities for Nf = 4, m = 0.01,on a 163 lattice. The data are plotted against arescaled reduced coupling t = g2c (1/g2 − 1/g2c).analysis, based on larger statistical samples, iscurrently in progress and we hope to report soonon the details of the spectrum for Nf = 4. Itis nonetheless worth to remark that the generalpattern is consistent with the existence of an UVfixed point. In particular, we see no sign for theunconventional spectrum in the symmetric phaseassociated to conformal phase transitions, as sug-gested in [4].0.50 0.60 0.70 0.80 0.900.00.20.40.60.8Figure 2. Fermion (circles), pion (diamonds) andscalar (triangles) masses vs. 1/g2, for Nf = 4,m = 0.01, on a 163 lattice.In order to distinguish between different sce-narios of chiral symmetry breaking (correspond-ing, e.g., to different solutions of the Schwinger–Dyson equations), several values of Nf need tobe studied. New preliminary results for the chi-ral condensate are reported in Fig. 3. The latticeused is larger than in our earlier works [2] and thebehaviour of the condensate as the bare mass mis decreased suggests the existence of a symmetrybroken phase at strong coupling. We will reportmore systematic results in the near future.0.20 0.30 0.400.000.050.100.150.20Figure 3. Chiral condensate vs. 1/g2 for Nf = 6,m = 0.01 (circles) on a 163 lattice and m = 0.02(squares) on a 123 lattice.Acknowledgements: Some of the numeri-cal simulations reported here are funded underHPCI, EPSRC grant GR/K41663 and PPARCgrant GR/K55745. Last, but not least, it is apleasure to thank the LOC for financial support.REFERENCES1. G. Parisi, Nucl. Phys. B 100 (1975) 368.2. L. Del Debbio, S.J. Hands, Phys. Lett. B 373(1996) 171; L. Del Debbio, Nucl. Phys. B(Proc. Suppl.) 53 (1997) 699; L. Del Debbio,J.C. Mehegan, S.J. Hands, hep-lat/9701016,accepted for publication in Nucl. Phys. B.3. J. Zinn-Justin, “Quantum Field Theory andCritical Phenomena”, second edition, Claren-don Press, Oxford 1993.4. V.A. Miranskii, K. Yamawaki, Phys. Rev. D55 5051 (1997).

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