Table Of Content

CRITICAL FLAVOR NUMBER IN THE THREE DIMENSIONAL THIRRING MODEL Stavros Christofia, Simon Handsb and Costas Strouthosc aFrederick Institute of Technology, CY-1303 Nicosia, Cyprus bDepartment of Physics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K. and c Institute of Chemical Sciences and Engineering, Ećole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland. WepresentresultsofaMonteCarlosimulation ofthethreedimensionalThirringmodelwiththe numberoffermion flavorsN variedbetween2and18. Byidentifyingthelatticecouplingatwhich f the chiral condensate peaks, simulations are be performed at couplings g2(N ) corresponding to f the strong coupling limit of the continuum theory. The chiral symmetry restoring phase transition is studied as N is increased, and the critical number of flavors estimated as N = 6.6(1). The f fc critical exponents measured at the transition do not agree with self-consistent solutions of the 7 Schwinger-Dyson equations; in particular there is no evidence for the transition being of infinite 0 order. Implications for thecritical flavornumberin QED3 are briefly discussed. 0 2 PACSnumbers: PACS:11.10.Kk,11.30.Rd,11.15.Ha n a J The study of quantum field theories in which the QED4 [10, 11]. Using different sequences of truncation, 2 groundstateshowsasensitivitytothenumberoffermion however,otherSDEapproacheshavefoundvaluesofN fc 2 flavors N is intrinsically interesting. According to cer- ranging between 3.24 [4] and ∞ [6]. f 1 tainapproximatesolutionsofSchwinger-Dysonequations SinceforNf <Nfcthechiralsymmetrybreakingtran- (SDEs), in d = 3 spacetime dimensions both quantum sition corresponds to a UV-stable RG fixed point at a v 6 electrodynamics(QED3)andthe Thirringmodeldisplay critical gc2, which can be analysed using finite volume 1 thisphenomenon. Bothmodelshavebeenproposedasef- scaling techniques, study using lattice Monte Carlo sim- 0 fectivetheoriesdescribingdifferentregionsofthecuprate ulation has proved possible with relatively modest com- 1 phase diagram, Thirring describing the superconduct- puter resources [12, 13, 14, 15]. It has been shown that 0 ing phase, while QED3 supposedly describes the non- for Nf ≤ 5 the model is in the chirally broken phase at 7 superconducting “pseudogap” behaviour seen in the un- strongenoughcouplingg2 <∞,providingalowerbound 0 derdoped regime [1, 2]. The Thirring model is a theory on N , and the critical exponents found to vary with / fc t of fermions interacting via a current contact interaction: N . This should be contrasted with numerical studies a f l of QED3: in this case the phase transition in the con- p- L=ψ¯(∂/ +m)ψ + g2 (ψ¯γ ψ )2, (1) tinuum limit at Nf = Nfc is an IR-stable fixed point e i i 2N i µ i (see, eg. [16, 17]), and SDE predictions for the separa- f h tionofscalesparametrisedbythedimensionlessquantity v: where ψi,ψ¯i are four-component spinors, m is a parity- hψ¯ψi/α2 aresosmallthatenormouslatticesarerequired Xi conserving bare mass, and the index i runs over Nf dis- to establish whether the model with a particular Nf lies tinct fermion flavors. In the chiral limit m → 0 the La- in the broken or symmetric phase [18]. A determination ar grangian(1)sharesthesameglobalU(1)chiralsymmetry of Nfc for QED3 (whose value may have profound im- ψ 7→ eiαγ5ψ, ψ¯ 7→ ψ¯eiαγ5 as QED3. Since the coupling plications for the cuprate phase diagram [2]) by purely g2hasmassdimension−1,naivepower-countingsuggests numerical means currently appears very difficult. that the model is non-renormalisable. However [3, 4, 5], As already noted, Thirring and QED3 share the same an expansion in powers of 1/N , rather than g2, is ex- globalsymmetries;moreoverin the strongcouplinglimit f actly renormalisable and the model has a well-defined the 1/Nf expansion predicts the existence of a massless continuumlimitcorrespondingtoaUV-stablefixedpoint spin-1 ψψ¯ bound state in the Thirring spectrum [5, 7]. of the renormalizationgroup(RG). The 1/N expansion The RG fixed point of both models is therefore a theory f may not, however, describe the true behaviour of the of massless fermions interacting via massless vector ex- model at smallN . Chiralsymmetry breaking,signalled change between conserved currents. It is therefore plau- f by a condensate hψ¯ψi =6 0, is forbidden at all orders in sible that the two models have the same Nfc, and that 1/N , and yet is predicted by a self-consistent SDE ap- simulation of the Thirring model is the most effective f proach [4, 6, 7, 8]. SDEs solved in the limit g2 →∞ [7] means to determine its value. Here, we present results show that chiral symmetry is spontaneously broken for based on numerical simulations with Nf = 2,...,18 in Nf <Nfc ≃4.32, close to certain predictions of Nfc for anefforttodetermineNfc withunprecedentedprecision, non-trivial IR behaviour in QED3 [9]. Based on these exploitingastrategyofidentifyingthevalueofthelattice results, at N = N the model is expected to undergo couplingg2 correspondingtothe strongcouplinglimitof f fc aninfiniteorderorconformalphasetransition,originally the continuum theory. discussed by Miranskii et al in the context of quenched The lattice actionwe haveused is based on[12] andis 2 as follows: where J(m) is the value of the integral contributed by 1 the second diagram in Fig. 1. The physics described by S = 2 χ¯i(x)ηµ(x)(1+iAµ(x))χi(x+µˆ)−h.c. continuum1/Nf perturbationtheoryoccursfortherange Xxµi of couplings g2 ∈ [0,∞), ie. for g2 ∈ [0,g2 ) where to R lim N leading order in 1/N g2 = 3 for m = 0. The strong +m χ¯ (x)χ (x)+ A (x)2 (2) lim 2 Xxi i i 4g2 Xxµ µ cgo2u>plgin2g ltihmeitauisxiltihaerryepforroeparegcaotvoerrbeedcoamt egs2n=egagtl2iimve.,aFnodr lim where χ,χ¯ are staggered fermion fields and the flavor the lattice model no longer describes a unitary theory. indexirunsoverN species. Wehaveintroducedanaux- As already discussed, chiral symmetry breaking is ab- iliary real-valuedlink field A , superficially resembling a sent in large-N calculations. It may well be that the µ f photon field, so that the lattice action can be expressed value of the second diagram in Fig. 1 is considerably al- as a bilinear form. The formulationis not unique; Gaus- tered in the chirally broken vacuum expected for N < f sian integration over the A in (2) results in a lattice N . In this study we use lattice simulations to estimate µ fc action resembling the continuum form (1) in that all in- the value of g2 as a function of N . Fig. 2 shows hψ¯ψi lim f teractionsremainof the formχ¯χχ¯χ for arbitraryN, but some ofwhich whenre-expressedinterms of continuum- 0.25 like Dirac spinors are of non-covariant form. As argued EoS fit in [13], in the 1/Nf expansion these unwanted contri- 0.2 123 butions are probably irrelevant, but more care may be 163 needed when discussing the UV fixed-point theory. m=0.02 0.15 m=0.03 For d = 3 N staggered fermion species describe Nf = <ψψ> N=4 f 2N continuum flavors. We employed the Hybrid Monte N=8 Carlo(HMC)algorithmtosimulateevenN [13]andthe 0.1 f f HybridMolecularDynamics(HMD)algorithmforoddor non-integerN [15]. FortheHMDsimulationsweuseda 0.05 f small enough fictitious time step ∆τ ≤ 0.0025 to ensure that the O(N2∆τ2) systematic errors in the molecular 0 0 0.2 0.4 0.6 0.8 dynamicsstepsweresmallerthanthe statisticalerrorsof -2 g the various observables. FIG. 2: Chiral condensate hψ¯ψi vs. g−2. N =6, m=0.01 f on a 163 lattice unless otherwise stated. The dashed line denotes a fit of the163, N =6, m=0.01 data to an EoS of f theform (5) taking finitevolume scaling into account. data as a function of g−2 for various N , m and lat- f tice volumes. The condensate is non-monotonic, show- ing a clear maximum for N = 6 flavors at g−2 ≃ 0.3. FIG. 1: Leading order vacuum polarisation in lattice QED. f The peak position at g2 = g2 , unlike the value of the max Next we review the discussion of [13] regarding the condensate itself, is independent of both lattice volume non-conservation of the vector current in the lattice andfermion mass,indicating that its originis at the UV Thirring model. The leading order 1/N quantum cor- scale. Since in an orthodox description of chiral sym- rections to the photon propagator in lattice QED3 are metry breaking |hψ¯ψi| is expected to increase with the sketchedinFig.1. The seconddiagram,arisingfromthe strength of the interaction, we interpret the peak as the gauge-invariantformχ¯xeiAµxχx+µˆ,ispeculiartothe lat- point where unitarity violation sets in, ie. gm2ax ≈ gl2im, tice regularisation,but is requiredto ensure transversity and hence identify the peak with the location of the of the vacuum polarisation tensor, ie: strong coupling limit. The figure also shows that g2 depends on N (Cf. [Π (x)−Π (x−µˆ)]=0. (3) max f µν µν Fig. 3 of Ref. [13]). For N < N , it is possible to fit f fc Xµ condensate data taken at g2 < g2 to an equation of max For the A propagator in the lattice Thirring model (2) state (EoS) of the form [13] µ however,the seconddiagramisabsent,andthe transver- sitycondition(3)violatedbyatermofO(a−1). Transver- m=A(g−2−gc−2)hψ¯ψip+Bhψ¯ψiδ, (5) sityofΠ iscrucialtotherenormalisabilityofthe1/N µν f whichdescribesacontinuoustransitioninthe limitm→ expansion; fortunately, the impact of the extra diver- 0atg2 =g2 toachirallysymmetricphase,characterised gencecanbeabsorbedbyawavefunctionrenormalisation c by critical exponents δ and a “magnetic” exponent β = of A and a coupling constant renormalisation µ (δ−p)−1. Onadatasettakenon123,...,323takingfinite g2 volume scaling into account we have found the best fit gR2 = 1−g2J(m), (4) for Nf = 6 given by gc−2 = 0.316(1), δ = 5.75(13), p = 3 1.18(2), to be compared with corresponding quantities shown in Fig. 3 and the predictions of the 1/N expan- f EtaobSulfiattefdorfoNrfs=ma6l,lesrhNowfninfor[1t5h].e 1S6in3cdeagtac2 b<∼ygam2daxa,shthede msioaninisnathdeispreagriimtyebNetfw>eenNgfm2c.ax(INnfp)aratnicdultahre, tlahregree-Nref- line in Fig. 2, is of borderline credibility. prediction gl2im = 23, which may be due to subleading Fig. 2 also shows that for Nf = 8 the peak moves to correctionsin 1/Nf, or to large finite-volume corrections the right, and is considerably less pronounced. In Fig. 3 described by the conformal field theory expected in the we plot gm−a2x vs. Nf for Nf ∈ [2,18]. For Nf <∼ 6, gm−a2x limit Nf →Nfc+. Fig.4isinterestingbecauseforthefirsttimeitpresents lattice data in a form suitable for direct comparison with SDE predictions. One such study of chiral sym- 0.8 metry breaking in the d-dimensional Thirring model with d ∈ (2,4) using this approach was by Itoh et al [7]. They calculated the dressed fermion propagator 0.6 S(p) = [A(p2)/p + B(p2)]−1, by first exploiting a hid- -2 g den local symmetry to fix a gauge in which A ≡ 1, max and then approximating the auxiliary propagator and 0.4 fermion-auxiliary vertex by their forms to leading order in 1/N . This enabled a solution for the self-energy f function Σ(x) = B(x)/Λ, with Λ a UV cutoff and x = (p/Λ)d−2, in the strong coupling limit g2 → ∞, 0.2 0 5 10 15 20 yielding a dynamically-generated fermion mass M: N f d−2 FIG. 3: gm2ax vs. Nf. M ∝exp −2π , (7) (cid:18) Λ (cid:19) Nfc −1 decreasessmoothlyandmonotonically,butintheinterval qNf  between 6.5 and6.6 there is a sudden sharpincrease;for 6.6 ≤ Nf ≤ 12, g−2 ≃ 0.43(2) is roughly constant, and where Nfc(d = 3) = 128/3π2 ≃ 4.32. The chiral order then increasesfor N stilllarger. With the identification parameter follows via the relation hψ¯ψi ∝ Σ′(x = 1), f of g2 with the strong coupling limit of the continuum leading to the strong coupling prediction max theory, as argued above,it is natural to ask whether the transitionatNf ≃6.5hasanycorrelationwiththechiral hψ¯ψi∝Λd−22Md2. (8) order parameter. Fig. 4 shows hψ¯ψ(g2 )i vs. N . This max f Spontaneous chiral symmetry breaking occurs for N < f N in qualitative, but not quantitative agreement fc with Fig. 4. Eqn. (7) predicts an infinite-order phase 3 0.3 16 transition. Its nature is further elucidated using the 3 32 anomalous scaling dimension of the ψ¯ψ bilinear γψ¯ψ = dlnhψ¯ψi/dlnΛ and the relations for the critical expo- 0.2 nents η and δ [19]: <ψψ> d+2−η 0.1 η =d−2γψ¯ψ ; δ = d−2+η. (9) From (8) we obtain γψ¯ψ = (d−2)/2 and hence η = 2, δ =1. Thisscenariohasbeentermeda“conformalphase 0 0 5 10 15 20 transition” [11], and has also been exposed by SDE ap- N f proachesinQED3[16]andquenchedQED4[10],thecon- trol parameter being respectively N and the fine struc- FIG. 4: hψ¯ψ(gm2ax)i vs. Nf for m=0.01. Lines denote EoS ture constant α. f fitsdiscussed in thetext. The SDE analysis of [7] was later extended to cover plot necessarilyputs anupper bound onthe condensate. g2 <∞ by Sugiura [8]. In this case the chiral transition It is difficult to draw any conclusion other than there takes place for N <N , and the solution for the order f fc being a chiral symmetry restoring phase transition at parameter is of the form Nfc =6.6(1). (6) hψ¯ψi∝Λd−2M. (10) Eqn. (6) is the main result of this Letter. The same chain of arguments leads to critical exponents Inordertorefinethis picture,furthertheoreticalanal- η = 4−d, δ = d−1, coincident with those of the d- ysis is needed to establish contact between the results dimensional Gross-Neveu model in the large-N limit f 4 [19]. The nature of the transition predicted by SDEs ted values for δ and ν together with those from EoS fits thus appearssensitiveto the orderofthe limits g2 →∞, to data from N <N compiled from Refs. [13, 14, 15], f fc N →N . as shown in Fig. 5, we see that the estimate for δ(N ) f fc fc We have been motivated by these considerations to lies within a general trend of δ increasing with N , with f attempt an EoS fit to the data of Fig. 4 of the form no evidence for non-commutativity ofthe limits g2 →∞ andN →N (there issomeindicationforajump from f fc m=A[(Nf −Nfc)+CL−ν1]hψ¯ψip+Bhψ¯ψiδ, (11) ν ≈ 0.5 to ν ≈ 1 as Nf increases from 6 to Nfc, which needs to be confirmed in a more refined study). In any where L is the linear extent of the system and the case, the fitted values of δ lie well above those predicted exponent ν is given by the hyperscaling relation ν = in the SDE approach. (δ+1)/d(δ−p). ThetermproportionaltoC accountsfor Insummary,forthe firsttime we havebeenable using finitevolumecorrections. OurbestfittodatawithN ∈ f lattice Monte Carlo simulation to study a chiral phase [4,8]yieldsN =6.89(2),δ =6.90(3),p=4.23(2),with fc transition as the number of fermion flavors N is var- χ2/dof=129/15. f ied continuously. We find a continuous phase transition Besides the relativelypoor quality,andsignificantdis- in the strong coupling limit at a critical flavor number agreement with Eq. (6), the fit should not be taken too N = 6.6(1), in qualitative, but not quantitative agree- fc seriouslyfortwofurtherreasons,namelythecombination mentwiththepredictionsofananalysisusingSchwinger- of“exact”HMCdatawithHMDcontainingasystematic Dysonequations. Inparticular,thecriticalexponentsare error due to ∆τ 6= 0, and the as yet unquantified errors notthoseeitherofaconformalphasetransitionorthe3d due to the identification of g2 with g2 . Nonetheless lim max Gross-Neveu model. It is plausible that our result may the disparity of the value of the exponent δ with the inform estimates of the corresponding critical number of SDE prediction is striking. Moreover, if we plot the fit- flavorsfor chiral symmetry breaking in QED3, where direct lattice simulations are hampered by a large separa- 8 tion of scales. Note that a recent perturbative analysis δ ν of RG flow equations in the large-Nf limit of QED3 predicts N = 6 [20]. In future it will be interesting to 6 fc check whether the same universal features of the strong couplinglimitemergeusingthealternativelatticeformu- 4 lations of the Thirring model reviewed in [13]. 2 Acknowledgments 0 Thesimulationswereperformedonaclusterof2.4GHz 2 3 4 5 6 7 N OpteronprocessorsattheFrederickInstituteofTechnol- f ogy, Cyprus. FIG. 5: Critical exponentsδ and ν vs. N . f [1] I.F. Herbut,Phys.Rev.Lett. 94 (2005) 237001 [11] V.A.MiranskiiandK.Yamawaki,Phys.Rev.D55(1997) [2] I.F. Herbut, Phys. Rev. B66 (2002) 094504; M. Franz, 5051. Z. Teˇsanović and O. Vafek, Phys. Rev. B66 (2002) [12] L. Del Debbio and S.J. Hands, Phys. Lett B373 (1996) 054535. 171. [3] G. Parisi, Nucl. Phys. B100 (1975) 368; S. Hikami and [13] L. Del Debbio, S.J. Hands, and J.C. Mehegan, Nucl. T. Muta, Prog. Theor. Phys. 57 (1977) 785; Z. Yang, Phys. B502 (1997) 269. Texas preprint UTTG-40-90 (1990). [14] L.DelDebbioandS.J.Hands,Nucl.Phys.B552(1999) [4] M.Gomes,R.S.Mendes,R.F.RibeiroandA.J.daSilva, 339. Phys.Rev.D43 (1991) 3516. [15] S.J. Handsand B. Lucini, Phys.Lett. B461 (1999) 263. [5] S.J. Hands, Phys.Rev. D51 (1995) 5816. [16] T.Appelquist,D.NashandL.C.R.Wijewardhana,Phys. [6] D.K.Hongand S.H.Park, Phys.Rev. D49(1994) 5507. Rev. Lett.60 (1988) 2575. [7] T. Itoh, Y. Kim, M. Sugiura and K. Yamawaki, Prog. [17] P. Maris, Phys.Rev. D54 (1996) 4049. Theor. Phys. 93 (1995) 417. [18] S.J. Hands,J.B. Kogut andC.G. Strouthos,Nucl.Phys. [8] M. Sugiura, Prog. Theor. Phys. 97 (1997) 311. B645 (2002) 321; S.J. Hands, J.B. Kogut, L. Scorzato [9] T.Ebihara,T.Iizuka,K.-I.KondoandE.Tanaka,Nucl. and C.G. Strouthos, Phys.Rev. B70 (2004) 104501. Phys.B434 (1995) 85. [19] S.J. Hands, A. Kocić and J.B. Kogut, Ann. Phys. 224 [10] P.I. Fomin, V.P. Gusynin, V.A. Miranskii and (1993) 29. Yu.A.Sitenko,Riv.NuovoCimento6(1983)1;V.A.Mi- [20] K. Kaveh and I.F. Herbut, Phys. Rev. B 71 (2005) ranskii, NuovoCimento 90A (1985) 149. 184519.