Critical flavor number in the three dimensional Thirring model

CRITICAL FLAVOR NUMBER IN THE THREE DIMENSIONAL THIRRING MODEL Stavros Christofia , Simon Handsb and Costas Strouthosc a arXiv:hep-lat/0701016v1 22 Jan 2007 b Frederick Institute of Technology, CY-1303 Nicosia, Cyprus Department of Physics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K. and c Institute of Chemical Sciences and Engineering, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland. We present results of a Monte Carlo simulation of the three dimensional Thirring model with the number of fermion flavors Nf varied between 2 and 18. By identifying the lattice coupling at which the chiral condensate peaks, simulations are be performed at couplings g 2 (Nf ) corresponding to the strong coupling limit of the continuum theory. The chiral symmetry restoring phase transition is studied as Nf is increased, and the critical number of flavors estimated as Nf c = 6.6(1). The critical exponents measured at the transition do not agree with self-consistent solutions of the Schwinger-Dyson equations; in particular there is no evidence for the transition being of infinite order. Implications for the critical flavor number in QED3 are briefly discussed. PACS numbers: PACS: 11.10.Kk, 11.30.Rd, 11.15.Ha The study of quantum field theories in which the ground state shows a sensitivity to the number of fermion flavors Nf is intrinsically interesting. According to certain approximate solutions of Schwinger-Dyson equations (SDEs), in d = 3 spacetime dimensions both quantum electrodynamics (QED3 ) and the Thirring model display this phenomenon. Both models have been proposed as effective theories describing different regions of the cuprate phase diagram, Thirring describing the superconducting phase, while QED3 supposedly describes the nonsuperconducting “pseudogap” behaviour seen in the underdoped regime [1, 2]. The Thirring model is a theory of fermions interacting via a current contact interaction: L = ψ̄i (∂/ + m)ψi + g2 (ψ̄i γµ ψi )2 , 2Nf (1) where ψi , ψ̄i are four-component spinors, m is a parityconserving bare mass, and the index i runs over Nf distinct fermion flavors. In the chiral limit m → 0 the Lagrangian (1) shares the same global U(1) chiral symmetry ψ 7→ eiαγ5 ψ, ψ̄ 7→ ψ̄eiαγ5 as QED3 . Since the coupling g 2 has mass dimension −1, naive power-counting suggests that the model is non-renormalisable. However [3, 4, 5], an expansion in powers of 1/Nf , rather than g 2 , is exactly renormalisable and the model has a well-defined continuum limit corresponding to a UV-stable fixed point of the renormalization group (RG). The 1/Nf expansion may not, however, describe the true behaviour of the model at small Nf . Chiral symmetry breaking, signalled by a condensate hψ̄ψi = 6 0, is forbidden at all orders in 1/Nf , and yet is predicted by a self-consistent SDE approach [4, 6, 7, 8]. SDEs solved in the limit g 2 → ∞ [7] show that chiral symmetry is spontaneously broken for Nf < Nf c ≃ 4.32, close to certain predictions of Nf c for non-trivial IR behaviour in QED3 [9]. Based on these results, at Nf = Nf c the model is expected to undergo an infinite order or conformal phase transition, originally discussed by Miranskii et al in the context of quenched QED4 [10, 11]. Using different sequences of truncation, however, other SDE approaches have found values of Nf c ranging between 3.24 [4] and ∞ [6]. Since for Nf < Nf c the chiral symmetry breaking transition corresponds to a UV-stable RG fixed point at a critical gc2 , which can be analysed using finite volume scaling techniques, study using lattice Monte Carlo simulation has proved possible with relatively modest computer resources [12, 13, 14, 15]. It has been shown that for Nf ≤ 5 the model is in the chirally broken phase at strong enough coupling g 2 < ∞, providing a lower bound on Nf c , and the critical exponents found to vary with Nf . This should be contrasted with numerical studies of QED3 : in this case the phase transition in the continuum limit at Nf = Nf c is an IR-stable fixed point (see, eg. [16, 17]), and SDE predictions for the separation of scales parametrised by the dimensionless quantity hψ̄ψi/α2 are so small that enormous lattices are required to establish whether the model with a particular Nf lies in the broken or symmetric phase [18]. A determination of Nf c for QED3 (whose value may have profound implications for the cuprate phase diagram [2]) by purely numerical means currently appears very difficult. As already noted, Thirring and QED3 share the same global symmetries; moreover in the strong coupling limit the 1/Nf expansion predicts the existence of a massless spin-1 ψ ψ̄ bound state in the Thirring spectrum [5, 7]. The RG fixed point of both models is therefore a theory of massless fermions interacting via massless vector exchange between conserved currents. It is therefore plausible that the two models have the same Nf c , and that simulation of the Thirring model is the most effective means to determine its value. Here, we present results based on numerical simulations with Nf = 2, . . . , 18 in an effort to determine Nf c with unprecedented precision, exploiting a strategy of identifying the value of the lattice coupling g 2 corresponding to the strong coupling limit of the continuum theory. The lattice action we have used is based on [12] and is 2 as follows: 1X χ̄i (x)ηµ (x)(1 + iAµ (x))χi (x + µ̂) − h.c. S = 2 xµi +m X χ̄i (x)χi (x) + xi N X Aµ (x)2 4g 2 xµ (2) where χ, χ̄ are staggered fermion fields and the flavor index i runs over N species. We have introduced an auxiliary real-valued link field Aµ , superficially resembling a photon field, so that the lattice action can be expressed as a bilinear form. The formulation is not unique; Gaussian integration over the Aµ in (2) results in a lattice action resembling the continuum form (1) in that all interactions remain of the form χ̄χχ̄χ for arbitrary N , but some of which when re-expressed in terms of continuumlike Dirac spinors are of non-covariant form. As argued in [13], in the 1/Nf expansion these unwanted contributions are probably irrelevant, but more care may be needed when discussing the UV fixed-point theory. For d = 3 N staggered fermion species describe Nf = 2N continuum flavors. We employed the Hybrid Monte Carlo (HMC) algorithm to simulate even Nf [13] and the Hybrid Molecular Dynamics (HMD) algorithm for odd or non-integer Nf [15]. For the HMD simulations we used a small enough fictitious time step ∆τ ≤ 0.0025 to ensure that the O(N 2 ∆τ 2 ) systematic errors in the molecular dynamics steps were smaller than the statistical errors of the various observables. where J(m) is the value of the integral contributed by the second diagram in Fig. 1. The physics described by continuum 1/Nf perturbation theory occurs for the range 2 2 of couplings gR ∈ [0, ∞), ie. for g 2 ∈ [0, glim ) where to 2 leading order in 1/N glim = 23 for m = 0. The strong 2 coupling limit is therefore recovered at g 2 = glim . For 2 g 2 > glim the auxiliary propagator becomes negative, and the lattice model no longer describes a unitary theory. As already discussed, chiral symmetry breaking is absent in large-Nf calculations. It may well be that the value of the second diagram in Fig. 1 is considerably altered in the chirally broken vacuum expected for Nf < Nf c . In this study we use lattice simulations to estimate 2 the value of glim as a function of Nf . Fig. 2 shows hψ̄ψi 0.25 EoS fit 3 12 3 16 m=0.02 m=0.03 Nf=4 Nf=8 0.2 0.15 <ψψ> 0.1 0.05 0 0 0.2 0.4 -2 0.6 0.8 g FIG. 2: Chiral condensate hψ̄ψi vs. g −2 . Nf = 6, m = 0.01 on a 163 lattice unless otherwise stated. The dashed line denotes a fit of the 163 , Nf = 6, m = 0.01 data to an EoS of the form (5) taking finite volume scaling into account. FIG. 1: Leading order vacuum polarisation in lattice QED. Next we review the discussion of [13] regarding the non-conservation of the vector current in the lattice Thirring model. The leading order 1/N quantum corrections to the photon propagator in lattice QED3 are sketched in Fig. 1. The second diagram, arising from the gauge-invariant form χ̄x eiAµx χx+µ̂ , is peculiar to the lattice regularisation, but is required to ensure transversity of the vacuum polarisation tensor, ie: X [Πµν (x) − Πµν (x − µ̂)] = 0. (3) µ For the Aµ propagator in the lattice Thirring model (2) however, the second diagram is absent, and the transversity condition (3) violated by a term of O(a−1 ). Transversity of Πµν is crucial to the renormalisability of the 1/Nf expansion; fortunately, the impact of the extra divergence can be absorbed by a wavefunction renormalisation of Aµ and a coupling constant renormalisation 2 gR = g2 , 1 − g 2 J(m) (4) data as a function of g −2 for various Nf , m and lattice volumes. The condensate is non-monotonic, showing a clear maximum for Nf = 6 flavors at g −2 ≃ 0.3. 2 The peak position at g 2 = gmax , unlike the value of the condensate itself, is independent of both lattice volume and fermion mass, indicating that its origin is at the UV scale. Since in an orthodox description of chiral symmetry breaking |hψ̄ψi| is expected to increase with the strength of the interaction, we interpret the peak as the 2 2 point where unitarity violation sets in, ie. gmax ≈ glim , and hence identify the peak with the location of the strong coupling limit. 2 The figure also shows that gmax depends on Nf (Cf. Fig. 3 of Ref. [13]). For Nf < Nf c , it is possible to fit 2 condensate data taken at g 2 < gmax to an equation of state (EoS) of the form [13] m = A(g −2 − gc−2 )hψ̄ψip + Bhψ̄ψiδ , (5) which describes a continuous transition in the limit m → 0 at g 2 = gc2 to a chirally symmetric phase, characterised by critical exponents δ and a “magnetic” exponent β = (δ−p)−1 . On a dataset taken on 123 , . . . , 323 taking finite volume scaling into account we have found the best fit 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 2 tabulated for smaller Nf in [15]. Since gc2 < ∼ gmax , the 3 EoS fit for Nf = 6, shown for the 16 data by a dashed line in Fig. 2, is of borderline credibility. Fig. 2 also shows that for Nf = 8 the peak moves to the right, and is considerably less pronounced. In Fig. 3 −2 −2 we plot gmax vs. Nf for Nf ∈ [2, 18]. For Nf < ∼ 6, gmax 0.8 -2 g 0.6 max 0.4 0.2 0 5 10 20 15 Nf 2 FIG. 3: gmax vs. Nf . decreases smoothly and monotonically, but in the interval between 6.5 and 6.6 there is a sudden sharp increase; for 6.6 ≤ Nf ≤ 12, g −2 ≃ 0.43(2) is roughly constant, and then increases for Nf still larger. With the identification 2 of gmax with the strong coupling limit of the continuum theory, as argued above, it is natural to ask whether the transition at Nf ≃ 6.5 has any correlation with the chiral 2 order parameter. Fig. 4 shows hψ̄ψ(gmax )i vs. Nf . This 3 16 3 32 0.3 0.2 <ψψ> shown in Fig. 3 and the predictions of the 1/Nf expansion in the regime Nf > Nf c . In particular, there re2 mains a disparity between gmax (Nf ) and the large-Nf 3 2 prediction glim = 2 , which may be due to subleading corrections in 1/Nf , or to large finite-volume corrections described by the conformal field theory expected in the limit Nf → Nf c+ . Fig. 4 is interesting because for the first time it presents lattice data in a form suitable for direct comparison with SDE predictions. One such study of chiral symmetry 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 S(p) = [A(p2 )p / + B(p2 )]−1 , by first exploiting a hidden local symmetry to fix a gauge in which A ≡ 1, and then approximating the auxiliary propagator and fermion-auxiliary vertex by their forms to leading order in 1/Nf . This enabled a solution for the self-energy function Σ(x) = B(x)/Λ, with Λ a UV cutoff and x = (p/Λ)d−2 , in the strong coupling limit g 2 → ∞, yielding a dynamically-generated fermion mass M :    d−2 −2π M , (7) ∝ exp  q Nf c Λ −1 Nf where Nf c (d = 3) = 128/3π 2 ≃ 4.32. The chiral order parameter follows via the relation hψ̄ψi ∝ Σ′ (x = 1), leading to the strong coupling prediction hψ̄ψi ∝ Λ d−2 2 (8) Spontaneous chiral symmetry breaking occurs for Nf < Nf c in qualitative, but not quantitative agreement with Fig. 4. Eqn. (7) predicts an infinite-order phase transition. Its nature is further elucidated using the anomalous scaling dimension of the ψ̄ψ bilinear γψ̄ψ = d lnhψ̄ψi/d ln Λ and the relations for the critical exponents η and δ [19]: η = d − 2γψ̄ψ ; δ = 0.1 d M2. d+2−η . d−2+η (9) plot necessarily puts an upper bound on the condensate. It is difficult to draw any conclusion other than there being a chiral symmetry restoring phase transition at From (8) we obtain γψ̄ψ = (d − 2)/2 and hence η = 2, δ = 1. This scenario has been termed a “conformal phase transition” [11], and has also been exposed by SDE approaches in QED3 [16] and quenched QED4 [10], the control parameter being respectively Nf and the fine structure constant α. The SDE analysis of [7] was later extended to cover g 2 < ∞ by Sugiura [8]. In this case the chiral transition takes place for Nf < Nf c , and the solution for the order parameter is of the form Nf c = 6.6(1). hψ̄ψi ∝ Λd−2 M. 0 0 5 10 15 20 Nf 2 FIG. 4: hψ̄ψ(gmax )i vs. Nf for m = 0.01. Lines denote EoS fits discussed in the text. (6) Eqn. (6) is the main result of this Letter. In order to refine this picture, further theoretical analysis is needed to establish contact between the results (10) The same chain of arguments leads to critical exponents η = 4 − d, δ = d − 1, coincident with those of the ddimensional Gross-Neveu model in the large-Nf limit 4 [19]. The nature of the transition predicted by SDEs thus appears sensitive to the order of the limits g 2 → ∞, Nf → Nf c . We have been motivated by these considerations to attempt an EoS fit to the data of Fig. 4 of the form 4 ted values for δ and ν together with those from EoS fits to data from Nf < Nf c compiled from Refs. [13, 14, 15], as shown in Fig. 5, we see that the estimate for δ(Nf c ) lies within a general trend of δ increasing with Nf , with no evidence for non-commutativity of the limits g 2 → ∞ and Nf → Nf c (there is some indication for a jump from ν ≈ 0.5 to ν ≈ 1 as Nf increases from 6 to Nf c , which needs to be confirmed in a more refined study). In any case, the fitted values of δ lie well above those predicted in the SDE approach. In summary, for the first time we have been able using lattice Monte Carlo simulation to study a chiral phase transition as the number of fermion flavors Nf is varied continuously. We find a continuous phase transition in the strong coupling limit at a critical flavor number Nf c = 6.6(1), in qualitative, but not quantitative agreement with the predictions of an analysis using SchwingerDyson equations. In particular, the critical exponents are not those either of a conformal phase transition or the 3d Gross-Neveu model. It is plausible that our result may inform estimates of the corresponding critical number of flavors for chiral symmetry breaking in QED3 , where direct lattice simulations are hampered by a large separation of scales. Note that a recent perturbative analysis of RG flow equations in the large-Nf limit of QED3 predicts Nf c = 6 [20]. In future it will be interesting to check whether the same universal features of the strong coupling limit emerge using the alternative lattice formulations of the Thirring model reviewed in [13]. 2 Acknowledgments 1 m = A[(Nf − Nf c ) + CL− ν ]hψ̄ψip + Bhψ̄ψiδ , (11) where L is the linear extent of the system and the exponent ν is given by the hyperscaling relation ν = (δ + 1)/d(δ − p). The term proportional to C accounts for finite volume corrections. Our best fit to data with Nf ∈ [4, 8] yields Nf c = 6.89(2), δ = 6.90(3), p = 4.23(2), with χ2 /dof=129/15. Besides the relatively poor quality, and significant disagreement with Eq. (6), the fit should not be taken too seriously for two further reasons, namely the combination of “exact” HMC data with HMD containing a systematic error due to ∆τ 6= 0, and the as yet unquantified errors 2 2 due to the identification of glim with gmax . Nonetheless the disparity of the value of the exponent δ with the SDE prediction is striking. 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