Field theory of bicritical and tetracritical points. IV. Critical dynamics including reversible terms
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Field theory of bicritical and tetracritical points. IV. Critical dynamics includingreversible terms.
R. Folk ∗ Institute for Theoretical Physics, Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040, Linz, Austria
Yu. Holovatch † Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine,1 Svientsitskii Str., UA–79011 Lviv, Ukraine andInstitute for Theoretical Physics, Johannes Kepler University Linz, Altenbergerstrasse 69, A-4040, Linz, Austria
G. Moser ‡ Department for Material Research and Physics, Paris LodronUniversity Salzburg, Hellbrunnerstrasse 34, A-5020 Salzburg, Austria (Dated: November 22, 2018)This article concludes a series of papers (R. Folk, Yu. Holovatch, and G. Moser, Phys. Rev. E , 041124 (2008); , 041125 (2008); , 031109 (2009)) where the tools of the field theoreticalrenormalization group were employed to explain and quantitatively describe different types of staticand dynamic behavior in the vicinity of multicritical points. Here, we give the complete two loopcalculation and analysis of the dynamic renormalization-group flow equations at the multicriticalpoint in anisotropic antiferromagnets in an external magnetic field. We find that the time scalesof the order parameters characterizing the parallel and perpendicular ordering with respect to theexternal field scale in the same way. This holds independent whether the Heisenberg fixed point orthe biconical fixed point in statics is the stable one. The non-asymptotic analysis of the dynamicflow equations shows that due to cancelation effects the critical behavior is described - in distancesfrom the critical point accessible to experiments - by the critical behavior qualitatively found in oneloop order. Although one may conclude from the effective dynamic exponents (taking almost theirone loop values) that weak scaling for the order parameter components is valid, the flow of the timescale ratios is quite different and they do not reach their asymptotic values. I. INTRODUCTION
Three component antiferromagnets in three spatial di-mensions in an external magnetic field in z direction con-tain in their phase diagram two second order transitionlines: (i) between the paramagnetic and the spin flopphase and (ii) between the antiferromagnetic and para-magnetic phase. The point where these two lines meet isa multicritical point which turned out to be either bicrit-ical or tetracritical. Within the renormalization group(RG) theory the stability and attraction region of thestatic fixed point (FP) of the RG transformation deter-mines, which kind of multicritical behavior is realized.For the bicritical point it is the Heisenberg FP, for thetetracritical point it is the biconical one. The stabilty ofa FP depends on the system’s global features as the spaceand order parameter (OP) dimensions d and n . In d = 3,the case considered here, the biconical FP is stable apartfrom a restricted attraction region of the Heisenberg FP. The static phase transition on each of the phase transi-tion lines belongs for (i) to an isotropic Heisenberg modelwith n ⊥ = 2 and for (ii) to Heisenberg model with n k = 1 ∗ Electronic address: [email protected] † [email protected] ‡ [email protected] [1, 2].Concerning the dynamical universality classes thetransition (i) belongs to the class described by model Fand (ii) belongs to the model C class (for the notationsee [3]). At the multicritical point the critical behavioris described by a new universality class both in staticsand dynamics. The interesting feature of these systemsis that all the different OPs characterizing the orderedphase are physically accessible. This is most importantfor the dynamical behavior since the only other examplebelonging to model F is the superfluid transition in Hewhere the OP (the complex macroscopic wave functionof the condensate [4]) is experimentally not accessible [5].Here the OPs are the components of the staggered mag-netization. Their correlations (static and dynamical) canbe measured by neutron scattering.A complete description of the critical dynamics nearthe multicritical point mentioned above has to take intoaccount the slow dynamical densities which are the OPsand the conserved densities present in the system. Dueto the external magnetic field the only conserved densitywhich has to be taken into account is the magnetizationin direction of the external field. A derivation of thedynamical equations follows along the usual steps cal-culating the reversible terms from the non-zero Poissonbrackets, introducing irreversible terms present also inthe hydrodynamic limit, dropping irrelevant terms andtaking into account terms arising in the renormalizationprocedure (see e.g. the review [3]). Such a dynamicalmodel has already been considered in [6–8] by RG the-ory and it was argued that due to nonanalytic terms in ε = 4 − d a FP in two loop order qualitative differentfrom the one loop FP is found. The result of the oneloop calculations is that the time scales of the paralleland perpendicular components of the staggered magne-tization scale differently whereas calculated in two looporder they scale similar although the FP value of thetimescale ratio of the two components cannot be foundby ε expansion and might be very small in d = 3 namelyof O (10 − ). It was argued that the terms leading tothe singular behavior in ε do not contribute to the FPvalue of the mode coupling. The calculations of the RG-functions in [6] where not complete in two loop order(they took into account only the terms which lead to thenonanalytic behavior in ε ). At that time also the Heisen-berg FP (named H ) was considered to be the stable staticone, whereas it turned out in two loop order (resummed)that it is the biconical FP (named B ) [2]. FP values in ... w ⋆ k w ′ ⋆ ⊥ v ′ ⋆ f ⋆ ⊥ B C [11] 0 . ≫ ∼ B .
555 0 1 . B ∼ . .
49 - - -model F [10] - 0 - 0 . β -functions) at d = 3 of different models for thetime scale ratios w ⋆ k , w ′ ⋆ ⊥ , v ′ ⋆ and the mode coupling constant f ⋆ ⊥ . The 2nd and 3rd lines quote results of this paper found inthe biconical static FP for the tetracritical behavior of the dy-namical model that takes into account reversible terms. Theyare compared with the two loop results found in the modelC multicritical point [11] as well as in the critical points ofmodel C for the one component OP [9] and of model F forthe two component order parameter [10]. A summary of the results obtained so far for the FPscharacterizing dynamical behavior is given in table I. Ne-glecting the reversible terms one is left with a purelyrelaxational dynamics. Then the asymptotic dynamicalcritical behavior is characterized by the FP values of theindependent time scale ratios of the system. These arethe following time scale ratios: (1) the ratio w k betweenthe relaxation rate of the staggered magnetization paral-lel to the external field and the diffusive transport coef-ficient of the magnetization parallel to the external field;(2) the ratio w ′⊥ between the real part of the relaxationrate of the staggered magnetization perpendicular to theexternal field and the diffusive transport coefficient of themagnetization parallel to the external field. In additionwe introduce the ratio v ′ between the two componentsof the real relaxation rates of the two OPs in order tocompare their dynamic scaling behavior. A non-zero fi-nite value of the time scale ratio means that the twoinvolved densities scale with the same exponent. If all time scale ratios are non-zero and finite, one speaks ofstrong dynamic scaling, otherwise of weak dynamic scal-ing. Especially of interest is the behavior of the scalingof the two components of the OP indicated by the FPvalue of v ′ . In the third line of table I the two loop orderresult shows weak dynamic scaling between the OPs andthe conserved density but strong scaling between the OPcomponents. However since the FP value time scale ratio v ′ is almost zero, the critical behavior is dominated bynon-asymptotic effects. For comparison the FP valuesfor the case of model C for the one component OP [9]and for model F for the two component order parameter[10] are included. They are the limiting cases when thetwo OPs characterizing the multicritical behavior woulddecouple in statics and dynamics.In the first line of table I the results for the multi-critical dynamical FP B C values taking into account thestatic coupling of the OP to the conserved density aredisplayed (see [11]). All time scale ratios are non-zeroand finite but since w ′ ⋆ ⊥ is very large ( v ′ ⋆ almost zero)the observable behavior in the vicinity of the multicriti-cal point is predicted to be dominated by non-asymptoticeffects and strong scaling is not observable [11]. In thesecond line the results of a one loop RG calculation withreversible terms for the biconical FP are given. The FPvalue of the mode coupling parameter f ⊥ is finite butsince w ⋆ k = 0 the critical dynamics is characterized byweak dynamic scaling and the two components of theOP scale different. A similar result for the HeisenbergFP was found in [6]. In the third line the results foundin this paper are shown, indicating weak scaling betweenthe conserved density and the components of the OP,but strong scaling between the parallel and perpendicu-lar components of the OP. Since the FP value of the timescale ratio between the component v ′ ⋆ is almost zero butdefinitively different from zero it is expected that non-asymptotic behavior is dominant.This article concludes a series of papers [2, 11, 12](henceforth cited as papers I, II, and III) where the toolsof the field theoretical RG were employed to explain andquantitatively describe different types of static and dy-namic behavior in the vicinity of multicritical points. Ashort account of the results presented here was given in[13]. The statics and dynamics were treated in Refs. [2]and [11, 12], respectively. First, purely relaxational dy-namics was considered (paper II) and later, in paper III,these results served as a basis to consider how an accountof magnetization conservation modifies dynamical behav-ior. The goal of the current study is more ambitious: wewill analyze a complete set of dynamical equations ofmotion taking into account reversible terms [14, 15] andgive a comprehensive description of dynamical behaviorin the vicinity of multicritical points in two loop order.The paper is organized as follows: In section II the dy-namic model is defined followed by a the definitions ofthe dynamical functions considered in section III. Therenormalization and corresponding RG-functions are pre-sented in section IV and V respectively. The two loop re-sults of our calculations for these dynamic RG-functionsare given in section VI. The one loop approximation forthe dynamic is discussed in section VII. In the next sec-tion VIII we consider the asymptotic properties of thetwo loop RG-functions leading to the general asymptoticresults in section IX. We then present the results ex-pected in the asymptotic subspace, section X. The non-asymptotic behavior, obtained by looking at the regionfurther away from the multicritical point, is shown in sec-tion XI, a conclusion XII ends the paper. In Appendicescalculational details for some intermediate steps of theRG calculation are presented. II. MODEL EQUATIONS OF THEANTIFERROMAGNET IN AN EXTERNALFIELD
The non-conserved OP ~φ of an isotropic antiferromag-net is given by the three dimensional vector ~φ = φ x φ y φ z (1)of the staggered magnetization, which is the differenceof two sublattice magnetizations. An external magneticfield applied to the ferromagnet induces an anisotropy tothe system. The OP splits into two OPs, ~φ ⊥ perpen-dicular to the field, and ~φ k parallel to the external field.Assuming the z -axis in direction of the external magneticfield, the two OPs are ~φ ⊥ = φ x φ y ! , φ k = φ z (2)In addition the z -component of the magnetization m hasto be taken into account for the dynamics and thereforehas to be included in statics although there it could beintegrated out and does not change the asymptotic staticcritical behavior. Thus the static critical behavior of thesystem is described by the functional H = Z d d x ( r ⊥ ~φ ⊥ · ~φ ⊥ + 12 d X i =1 ∇ i ~φ ⊥ · ∇ i ~φ ⊥ + 12˚ r k φ k φ k + 12 d X i =1 ∇ i φ k ∇ i φ k + ˚ u ⊥ (cid:16) ~φ ⊥ · ~φ ⊥ (cid:17) +˚ u k (cid:16) φ k φ k (cid:17) + 2˚ u × (cid:16) ~φ ⊥ · ~φ ⊥ (cid:17)(cid:16) φ k φ k (cid:17)) (3)+ 12 m + 12 ˚ γ ⊥ m ~φ ⊥ · ~φ ⊥ + 12 ˚ γ k m φ k φ k − ˚ hm ) , with familiar notations for bare couplings { ˚ u, ˚ γ } , masses { ˚ r } and field ˚ h [2, 12]. The critical dynamics of relaxing OPs coupled to a diffusing secondary density is governedby the following equations of motion [6]: ∂φ α ⊥ ∂t = − ˚Γ ′⊥ δ H δφ α ⊥ + ˚Γ ′′⊥ ǫ αβz δ H δφ β ⊥ + ˚ g ǫ αβz φ β ⊥ δ H δm + θ αφ ⊥ , (4) ∂φ k ∂t = − ˚Γ k δ H δφ k + θ φ k , (5) ∂m ∂t = ˚ λ ∇ δ H δm + ˚ g ǫ αβz φ α ⊥ δ H δφ β ⊥ + θ m , (6)with the Levi-Civita symbol ǫ αβz . Here α, β = x, y andthe sum over repeated indices is implied.The dynamical equations describe the dynamics of anantiferromagnet with the usual Lamor precession termsfor the alternating magnetization and relaxational terms.Due to the static coupling to the conserved magnetiza-tion additional Lamor terms arise together with a diffu-sive term for the magnetization. Renormalization con-siderations lead on one hand to a neglection of severalLamor terms and on the other hand create an additionalreversible term (the second term on the right hand sideof Eq. 5)) not present in the usual dynamics of antifer-romagnets [16].Combining the kinetic coefficients of the OP to a com-plex quantity, ˚Γ ⊥ = ˚Γ ′⊥ + i˚Γ ′′⊥ , the imaginary part con-stitutes a precession term created by the renormalizationprocedure even if it is absent in the background. The ki-netic coefficient ˚ λ and the mode coupling ˚ g are real. Thestochastic forces ~θ φ ⊥ , ~θ φ k and θ m fulfill Einstein relations h θ αφ ⊥ ( ~x, t ) θ βφ ⊥ ( ~x ′ , t ′ ) i = 2˚Γ ′⊥ δ ( ~x − ~x ′ ) δ ( t − t ′ ) δ αβ , (7) h θ φ k ( ~x, t ) θ φ k ( ~x ′ , t ′ ) i = 2˚Γ k δ ( ~x − ~x ′ ) δ ( t − t ′ ) , (8) h θ m ( ~x, t ) θ m ( ~x ′ , t ′ ) i = − λ ∇ δ ( ~x − ~x ′ ) δ ( t − t ′ ) . (9)In view of dynamical calculations it is more convenientto deal with a scalar complex order parameter ψ = ψ ′ +i ψ ′′ instead of the real two-dimensional OP ~φ ⊥ in (2).Thus we may introduce ψ = φ x − i φ y , ψ +0 = φ x + i φ y (10)as OP of the perpendicular components. The superscript + denotes complex conjugated quantities also in the fol-lowing equations. In addition to the two OPs the z -component of the magnetization, which is the sum of thetwo sublattice magnetizations, has to be considered asconserved secondary density m .Expressed in terms of the above densities the dynamicequations take the form ∂ψ ∂t = − ⊥ δHδψ +0 + i ψ ˚ g δHδm + θ ψ , (11) ∂ψ +0 ∂t = − + ⊥ δHδψ − i ψ +0 ˚ g δHδm + θ + ψ , (12) ∂φ k ∂t = − ˚Γ k δHδφ k + θ φ k , (13) ∂m ∂t = ˚ λ ∇ δHδm − g ℑ [ ψ +0 ∇ ψ ] + θ m . (14)Due to the fact that the stochastic forces θ αφ ⊥ in (4) are δ -correlated and fulfil the Einstein relations, similar prop-erties hold also for the stochastic forces θ ψ : h θ ψ ( x, t ) θ + ψ ( x ′ , t ′ ) i = 4˚Γ ′⊥ δ ( x − x ′ ) δ ( t − t ′ ) . (15)The critical behavior of the thermodynamic derivativesfollows from the extended static functional (the func-tional (3) written in the variables intoduced in (10)) H = H (0) + H ( int ) (16)with a Gaussian part H (0) = Z d d x ( r ⊥ ψ +0 ψ + 12 ( ∇ ψ +0 )( ∇ ψ )+ 12˚˜ r k φ k + 12 ( ∇ φ k ) + 12 m − ˚ hm ) , (17)and an interaction part H ( int ) = Z d d x ( ˚˜ u ⊥
4! ( ψ +0 ψ ) + ˚˜ u k φ k + 2˚˜ u × ψ +0 ψ φ k + 12˚ γ ⊥ m ψ +0 ψ + 12˚ γ k m φ k ) . (18)The above static functional may be reduced to theGinzburg-Landau-Wilson (GLW) functional with com-plex OP by considering an appropriate Boltzmann dis-tribution and integrating out the secondary density. Oneobtains H GLW = Z d d x ( r ⊥ ψ +0 ψ + 12 ( ∇ ψ +0 )( ∇ ψ )+ 12˚ r k φ k + 12 ( ∇ φ k ) +˚ u ⊥
4! ( ψ +0 ψ ) + ˚ u k φ k + 2˚ u × ψ +0 ψ φ k ) . (19)The parameters { ˚ r } ≡ ˚ r ⊥ , ˚ r k and { ˚ u } ≡ ˚ u ⊥ , ˚ u k , ˚ u × in(19) are related to the corresponding parameters of the extended static functional (16) by˚ r ⊥ = ˚˜ r ⊥ + ˚ γ ⊥ ˚ h , ˚ u ⊥ = ˚˜ u ⊥ − γ ⊥ , (20)˚ r k = ˚˜ r k + ˚ γ k ˚ h , ˚ u k = ˚˜ u k − γ k , (21)˚ u × = ˚˜ u × − γ ⊥ ˚ γ k . (22)The property that the static critical behavior does notdepend on the secondary densities, which can be inte-grated out in (16), leads to relations between the corre-lation functions of the secondary densities and the OPcorrelation functions. These relations and their deriva-tions have been extensively discussed in paper III withreal OP functions ~φ ⊥ and φ k . Because the derivationof the relations is independent of the type of OP (realor complex), all of the relations remain valid and can betaken over from paper III. Therefore we will not repeatthem here. III. DYNAMIC CORRELATION AND VERTEXFUNCTIONS
The Fourier transformed dynamic correlation functionsof the two OPs are usually introduced as˚ C ψψ + ( { ξ } , k, ω ) = Z d d x Z dte − ikx + iωt h ψ ( x, t ) ψ +0 (0 , i c (23)˚ C φ k φ k ( { ξ } , k, ω ) = Z d d x Z dte − ikx + iωt h φ k ( x, t ) φ k (0 , i c (24)All functions depend on the two correlation lengths ξ ⊥ and ξ k , which is indicated by { ξ } in a short notation. h AB i c = h AB i − h A ih B i denotes the cumulant. The av-erages are calculated with a propability density includ-ing a dynamic functional, which can be constituted fromthe dynamic equations (11) - (14). In the consideredapproach of [17] for every density auxiliary densities areintroduced accordingly. They are denoted as ˜ ψ +0 , ˜ ψ , ˜ φ k and ˜ m . The dynamic correlation functions of the orderparameters are connected to dynamic vertex functionsvia ˚ C ψψ + ( { ξ } , k, ω ) = − ˚Γ ˜ ψ ˜ ψ + ( { ξ } , k, ω ) (cid:12)(cid:12)(cid:12) ˚Γ ψ ˜ ψ + ( { ξ } , k, ω ) (cid:12)(cid:12)(cid:12) , (25)˚ C φ k φ k ( { ξ } , k, ω ) = − ˚Γ ˜ φ k ˜ φ k ( { ξ } , k, ω ) (cid:12)(cid:12)(cid:12) ˚Γ φ k ˜ φ k ( { ξ } , k, ω ) (cid:12)(cid:12)(cid:12) , (26)where the two-point vertex functions appearing on theright hand side in the above expression have to be calcu-lated within perturbation expansion. They are obtainedby collecting all 1-particle irreducible Feynman graphswith corresponding external legs. A closer examinationof the loop expansion reveals [18] that the dynamic re-sponse vertex functions ˚Γ ψ ˜ ψ + and ˚Γ φ k ˜ φ k have the generalstructure˚Γ ψ ˜ ψ + ( { ξ } , k, ω ) = − i ω ˚Ω ψ ˜ ψ + ( { ξ } , k, ω )+ ˚Γ ψψ + ( { ξ } , k )˚Γ ( d ) ψ ˜ ψ + ( { ξ } , k, ω ) , (27)˚Γ φ k ˜ φ k ( { ξ } , k, ω ) = − i ω ˚Ω φ k ˜ φ k ( { ξ } , k, ω )+ ˚Γ φ k φ k ( { ξ } , k )˚Γ k , (28)where ˚Γ ψψ + ( { ξ } , k ) and ˚Γ φ k φ k ( { ξ } , k ) are the well knownstatic two point vertex functions of the bicritical GLW-model with a complex OP. We want to remark that thestatic vertex functions in (27) and (28) are related by˚Γ ψψ + ( { ξ } , k ) = 12˚Γ (2 , ⊥⊥ ( { ξ } , k ) (29)and ˚Γ φ k φ k ( { ξ } , k ) = ˚Γ (2 , kk ( { ξ } , k ) (30)to the static vertex functions introduced in papers I–IIIfor the model with real OPs. Thus the correlation lengths ξ ⊥ and ξ k are now determined by ξ ⊥ ( { ˚ r } , { ˚ u } ) = ∂ ln ˚Γ ψψ + ( k, { ˚ r } , { ˚ u } ) ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 , (31) ξ k ( { ˚ r } , { ˚ u } ) = ∂ ln ˚Γ φ k φ k ( k, { ˚ r } , { ˚ u } ) ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 . (32)˚Ω ψ ˜ ψ + , ˚Γ ( d ) ψ ˜ ψ + and ˚Ω φ k ˜ φ k are purely dynamic functions.The explicit expressions of these functions are given inappendix A, Eqs. (A1) - (A3). They determine alsothe dynamic vertex functions ˚Γ ˜ ψ ˜ ψ + and ˚Γ ˜ φ k ˜ φ k in (25)and (26). A proper rearrangement of the perturbativecontributions shows that the relations˚Γ ˜ ψ ˜ ψ + ( { ξ } , k, ω ) = − ℜ h ˚Ω ψ ˜ ψ + ( { ξ } , k, ω )˚Γ ( d ) ψ ˜ ψ + ( { ξ } , k, ω ) i , (33)˚Γ ˜ φ k ˜ φ k ( { ξ } , k, ω ) = − k ℜ h ˚Ω φ k ˜ φ k ( { ξ } , k, ω ) i (34)hold. ℜ [ . ] is the real part of the expression in the brack-ets.Analogous to (23) and (24) the Fourier transformeddynamic correlation function of the secondary density isintroduced as˚ C mm ( { ξ } , k, ω ) = Z d d x Z dte − ikx + iωt h m ( x, t ) m (0 , i c (35) The connection to the dynamic vertex functions is anal-ogous to the case of the OP Eqs. (25 and (26):˚ C mm ( { ξ } , k, ω ) = − ˚Γ ˜ m ˜ m ( { ξ } , k, ω ) (cid:12)(cid:12)(cid:12) ˚Γ m ˜ m ( { ξ } , k, ω ) (cid:12)(cid:12)(cid:12) . (36)The dynamic response vertex function of the secondarydensity has the general structure˚Γ m ˜ m ( { ξ } , k, ω ) = − i ω ˚Ω m ˜ m ( { ξ } , k, ω )+ ˚Γ mm ( { ξ } , k )˚Γ ( d ) m ˜ m ( { ξ } , k, ω ) (37)where ˚Γ mm ( { ξ } , k ) is the static two point vertex func-tion calculated with the extended static functional (16)which already has been introduced in paper III. A re-lation corresponding to (33) holds also for the dynamicvertex function of the secondary density. We have˚Γ ˜ m ˜ m ( { ξ } , k, ω ) = − ℜ h ˚Ω m ˜ m ( { ξ } , k, ω )˚Γ ( d ) m ˜ m ( { ξ } , k, ω ) i . (38) IV. RENORMALIZATIONA. Static renormalization
The renormalization of the GLW-functional (19) hasbeen extensively discussed in paper I. The only differencein the present paper is that we now have to renormalizethe complex OP ψ instead of the real vector OP ~φ ⊥ .We introduce the following renormalization factor ψ = Z / ψ ψ , ψ +0 = Z / ψ ψ + (39)where Z ψ is a real quantity and identical to Z φ ⊥ in paperI taken at n ⊥ = 2 and n k = 1. This means Z ψ = Z φ ⊥ (cid:12)(cid:12)(cid:12) n ⊥ =2 n k =1 . (40)The renormalization of the parameters ˚ r ⊥ , ˚ r k and thecouplings ˚ u ⊥ , ˚ u k , ˚ u × appearing in (19) is given in paperI (see Eqs. (16), (17) and (5)-(7)). In all relations onehas to replace Z φ ⊥ by Z ψ . All renormalization factorsremain valid if one sets n ⊥ = 2 and n k = 1. This is alsotrue for the Z -matrix Z φ introduced in Eq.(10) of paperI and the additive renormalization A ( { u } ) defined in Eq.(15) of paper I.The renormalization of the parameters in the extendedstatic functional (16) has been presented in paper III. Asin the case of the bicritical GLW-model all Z -factors andrelations between them remain valid if Z φ ⊥ therein isreplaced by Z ψ , and if one sets n ⊥ = 2 and n k = 1 inexplicit expressions. B. Dynamic renormalization
Within dynamics auxiliary densities ˜ ψ , ˜ φ k and ˜ m corresponding to the two OPs and the secondary densityare introduced [17]. Instead of renormalization condi-tions we use the minimal subtraction scheme [19] as inthe preceding papers II and III. The auxiliary densityof the complex OP is multiplicatively renormalizable byintroducing complex Z -factors:˜ ψ = Z / ψ ˜ ψ , ˜ ψ +0 = Z / ψ + ˜ ψ + (41)The complex renormalization factors in (41) fulfill therelation Z ˜ ψ + = Z +˜ ψ . For the auxiliary densities of thesingle-component real OP and the secondary density thecorresponding renormalization factors are introduced:˜ φ k = Z / φ k ˜ φ k , ˜ m = Z ˜ m ˜ m , (42)where Z ˜ φ k and Z ˜ m are real. Within the minimal subtrac-tion scheme the Z -factors of the auxiliary densities of thenon-conserved OPs Z ˜ ψ + and Z ˜ φ k are determined by the ε -poles of the functions ˚Ω ψ ˜ ψ + and ˚Ω φ k ˜ φ k introduced in(27) and (28). The corresponding function of the con-served secondary density ˚Ω m ˜ m in (37) does not containnew poles. Therefore one has Z ˜ m = Z − m (43)where Z m has been introduced in Eq. (30) in paper III.The kinetic coefficients renormalize as˚Γ ⊥ = Z Γ ⊥ Γ ⊥ , ˚Γ k = Z Γ k Γ k , ˚ λ = Z λ λ . (44)The renormalization of the complex kinetic coefficient Γ ⊥ in (44) leads to a complex Z Γ ⊥ , while the other two renor-malization factors in (44) are real valued. Z Γ ⊥ can beseparated into Z Γ ⊥ = Z / ψ Z − / ψ + Z ( d )Γ ⊥ , (45)where Z ( d )Γ ⊥ contains the singular contributions of the dy-namic function ˚Γ ( d ) ψ ˜ ψ + , appearing in (27).The dynamic equation (13) for the OP φ k contains nomode coupling term. As a consequence only the kineticcoefficient ˚Γ k appears in the dynamic vertex function (28)instead of a function ˚Γ ( d ) φ k ˜ φ k . Therefore Z ( d )Γ k = 1 and wecan write Z Γ k = Z / φ k Z − / φ k . (46)Using Eq.(43) the kinetic coefficient of the secondary den-sity renormalizes as Z λ = Z m Z ( d ) λ (47) where Z ( d ) λ contains only the poles of the k derivative of˚Γ ( d ) m ˜ m taken at zero frequency and wave vector modulus.The mode coupling coefficient needs no independentrenormalization, so we simply have˚ g = κ ε/ Z m gA − / d . (48)The geometric factor A d [20] already used in the staticrenormalization has been given in paper I Eq. (8). V. RENORMALIZATION GROUP FUNCTIONS
In order to obtain the temperature dependence of themodel parameters, as well as the asymptotic dynamicexponents, the RG functions, which are usually denotedas ζ - and β -functions have to be introduced. A. General definitions
In order to simplify the general handling of the RGfunctions we will use the uniform definition ζ a i ( { α j } ) = d ln Z − a i ( { α j } ) d ln κ (49)for all ζ -functions in statics and dynamics. The deriva-tive is taken at fixed bare parameters. { α j } denotes theset of static and dynamic model parameters which in-clude the static couplings { u } and { γ } , the mode cou-pling g , and all kinetic coefficients Γ ⊥ , Γ + ⊥ , Γ k , λ . The ζ -function ζ a i is calculated from the renormalization fac-tor Z a i introduced in the previous section. Thus a i maydenote a model parameter from the set { α j } , a density φ ⊥ , φ k , m , or a composite operator φ ⊥ , φ k . The ap-proach of the model parameters α i ( l ) to their FP valuesin the vicinity of the multicritical point is determined bythe flow equations with the flow parameter ll dα i ( l ) dl = β α i ( { α j ( l ) } ) (50)with β -functions β α i ( { α j ( l ) } ) = α i ( l ) (cid:16) − c i + ζ α i ( { α j ( l ) } ) (cid:17) (51) c i is the naive dimension of the corresponding parameter α i obtained by power counting. For the static couplings u ⊥ , u × or u k the naive dimension c i is equal to ε , while for γ ⊥ or γ k and the mode coupling g it is ε/ ⊥ , Γ + ⊥ , Γ k and λ , aredimensionless quantities, which means c i = 0.The flow equations (50) have fixed points at the zerosof the β -functions. The FP values of the model parame-ters { α ⋆j } are defined by the equations β α i ( { α ⋆j } ) = 0 . (52)The FP is stable if all eigenvalues of the matrix ∂β α i /∂α k are positive or possesses positive real parts. Starting atvalues { α j ( l ) } at an initial flow parameter value l , theflow equations can be solved numerically. The asymp-totic critical values of the parameters are obtained in thelimit l →
0. If a stable FP is present the flow of theparameters has the propertylim l → { α j ( l ) } = { α ⋆j } . (53)A set of FP values { α ⋆j } determines all static and dynamicexponents. The static relations between ζ -functions andcritical exponents have been extensively discussed in pa-pers I and III. The dynamic exponents are related by z φ ⊥ = 2 + ζ ⋆ Γ ′⊥ , z φ k = 2 + ζ ⋆ Γ k , z m = 2 + ζ ⋆λ (54)to the dynamic ζ -functions (see [3]). In (54) the shortnotation ζ ⋆α i ≡ ζ α i ( { α ⋆j } ) has been introduced. In thenon-asymptotic background region effective dynamic ex-ponents are defined as z ( eff ) ⊥ ( l ) = 2 + ζ Γ ′⊥ (cid:0) { α j ( l ) } (cid:1) , (55) z ( eff ) k ( l ) = 2 + ζ Γ k (cid:0) { α j ( l ) (cid:1) , (56) z ( eff ) m ( l ) = 2 + ζ λ (cid:0) { α j ( l ) (cid:1) . (57)where the flow of the parameters is inserted into the ζ -functions instead of the FP values. The effective expo-nents depend on the flow parameter, or reduced tem-perature accordingly. Relation (53) makes sure that theeffective exponents turn into the asymptotic exponentsin the critical limit, that islim l → z ( eff ) k ( l ) = z k with k = ⊥ , k , m (58) B. Time scale ratios and mode coupling parameters
It is convenient to introduce ratios of the kinetic co-efficients or mode couplings, which may have finite FPvalues. The following ratios will be used in the subse-quent sections:(i) The time scale ratios between the order parametersand the secondary density w ⊥ ≡ Γ ⊥ λ , w k ≡ Γ k λ . (59)From this we may also define the ratio between ki-netic coefficients of the two order parameters v ≡ Γ k Γ ⊥ = w k w ⊥ (60)which already previously has been used in the bi-critical model A and model C. Note that in con-trast to the two models mentioned, w ⊥ and v are now complex quantities. The ratios in Eqs.(59) and(60) are of course not independent as shown bythe equality in (60). The structure of the dynamic ζ -functions presented subsequently further impliesthe introduction of the complex ratio v ⊥ ≡ Γ ⊥ Γ + ⊥ = w ⊥ w + ⊥ = v + v . (61)(ii) The mode coupling parameter F ≡ gλ . (62)The above ratio does not necessarily have a finiteFP value. Thus it may be more appropriate to usethe ratio f ⊥ ≡ g p Γ ′⊥ λ = F p w ′⊥ (63)in several cases, especially in the discussion of theflow equations and the fixed points.The flow equations for the ratios defined above can befound from the ζ - and β -functions introduced in the pre-vious subsection. From the definition of the parametersin (59), (63) and the renormalization (44) and (48) weobtain together with (49) the flow equations l dw ⊥ dl = w ⊥ ( ζ Γ ⊥ − ζ λ ) , (64) l dw k dl = w k (cid:0) ζ Γ k − ζ λ (cid:1) , (65) l df ⊥ dl = − f ⊥ (cid:18) ε + ζ λ − ζ m + ℜ (cid:20) w ⊥ w ′⊥ ζ Γ ⊥ (cid:21)(cid:19) . (66)From (64) and (65) follows immediately the flow equationfor the ratio l dvdl = v (cid:0) ζ Γ k − ζ Γ ⊥ (cid:1) , (67)which has been defined in (60).The remaining task is to calculate the explicit expres-sions of the dynamic functions ζ Γ ⊥ , ζ Γ k and ζ λ in twoloop order. VI. DYNAMIC RG-FUNCTIONS IN TWO LOOPORDER
The perturbation expansion of the dynamic vertexfunctions and the structures therein are outlined in detailin appendix A. The outcoming expressions for the dy-namic renormalization factors in two loop order are pre-sented in appendix B. With these expressions at hand weare in the position to obtain explicit two loop expressionsfor the RG ζ -functions as expressed in the following. A. Dynamic ζ -functions of the OPs Relation (46) between the Z-factors implies the rela-tions between the corresponding ζ -functions ζ Γ ⊥ = ζ ( d )Γ ⊥ − ζ ˜ ψ + + 12 ζ ψ , (68) ζ Γ k = − ζ ˜ φ k + 12 ζ φ k . (69)The static ζ -functions ζ ψ = ζ φ ⊥ has been presentedEqs.(20) in paper I. Inserting (B1) and (B2) into (49)and (68) we obtain the dynamic ζ -function for the ki-netic coefficient of the perpendicular components as ζ Γ ⊥ = D ⊥ w ⊥ (1 + w ⊥ ) − u ⊥ D ⊥ w ⊥ (1 + w ⊥ ) A ⊥ − D ⊥ w ⊥ (1 + w ⊥ ) B ⊥ − γ k D ⊥ w ⊥ (cid:18) u × γ k D ⊥ w ⊥ (cid:19) X ⊥ + ζ ( A )Γ ⊥ (cid:0) { u } , v ⊥ , v (cid:1) (70)where we have introduced the coupling D ⊥ ≡ w ⊥ γ ⊥ − i F . (71)The functions A ⊥ , B ⊥ and X ⊥ are defined as A ⊥ ≡ w ⊥ γ ⊥ (1 − x L ) + i F x − x L − D ⊥ L (72) B ⊥ ≡ w ⊥ γ ⊥ (1 − x L ) + F (2 x − L + L R )+2 w ⊥ γ ⊥ i F (1 + 2 x − x L ) − L D ⊥ − D ⊥ w ⊥ w ⊥ + (1 + 2 w ⊥ ) ln (1 + w ⊥ ) w ⊥ ! (73) X ⊥ ≡ v v − (cid:18) v (cid:19) ln 2(1 + v )2 + v (74)with L R ≡ h x + + v ⊥ + x ( x + 2 v ⊥ ) i L x + − v ⊥ . (75)We have used the following definitions in the above ex-pressions: x ± ≡ ± v ⊥ , x ≡ v ⊥ , (76) L ≡ v ⊥ , L ≡ ln (cid:16) v ⊥ (cid:17) v ⊥ . (77) ζ ( A )Γ ⊥ (cid:0) { u } , v ⊥ , v (cid:1) is the ζ -function of the kinetic coefficientof the perpendicular components in the bicritical model A, but now with a complex kinetic coefficient Γ ⊥ . Itreads in two loop order ζ ( A )Γ ⊥ (cid:0) { u } , v ⊥ , v (cid:1) = u ⊥ (cid:18) L + x L − (cid:19) + u × (cid:18) L ( × ) ⊥ − (cid:19) (78)with L ( × ) ⊥ ≡ ln (1 + v ) v (2 + v ) + 2 v ln 2(1 + v )2 + v . (79)The dynamic ζ -function of the parallel component is ob-tained by inserting Eq.(21) of paper I and (B3) into (49)and (69). The result is ζ Γ k = w k γ k w k − w k γ k w k " u k γ k (cid:18) − (cid:19) + w k γ k w k (cid:18) − (cid:19) − w k w k − w k w k ln (1 + w k ) w k ! + (cid:18) u × + w k γ k w k γ ⊥ (cid:19) ℜ h T w ′⊥ i − γ k F w ′⊥ (1+ w k ) ℑ h T w ′⊥ i + ζ ( A )Γ k (cid:0) { u } , v ⊥ , v (cid:1) . (80)The functions T and T are defined as T ≡ D ⊥ " v ⊥ v − (cid:18) v + 1 v ⊥ (1+ v ) (cid:19) ln (1+ v ) (cid:16) v ⊥ (cid:17) v + v ⊥ (1+ v ) , (81) T ≡ w + ⊥ D ⊥ " (1 + v ⊥ ) v − ln 1 + v ⊥ v − (cid:18) v + 1 v ⊥ (1+ v ) (cid:19) (cid:0) v + v ⊥ (1+ v ) (cid:1) × ln (1+ v ) (cid:16) v ⊥ (cid:17) v + v ⊥ (1+ v ) , (82) ζ ( A )Γ k (cid:0) { u } , v ⊥ , v (cid:1) is the ζ -function of the kinetic coefficientof the parallel component in the bicritical model A. Witha complex Γ ⊥ it reads ζ ( A )Γ k (cid:0) { u } , v ⊥ , v (cid:1) = u k (cid:18) ln 43 − (cid:19) + u × (cid:18) L ( × ) k − (cid:19) (83)with L ( × ) k ≡ ln (1+ v ) (cid:16) v ⊥ + v (cid:17) v + v ⊥ (1+ v ) + vv ⊥ ln (cid:16) v ⊥ (cid:17) (cid:16) v ⊥ + v (cid:17) v + v ⊥ (1+ v )+ v ln (cid:16) v ⊥ (cid:17) (1+ v ) v + v ⊥ (1+ v ) . (84) B. Dynamic ζ -functions of the secondary density With relation (47) we can separate the static contri-butions to the ζ -function ζ λ . Thus we have ζ λ = 2 ζ m + ζ ( d ) λ (85)By separating the static from the dynamic parts in the ζ -functions one can take advantage of the general struc-tures appearing in the purely dynamic ζ -function ζ ( d ) λ aswell as in the static ζ -function ζ m . Inserting n ⊥ = 2 and n k = 1 into relation (40) in paper III ζ m can be writtenas ζ m = 12 γ ⊥ + 14 γ k (86)which is valid up to two loop order. From the diagram-matic structure of the dynamic perturbation theory fol-lows ζ ( d ) λ = − f ⊥ (cid:16) Q (cid:17) . (87)The real function Q contains all higher order contri-butions beginning with two loop order. Setting Q = 0in (87) reproduces the one loop expressions of this func-tion. The function Q in the dynamic ζ -function of thesecondary density (87) has the structure Q = 12 ℜ [ X ] (88)from which immediately follows that it is a real quantity. X reads X = D ⊥ w ′⊥ (1 + w ⊥ ) " D ⊥ (cid:18)
12 + ln 1 + w ⊥ w + ⊥ (cid:19) (89)+ D + ⊥ (1 + w ⊥ ) − (cid:16) W ( m ) ⊥ γ ⊥ + w ⊥ i F (cid:17) W ( m ) ⊥ L ( m ) ⊥ where we have introduced the definitions L ( m ) ⊥ = ln W ( m ) ⊥ ! , (90) W ( m ) ⊥ = w ⊥ + w + ⊥ + w ⊥ w + ⊥ . (91)Note that X coincides with the corresponding functionin model F in [3, 21]. VII. CRITICAL BEHAVIOR IN ONE LOOPORDER
Although the one loop critical behavior of the consid-ered system has already been discussed in [6] we wantto summarize the results in order to compare it with theconsiderably differing results of the two loop calculation.In one loop order the ζ -functions (70), (80) and (87) re-duce to ζ Γ ⊥ = D ⊥ w ⊥ (1 + w ⊥ ) , ζ Γ k = w k γ k w k , ζ ( d ) λ = − f ⊥ . (92)Inserting (92) into the right hand sides of (64)-(66) leadsto a set of equations in which the zeros determine thedynamical FPs. The only stable FP is found for w ⋆ k = 0and w ′ ⋆ ⊥ , w ′′ ⋆ ⊥ , f ⋆ ⊥ finite. The corresponding values arepresented in Tab. II. As a consequence we have v ⋆ = 0.We want to note that the static FP values of the two loopcalculation in paper I and III have been used. We wereinterested in the non-asymptotic properties described byflow equations and since no real FP in statics is reachedin two loop order we had to resum the static β -functionsin order to get real FP values [2]. To each type of staticFP (biconical or Heisenberg) two equivalent static fixedpoints exist differing in the signs of γ ⊥ and γ k [11]. Ac-cordingly four equivalent dynamic fixed points exist withdifferent signs in w ′′⊥ and f ⊥ . They correspond to thedirections of the external fields of the parallel and per-pendicular OP.The finite value of w ⋆ ⊥ implies the relations ζ ′ ⋆ Γ ⊥ = ζ ⋆λ , ζ ′′ ⋆ Γ ⊥ = 0 , ε + ζ ′ ⋆ Γ ⊥ + ζ ⋆λ − ζ ⋆m = 0 , (93)which follow from (64) and (66). The vanishing w ⋆ k leadsto ζ ⋆ Γ k = 0 as can immediately be seen from (92). Usingthe first relation in (93) and the third one, we obtain ζ ′ ⋆ Γ ⊥ = ζ ⋆λ = 12 (2 ζ ⋆m − ε ) . (94)Inserting the FP value of the static ζ -function ζ ⋆m (seerelation (105) in paper III) ζ ⋆m = φν − d ζ ′ ⋆ Γ ⊥ = ζ ⋆λ = φν − . (96)The dynamic critical exponents (54) in one loop orderare therefore completely expressed in terms of the staticexponents: z ⊥ ≡ z φ ⊥ = z m = φν , z k ≡ z φ k = 2 . (97)0These static exponents might also be taken from staticexperiments. All our numerical calculations are per-formed in d = 3 ( ǫ = 1). The numerical values of thestatic exponents φ and ν have been calculated in two looporder in paper I and are given there in Tab.III ( ν = ν + therein) in two loop order resummed. In one loop or-der the two OPs have different dynamic critical expo-nents. Scaling is fulfilled only between the perpendicularOP and the secondary density. The parallel OP behaveslike the van Hove model. This is demonstrated in Fig.1,where the effective exponents defined in (55) - (57) havebeen calculated by using the flow equations in one looporder. At a flow parameter about l ∼ e − for both, thebiconical FP (solid lines) and the Heisenberg FP (dashedlines), the asymptotic values of the dynamic exponents z ⊥ and z m are reached. The classical value z k = 2, validfor both static fixed points, also is indicated by a straightline. The corresponding flow is presented in Fig.2, whichproofs that the dynamic exponents in Fig.1 have reachedtheir asymptotic behavior because the dynamic parame-ters are at their FP values at ln l = − VIII. LIMITING BEHAVIOR OF THEDYNAMICAL ζ -FUNCTIONS IN 2-LOOP ORDER The appearance of ln v -terms in the two loop contribu-tion to the ζ Γ ⊥ -function, Eq (70), changes the discussionof the fixed points considerably compared to the one loopcase. In order to determine the dynamical fixed pointsof the current model in two loop order it is necessaryto know something about the limiting behavior of the ζ -functions. For this reason we will present the ζ -functions -25 -20 -15 -10 -5 01.61.82.02.22.4 (eff) z (eff) z m z || (eff) z (eff) z m ln l FIG. 1: Effective dynamic exponents at d = 3 calculatedin one loop order using the one loop expression for the flowequations (64), (66). The effective exponents are calculated atthe biconical FP (full lines) and at the Heisenberg FP (dashedline). z k is valid for both FPs. -25 -20 -15 -10 -5 00.51.0 f ln l -1.0-0.50.00.5 s w ’ FIG. 2: Flow of the parameters w ′⊥ , s = w ′′⊥ /w ′⊥ and f ⊥ inone loop order at d = 3. The calculation has been performedfor the biconical (solid lines) and the Heisenberg (dashedlines) FP. in cases where one or several dynamical parameters go tozero or infinity under definite conditions. This is neces-sary because some ζ -functions exhibit singular behaviorunder these conditions, which influences the discussionof possible fixed points. It is anticipated that the criticalexponents defined by the values of the ζ -functions at theFP are finite and real.The auxiliary functions X ⊥ , L ( × ) ⊥ , T , T and L ( × ) k ,which appear in the ζ -functions (70) and (80) behave sin-gularly in several limits of the parameters. Thus severalFP values of the different parameters can be excludeddue to diverging ζ -functions. For a summary of the sub-sequent analysis of the ζ -function on the time scale ratiossee Tab. III.i) At first we will consider the two functions X ⊥ and L ( × ) ⊥ in (74) and (79), which appear in ζ Γ ⊥ and depend on v only. These two functions remain regular if v grows toinfinity. In this case one simply has X ⊥ ( v → ∞ ) = 1 and L ( × ) ⊥ ( v → ∞ ) = 0. But for vanishing v both functionsevolve a term proportional to ln v . One gets X ⊥ ( v →
0) = ln(2 v ) , L ( × ) ⊥ ( v →
0) = 1 − ln(2 v ) . (98)Thus divergent ln v terms appear in ζ Γ ⊥ ( v →
0) indepen-dent from the individual behavior of w ⊥ and w k because1 FP u ⋆ k u ⋆ ⊥ u ⋆ × γ ⋆ k γ ⊥ ⋆ w ′ ⋆ ⊥ w ′′ ⋆ ⊥ f ⋆ ⊥ B ∓ ± B ± ± H ∓ ± H ± ± n k = 1, n ⊥ = 2 at d = 3. B indicates the biconical, H the HeisenbergFP. There are always two equivalent static FPs depending on the signs of the couplings γ . The FP values of the static couplings { u } and { γ } are taken from the resummed two-loop results [2], whereas w ⋆ ⊥ and f ⋆ ⊥ are calculated from the one-loop β -functions. w ⋆ k = v ⋆ = 0 is valid in all cases. Corresponding to the two equivalent cases in statics and the sign of w ′′ ⋆ ⊥ there are equivalentdynamic FPs with corresponding signs of the FP value of the mode coupling f ⋆ ⊥ . only the ratio v enters the function.ii) The dynamic ζ -function (80) of the parallel compo-nent contains the three functions T , T and L ( × ) k definedin (81), (82) and (84) which contain the ratio v . Thesefunctions, and therefore also ζ Γ k , remain non-divergentfor vanishing v . One obtains T ( v → D ⊥ (cid:18) (cid:18) v ⊥ (cid:19) − v ⊥ ln(1 + v ⊥ ) (cid:19) ,T ( v → − w ⋆ ⊥ D ⊥ (cid:18) ln (cid:18) v ⊥ (cid:19) + ln(1 + v ⊥ ) (cid:19) ,L ( × ) k ( v → . (99)But they diverge when v is growing to infinity: T ( v → ∞ ) = D ⊥ ln (cid:16) v ⊥ (cid:17) v ,T ( v → ∞ ) = − w ⋆ ⊥ D ⊥ ln (cid:16) v ⊥ (cid:17) v ,L ( × ) k ( v → ∞ ) = 1 + ln v (cid:16) v ⊥ (cid:17) . (100)In contrast to the case i) the function ζ Γ k in the paral-lel subspace evolves logarithmic terms ln v in the limit v → ∞ and stays finite in the limit v →
0. The abovediscussion is also independent of the individual behaviorof w ⊥ and w k because only the ratio v ⊥ stays always fi-nite and the three functions T , T and L ( × ) k remain finiteeven for diverging time scale ratios if their prefactors aretaken into account. Limit ζ Γ ⊥ ζ Γ k ζ λ v → ∼ ln v regular unaffected v → ∞ regular ∼ ln v unaffected w ⊥ → ∞ ∼ ln w ⊥ regular ∼ w ′ ⊥ w k → ∞ regular ∼ ln w k unaffected w ⊥ → ∼ ln v regular w k → ∼ ln v regular unaffectedTABLE III: Limiting behavior of the dynamic ζ -functions iii) Additional logarithmic singularities may arise inthe dynamic ζ -functions if the time scale ratios w ⊥ and w k grow individually to infinity independent of the be-havior of v . A closer examination of (70) reveals that inthe limit w ⊥ → ∞ the ζ -function is proportional to ζ Γ ⊥ ( w ⊥ → ∞ ) ∼ γ ⊥ ln w ⊥ v . Quite analogously thesame happens in (80) when w k grows to infinity. Oneobtains ζ Γ k ( w k → ∞ ) ∼ γ k ln w k . (102)Supposing a finite (different from zero or infinity) FPvalue f ⋆ ⊥ for the mode coupling parameter we may con-clude the following concerning the allowed FP values ofthe remaining parameters:a) From i) and ii) follows that v ⋆ has to be also differentfrom zero or infinity, otherwise ln v contributions wouldlead to divergent ζ -functions.b) From iii) follows that the finite v ⋆ only can be real-ized either by w k and w ⊥ both finite, or w k and w ⊥ bothgoing to zero in the same way. The possibility that bothtime scale ratios are going to infinity in the same way isexcluded because of the ln w k and ln w ⊥ terms appearingin this case. IX. GENERAL ASYMPTOTIC RELATIONS
The FP values { α ⋆j } of the model parameters are foundfrom the zeros of the β -functions in Eqs.(64)-(66). Fromthe right hand side of the equations one obtains w ⋆ ⊥ (cid:0) ζ ⋆ Γ ⊥ − ζ ⋆λ (cid:1) = 0 , (103) w ⋆ k (cid:16) ζ ⋆ Γ k − ζ ⋆λ (cid:17) = 0 , (104) f ⋆ ⊥ (cid:18) ε + ζ ⋆λ − ζ ⋆m + ℜ (cid:20) w ⋆ ⊥ w ′ ⋆ ⊥ ζ ⋆ Γ ⊥ (cid:21)(cid:19) = 0 . (105)A FP which fulfills Eqs.(103)-(105) has to be also a so-lution of v ⋆ (cid:16) ζ ⋆ Γ k − ζ ⋆ Γ ⊥ (cid:17) = 0 (106)2which follows from (67). The ζ -function ζ Γ ⊥ for the per-pendicular component of the OP relaxation is a complexfunction. Separating real and imaginary part leads to ζ Γ ⊥ = ζ ′ Γ ⊥ + iζ ′′ Γ ⊥ . (107)As a consequence also the equations (103) for w ⊥ and(106) for v are complex expressions. The ζ -function forΓ ′⊥ is ζ Γ ′⊥ = ζ ′ Γ ⊥ − sζ ′′ Γ ⊥ (108)with s defined in (121).We anticipate that in a real physical system definitedynamical exponents exist, and therefore the dynamic ζ -functions have to be finite at the stable FP. As alreadymentioned in subsection VIII, the ζ -functions contain ln v terms requiring a finite FP value v ⋆ in order to obtainfinite dynamical exponents. Separating (106) into realand imaginary part one has v ′ ⋆ ( ζ ′ ⋆ Γ ⊥ − ζ ⋆ Γ k ) − v ′′ ⋆ ζ ′′ ⋆ Γ ⊥ = 0 , (109) v ′ ⋆ ζ ′′ ⋆ Γ ⊥ + v ′′ ⋆ ( ζ ′ ⋆ Γ ⊥ − ζ ⋆ Γ k ) = 0 . (110)From these two equations the FP relations ζ ′ ⋆ Γ ⊥ = ζ ⋆ Γ k , ζ ′′ ⋆ Γ ⊥ = 0 (111)immediately follow. The second equation in (111) impliesthat the dynamical exponent z φ ⊥ in (54) can be writtenas z φ ⊥ = 2 + ζ ′ ⋆ Γ ⊥ . (112)From the first relation in (111) follows z φ ⊥ = z φ k . (113)This means that in the case of a finite FP value v ⋆ scalingbetween the OPs is valid. In order to obtain a criticalbehavior different from model C, the FP value f ⋆ ⊥ of themode coupling parameter also has to be finite and differ-ent from zero. Then from Eq.(105) follows ε + ζ ′ ⋆ Γ ⊥ + ζ ⋆λ − ζ ⋆m = 0 , (114)where the second relation of (111) already has been used.Inserting (85) and (95) into (114) one obtains the relation z φ ⊥ + z m = 2 φν (115)between the exponents. In summary, the condition thatboth v ⋆ and f ⋆ ⊥ have to be finite leads to the two rela-tions (113) and (115) between the exponents. Furtherrelations are dependent whether the FP values of thetime scale ratios w ⊥ and w k are finite or zero and leadto the following cases. (i) Dynamical strong scaling FP:
In the case that w ⊥ and w k are finite at the FP,from (103) and (104) the relation ζ ′ ⋆ Γ ⊥ = ζ ⋆ Γ k = ζ ⋆λ (116)is obtained, where (111) already has been used.From (54) it follows immediately that the dynami-cal exponents have to fulfill the relations z φ ⊥ = z φ k = z m ≡ z . (117)Thus in the case of strong scaling one dynamicalexponent z exists only. The exact value of this ex-ponent can be found by inserting (117) into (115).One obtains z = φν . (118)(ii) Dynamical weak scaling FP:
In the case that w ⊥ and w k are zero with v finiteat the FP, Eqs.(103) and (104) are trivially fulfilledand no additional relation between the ζ -functions,and dynamical exponents respectively, arises. As aconsequence two dynamical exponents exist. Thefirst one z φ k = z φ ⊥ = z OP (119)for the OPs, follows from relation (113). The sec-ond one z m = 2 φν − z OP (120)for the secondary density, is obtained from (115).A closer examination of the β -functions (103) - (105),also with numerical methods in d = 3, reveals that noFP solution can be found where both, w ⊥ and w k , arefinite. Thus the only solution in d = 3 which remains is w ′ ⋆ ⊥ = w ′′ ⋆ ⊥ = w ⋆ k = 0 with v ⋆ and v ⋆ ⊥ finite. This result ofcourse depends also on the specific numerical values [22]of the static FPs (given in Tab II) used in the dynamicalequations. The stable FP lies then in the subspace wherethe time scale ratios w k and w ⊥ approach zero in sucha way that their ratios v and v ⊥ remain finite and ingeneral complex quantities. In order to obtain the finiteFP values for v and v ⊥ the two loop ζ -functions may bereduced by setting w ⊥ and w k equal to zero by keepingtheir ratios finite. This will be performed in the followingsection. X. CRITICAL BEHAVIOR IN THEASYMPTOTIC SUBSPACE
Since the asymmetric couplings γ α always appear to-gether with the time scale ratios w α all terms propor-tional to these couplings drop out in the asymptotic limit3where w α →
0. It is convenient to introduce the real ra-tios s ≡ w ′′⊥ w ′⊥ = Γ ′′⊥ Γ ′⊥ , q ≡ w k w ′⊥ = Γ k Γ ′⊥ . (121)Thus only s , q and f ⊥ remain as independent dynamicalvariables.The ratio s determines the behavior of the imaginarypart of w ⊥ with respect to the real part, while the ratio q indicates the behavior of w k with respect to the real partof w ⊥ . The complex parameters v ⊥ and v , introduced in(61) and (60), are expressed by s and q as v ⊥ = 1 + is − is , v = q is (122)in the following expressions. A. ζ -functions We discuss the behavior of the ζ -functions in the limit w ⊥ → w k → s and q constant. Case s = 0 : For w ⊥ = 0 and w k = 0 the ζ -function (70), reduces to ζ ( as )Γ ⊥ (cid:0) { u } , s, q, f ⊥ (cid:1) = − f ⊥ is ( u ⊥ (cid:16) L ( s )+ x − ( s ) x ( s ) L ( s ) (cid:17) − f ⊥ is (cid:16) x − ( s ) L ( s ) − L ( s ) + L R ( s ) (cid:17)) + ζ ( A )Γ ⊥ (cid:0) { u } , s, q (cid:1) . (123)The functions x − ( s ), x ( s ), L ( s ), L ( s ) and L R ( s ) arethe same as in (75) - (77) with v ⊥ replaced by (122). Thesame is true for ζ ( A )Γ ⊥ (cid:0) { u } , s, q (cid:1) , which has been definedin (78), and where also (122) has been used to replace v ⊥ and v .Performing the limit in the dynamical ζ -function (80)it reduces to ζ ( as )Γ k (cid:0) { u } , s, q (cid:1) = ζ ( A )Γ k (cid:0) { u } , s, q (cid:1) (124)where ζ ( A )Γ k (cid:0) { u } , s, q (cid:1) is the model A function (83) withrelation (122) inserted into (84).Finally the function X in (89) simplifies for vanishingtime scale ratios to X ( as )2 (cid:0) f ⊥ (cid:1) = f ⊥ . (125)Inserting this expression into (88) and (87), the dynam-ical ζ -function (85) reads ζ ( as ) λ (cid:0) { γ } , f ⊥ (cid:1) = γ ⊥ + 12 γ k − f ⊥ (cid:18) f ⊥ (cid:19) . (126) The value of v ⊥ at the FP depends on how w ′⊥ goes tozero in the critical limit l → w ′′⊥ . Thereare three possible scenarios:i) w ′′⊥ goes to zero faster than w ′⊥ so that s →
0. Then v ⊥ is turning to the real value 1.ii) w ′⊥ and w ′′⊥ behave in the same way so that the ratio s = s is constant. v ⊥ is in this case a complex constant v ⊥ = 1 + is − is . (127)iii) w ′⊥ goes to zero faster than w ′′⊥ so that s → ∞ .Then v ⊥ is turning to the real value − ζ -functions donot stay finite for v ⊥ = −
1. Finite ζ -functions at theFP and therefore well defined critical exponents may beobtained only in the first two scenarios.The ζ -function for scenario ii) are already given in(123) - (126) when (127) is inserted. Case s = 0 : For v ⊥ = 1 ( s = 0) the ζ -functions (123) and (124)simplify to ζ ( as ⊥ (cid:0) { u } , q, f ⊥ (cid:1) = − f ⊥ ( − f ⊥ (cid:16)
272 ln 43 − (cid:17)) + ζ ( A ⊥ (cid:0) { u } , q (cid:1) , (128) ζ ( as k (cid:0) { u } , q (cid:1) = ζ ( A k (cid:0) { u } , q (cid:1) . (129)The model A functions (78) and (83) are now ζ ( A ⊥ (cid:0) { u } , q (cid:1) = u ⊥ (cid:18) − (cid:19) + u × (cid:18) L ( × ) ⊥ ( q ) − (cid:19) (130)and ζ ( A k (cid:0) { u } , q (cid:1) = u k (cid:18) ln 43 − (cid:19) + u × (cid:18) L ( × ) k ( q ) − (cid:19) , (131)where in L ( × ) i ( q ), introduced in (79) and (84), the rela-tions (122) with s = 0 have been inserted. B. Fixed points in the asymptotic subspace
Inserting the ζ -functions of the previous subsectioninto (103) - (105) one obtains the FP values in the asymp-totic subspace for s ⋆ finite, or s ⋆ = 0. The results arepresented in Tab.IV for the biconical ( B ) and the Heisen-berg ( H ) FP. It turns out that especially at the biconical4FP the values of the ratio q are extremely small, but def-initely not zero. Thus the asymptotic critical behaviorin two loop order changes considerably compared to oneloop (see section VII). Weak scaling as discussed in sec-tion IX is valid. The two order parameters scale with thesame dynamic exponent z OP from relation (119), whilethe secondary density scales with a different dynamic ex-ponent z m given in (120). The numerical values of thesetwo dynamic exponents are also given in Tab.IV in twoloop order. Note that the values of the dynamical expo-nents to the accuracy shown are independent wether theFP value of s is zero or not. f ⋆ ⊥ q ⋆ s ⋆ z OP z m C [9] - - 0 2 .
18 2 .
18F [10] 0 .
83 - 0 ∼ . ∼ . B .
232 1 . · − .
048 1 . H .
211 3 . · − .
003 1 . B .
232 2 . · − .
705 2 .
048 1 . H .
211 3 . · − .
698 2 .
003 1 . f ⊥ and the ratios q = w k /w ′⊥ and s = w ′′⊥ /w ′⊥ in the subspace w k = 0, w ⊥ = 0and finite v = q/ (1 + is ) for different cases of the biconical B and Heisenberg H FP in d = 3. For comparison results formodel C and model F FPs are shown at n = 1 and n = 2,correspondingly. The comparison with the dynamical critical exponentsin the cases when the OPs decouple statically and dy-namically into model C and model F shows the changesin the multicritical case where the exponents are changedbut each component reflects the decoupled values accord-ingly.
C. Effective exponents in the asymptotic subspace
The flow of the parameters q , s and f ⊥ can be foundby solving the equations l dqdl = q (cid:16) ζ ( as )Γ k − ℜ [ ζ ( as )Γ ⊥ ] + s ℑ [ ζ ( as )Γ ⊥ ] (cid:17) , (132) l dsdl = (1 + s ) ℑ [ ζ ( as )Γ ⊥ ] , (133) l df ⊥ dl = − f ⊥ (cid:16) ε + ζ ( as ) λ − ζ m + ℜ [ ζ ( as )Γ ⊥ ] − s ℑ [ ζ ( as )Γ ⊥ ] (cid:17) . (134)The ζ -functions in the above flow equations are the re-duced expressions (123), (124) and (126), which are func-tions of q, s, f ⊥ . We consider the case s = 0 since the FP s ⋆ = 0 is reached only starting with s = 0. From thesolution of Eqs.(133)-(134) the flow q ( l ), s ( l ), f ⊥ ( l ) isobtained, which is used to calculate asymptotic effective -20000 -15000 -10000 -5000 01.01.21.41.61.82.0 (eff) z || (eff) z (eff) z m ln l -2000 -1500 -1000 -500 01.41.61.82.02.22.4 (eff) z (eff) z || (eff) z m ln l FIG. 3: Effective dynamic exponents in the subspace w k = w ⊥ = 0 with q and s finite in d = 3. The static valuesare taken for the Heisenberg FP and for the biconical FP.The non-asymptotic region is extended by a factor 10 at thebiconical FP. For the static FP values see Tab. II, for thedynamic FP values Tab. IV. dynamic exponents z ( as ) ⊥ ( l ) = 2 + ℜ (cid:2) ζ ( as )Γ ⊥ (cid:0) q ( l ) , s ( l ) , f ⊥ ( l ) (cid:1)(cid:3) − s ( l ) ℑ (cid:2) ζ ( as )Γ ⊥ (cid:0) q ( l ) , s ( l ) , f ⊥ ( l ) (cid:1)(cid:3) , (135) z ( as ) k ( l ) = 2 + ζ ( as )Γ k (cid:0) q ( l ) , s ( l ) (cid:1) , (136) z ( as ) m ( l ) = 2 + ζ ( as ) λ (cid:0) q ( l ) , s ( l ) , f ⊥ ( l ) (cid:1) . (137)They can be calculated for different static fixed points,i.e. biconical or Heisenberg FP, as presented in Fig.3.The values of u ⋆ ⊥ , u ⋆ k , u ⋆ × , as well as γ ⋆ ⊥ , γ ⋆ k , used inthe current calculations can be found in Tab.II. At bothfixed points weak scaling, as discussed in section IX, isfulfilled. The difference to the one loop result is nowthat the dynamic exponents z ⊥ and z k of the OPs areequal in the asymptotic region, while z m stays different.Moreover the transient exponents in two loop order arevery small compared to one loop. There the effective ex-ponents reach their asymptotic values about l ∼ e − as5can be seen from Fig.1. In two loop order the asymptoticregion is of magnitudes smaller. From Fig.3 one can seethat at the Heisenberg FP the flow parameter has to beof the order l ∼ e − to obtain the asymptotic valuesof the dynamic exponents. At the biconical FP l has tobe even of the order l ∼ e − (note that there is a fac-tor 10 between the x -scales in Fig.3) to reach asymptoticvalues. However as will be seen in the next section thesubspace will not be reached by the flow in the completeparameter space for reasonable values of l . XI. GENERAL FLOW ANDPSEUDO-ASYMPTOTICS
Although in general the dynamic flow equations haveto be solved in the full parameter space, the results for theeffective exponents presented in the previous subsectionare obtained from the flow equations which already havebeen reduced to the subspace w ⊥ = w k = 0. The reasonto do this is that the flow and the ζ -functions in the fullparameter space shows some peculiar behavior.In the non-asymptotic region the flow is generated bythe system of equations for four parameters which are w ′⊥ , s , w k and f ⊥ obtained from Eqs. (64) - (66) and(121). The static parameters are taken at their FP val-ues given in Tab. II. Due to the presence of the staticasymmetric couplings γ i and the mode coupling f ⊥ animaginary part of w ⊥ is produced even if one starts witha zero initial value. Starting with a typical set of initialvalues, i.e. w k ( l ) = 0 . w ′⊥ ( l ) = 0 . s ( l ) = 0 . f ⊥ ( l ) = 0 . l = −
1, theeffective exponents in the complete parameter space havebeen calculated in d = 3. The result is presented in Fig.4for both static fixed points, where the solid lines are theresults of the two loop calculation and the dashed lineis a result of a complete (flow and effective exponent)one loop calculation. However the static FP values fromTab. II have been used also in the dynamic one loopflow. There it seems that in two loop order the sameresults as in the one loop calculation are obtained. z eff ⊥ and z effm are getting close together (solid lines) for flowparameters l < e − and seem to coincide even numer-ically with the corresponding results in one loop order(dashed line). This is the type of weak scaling in oneloop order, which also can be seen from Fig.1 and inqualitative contradiction to the discussion in the previoussections (see Fig.3). But the examination of the flow ofthe dynamic parameters reveals a fundamental differencebetween the one and two loop calculation. In one looporder the dynamic parameters w ′⊥ , s , w k and f ⊥ mergeto the FP values when the effective exponents turns overin their constant asymptotic values, which is presentedin Fig.2. This happens in the region about l < e − andthe dynamic parameters stay constant for all lower flowparameter values. In two loop order the situation is dif-ferent. Although the two loop results for the effectiveexponents look like one has reached the asymptotic re- -300 -200 -100 01.41.61.82.0 (eff) z m in one loop (eff) z || (eff) z (eff) z m ln l (eff) z m in one loop (eff) z (eff) z || (eff) z m FIG. 4: Effective dynamic exponents in the background us-ing the flow equations (64), (66) in two loop order in d = 3in the complete dynamical parameter space (full lines). Forcomparison the effective dynamic exponent z ( eff ) m in one looporder is shown (dashed line). gion (the exponents seem to be constant), the dynamicparameters in contrast are far from their asymptotic FPvalues. This is presented in Fig.5. The parameters w ′⊥ and s are still increasing and obviously have not reacheda FP value. At the first glance the flow of f ⊥ seems tohave reached a FP value (see lowest plot in Fig.5). Buta closer examination shows that this is not the case. f ⊥ is constantly increasing with a very small slope as can beseen from the inserted small figure, where both axes hasbeen enlarged. Actually the set of two loop β -functionsdoes not have a zero for finite w ⊥ and w k , and thereforeno FP exists in the parameter region of Fig.5. Thus theeffective exponents in two loop order in Fig.4 only showa pseudo-asymptotic behavior completely different fromreal asymptotic behavior (there z eff ⊥ and z eff k have tobe equal) discussed in section X C (see Fig. 3). Even ifone draws the x -axis in Fig.4 down to ln l = − -20000 -15000 -10000 -5000 00.81.01.21.4 f ln l -20000 -19000 -180001.09191.09201.0921 s w ’ FIG. 5: Flow of the parameters w ′⊥ , s and f ⊥ in the fullparameter space changes initial conditions of the parameters the qualita-tive result remains the same. The different flows mergewithin a region of ln l = −
150 to the same result.In order to get some insight how this pseudo-asymptotic behavior is possible, in Fig.6 the relativeslopes 1 α i dα i d ln l = β α i α i (138)for the parameters α i chosen to be w ′⊥ and f ⊥ have beencalculated. One can see that the relative changes in theseparameters drop down to very small values. As a firstconsequence one has to calculate down to extremely smallflow parameter values ln l < − where one can expectto leave the pseudo-asymptotic region. But although the β -functions cannot be zero in the considered parameterrange they may reach values which are so small that theycannot longer be separated numerically from zero. Thismeans that coming from the background it is impossibleto pass the pseudo-asymptotic region numerically intothe real asymptotic region. This is the reason why in theprevious section the flow has been started in an asymp-totic subspace.Due to the presence of logarithmic terms in thetimescale ratio v the FP value of v has to be finite. Asfollows from Tab. IV via Eq. (122) it turns out to be -20000 -15000 -10000 -5000 0-1x10 -5 -8x10 -6 -6x10 -6 -4x10 -6 -2x10 -6 f /f ln l -1x10 -3 -8x10 -4 -6x10 -4 -4x10 -4 -2x10 -4 w /w FIG. 6: Relative slope of the parameters w ′⊥ and f ⊥ in thefull parameter space very small leading to very large (negative) values of ln v .So one expects that the ln v -terms begin to dominate ζ Γ ⊥ in a certain region of ln l near the asymptotics. Makingthe ln v -terms explicit one may rewrite ζ Γ ⊥ , given in Eq.(70), as ζ Γ ⊥ = − γ k D ⊥ w ⊥ (cid:18) u × γ k D ⊥ w ⊥ (cid:19) ln v − u ×
36 ln v + remaining terms . (139)Inserting Eq. (122), the essential term is the real part V ′ of the prefactor of ln q . One obtains ζ Γ ⊥ = V ′ ln q + remaining terms . (140)The prefactor V ′ is V ′ = − (cid:20) u ×
36 + 12 γ k A (cid:18) u × γ k A (cid:19) − γ k B (cid:21) (141)with A = w ′⊥ [(1 + w ′⊥ + w ′⊥ s ) γ ⊥ − sF ](1 + w ′⊥ ) + ( w ′⊥ s ) , (142) B = w ′⊥ sγ ⊥ − (1 + w ′⊥ ) F ](1 + w ′⊥ ) + ( w ′⊥ s ) . (143)In Fig.7 the behavior of the ln q -contributions to ζ Γ ⊥ atthe biconical FP is presented. Although the ln q term al-ready reaches very large negative values ( q is very small),7 (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) -20000(cid:13) -15000(cid:13) -10000(cid:13) -5000(cid:13) 0(cid:13)-0.6(cid:13)-0.5(cid:13)-0.4(cid:13)-0.3(cid:13) V ln(q)(cid:13) (cid:13) (cid:13) (cid:13) ln l(cid:13) -5(cid:13)-4(cid:13)-3(cid:13)-2(cid:13)-1(cid:13) -log(-ln(q))(cid:13)log(V)(cid:13) (cid:13) (cid:13) (cid:13)
FIG. 7: Contribution of the V ′ ln q term in (140) (lower fig-ure). It is of the same magnitude as the other terms in the ζ -function. V ′ and ln q have the same behavior with oppositeexponents. This is demonstrated in the upper figure, wherethe decadic logarithm of V ′ (solid curve) and the negativedecadic logarithm of − ln q has been drawn (dashed curves). as expected, this is compensated by the prefactor V ′ which has very small values in the considered region. Inthe upper part of Fig.7 the decadic logarithm of V ′ and − ln q have been plotted. Both curves show a similar be-havior and a small difference. In the lower part V ′ ln q iscalculated from (141) - (143). As a consequence of theresults in the upper part of the figure the numerical val-ues are about − . ζ Γ ⊥ which have the same magnitude. Thusthe ζ -function is not in the asymptotic region as has beenalso indicated by the flow in Fig.5.Thus one expects in the experimentally accessible re-gion non-universal effective dynamical critical behavior.This is described in the crossover region to the back-ground by the flow equations together with a suitablematching condition related to the temperature distance,the wave vector modulus etc. The initial conditions haveto be found by comparison with experiment. XII. CONCLUSION
Our two loop calculation for the dynamics at the mul-ticritical point in anisotropic antiferromagnets in an ex-ternal magnetic field leads to a FP where the OPs charac- terizing the parallel and perpendicular ordering with re-spect to the external field scale in the same way (strongdynamic scaling). This holds independent wether theHeisenberg FP or the biconical FP in statics is the stableone. The non-asymptotic analysis of the dynamic flowequations show that due to cancelation effects the criti-cal behavior is described - in distances from the criticalpoint accessible to experiments - by the critical behav-ior qualitatively found in one loop order. That meansthe time scales of the two OP components become al-most constant in a so called pseudo-asymptotic regionand scale differently.So far we have not included the non-asymptotic flowof the static parameters which are expected to lead tominor deviations from the overall picture. Another itemwould be the study of the decoupled FP since in the non-asymptotic region the OPs remain statically and dynam-ically coupled and the behavior depend on the stabilityexponents how fast these effects decay. This in turn de-pends on the distance of the system in dimensional spaceand the space of the OP components from the stabilityborder line to other FPs than the decoupled FP (see Fig.1 in paper I).The numerical results presented in this series of papershave been calculated for dimension d = 3. One mightspeculate that the peculiar behavior found is specific tothe dimension of the physical space rather than to themulticritical character of the specific point. This aspectwas out of the scope of this series of papers. We note thatthe two critical lines meet at the multicritical point (bi-critical or tetracritical) tangential. This has been takeninto account for the nonsaymptotic behavior by choosinga path approaching the multicritical point without meet-ing one of the two critical lines [23]. The nonasymptoticbehavior in fact is more complicated since two criticallength scales are present in the system. This has to betaken into account when studying the crossover behaviorin approaching one of the critical lines [24].Only recently a bicritical point has been identified bycomputer simulation [25]. The corresponding FP hasbeen identified as the Heisenberg FP which correspondsto the type of phase diagram obtained. It seems to bedifficult to look for situations where a phase diagramcontaining a tetracritical point is present. Even morecomplicated would it be to identify the dynamical char-acteristic of this multicritical point, where - coming fromthe disordered phase - two lines belonging to differentdynamic universality classes meet. The dynamical uni-versality class of the case with a n = 1 OP (model C) hasbeen studied in [26, 27] with different results leading tocritical exponents larger than expected. The dynamicaluniversality class of the case with a n = 2 OP (model F)case has been studied by computer simulations in [28, 29].The methods of these simulations might be extended inorder to be used also in the case of the multicritical pointstudied in this paper. Acknowledgment
This work was supported by the Fonds zur F¨orderung der8wissenschaftlichen Forschung under Project No. P19583-N20.Yu.H. acknowledges partial support from the FP7 EUIRSES project N269139 ’Dynamics and Cooperative Phe-nomena in Complex Physical and Biological Media’.We thank one of the referees for the exceptionally crit-ical and precise remarks which have improved the pre-sentation.
Appendix A: Calculation of the dynamic vertexfunctions of the OPs
In perturbation expansion up to two loop order thefunctions ˚Ω ψ ˜ ψ + , ˚Γ ( d ) ψ ˜ ψ + and ˚Ω φ k ˜ φ k , which appear in (27)and (28), can be written as˚Ω ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) = 1 + ˚Ω (1 L ) ψ ˜ ψ + ( ξ ⊥ , k, ω )+˚Ω (2 L ) ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) , (A1)˚Γ ( d ) ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) = 2 h ˚Γ ⊥ + ˚ G (1 L ) ψ ˜ ψ + ( ξ ⊥ , k, ω )+ ˚ G (2 L ) ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) i , (A2)˚Ω φ k ˜ φ k ( ξ ⊥ , ξ k , k, ω ) = 1 + ˚Ω (1 L ) φ k ˜ φ k ( ξ k , k, ω )+˚Ω (2 L ) φ k ˜ φ k ( ξ ⊥ , ξ k , k, ω ) . (A3)The superscript ( iL ) indicates the loop order. Of courseall functions considered depend on all model parameters(couplings and kinetic coefficients), but only the indepen-dent lengths ξ ⊥ , ξ k , k and ω will be mentioned explicitlyin the following. The one loop contributions are˚Ω (1 L ) ψ ˜ ψ + ( ξ ⊥ , k, ω ) = (cid:0) ˚Γ ⊥ ˚ γ ⊥ − i˚ g (cid:1) ˚ γ ⊥ I ⊥ ( ξ ⊥ , k, ω ) , (A4)˚ G (1 L ) ψ ˜ ψ + ( ξ ⊥ , k, ω ) = (cid:0) ˚Γ ⊥ ˚ γ ⊥ − i˚ g (cid:1) i˚ gI ⊥ ( ξ k , k, ω ) , (A5)˚Ω (1 L ) φ k ˜ φ k ( ξ k , k, ω ) = ˚Γ k ˚ γ k I k ( ξ k , k, ω ) . (A6)The one loop integrals I ⊥ and I k in (A4)-(A6) read I ⊥ ( ξ ⊥ , k, ω ) = Z k ′ (cid:0) ξ − ⊥ +( k + k ′ ) (cid:1) ( − i ω + α ′⊥ ) , (A7) I k ( ξ k , k, ω ) = Z k ′ (cid:0) ξ − k +( k + k ′ ) (cid:1) ( − i ω + α ′k ) . (A8) The dynamic propagators α ′⊥ and α ′k are defined as α ′⊥ ≡ ˚Γ ⊥ (cid:0) ξ − ⊥ +( k + k ′ ) (cid:1) + ˚ λk ′ , (A9) α ′k ≡ ˚Γ k (cid:0) ξ − k +( k + k ′ ) (cid:1) + ˚ λk ′ . (A10)The two loop contributions to the dynamic vertex func-tion of the orthogonal components (A1) and (A2) havethe structure˚Ω (2 L ) ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) = 29˚Γ ⊥ ˚ u ⊥ ˚ W ( A ⊥ ) ψ ˜ ψ + ( ξ ⊥ , k, ω )+ 118˚Γ ⊥ ˚ u × ˚ W ( A × ) ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) − (cid:0) ⊥ ˚ γ ⊥ − i˚ g (cid:1) ˚ u ⊥ ˚ F ( T ⊥ ) ψ ˜ ψ + ( ξ ⊥ , k, ω ) − (cid:0) ⊥ ˚ γ ⊥ − i˚ g (cid:1) ˚ u × ˚ F ( T × ) ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω )+ (cid:0) ˚Γ ⊥ ˚ γ ⊥ − i˚ g (cid:1) ˚ γ ⊥ ˚ F ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) (A11)and˚ G (2 L ) ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) = − ⊥ ˚ u ⊥ i˚ g ˚ F ( T ⊥ ) ψ ˜ ψ + ( ξ ⊥ , k, ω ) − ⊥ ˚ u × i˚ g ˚ F ( T × ) ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω )+ (cid:0) ˚Γ ⊥ ˚ γ ⊥ − i˚ g (cid:1) i˚ g ˚ F ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) . (A12)Note that both two loop functions differ only in termscontaining the static fourth order couplings ˚ u ⊥ and ˚ u × .The remaining contributions are the same in both func-tions apart from a factor ˚ γ ⊥ and i ˚ g respectively. Thefunction ˚ F ψ ˜ ψ + is defined as˚ F ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) ≡ (cid:0) ˚Γ ⊥ ˚ γ ⊥ − i˚ g (cid:1) ˚ F ( T ⊥ ) ψ ˜ ψ + ( ξ ⊥ , k, ω )+ ˚ F ( T ⊥ ) ψ ˜ ψ + ( ξ ⊥ , k, ω ) − ˚ γ ⊥ ˚ F ( T ⊥ ) ψ ˜ ψ + ( ξ ⊥ , k, ω )+ 12 (cid:16) ˚ F ( T × ) ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) − ˚ γ k ˚ F ( T × ) ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) (cid:17) + ˚ F ( T ⊥ ) ψ ˜ ψ + ( ξ ⊥ , k, ω ) − ˚ γ ⊥ ˚ F ( T ⊥ ) ψ ˜ ψ + ( ξ ⊥ , k, ω ) . (A13)The first two loop contributions in (A11) come from thebicritical model A. The integrals ˚ W ( A i ) ψ ˜ ψ + are defined by9˚ W ( A ⊥ ) ψ ˜ ψ + ( ξ ⊥ , k, ω ) = Z k ′ Z k ′′ (cid:0) ξ − ⊥ +( k + k ′ ) (cid:1) ( ξ − ⊥ + k ′′ ) (cid:0) ξ − ⊥ +( k ′ + k ′′ ) (cid:1) ( − i ω + A ⊥⊥ + ⊥ ) , (A14)˚ W ( A × ) ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) = Z k ′ Z k ′′ (cid:0) ξ − ⊥ +( k + k ′ ) (cid:1) ( ξ − k + k ′′ ) (cid:0) ξ − k +( k ′ + k ′′ ) (cid:1) ( − i ω + A ⊥kk ) (A15)with A ⊥⊥ + ⊥ ≡ ˚Γ ⊥ (cid:0) ξ − ⊥ +( k + k ′ ) (cid:1) + ˚Γ + ⊥ ( ξ − ⊥ + k ′′ )+˚Γ ⊥ (cid:0) ξ − ⊥ +( k ′ + k ′′ ) (cid:1) (A16)and A ⊥kk ≡ ˚Γ ⊥ (cid:0) ξ − ⊥ +( k + k ′ ) (cid:1) + ˚Γ k ( ξ − k + k ′′ )+˚Γ k (cid:0) ξ − k +( k ′ + k ′′ ) (cid:1) . (A17) The further two loop contributions in (A11)-(A13) aremarked with superscripts ( T i ), which indicate the differ-ent graph topologies. The explicit expressions are˚ F ( T ⊥ ) ψ ˜ ψ + ( ξ ⊥ , k, ω ) = Z k ′ Z k ′′ (cid:0) ξ − ⊥ +( k + k ′ ) (cid:1) ( − i ω + α ′⊥ )( − i ω + A ⊥⊥ + ⊥ ) ˚Γ ⊥ ˚ γ ⊥ − i˚ gξ − ⊥ + k ′′ + ˚Γ + ⊥ ˚ γ ⊥ +i˚ gξ − ⊥ +( k ′ + k ′′ ) ! , (A18)˚ F ( T × ) ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) = Z k ′ Z k ′′ k ˚ γ k (cid:0) ξ − ⊥ +( k + k ′ ) (cid:1) ( ξ − k + k ′′ )( − i ω + α ′k )( − i ω + A ⊥kk ) , (A19)˚ F ( T ⊥ ) ψ ˜ ψ + ( ξ ⊥ , k, ω ) = Z k ′ Z k ′′ (cid:0) ξ − ⊥ +( k + k ′ + k ′′ ) (cid:1) ( − i ω + α ′⊥ ) ( − i ω + β ⊥ ) , (A20)˚ F ( T ⊥ ) ψ ˜ ψ + ( ξ ⊥ , k, ω ) = Z k ′ Z k ′′ ˚ γ ⊥ ˚ λk ′ − i˚ g [( k ′ + k ′′ ) − k ′′ ] (cid:0) ξ − ⊥ +( k + k ′ ) (cid:1) ( − i ω + α ′⊥ ) ( − i ω + A ⊥⊥ + ⊥ ) ˚Γ ⊥ ˚ γ ⊥ − i˚ gξ − ⊥ + k ′′ + ˚Γ + ⊥ ˚ γ ⊥ +i˚ gξ − ⊥ +( k ′ + k ′′ ) ! , (A21)˚ F ( T × ) ψ ˜ ψ + ( ξ ⊥ , ξ k , k, ω ) = Z k ′ Z k ′′ k ˚ γ k ˚ λk ′ (cid:0) ξ − ⊥ +( k + k ′ ) (cid:1) ( ξ − k + k ′′ )( − i ω + α ′⊥ ) ( − i ω + A ⊥kk ) , (A22)˚ F ( T ⊥ ) ψ ˜ ψ + ( ξ ⊥ , k, ω ) = Z k ′ Z k ′′ ˚ γ ⊥ ˚ λk ′′ +i˚ g [( k + k ′ + k ′′ ) − ( k + k ′ ) )] (cid:0) ξ − ⊥ +( k + k ′ ) (cid:1) ( − i ω + α ′⊥ )( − i ω + α ′′⊥ )( − i ω + S ⊥⊥ + ⊥ ) ˚Γ ⊥ ˚ γ ⊥ − i˚ gξ − ⊥ +( k + k ′ + k ′′ ) + ˚Γ + ⊥ ˚ γ ⊥ +i˚ gξ − ⊥ +( k + k ′′ ) ! + Z k ′ Z k ′′ ˚Γ ⊥ ˚ γ ⊥ − i˚ g (cid:0) ξ − ⊥ +( k + k ′ ) (cid:1) ( − i ω + α ′′⊥ )( − i ω + β ⊥ ) ˚Γ ⊥ ˚ γ ⊥ − i˚ g − i ω + α ′⊥ + ˚ γ ⊥ ξ − ⊥ +( k + k ′ + k ′′ ) +˚ γ ⊥ ˚ λk ′′ +i˚ g [( k + k ′ + k ′′ ) − ( k + k ′ ) )] (cid:0) ξ − ⊥ +( k + k ′ + k ′′ ) (cid:1) ( − i ω + α ′⊥ ) ! , (A23)with the dynamic propagators β ⊥ ≡ ˚Γ ⊥ (cid:0) ξ − ⊥ +( k + k ′ + k ′′ ) (cid:1) + ˚ λ (cid:0) k ′ + k ′′ (cid:1) , (A24) S ⊥⊥ + ⊥ ≡ ˚Γ ⊥ (cid:0) ξ − ⊥ +( k + k ′ ) (cid:1) + ˚Γ + ⊥ (cid:0) ξ − ⊥ +( k + k ′ + k ′′ ) (cid:1) + ˚Γ ⊥ (cid:0) ξ − ⊥ +( k + k ′′ ) (cid:1) (A25)0which are both invariant under an interchange of k ′ and k ′′ .The two loop contributions to the dynamic vertex function of the parallel component (A3) has the structure˚Ω (2 L ) φ k ˜ φ k ( ξ ⊥ , ξ k , k, ω ) = 16˚Γ k ˚ u k ˚ W ( A k ) φ k ˜ φ k ( ξ k , k, ω ) + 19˚Γ k ˚ u × ˚ W ( A × ) φ k ˜ φ k ( ξ ⊥ , ξ k , k, ω ) − ˚Γ k ˚ u k ˚ γ k ˚ F ( T k ) φ k ˜ φ k ( ξ k , k, ω ) − k ˚ u × ˚ γ k ˚ F ( T × ) φ k ˜ φ k ( ξ ⊥ , ξ k , k, ω ) + ˚Γ k ˚ γ k ˚ F φ k ˜ φ k ( ξ ⊥ , ξ k , k, ω ) . (A26)The function ˚ F φ k ˜ φ k is defined as˚ F φ k ˜ φ k ( ξ ⊥ , ξ k , k, ω ) ≡ ˚Γ k ˚ γ k ˚ F ( T k ) φ k ˜ φ k ( ξ k , k, ω ) + 12 (cid:16) ˚ F ( T k ) φ k ˜ φ k ( ξ k , k, ω ) − ˚ γ k ˚ F ( T k ) φ k ˜ φ k ( ξ k , k, ω ) (cid:17) + ˚ F ( T × ) φ k ˜ φ k ( ξ ⊥ , ξ k , k, ω ) − ˚ γ ⊥ ˚ F ( T × ) φ k ˜ φ k ( ξ ⊥ , ξ k , k, ω ) + ˚ F ( T k ) φ k ˜ φ k ( ξ k , k, ω ) − ˚ γ k ˚ F ( T k ) φ k ˜ φ k ( ξ k , k, ω ) . (A27)The integrals ˚ W ( A i ) φ k ˜ φ k in the two loop contributions from the bicritical model A are˚ W ( A k ) φ k ˜ φ k ( ξ k , k, ω ) = Z k ′ Z k ′′ (cid:0) ξ − k +( k + k ′ ) (cid:1) ( ξ − k + k ′′ ) (cid:0) ξ − k +( k ′ + k ′′ ) (cid:1) ( − i ω + A kkk ) , (A28)˚ W ( A × ) φ k ˜ φ k ( ξ ⊥ , ξ k , k, ω ) = Z k ′ Z k ′′ (cid:0) ξ − k +( k + k ′ ) (cid:1) ( ξ − ⊥ + k ′′ ) (cid:0) ξ − ⊥ +( k ′ + k ′′ ) (cid:1) ( − i ω + A k⊥ + ⊥ ) (A29)with the propagators A kkk ≡ ˚Γ k (cid:0) ξ − k +( k + k ′ ) (cid:1) + ˚Γ k ( ξ − k + k ′′ ) + ˚Γ k (cid:0) ξ − k +( k ′ + k ′′ ) (cid:1) (A30)and A k⊥ + ⊥ ≡ ˚Γ k (cid:0) ξ − k +( k + k ′ ) (cid:1) + ˚Γ + ⊥ ( ξ − ⊥ + k ′′ ) + ˚Γ ⊥ (cid:0) ξ − ⊥ +( k ′ + k ′′ ) (cid:1) . (A31)The remaining two loop contributions in (A26) and (A27) are˚ F ( T k ) φ k ˜ φ k ( ξ k , k, ω ) = Z k ′ Z k ′′ k ˚ γ k (cid:0) ξ − k +( k + k ′ ) (cid:1) ( ξ − k + k ′′ )( − i ω + α ′k )( − i ω + A kkk ) , (A32)˚ F ( T × ) φ k ˜ φ k ( ξ ⊥ , ξ k , k, ω ) = Z k ′ Z k ′′ (cid:0) ξ − k +( k + k ′ ) (cid:1) ( − i ω + α ′k )( − i ω + A k⊥ + ⊥ ) ˚Γ ⊥ ˚ γ ⊥ − i˚ gξ − ⊥ + k ′′ + ˚Γ + ⊥ ˚ γ ⊥ +i˚ gξ − ⊥ +( k ′ + k ′′ ) ! , (A33)˚ F ( T k ) φ k ˜ φ k ( ξ | , k, ω ) = Z k ′ Z k ′′ (cid:0) ξ − k +( k + k ′ + k ′′ ) (cid:1) ( − i ω + α ′k ) ( − i ω + β k ) , (A34)˚ F ( T k ) φ k ˜ φ k ( ξ | , k, ω ) = Z k ′ Z k ′′ k ˚ γ k ˚ λk ′ (cid:0) ξ − k +( k + k ′ ) (cid:1) ( ξ − k + k ′′ )( − i ω + α ′k ) ( − i ω + A kkk ) , (A35)˚ F ( T × ) φ k ˜ φ k ( ξ ⊥ , ξ k , k, ω ) = Z k ′ Z k ′′ ˚ γ ⊥ ˚ λk ′ − i˚ g [( k ′ + k ′′ ) − k ′′ ] (cid:0) ξ − k +( k + k ′ ) (cid:1) ( − i ω + α ′k ) ( − i ω + A k⊥ + ⊥ ) ˚Γ ⊥ ˚ γ ⊥ − i˚ gξ − ⊥ + k ′′ + ˚Γ + ⊥ ˚ γ ⊥ +i˚ gξ − ⊥ +( k ′ + k ′′ ) ! , (A36)˚ F ( T k ) φ k ˜ φ k ( ξ k , k, ω ) = Z k ′ Z k ′′ ˚Γ k ˚ γ k ˚ λk ′′ (cid:0) ξ − k +( k + k ′ ) (cid:1) ( − i ω + α ′k )( − i ω + α ′′k )( − i ω + S kkk ) ξ − k +( k + k ′ + k ′′ ) + 1 ξ − ⊥ +( k + k ′′ ) ! + Z k ′ Z k ′′ ˚Γ k ˚ γ k (cid:0) ξ − k +( k + k ′ ) (cid:1) ( − i ω + α ′′k )( − i ω + β k ) ˚Γ k − i ω + α ′k + 1 ξ − k +( k + k ′ + k ′′ ) λk ′′ − i ω + α ′k ! ! . (A37)1The additional dynamic propagators are S kkk ≡ ˚Γ k (cid:0) ξ − k +( k + k ′ ) (cid:1) + ˚Γ k (cid:0) ξ − k +( k + k ′ + k ′′ ) (cid:1) + ˚Γ k (cid:0) ξ − k +( k + k ′′ ) (cid:1) and β k ≡ ˚Γ k (cid:0) ξ − k +( k + k ′ + k ′′ ) (cid:1) + ˚ λ (cid:0) k ′ + k ′′ (cid:1) . (A38)The integrals contained in (A14) - (A23) and (A28) - (A37) are of the same type as already has been presented in[30] (see Eqs.(A19) - (A26) in the appendix therein). The ε -poles of these integrals can be found in Eqs.(C2) - (C9)of the same reference. Appendix B: Dynamic Z-factors in two loop order
Within the minimal subtraction scheme of the renormalization group calculation one has to collect in two loop orderthe pole terms of order 1 /ε and 1 /ε in the functions ˚Ω ψ ˜ ψ + and ˚Γ ( d ) ψ ˜ ψ + in (27). The resulting dynamic renormalizationfactors are Z / ψ ∗ = 1 − ε γ ⊥ D ⊥ w ⊥ − ε (cid:20) u ⊥ (cid:18) L + x L − (cid:19) + u × (cid:18) L ⊥ − (cid:19)(cid:21) + 14 ε (cid:20) u ⊥ ( w ⊥ γ ⊥ + D ⊥ ) w ⊥ (1 + w ⊥ ) A ⊥ + γ ⊥ D ⊥ w ⊥ (1 + w ⊥ ) B ⊥ + γ k w ⊥ ) (cid:18) u × w ⊥ γ ⊥ + D ⊥ ) + w ⊥ γ ⊥ γ k D ⊥ w ⊥ (cid:19) X ⊥ (cid:21) + 12 ε " − w ⊥ γ ⊥ + D ⊥ w ⊥ (cid:18) u ⊥ γ ⊥ + u × γ k (cid:19) + γ ⊥ D ⊥ (1 + w ⊥ ) D ⊥ w ⊥ − w ⊥ (cid:16) γ ⊥ + γ k (cid:17) − f ⊥ ! (B1) Z ( d )Γ ⊥ = 1 − ε i F D ⊥ w ⊥ (1 + w ⊥ ) + 14 ε (cid:20) u ⊥ i Fw ⊥ (1 + w ⊥ ) A ⊥ + i F D ⊥ w ⊥ (1 + w ⊥ ) B ⊥ + γ k i F w ⊥ ) (cid:18) u × γ k D ⊥ w ⊥ (cid:19) X ⊥ (cid:21) + 12 ε " − i F w ⊥ (cid:18) u ⊥ γ ⊥ + u × γ k (cid:19) + i F D ⊥ w ⊥ (1 + w ⊥ ) D ⊥ w ⊥ − w ⊥ (cid:16) γ ⊥ + γ k (cid:17) − f ⊥ ! . (B2)The coupling D ⊥ and the functions A ⊥ , and B ⊥ and X ⊥ are defined in (71)-(74). The pole terms of the function˚Γ ( d ) φ k ˜ φ + k are collected in the renormalization factor Z Γ k = 1 + 1 ε w k γ k w k + 1 ε " u k (cid:18) ln 43 − (cid:19) + u × (cid:18) vT A − (cid:19) − ε " w k γ k w k u k (cid:18) − (cid:19) + w k γ k w k ! (cid:18) − (cid:19) − w k w k − w k w k ln (1 + w k ) w k ! + (cid:18) u × + w k γ k w k γ ⊥ (cid:19) ℜ (cid:20) T w ′⊥ (cid:21) − γ k F w ′⊥ (1 + w k ) ℑ (cid:20) T w ′⊥ (cid:21) + 12 ε w k γ k w k " u k γ k + 23 u × γ ⊥ + γ k w k (cid:18) w k γ k (cid:18) − w k w k (cid:19) + w k γ ⊥ + f ⊥ (cid:19) + 2 w k γ k w k . (B3)The functions T , T and T A have been introduced in (81)-(84).The renormalization factor Z λ is identical to the one of model F [21] with all parameters of the perpendicularsubsystem. With Q defined in (88), one gets Z ( d ) λ = 1 − ε f ⊥ ( Q − ε w ′⊥ (cid:20) D ⊥ w ⊥ + D +2 ⊥ w + ⊥ (cid:21) ) (B4)2 [1] J. M. Kosterlitz, D. Nelson, and M. E. Fisher, Phys. Rev.B , 412 (1976).[2] R. Folk, Yu. Holovatch, and G. Moser, Phys. Rev. E ,041124 (2008); henceforth called paper I[3] R. Folk and G. Moser, J. Phys. A: Math. Gen. , R207(2006)[4] B. D. Josephson, Phys. Letters , 608 (1966)[5] B. I. Halperin and P. C. Hohenberg, Phys. Rev. , 952(1969)[6] V. Dohm and H.-K. Janssen, Phys. Rev. Lett. , 946(1977); J. Appl. Phys. , 1347 (1978)[7] V. Dohm in Multicritical Phenomena , ed. Plenum, NewYork and London 1983 page 81[8] V. Dohm,
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