Six-loop ε expansion study of three-dimensional n -vector model with cubic anisotropy
L.Ts. Adzhemyan, E.V. Ivanova, M.V. Kompaniets, A. Kudlis, A.I. Sokolov
SSix-loop ε expansion study of three-dimensional n -vector modelwith cubic anisotropy L. Ts. Adzhemyan a , E. V. Ivanova a , M. V. Kompaniets a , A. Kudlis a, ∗ , A. I. Sokolov a a St. Petersburg State University, 7 / Abstract
The six-loop expansions of the renormalization-group functions of ϕ n -vector model with cubicanisotropy are calculated within the minimal subtraction (MS) scheme in 4 − ε dimensions. The ε expansions for the cubic fixed point coordinates, critical exponents corresponding to the cubicuniversality class and marginal order parameter dimensionality n c separating di ff erent regimesof critical behavior are presented. Since the ε expansions are divergent numerical estimates ofthe quantities of interest are obtained employing proper resummation techniques. The numbersfound are compared with their counterparts obtained earlier within various field-theoretical ap-proaches and by lattice calculations. In particular, our analysis of n c strengthens the existingarguments in favor of stability of the cubic fixed point in the physical case n = Keywords: renormalization group, cubic anisotropy, multi-loop calculations, ε expansion,critical exponents.
1. Introduction
As is well known, the systems undergoing continuous phase transitions demonstrate the uni-versal critical behavior. This leads to the concept of classes of universality introduced decadesago. They are determined by the general properties of the system such as spatial dimensional-ity, symmetry, and the number of order parameter components, thereby its microscopic naturedoes not play any role in the vicinity of phase transition temperature. There is a set of universalparameters such as critical exponents, critical amplitude ratios, etc. that characterize the criticalbehavior of the systems belonging to the same universality class.The analysis of critical phenomena in a broad variety of materials can be performed on thebase of three-dimensional O ( n )-symmetric ϕ field model. In case of one-component – scalar –order parameter ( n =
1) one deals with the Ising model describing phase transitions in uniaxialferromagnets, simple fluids, binary mixtures, and many other systems. There is also a greatnumbers of substances with the vector ordering, e.g. easy-plane ferromagnets, superconductorsand superfluid helium-4 ( n = n = n = n =
18) and the neutron star ∗ Corresponding author
Email address: [email protected] (A. Kudlis)
Preprint submitted to Nuclear Physics B February 1, 2019 a r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n atter ( n = O (3)-symmetric theory ne-glecting crystal anisotropy has been used. The detailed analysis performed later within therenormalization-group (RG) approach has shown, however, that for proper description of the crit-ical behavior of real cubic crystals one should take into account the presence of the anisotropy,i. e. add to the Landau-Wilson Hamiltonian an extra term invariant with respect to the cubicgroup of transformations. It looks as g (cid:80) n α = ϕ α , where ϕ α is n -vector ordering field and g –anisotropic coupling constant. This new quartic coupling, in particular, accounts for the fact thatin real ferromagnets ( n =
3) the vector of magnetization ”feels” the crystal anisotropy and canlie only along the axes or spatial diagonals of cubic unit cell in the ordered phase.This model with two coupling constants – g (isotropic) and g – was carefully examined since1972 [1] by many researches. As was found, its RG equations describing evolution of quarticcouplings under T → T c possess four fixed points: Gaussian (0 , , g ∗ I ), Heisenberg( g ∗ H ,
0) and cubic( g ∗ , g ∗ ). One of the most important issues involved in the study is the determi-nation of the stability of these fixed points or, in other words, what critical regime takes placein real ferromagnets. Analyzing the RG flows it was shown that the first two points are alwaysunstable for arbitrary values of order parameter dimensionality n whereas the last two of themcorresponding to the Heisenberg (isotropic) and cubic (anisotropic) modes of critical behaviorcompete with each other. Which regime turns out to be stable depends on n . For n < n c , where n c is some marginal value of spin dimensionality, the isotropic (Heisenberg) critical regime is sta-ble while for n > n c the cubic critical behavior is realized. If initial (”bare”) values of couplingconstants lie outside the regions of fixed points attraction critical fluctuations strongly modifythe behavior of the system converting the second-order phase transition into the first-order one.Figure 1 illustrates the situation. g g G HI C g g G HI C
Figure 1: RG flows of renormalized coupling constants. The left picture corresponds to n < n c , the right one – to n > n c .Symbols in boxes mark Gaussian, Ising, Heisenberg and cubic fixed points. Thus, in the case n > n c the cubic quartic term is certainly relevant and has to be takeninto account. This results in the emergence of new class of universality corresponding to theanisotropic – cubic – critical behavior. So, the value of n c becomes of prime physical importancesince it determines the true regime of the critical behavior in real cubic ferromagnets and of someother systems of interest.Detailed study of the n -vector cubic model including evaluation of critical exponents and n c n c obtained in the lower-order approximations withinthe ε expansion approach [2, 4, 5, 6] and in the frame of 3D RG machinery [12, 15, 16] turnedout to be in favor of the conclusion that n c >
3, while lattice calculations implied n c is practicallyequal to 3 [13]. This made the study of the cubic class of universality less interesting fromthe physical point of view. Later, however, the higher-order analysis including resummationof RG perturbative series was performed and shown that numerical value of n c falls below 3[17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 32]. To date, the most advanced estimates of n c obtainedwithin the ε expansion, 3D RG and pseudo- ε expansion approaches are n c = . , .
87 [21, 26], n c = . , .
91 [24, 26] and n c = .
86 [28, 32], respectively.These numbers di ff er from each other appreciably what may be considered as a stimulus to findthe value of n c with higher accuracy. On the other hand, recently the ε expansions of record length– six-loop – for O ( n )-symmetric ϕ field theory [33, 34, 35] were calculated. This paves the wayto analysis of the critical behavior of the cubic model within the highest-order ε approximationincluding getting precise numerical estimates for critical exponents and n c . Such an analysis isthe aim of this work.The paper is organized as follows. In Sec. 2 we write down the fluctuation Hamiltonian(Landau-Wilson action) of n -vector cubic model and describe the renormalization procedure. InSec. 3 the six-loop ε expansions for β functions, critical exponents and n c are calculated. Thesix-loop ε series for cubic fixed point coordinates and critical exponents are also presented herefor the physically interesting case n =
3. In Sec. 4 the ε expansions for ”observables” – n c andcritical exponents – are resummed and corresponding numerical estimates are found. In Sec. 5the numbers obtained are discussed and compared with their counterparts given by alternativefield-theoretical approaches and extracted from the lower-order approximations. Sec. 6 containsthe summary of main results and concluding remarks.
2. Model and renormalization
In this work we address the field-theoretical RG approach in spatial dimensionality D = − ε . The critical behavior of the cubic model is governed by the well-known Landau-Wilson actionwith two coupling constants S = (cid:90) d D x (cid:40) (cid:104) ( ∂ϕ α ) + m ϕ α (cid:105) + (cid:104) g T (1) αβγδ + g T (2) αβγδ (cid:105) ϕ α ϕ β ϕ γ ϕ δ (cid:41) , (1)where ϕ α is n -component bare field, g and g being the bare coupling constants. The tensorfactors T (1) and T (2) entering the O ( n )-invariant and cubic terms respectively are as follows T (1) αβγδ =
13 ( δ αβ δ γδ + δ αγ δ βδ + δ αδ δ γβ ) , T (2) αβγδ = δ αβγδ , δ α ...α n = , α = α = . . . = α n , otherwise . (2) Original six-loop calculations [33, 34, 35] were performed in space dimension D = − ε which is more commonfor high energy physics.
3n particular, T (1) αβγδ T (1) αβγδ = n ( n + , T (1) αβγδ T (2) αβγδ = n , T (2) αβγδ T (2) αβγδ = n . (3)The action (1) is seen to be physical (positively defined) if g > − g for g > g > − ng for negative g .The model is known to be multiplicatively renormalizable. The bare parameters g , g , m , ϕ can be expressed via the renormalized ones g , g , m , ϕ by means of the following relations m = m Z m , g = g µ ε Z g , g = g µ ε Z g , ϕ = ϕ Z ϕ , Z = Z ϕ , Z = Z m Z ϕ , Z = Z g Z ϕ , Z = Z g Z ϕ . (4)Using these relations we arrive to the renormalized action S R = (cid:90) d D x (cid:40) (cid:104) Z ( ∂ϕ α ) + Z m ϕ α (cid:105) + (cid:104) Z g µ ε T (1) αβγδ + Z g µ ε T (2) αβγδ (cid:105) ϕ α ϕ β ϕ γ ϕ δ (cid:41) , (5)where µ is an arbitrary mass scale introduced to make couplings g and g dimensionless. Renor-malization constants are defined in a way enabling to absorb divergences from all Green func-tions, so that renormalized Green functions are free of divergences. Due to multiplicative renor-malizability of the model it is enough to remove divergences in two- and four-point one-particleirreducible Green functions: Γ (2) αβ = Γ (2) δ αβ , Γ (4) αβγδ = Γ (4)1 T (1) αβγδ + Γ (4)2 T (2) αβγδ , (6) Γ (4)1 = T (1) αβγδ − T (2) αβγδ ) n ( n − Γ (4) αβγδ , Γ (4)2 = ( n + T (2) αβγδ − T (1) αβγδ n ( n − Γ (4) αβγδ . (7)In this paper we employ the Minimal Subtraction (MS) scheme where renormalization con-stants acquire only pole contributions in ε and depend only on ε and coupling constants: Z i ( g , g , ε ) = + ∞ (cid:88) k = Z ( k ) i ( g , g ) ε − k . (8)Renormalization constants can be found from the requirement of the finiteness of renormalizedtwo- and four-point one-particle irreducible Green functions. Another way to calculate renor-malization constants is use of Bogolubov-Parasiuk R (cid:48) operation: Z i = + KR (cid:48) ¯ Γ i , (9)where R (cid:48) – incomplete Bogoludov-Parasiuk R -operation, K – projector of the singular part of thediagram and ¯ Γ i – normalized Green functions of the basic theory (see e.g. [36, 37]) defined bythe following relations:¯ Γ = ∂∂ m Γ (2) | p = , ¯ Γ = (cid:32) ∂∂ p (cid:33) Γ (2) | p = ¯ Γ = g µ ε Γ (4)1 | p = , ¯ Γ = g µ ε Γ (4)2 | p = . (10)One of the most important advantages of the Bogoludov-Parasiuk approach is that countert-erms of the diagrams computed for O(1)-symmetric (scalar) model can be easily generalized toany theory with non-trivial symmetry due to the factorization of the tensor structures (see e.g.[38, 39, 40]). To calculate tensor factors for particular diagrams of the cubic model (1) one shouldapply projectors (7) to it. Such an operation can be automated with FORM [41] and GraphState[42] while counterterm values can be taken from data obtained in the course of recent 6-loopcalculations for O ( n )-symmetric model [35]. 4 . Six-loop expansions for RG functions, cubic fixed point coordinates, critical exponentsand n c The RG functions, i. e. β functions and anomalous dimensions γ ϕ , γ m are related to renor-malization constants Z i by the following relations: β i ( g , g , ε ) = µ ∂ g i ∂µ | g , g = − g i ε − g ∂ Z (1) g i ∂ g − g ∂ Z (1) g i ∂ g , i = , ,γ j ( g , g ) = µ ∂ log Z j ∂µ | g , g = − g ∂ Z (1) j ∂ g − g ∂ Z (1) j ∂ g , j = ϕ, m , (11)where Z (1) i – coe ffi cients at first pole in ε from (8).We calculated the RG functions as series in renormalized coupling constants up to six-looporder. They are found analytically and presented in Tables 1, 2, 3 and 4 of Supplementarymaterials (see Appendix A) in the form β i = g i − ε + (cid:88) l = l (cid:88) k = C k , ( l − k ) β i g k g l − k , i = , , (12) γ j = (cid:88) l = l (cid:88) k = C k , ( l − k ) γ j g k g l − k , j = ϕ, m . (13)The critical regimes of the system are controlled by the fixed points ( g ∗ , g ∗ ) of RG equations thatare zeroes of β functions: β ( g ∗ , g ∗ , ε ) = , β ( g ∗ , g ∗ , ε ) = . (14)As was already mentioned, for the model under consideration there are four fixed points: Gaus-sian (0 , , g ∗ I ), Heisenberg ( g ∗ H ,
0) and cubic ( g ∗ , g ∗ ). Since six-loop ε expansions anal-ysis of Ising and Heisenberg models have been performed earlier [33, 34, 35] we concentrate onthe cubic critical behavior. To calculate ε expansions for critical exponents we have to find thosefor coordinates of the cubic fixed point. Solving (14) by means of iterations in ε for the cubicfixed point we find: g ∗ = ε n + ε (cid:18) − n + n − n (cid:19) + (cid:88) k = C ( k ) g ε k + O (cid:16) ε (cid:17) , g ∗ = ε ( n − n + ε (cid:18) n − n + n + (cid:19) + (cid:88) k = C ( k ) g ε k + O (cid:16) ε (cid:17) , (15)where higher-order coe ffi cients C ( k ) g , C ( k ) g are presented in Tables 5 and 6 of Supplementary ma-terials (see Appendix A).To fully characterize the cubic class of universality, we need to calculate the critical exponents α , β , γ , η , ν and δ . They can be expressed via γ ∗ m ≡ γ m ( g ∗ , g ∗ ) and γ ∗ ϕ ≡ γ ϕ ( g ∗ , g ∗ ) in the5ollowing way: α = − D + γ ∗ m , β = D / − + γ ∗ ϕ + γ ∗ m , γ = − γ ∗ ϕ + γ ∗ m , η = γ ∗ ϕ ,ν = + γ ∗ m , δ = D + − γ ∗ ϕ D − + γ ∗ ϕ . (16)The critical exponents are related to each other by well-known scaling relations and only two ofthem may be referred to as independent.It is instructive to present ε expansions of cubic fixed point coordinates for physically impor-tant case n =
3. They are as follows: g ∗ = ε + ε + ε (cid:34) − ζ (3)729 − (cid:35) + ε (cid:34) ζ (3)177147 + ζ (4)729 + ζ (5)2187 − (cid:35) ++ ε (cid:34) + ζ (3)114791256 + ζ (4)708588 − ζ (6)39366 − ζ (7)6561 −− ζ (3) − (cid:35) + ε (cid:34) ζ (3)27894275208 + ζ (4)1721868840 −− ζ (5)645700815 − ζ (6)9565938 + ζ (7)7971615 + ζ (8)13286025 ++ ζ (9)4782969 − ζ (3) + ζ (3) − ζ (4) ζ (3)177147 ++ ζ (5) ζ (3)177147 + ζ (3 , − (cid:35) + O (cid:16) ε (cid:17) , (17) g ∗ = − ε + ε + ε (cid:34) − ζ (3)2187 (cid:35) ++ ε (cid:34) − ζ (3)531441 − ζ (4)2187 + ζ (5)2187 + (cid:35) ++ ε (cid:34) − ζ (3)344373768 − ζ (4)2125764 + ζ (5)531441 + ζ (6)39366 − ζ (7)19683 ++ ζ (3) + (cid:35) + ε (cid:34) − ζ (3)418414128120 − ζ (4)1033121304 ++ ζ (5)645700815 + ζ (6)9565938 − ζ (7)23914845 − ζ (8)39858075 ++ ζ (9)14348907 + ζ (3) + ζ (3) + ζ (4) ζ (3)2657205 −− ζ (5) ζ (3)1594323 + ζ (3 , + (cid:35) + O (cid:16) ε (cid:17) , (18)where ζ (3 ,
5) is double zeta value [35]: ζ (3 , = (cid:88) < n < m n m (cid:39) . . (19)6o give an idea about the numerical structure of these expansions we present them also with thecoe ffi cients in decimals: g ∗ = . ε + . ε − . ε + . ε − . ε + . ε + O (cid:16) ε (cid:17) , g ∗ = − . ε + . ε + . ε + . ε − . ε + . ε + O (cid:16) ε (cid:17) . (20)The character of a fixed point and, in particular, its stability is determined by the eigenvalues ω , ω of the matrix Ω = ∂β ( g , g ) ∂ g ∂β ( g , g ) ∂ g ∂β ( g , g ) ∂ g ∂β ( g , g ) ∂ g (21)taken at g = g ∗ , g = g ∗ . If both eigenvalues are positive the fixed point is stable and describestrue critical behavior. At the same time, the roles of ω and ω in governing the cubic criticalbehavior are quite di ff erent. The eigenvalue ω determines the rate of flow to the cubic fixedpoint along the radial direction in the plane ( g , g ), while ω controls approaching this pointnormally to the radial ray. In particular, when n → n c the cubic fixed point tends to coincide withHeisenberg one and ω goes to zero. So, the dependence of ω on n and its numerical value at n = ε expansion for ω only. It reads: ω = ε n − n + ε ( n − − + n + n − n )81 n ( n + + (cid:88) k = C ( k ) ω ε k + O (cid:16) ε (cid:17) , (22)where coe ffi cients C ( k ) ω , along with those for ω , are presented in Tables 7 and 8 of Supplementarymaterials (see Appendix A).With ε expansion for ω in hand we can find ε series for the marginal dimensionality of thefluctuating field n c . It may be extracted from the equation ω ( n c , ε ) = . (23)Solving it by iterations in ε we obtain: n c = − ε + ε (cid:34) ζ (3)2 − (cid:35) + ε (cid:34) ζ (4)8 + ζ (3)8 − ζ (5)3 − (cid:35) ++ ε (cid:34) ζ (3)128 + ζ (4)32 − ζ (5)1728 − ζ (6)12 + ζ (7)384 − ζ (3) − (cid:35) ++ ε (cid:34) ζ (3)6912 − ζ (3 , + ζ (4)512 + ζ (5)20736 − ζ (6)6912 + ζ (7)41472 ++ ζ (8)2560 − ζ (9)23328 − ζ (3) − ζ (3) − ζ (4) ζ (3)96 −− ζ (5) ζ (3)216 + (cid:35) + O (cid:16) ε (cid:17) (24)7r, in decimals, n c = − ε + . ε − . ε + . ε − . ε + O (cid:16) ε (cid:17) . (25)Six-loop ε expansions for critical exponents η and ν corresponding to the cubic class of univer-sality result directly from those for anomalous dimensions and scaling relations (16). In its turn,six-loop ε expansions for γ ϕ and γ m originate from RG series (13) and ε expansions for the cubicfixed point coordinates. Since ε expansions for the critical exponents under arbitrary n are ex-tremely lengthy they are presented in Tables 9 and 10 of Supplementary materials (see AppendixA). Here we write down them only for physically interesting case n = η = ε + ε + ε (cid:34) − ζ (3) (cid:35) + ε (cid:34) − ζ (3)19131876 − ζ (4)19683 ++ ζ (5)19683 + (cid:35) + ε (cid:34) − ζ (3)55788550416 − ζ (4)25509168 ++ ζ (5)3188646 + ζ (6)19683 − ζ (7)531441 + ζ (3) − (cid:35) + O (cid:16) ε (cid:17) == . ε + . ε − . ε + . ε − . ε + O (cid:16) ε (cid:17) , (26) ν − = − ε − ε + ε (cid:34) ζ (3) − (cid:35) ++ ε (cid:34) ζ (3)531441 + ζ (4)729 − ζ (5)6561 − (cid:35) ++ ε (cid:34) ζ (3)172186884 + ζ (4)708588 − ζ (5)1594323 − ζ (6)6561 + ζ (7)19683 −− ζ (3) + (cid:35) + ε (cid:34) − ζ (3)167365651248 + ζ (4)229582512 ++ ζ (5)1549681956 − ζ (6)3188646 + ζ (7)19131876 + ζ (8)15943230 −− ζ (9)14348907 − ζ (3) − ζ (3) − ζ (4) ζ (3)59049 −− ζ (5) ζ (3)1594323 − ζ (3 , + (cid:35) + O (cid:16) ε (cid:17) == − . ε − . ε + . ε − . ε ++ . ε − . ε + O (cid:16) ε (cid:17) . (27)Of significant interest is also the critical exponent of susceptibility γ which is usually measuredin experiments and extracted from lattice calculations. Coe ffi cients of its ε expansion at the cubicfixed point under arbitrary n are presented in Table 11 of Supplementary materials (see Appendix8). For n = γ = + ε + ε + ε (cid:34) − ζ (3)2187 + (cid:35) ++ ε (cid:34) − ζ (3)1062882 − ζ (4)729 + ζ (5)6561 + (cid:35) ++ ε (cid:34) − ζ (3)172186884 − ζ (4)1417176 + ζ (5)1594323 + ζ (6)6561 − ζ (7)19683 ++ ζ (3) + (cid:35) + ε (cid:34) − ζ (3)83682825624 − ζ (4)229582512 − + ζ (5)3099363912 + ζ (6)6377292 − ζ (7)38263752 − ζ (8)31886460 + ζ (9)14348907 + ζ (3) + ζ (3) ζ (4)59049 + ζ (3) ζ (5)1594323 + ζ (3) + ζ (3 , − (cid:35) + O (cid:16) ε (cid:17) == + . ε + . ε − . ε + . ε + − . ε + . ε + O (cid:16) ε (cid:17) . (28)All calculated ε expansions are rather complicated and need to be checked up. We comparedthem with known five-loop series [20] and found complete agreement. In the Ising ( g →
0) andHeisenberg ( g →
0) limits our ε expansions are found to reduce to their counterparts for O ( n )-symmetric model [35] under n = n respectively. Our ε expansions should alsoobey some exact relations appropriate to the cubic model with n =
2. Such a system possesses aspecific symmetry: if the field ϕ α undergoes the transformation ϕ → ϕ + ϕ √ , ϕ → ϕ − ϕ √ , (29)the coupling constants are also transformed: g → g + g , g → − g , (30)but the structure of the action itself remains unchanged [1]. Since the RG functions are com-pletely determined by the structure of the action, the RG equations should be invariant with re-spect to any transformation conserving this structure [10]. It means that under the transformation(30) the β functions should transform in an analogous way while all the observables includingcritical exponents should be invariant with respect to above replacement (see [10, 29, 43] fordetails and extra examples). The expansions (12) and (13) do satisfy these symmetry require-ments. Moreover, transformation (30) converts the Ising fixed point into cubic one and vice versamaking them dual under n =
2. Six-loop ε expansions (15) reproduce this duality.
4. Resummation and numerical estimates
With six-loop ε expansions in hand we can obtain advanced numerical estimates for all thequantities of interest. It is well known that ε expansions as other field-theoretical perturbative9eries are divergent and for getting proper numerical results some resummation procedures haveto be applied. In this paper we address the methods of resummation based upon Pad´e approxi-mants [L / M] which are the ratios of polynomials of orders L (numerator) and M (denominator)and Borel-Leroy transformation. The Pad´e-Borel-Leroy technique enables one to optimize theresummation procedure by tuning the shift parameter b and proved to yield accurate numericalestimates for basic models of phase transitions. Much simpler Pad´e technique that is certainlyless powerful will be also used, mainly in order to clear up to what extent the numerical resultsdepend on the resummation procedure. Note that both approaches do not require a knowledgeof higher-order (Lipatov’s) asymptotics of the ε expansions coe ffi cients finding of which is aseparate non-trivial problem. Application of Pad´e approximants and use of Pad´e-Borel-Leroy resummation technique arerather straightforward and were described in detail in a good number of papers and books. Atthe same time, the determination of the final estimate of the quantity to be found and evaluationof corresponding error bar (apparent accuracy) are somewhat ambiguous procedures. The pointis that the choice of a subset of approximants which can be accepted as working and used to getthe asymptotic or averaged estimate of a given order usually may be tuned within a very widerange what may lead to unreliable (unstable) results and overestimation of the accuracy.Here we suggest clear and consistent strategy for calculating estimates with Pad´e approximantsand Pad´e-Borel-Leroy technique which is aimed to yield the stable results and reasonable errorestimates from order to order. While finding numerical values of physical quantities with Pad´eapproximants we use the following procedure. To estimate the value in k -th order of perturbationtheory we take into consideration approximants of k and k − L / M ]depend on the observable). The reason of accounting for such a subset is to provide the resultsstable from order to order while keeping the contribution from k -th order dominant. From thisset of approximants we exclude ”maximally o ff -diagonal” ones, in particular [0 / M ] and [ L /
0] asthey are known to possess bad approximating properties. We exclude also approximants whichhave poles in the interval ε ∈ [0 , ε phys ] (in our case ε phys = ε ∈ [0 , ε phys ] the approximant simply cannot be used to estimate the value at ε phys =
1, but even if the pole lying outside this area is still close to ε phys = ε phys ), namely multiplier 2 is based on our experience and tries tokeep a balance between dropping out unsuitable approximants and keeping a total number ofworking approximants as large as possible.To estimate the error bar (apparent accuracy) we consider values given by di ff erent approxi-mants as ”independent measurements” of the quantity and use t -distribution t p , n with p = . (cid:104) x (cid:105) = x + . . . + x n n , ∆ x = t . , n (cid:115) ( (cid:104) x (cid:105) − x ) + . . . + ( (cid:104) x (cid:105) − x n ) n ( n − . (31)In the case of Pad´e-Borel-Leroy resummation the procedure is almost the same except the factthat we have an additional – tuning – parameter b . For each particular value of b we performBorel-Leroy transformation of the original series, construct Pad´e approximants of k and k − / M ], [ L /
0] and those spoiled10y pole(s) on positive real axis. To find the optimal value of b we perform discrete scan over b ∈ [0 ,
20] with ∆ b = .
01 and search for the value of b which minimizes the standard deviation.The final estimate and error bar are then calculated with (31) for this value of b . c Let us start from the estimation of the fluctuating field marginal dimensionality n c . As seenfrom (25) ε expansion for n c is alternating and its coe ffi cients rapidly grow in modulo. Theformer property makes employing Pad´e approximants not meaningless. The results of Pad´eresummation of the series (25) under the physical value ε = Table 1: Pad´e triangle for the ε expansion of n c . Here Pad´e estimate of k -th order (lower line, RoC) is the numbergiven by corresponding diagonal approximant [L / L] or by a half of the sum of the values given by approximants [L / L − − / L] when a diagonal approximant does not exist. Three estimates are absent because corresponding Pad´eapproximants have poles close to the physical value ε = M \ L − − n (4) c = . ± . n (5) c = . ± .
12 and n (6) c = . ± .
14 as the four-loop, five-loop and six-loop estimates respectively.These estimates are seen to converge to the value close to 2.9 but the rate of convergence and theaccuracy are certainly very low.Since higher-order coe ffi cients of the ε expansion for n c are big and rapidly grow use of Borel-Leroy transformation that factorially weakens such a growth should significantly accelerate theconvergence and refine the estimate itself. This transformation looks as follows f ( x ) = ∞ (cid:88) i = c i x i = ∞ (cid:90) e − t t b F ( xt ) dt , F ( y ) = ∞ (cid:88) i = c i Γ ( i + b + y i . (32)Pad´e-Borel-Leroy resummation procedure consists of transformation (32) and analytical contin-uation of the Borel transform F ( y ) by means of Pad´e approximants. It includes also the choice(tuning) of the shift parameter b enabling one to achieve the fastest convergence of the itera-tion scheme. The results of the Pad´e-Borel-Leroy resummation of the six-loop series for n c arepresented in Fig. 2 and Table 2. The figure shows the behavior of relevant six-loop and five-loop Pad´e-Borel-Leroy estimates as functions of the parameter b and illustrates, in particular,the emergence of the optimal value b opt . Note that the curves in Fig. 2 are drawn only withinthe regions where Pad´e approximants of the Borel-Leroy transform have no positive axis poles.Pad´e-Borel-Leroy estimates of various approximants obtained under the optimal value of b whichwas found to be b opt = .
845 are collected in Table 2.As is seen the application of Pad´e-Borel-Leroy machinery indeed makes the iteration fasterconvergent and corresponding estimates much less oscillating. Being processed according to11 opt
P[1,4]P[2,3]P[3,2]P[4,1]P[1,3]P[2,2]P[3,1]
Figure 2: Pad´e-Borel-Leroy estimates of n c based upon approximants [1 / / / / / /
2] and [3 /
1] asfunctions of the parameter b. The curves are depicted only within the intervals where corresponding Pad´e approximantsare free from the ”dangerous” (positive axis) poles.Table 2: Pad´e-Borel-Leroy estimates of n c obtained from ε expansion (25) under the optimal value of the shift parameter b opt = . k -th order (lower line, RoC) is the number given by corresponding diagonal approximant[L / L] or by a half of the sum of the values given by approximants [L / L −
1] and [L − / L] when a diagonal approximantdoes not exist. Two estimates are absent because corresponding Pad´e approximants turn out to be spoiled by dangerouspoles. M \ L n (4) c = . ± . n (5) c = . ± .
03 and n (6) c = . ± .
003 at the four-, five- and six-loop levels. The last, highest-ordervalue n c = n (6) c = . ± .
003 (33)we accept as a final result of our calculations.
Since the coordinates of the fixed points depend on the normalization conditions adopted theirnumerical values being non-universal are not interesting from the physical point of view. Thatis why further we proceed directly to evaluation of critical exponents characterizing the cubicclass of universality at n =
3. Starting from the six-loop ε expansions for η and ν − and usingwell-known scaling relation we obtain ε expansions for exponents α , β , γ , ν and δ . Then we12erform Pad´e and Pad´e-Borel-Leroy resummation of all the series in hand. As the Pad´e-Borel-Leroy resummation procedure turns out to be most e ff ective (regular and fast convergent) for β and γ we present here details of evaluation of these two exponents. Numerical values of β and γ obtained within Pad´e and Pad´e-Borel-Leroy resummation approaches are collected in Tables 3,4, 5 and 6. Similar tables were calculated for the exponents α , δ , η and ν . All the final estimates Table 3: Pad´e triangle for the ε expansion of β . Five estimates are absent because corresponding Pad´e approximants havepoles lying between ε = ε = M \ L Table 4: Pad´e-Borel-Leroy estimates of β obtained from corresponding ε expansion under the optimal value of the shiftparameter b opt = . M \ L Table 5: Pad´e triangle for the ε expansion of γ . Five estimates are absent because corresponding Pad´e approximants havepoles close to the physical value ε = M \ L ff erences between the Pad´e-Borel-Leroy and Pad´e estimates presented in Table 7.13 able 6: Pad´e-Borel-Leroy estimates of γ obtained from six-loop ε expansion under the optimal value of the shift param-eter b opt = . M \ L Table 7: The values of critical exponents for the cubic class of universality obtained by means of Pad´e-Borel-Leroyresummation of the six-loop ε expansions. Corresponding Pad´e estimates and the di ff erences between Pad´e-Borel-Leroyestimates and their Pad´e counterparts are also presented. n = α β γ δ η ν PBL resum. − − ff erence 0.02(11) 0.0004(33) − . − . − . ff erent ways.We choose the next set of independent relations:1) γν (2 − η ) − = ,
2) 2 βν (1 + η ) − = ,
3) 5 − ηδ (1 + η ) − = , β + α + γ − = , (34)that are ”normalized to unity” to get the estimates of accuracy more uniform. Since the calculatedvalues of critical exponents are approximate they can not meet the scaling relations precisely andemerging discrepancies may be considered as a measure of achieved accuracy. The discrepanciesrelevant to scaling relations (34) along with their error bars originating from the estimates of thecritical exponents themselves (Table 7, upper line) are presented in Table 8. As is seen the Table 8: Six-loop estimates of critical exponents versus scaling relations
Scaling relation: 1 2 3 4Deviation from zero -0.005(14) 0.016(13) 0.0121(36) 0.007(45)deviations from exact scaling relations are small demonstrating the consistency of our approachand indicating that actual computational uncertainty of found numerical estimates is of order of0.01.To finalize this section, in Table 9 we present, for completeness, the values of correction-to-scaling exponents ω and ω obtained by resummation of corresponding ε expansions. Despitethe fact that zero lies inside the error bar for ω the median value of this exponent, being very14mall, turns out to be positive. Moreover, keeping in mind the results of independent evalua-tion of n c we may state that the value of ω given by six-loop ε expansion analysis is certainlypositive. More accurate estimates for ω can be obtained within the higher-order (seven-loop,etc.) approximations or by means of more sophisticated resummation procedure such as Boreltransformation combined with conformal mapping which will be a subject of a separate paper. Table 9: The values of correction-to-scaling exponents ω and ω for the cubic class of universality obtained by means ofPad´e-Borel-Leroy resummation of the six-loop ε expansions. Corresponding Pad´e estimates and the di ff erences betweenPad´e-Borel-Leroy estimates and their Pad´e counterparts are also presented. n = ω ω PBL resum. 0.799(4) 0.005(5)Pade resum. 0.78(11) 0.008(38)Di ff erence 0.02(11) − .
5. Discussion
In this section we will compare our results with those obtained earlier within the lower-orderapproximations and by alternative methods.The first quantity of interest is the marginal spin dimensionality for which we get the value n c = . ε expansion for this quantity has rapidly growingcoe ffi cients (see eq. (25)) what prevents Pad´e approximants from giving accurate enough numer-ical results while Pad´e-Borel-Leroy approach yields stable estimates with an accuracy increasingfrom order to order. The results of previous studies performed within the ε expansion approachand RG machinery in fixed dimensions (3D RG) as well as the numbers extracted from the MonteCarlo simulations and the six-loop pseudo- ε expansion are aggregated in the Table 10. Table 10: Marginal order parameter dimensionality n c given by the ε expansion technique, 3D RG approach, MonteCarlo simulations and the pseudo- ε expansion machinery. By the number of loops we mean the order of approximation. Numberof loops n c Paper n c Paper n c Paper ε expansion 3D RG Others1 4 [5]-1974 Monte Carlo2 2.333 [5]-1974 2.0114 [44]-1983 3 [23]-19983 3.128 [5]-1974 3.003 [45]-19844 2.918 [21]-1997 2.9 [17]-19892.96(11) This work-2019 2.89(2) [25] -20002.958 [20]-19955 < ε expansion6 2.915(3) This work-2019 2.89(4) [26]-2000 2.86(1) [32]-20162.862(5) [28]-200015n addition, the values of n c collected in Table 10 are depicted at Fig. 3 to visualize the trend thesevalues demonstrate under increasing order of approximation. This trend enable us to concludethat n c is certainly less than 3 for the 3D cubic model that justifies the significance of studyingthe cubic class of universality. . . . . . ε -expansion3D RGThis work . . . . . Figure 3: Dependence of the marginal spin dimensionality value on the order of RG approximation. The upper curve(” ε expansion”) represents the estimates obtained earlier from the five-loop ε expansion for n c . The other quantities of prime physical importance are critical exponents of the cubic universal-ity class. We should stress that to get estimates for critical exponents we perform resummation ofthe series for each exponent separately and afterwards checked a validity of several scaling rela-tions (34). Despite the fact that sometimes the relations are satisfied with inaccuracies exceedingcorresponding error bar estimates, these deviations are not too large lying within 3 σ interval.This may be considered as a proof of the consistency of the results obtained and a demonstrationof the numerical power of the ε expansion approach.It is worthy to compare our estimates with their analogs given by the lower-order approxi-mations and with the results of multi-loop 3D RG analysis. The data enabling one to do such acomparison are collected in Table 11. The numbers presented in both columns are seen to rapidlyconverge to the asymptotic values that di ff er from each other only tiny coinciding in fact withinthe declared error bars. It confirms the conclusion that the field theory is a powerful instrumentenabling one to get precise numerical results provided the calculations are performed in high16nough pertubative order. On the other hand, addressing the six-loop ε approximation shifts theestimates only slightly indicating that they should be very close to the exact values still unknown. Table 11: Critical exponent values given by multi-loop ε expansion calculations versus those resulting from 3D RGanalysis. Error bar for four loop estimate of η is absent because it can not be evaluated within approach described inSec. 4.1. Numberof loops η ν
Paper η ν
Paper ε expansion 3D RG3 – 0.700 [45]-19844 0.034 0.68(3) This work-2019 0.0331 0.6944 [17]-19890.0332 0.6996 [25] -20005 0.0375(5) 0.6997(24) [46]-1998 0.025(10) 0.671(5) [24]-20000.0374(22) 0.701(4) [26]-20000.0353(21) 0.686(13) This work-20196 0.036(3) 0.700(8) This work-2019 0.0333(26) 0.706(6) [26]-2000Another point to be discussed is to what extent – quantitatively – the critical exponents ofthe cubic class of universality di ff er from those of the 3D Heisenberg model. Since for n = ff erences are known to berather small. In Table 12 we present the estimates of critical exponents for cubic and Heisen-berg classes of universality obtained in the six-loop approximation. As expected, the di ff erencesbetween numerical values of critical exponents for these two classes are really small. So, it ishardly believed that measuring critical exponents in physical or computer experiments one candistinguish between cubic and Heisenberg critical behaviors. Table 12: Comparison of critical exponents for cubic (this work) and Heisenberg ([35]) classes of universality for n = ε expansion estimates for η and ν via scaling relations. n = α β γ δ η ν Cubic − − Conclusion
To summarize, we performed six-loop RG analysis of the critical behavior of n -vector ϕ model with cubic anisotropy in the framework of ε expansion approach employing the minimalsubtraction scheme. We calculated ε expansions for marginal spin dimensionality n c and crit-ical exponents α , β , γ , δ , η , ν , ω , ω for the cubic class of universality. We resummed thesediverging series with Pad´e approximants and using Pad´e-Borel-Leroy technique. Obtained nu-merical estimates for critical exponents turn out to be self-consistent in the sense that they are in17ccord, within the computational uncertainties, with the scaling relations. Six-loop contributionsare found to shift five-loop estimates only slightly but they improve numerical results consider-ably diminishing their error bars. Our results confirm and strengthen the conclusion that cubicferromagnets (D =
3, n =
3) belong to cubic class of universality and their critical behavior is de-scribed by critical exponents di ff ering from those of 3D Heisenberg model. At the same time, thecritical exponents of 3D cubic and Heisenberg models are numerically so close to each other thatit makes their behaviors practically indistinguishable if one limits himself by measuring criticalexponents only. Acknowledgment
It is a pleasure to thank Professor M. Hnatiˇc and Professor M.Yu. Nalimov for fruitful discus-sions. E.I. and A.K. are especially grateful to the Professor Hnatiˇc for support and hospitalityduring their stay in Slovakia. This work has been supported by Foundation for the Advancementof Theoretical Physics ”BASIS” (grant 18-1-2-43-1).
Appendix A. Supplementary materials
In Supplementary materials we present expansions of RG functions and critical exponents forarbitrary n . In rg expansion coe ffi cients.pdf we list coe ffi cients C i , jk of the expansions of betafunctions β , β (12), anomalous dimensions γ φ , γ m (13) and ε expansions of coordinates of thecubic fixed point (15), correction-to-scaling exponents ω , ω (22) and critical exponents η (26),1 /ν (27) and γ (28) corresponding to cubic universality class.Additionally, for RG functions ( β ( g , g ), β ( g , g ), γ φ ( g , g ), γ m ( g , g )) we provide Math-ematica file with their expansions ( rg expansion.m ). For critical exponents we present Mathe-matica files for all non-trivial fixed points: cubic ( cubic crit exp.m ), Ising ( ising crit exp.m ) andHeisenberg ( heisenberg crit exp.m ). Each file contains ε expansion for exponents α , β , γ , δ , η , ν as well as for 1 /ν , correction-to-scaling exponents ω , ω and coordinates of fixed points g ∗ , g ∗ . 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