Critical behavior at m-axial Lifshitz points: Field-theory analysis and -expansion results

Transcription

1 PHYSICAL REVIEW B VOLUME 62, NUMBER 8 NOVEMBER 2-II Critical behavior at m-axial Lifshitz points: Field-theory analysis -expansion results H. W. Diehl Fachbereich Physik, Universität-Gesamthochschule Essen, D-57 Essen, Federal Republic of Germany Department of Physics, Virginia Polytechnic Institute State University, Blacksburg, Virginia 26 M. Shpot* Fachbereich Physik, Universität-Gesamthochschule Essen, D-57 Essen, Federal Republic of Germany Received 3 May 2 The critical behavior of d-dimensional systems with an n-component order parameter is reconsidered at (m,d,n)-lifshitz points, where a wave-vector instability occurs in an m-dimensional subspace of R d. Our aim is to sort out which ones of the previously published partly contradictory -expansion results to second order in m/2d are correct. To this end, a field-theory calculation is performed directly in the position space of dm/2 dimensions, using dimensional regularization minimal subtraction of ultraviolet poles. The residua of the dimensionally regularized integrals that are required to determine the series expansions of the correlation exponents l2 l of the wave-vector exponent q to order 2 are reduced to single integrals, which for general m,...,d can be computed numerically, for special values of m, analytically. Our results are at variance with the original predictions for general m. For m2 m6, we confirm the results of Sak Grest Phys. Rev. B 7, Mergulhão Carneiro s recent field-theory analysis Phys. Rev. B 59, I. INTRODUCTION A Lifshitz point is a critical point at which a disordered phase, a spatially homogeneous ordered phase, a spatially modulated phase meet. In the case of a d-dimensional system with an n-component order parameter, it is called an (m,d,n)-lifshitz point or m-axial Lifshitz point if a wavevector instability occurs in an m-dimensional subspace. Such multiphase points are known to occur in a variety of distinct physical systems, including magnetic ones, 5,6 ferroelectric crystals, 7 charge-transfer salts, 8,9 liquid crystals, systems undergoing structural phase transitions or having domainwall instabilities, 2 the ANNNI model. 3, A survey covering the work related to them till 992 has been given by Selke, which complements updates an earlier review by Hornreich. 3 Recently there has also been renewed interest in various aspects of the problem, 5 9 including the effects of surfaces on the critical behavior at Lifshitz points From a general vantage point, critical behavior at Lifshitz points is an interesting subject in that it presents clear simple examples of anisotropic scale invariance. Epitomized also by dynamic critical phenomena near thermal equilibrium, 23 known to occur as well in other static equilibrium systems e.g., uniaxial dipolar ferromagnets, this kind of invariance has gained increasing attention in recent years since it was found to be realized in many nonequilibrium systems such as driven diffusive systems 2 in growth processes. 25 Systems at Lifshitz points are good cidates for studying the general aspects of anisotropic scale invariance. 26,27 For one thing, the continuum theories representing the universality classes of systems with short-range interactions at (m,d,n)-lifshitz points are conceptually simple; second, they involve the degeneracy m as a parameter, which can easily be varied between d. A thorough understing of critical behavior at such Lifshitz points is clearly very desirable. The problem has been studied decades ago by means of an expansion about the upper critical dimension,28 3 d*m m, m8. 2 Other investigations employed the dimensionality expansion about the lower critical dimension 3 d * (m)2m/2 for n 3, or the /n expansion. 2,32,33 Unfortunately, the -expansion results to order 2 one group of authors,28,29 obtained for the correlation exponents l2 l the wave-vector exponent q are in conflict with those of Sak Grest 3 for the cases m2 m6. In order to resolve this long-sting controversy, Mergulhão Carneiro 7,8 recently presented a reanalysis of the problem based on renormalized field theory dimensional regularization. Exploiting the form of the resulting renormalization-group equations, they were able to derive various previously given general scaling laws one expects to hold according to the phenomenological theory of scaling. However, their calculation of critical exponents was limited in a twofold fashion: They treated merely the special cases m2 m6, in which considerable simplifications occur. Their results for l2 l to order 2, agree with Sak Grest s 3 but disagree with Mukamel s. 28 Second, the exponent q an independent exponent that does not follow from these correlation exponents via a scaling law was not considered at all by them. Thus it is an open question whether Sak Grest s or Mukamel s O( 2 ) results for q with m2 m6 are correct. Furthermore, for other values of m, the published O( 2 ) results 28,29 for the exponents l2, l, q remain unchecked. It is the purpose of this /2/628/23382/$5. PRB The American Physical Society

2 PRB 62 CRITICAL BEHAVIOR AT m-axial LIFSHITZ work to fill these gaps to determine the expansion of the critical exponents l2, l, q for general values of m to order 2. Technically, we employ dimensional regularization in conjunction with minimal subtraction of poles in. This way of fixing the counterterms appears to us somewhat more convenient than the use of normalization conditions as was done in Refs In order to overcome the rather deming technical challenges, we have found it useful to work directly in position space. Thus the Laurent expansion of the distributions to which the Feynman graphs of the primitively divergent vertex functions correspond in position space must be determined to the required order in. The source of the technical difficulties is that these Feynman graphs, at criticality, involve a free propagator G(x) which is a generalized homogeneous rather than a homogeneous function, because of the anisotropic scale invariance of the free theory. While such a situation is encountered also in other cases of anisotropic scale invariance, the scaling function associated with G(x) turns out to be a particularly complicated function in the present case of a general (m,d,n)-lifshitz point. For general values of m, itisasum of two generalized hypergeometric functions. In the next section, we recall the familiar continuum model describing the critical behavior at a (m,d,n)-lifshitz point discuss its renormalization. In Sec. III details of our calculation are presented, our results for the renormalization factors are derived. Then renormalization-group equations are given in Sec. IV, which are utilized to deduce the general scaling form of the correlation functions, to identify the critical exponents, to derive their scaling laws as well as the anticipated multi-scale-factor universality. This is followed by a presentation of our -expansion results for the critical exponents l2, l, q. Section V contains a brief summary concluding remarks. Finally, there are two Appendixes to which some computational details have been relegated. II. THE MODEL AND ITS RENORMALIZATION We consider the stard continuum model representing the universality class of a (m,d,n)-lifshitz point with the Hamiltonian H 2 d x d u 2. 2 Here (x)(,..., n ) is an n-component orderparameter field. The coordinate xr d has an m-dimensional parallel component, x,a(dm)-dimensional perpendicular one, x. Likewise, denote the associated parallel perpendicular components of the gradient operator, while means the Laplacian 2. At the level of Lau theory, the Lifshitz point is located at. The Hamiltonian is invariant under the transformation x a x, x x, a m/2, a, a 2,, u a m u. 3 Thus, appropriate invariant interaction constants are u m/, /2,, the dependence on the parallel coordinates is through the invariant combination / x. Dimensional analysis yields the dimensions : x / /2, x, 2, /2, u m/ with d*md, i x m/8 (d2m/2)/2, where is an arbitrary momentum scale. Let (N) G i,...,i N x,...,x N i x... in x N cum 5 denote the connected N-point correlation functions cumulants i,...,i (x (N) N,...,x N ) denote the corresponding vertex functions. Using power counting one concludes that the ultraviolet uv singularities of these functions can be absorbed through the reparametrizations where Z /2 ren, c 2 Z, Z, u m/ A d,m Z u u, c /2 Z, A d,m S dm S m d/2 dm 2 m/2 6a 6b 6c 6d 6e is a convenient normalization factor we absorb in the renormalized coupling constant. Here S D 2 D/2 8 D/2 is the surface area of a D-dimensional unit sphere. The quantities c c correspond to shifts of the Lifshitz point. In our perturbative approach based on dimensional regularization they vanish. If we wanted to regularize the uv singularities via a cutoff restricting the integrations / over parallel perpendicular momenta by q q ), they would be needed to absorb uv singularities quadratic linear in, respectively. In the renormalization scheme we use, the renormalization factors Z, Z, Z, Z, Z u, for given values of the parameters, n, m, depend just on the dimensionless renormalized coupling constant u; that is, they are independent of,,. This follows from the fact that the primitive divergences of the momentum-space vertex functions (2) (q) () (q,...,q ), at any order of u m/, are poles in whose residua depend linearly on 7

3 2 3 H. W. DIEHL AND M. SHPOT PRB 62 q 2, q 2, q, in the case of the former are independent of these momenta mass parameters in the case of the latter. Subtracting these poles minimally as usual implies that these renormalization factors differ from through Laurent series in : Z p r a (r),p u;m,n p r a,p p (r) m,n ur,,u,,,. 9 p (m2)/2 dq q 2 J 2 d/2 (m2)/2 qk (dm)/2 q 2. 5 The integral remaining in Eq. 5 can be expressed as a combination of generalized hypergeometric functions F 2 see Appendix A. For special values of m d, the result reduces to simple expressions, which we have gathered in Appendix A. III. OUTLINE OF COMPUTATION AND PERTURBATIVE RESULTS We compute the leading nontrivial contributions to these renormalization factors. In the cases of Z, Z, Z, whose O(u) contributions vanish, these are of order u 2 ; for Z u Z they are of first order in u. To this end we exp about the Lifshitz point, using the free propagator e Gx i(q x q x ). q q 2 q Here the dimensionally regularized momentum-space integral is defined through q... q q... R m d m q d dm q m R dm dm 2 Let r x r x. Then the free propagator can be written in the scaling form with Gxr 2 m/ / r r /2 2 e ;m,d i(q q e ), 3 q q 2 q where R m is a vector of length arbitrary orientation, while e means the unit vector x /r. Note that the scaling function depends parametrically on m d. For the sake of brevity, we will usually suppress these variables, writing (;m,d) only when special values of m d are chosen or when we wish to stress the dependence on these parameters. The integration over q in Eq. 3 yields 2 (dm)/2 q q dm2 K (dm)/2 q 2 e i q. G s, r s(2) s r r /2, d d xr s(2) s r r /2 x d dm x r s(2)m/2 s x, 6 where the functions s (x ) s (r, ) are defined through s x d m x s r r r, ;r,. 7 The final result in Eq. 6 is the linear functional (r s(2)m/2, s ). Generalized functions such as r (...) their Laurent expansions are discussed in Ref. 3. Let (x )(r, ) be a smooth (C ) test function on R dm r S dm d r, its spherical average. Then we have r s(2)m/2, d dm x r s(2)m/2 x S dm dr r 32s(s) r S dm r 32 s(s),. 8 9 Here r is a stard generalized function in the notation of Ref. 3. Its Laurent expansion about the pole at p,2,... reads Upon introducing spherical coordinates qq (m) (,..., m ) for q, with d (m) sin m2 m d m d (m), one can perform the angular integrations. This gives r p p! (p) r r p p Op, where the generalized function r p is defined by 2

4 PRB 62 CRITICAL BEHAVIOR AT m-axial LIFSHITZ p2 r j r p,r dr r p r j j! ( j) r p p! (p) r. 2 Using these results, the leading terms of the Laurent expansions of (G s,) can be determined in a straightforward manner. However, it should be noted that the functions s (x ) introduced in Eq. 7 are not a priori guaranteed to have the usually required strong properties of test functions continuous partial derivatives of all orders sufficiently fast decay as x ). In particular, one may wonder whether the dependence on the variable r r of in Eq. 7 does not imply that derivatives such as s become singular at the origin. Closer inspection reveals that this is not the case since the problematic term r involves the vanishing angular integral d x (...). One obtains G 2, S dm G 3, S dm 2 r, 2 r O 3 r 3, 3 r O From its definition in Eq. 7 we see that the residuum 2 () on the right-h side of Eq. 22 reduces to a simple expression (). We thus arrive at the expansion G 2 x J,2m,d* xo, A d,m where J,2 is a particular one of the integrals J p,s m,d mp s ;m,dd G* Gx m2/ r / r r /2, 28 where q ;m,d 2 e i(q q e ) 29 q q q 2 2 is the analog of the scaling function () cf. Eq. 2. Proceeding as in the case of the latter, one obtains q dm2 22 (dm)/2 K (dm)/2 q 2 e i q q (m2)/2 dq q 2 J 22 d/2 (m2)/2 qk (dm)/2 q 2. 3 The remaining single integral can again be expressed in terms of generalized hypergeometric functions. The corresponding general expression, as well as the simpler ones to which this reduces for special values of m d, may be found in Appendix A. DxG 2 x G* Gx. 3 whose pole term can be worked out in a straightforward fashion by the techniques employed above. One finds with G 2 x G* Gx A d,m I m,d* x O m 32 In order to convert the Laurent expansion 23 into one for G 3 (x), we must compute 3 (). This in turn requires the calculation of the following angular average: I m,d m 2 ;m,d;m,dd r 2 r r, ;r,,r 2! x, e 2, r 2 mm2 dm. Using this in conjunction with Eq. 23 gives G 3 x A d,m J,3m,d* 2 x 6 mm2 26 J,3m,d* x O. d*m 27 Let us introduce coefficients b (m) for the leading nontrivial contributions to the renormalization factors Z, writing these in the form Z u b u m n8 9 Z b m n2 3 u Ou2, u Ou2, 3 35

5 2 32 H. W. DIEHL AND M. SHPOT PRB 62 Z b m n2 3 u 2 Ou3,,,. 36 From the pole terms of G 2 (xy) given in Eq. 2 one easily deduces that b u m3b m 3 2 J,2m,d*. 37 The pole terms proportional to (x), 2 (x), (x) of the two-loop graphs considered above are absorbed by counterterms involving the renormalization factors Z, Ž Z Z, Ž Z Z Z /2, respectively. Utilizing the Laurent expansions 27 3, one finds that their coefficients are given by b m J,3 m,d*, 38 2 d*m A d *,m with D N 2 (N) ren 5 D u u 2, 6 where the beta eta functions are defined by u uu u u 7 ln Z,,,,,u, 8 respectively. Here means a derivative at fixed bare variables,,,. Owing to our use of the minimal subtraction procedure, the functions can be expressed in terms of the residua a, (u;m,n) as uu da,,,,,,u. 9 du bˇ J,3 m,d* m, mm2 bˇ m 8m A d *,m I m,d*. A d *,m To solve the renormalization-group RG equations 3 via characteristics, we introduce flowing variables through l d dl ūl uūl, ūu, 5 l d dl l ū,, 5 The coefficients b b are related to these via b mbˇ mb m l d dl l ū,, 52 b mbˇ m 2 b m 2 bˇ m. 2 l d dl l2 ū,. 53 IV. RENORMALIZATION-GROUP EQUATIONS AND -EXPANSION RESULTS The reparametrizations 6 yield the following relations between bare renormalized correlation vertex functions: G (N) x,x Z N/2 G (N) ren x,x, 3a (N) x,x Z N/2 (N) ren x,x, 3b where x x st for the set of all parallel perpendicular coordinates on which G (N) (N) depend. For conciseness, we have suppressed the tensorial indices i,...,i N of these functions will generally do so below. Upon exploiting the invariance of the bare functions under a change (l) l of the momentum scale in the usual fashion, one arrives at the renormalization-group equations D N 2 G (N) ren, The flow equation 5 for the running coupling constant ū(l) can be solved for l to obtain ū dx ln l u u x. 5 For, the beta function u (u) is known to have a nontrivial zero u*, corresponding to an infrared-stable fixed point. Exping about this fixed point gives the familiar asymptotic form ūl u*uu*l uol 2 u l in the infrared limit l, where 55 u d u du u* 56 is positive. The solutions to the other flow equations, 5 53, can be conveniently written in terms of the anomalous dimensions * (u*) the renormalization-group-trajectory integrals

6 PRB 62 CRITICAL BEHAVIOR AT m-axial LIFSHITZ E ū,uexp u ū(l) dx * x u x,,,,, which approach nonuniversal constants 57 E*uE u*,u,,,,, 58 in the infrared limit l. One finds * ll E ūl,u l * E*u, l 59 ( ll *) E ūl,u l (*) E*u, 6 l (2 ll *) E ūl,u l (2*) E*u. 6 l Solving the RG equation 3a in terms of characteristics yields G (N) ren x,x ;,,u,, l * N/2 E ū,u m ld2 m/ E ū,u G (N) ren x,x ;,,ū,, l 2 l * N/2 G (N) ren 2 l 2 / / x, l x ;,,ū,,. 62 To obtain the second equality, we have used the relation G (N) ren x,x ;,,ū,, d2m/2 m/ N/2 G (N) / /2 ren x, x ;,,ū,,, 63 implied by our dimensional considerations. (N) Let us assume that the function G ren on the right-h side of Eq. 62 has a nonvanishing limit ū u* for. This assumption is in conformity with, can be checked by, RG-improved perturbation theory. We choose ll such that l for 6 consider the limit. To write the resulting asymptotic form of G (N) ren in a compact fashion, we introduce the correlation-length exponents l2 2 * l 2 * 2 *, the crossover exponent l2 *, as well as the correlation lengths l E *u l2 l / 2 2 l /2 E *u / E*u l In terms of these quantities the asymptotic critical behavior of G (N) ren becomes with G (N) ren x,x ;,,u,, * N/2 E* (dm2*) m G (N) x, x ;E * / l2 7 G (N) x,x ;G (N) ren x,x ;,,u*,,. 7 The result is the scaling form expected according to the phenomenological theory of scaling. As it shows, the scaling function G (N) is universal, up to a redefinition of the nonuniversal metric factors associated with the relevant scaling fields, i.e., E*, E*, E*, E*. Note that E*, whose change would affect the overall amplitude of G (N), as usual corresponds to a metric factor associated with the magnetic scaling field; see, e.g., Ref. 35. The correlation exponents l2 l are given by l2 * 72 l ** 2*. 73 This can be seen either by taking the Fourier transform of the above result 7 with N2 or else by solving directly the renormalization-group equation of (2) ren (q,q ). In order to identify the wave-vector exponent q, we utilize the scaling form (2) ren q,q ;,,u q,q ; 7 of the inverse susceptibility (2) argue as in Ref. 28: On the helical branch T hel () of the critical line, the inverse susceptibility vanishes at q c (q c,). Hence in the scaling regime, the line T hel () is determined by the zeroes of the scaling function (p,,ϱ). Denoting these as p c ϱ c, we obtain the relations q c p c p c l 75

7 2 3 H. W. DIEHL AND M. SHPOT PRB 62 which yield with ϱ c, q c q q l 2* *, 78 where the last equality follows upon substitution of Eqs for l, respectively. To compute the exponent functions 8 the function 7, we insert the residua of the renormalization factors 3 36 into Eq. 9 express b in terms of b u using Eq. 37. We thus obtain u2 n2 3 b mu 2 Ou 3,,,, 79 u 3 n2 3 b umuou 2, 8 u uu n8 9 b umuou 2. 8 From the last equation we can read off the expansion of u*, the nontrivial zero of u : u* 9 n8 b u m O2. 82 Evaluation of the above exponent functions at this fixedpoint value gives us the expansions of the anomalous dimensions *. Substituting these into the expressions 65 67, 72, 73, 78 for the critical exponents yields l2 2 n2 n8 O2,.252 for m,.295 for m2, l 27n2 b mbˇ m l2 2 n8 2 2 O 3 2 b u m 2 2 O3 27n for m3, n for m,.229 for m5,.233 for m6, l2 27n2 n8 2 l2 2 27n2 n8 2 l 27n2 n8 2 b m b u m 2 2 O 3 O 3 27n2 bˇ m b u m 2 2 O 3 O 3 27n for m,.66 for m2,.56 for m3, n for m,.8 for m5,.353 for m6, for m,.97 for m2,.33 for m3, n for m,.73 for m5,.92 for m6, for m,.587 for m2, b m2bˇ mbˇ m 2 O 3 O 3 27n for m3, b u m 2 n for m,.9 for m5,.3658 for m6,

8 PRB 62 CRITICAL BEHAVIOR AT m-axial LIFSHITZ q 27n2 2 n for m..295 for m2. bˇ mbˇ m 2 O 3 b u m 2 2 O3 27n2.337 for m3. n8 2.2 for m for m for m6. 88 We have expressed the results in terms of the coefficients b u (m), b (m), bˇ (m), bˇ (m), which according to Eqs. 37 are proportional to the integrals J,2 (m,d*), J,3 (m,d*), J,3 (m,d*), I (m,d*), respectively. These integrals are defined by Eqs The first one of them the one-loop integral J,2 (m,d) is analytically computable 36 for general values of d m. The result is giving J,2 m,d 22 2 d 2 b u m m 2 m 2, 89 2 m m 2 m 2. 9 The fixed-point value that results when this value of b u (m) is inserted into Eq. 82 is consistent with the one found in calculations based on Wilson s momentum-shell integration method. 28 The integrals J,3 (m,d*), J,3 (m,d*), I (m,d*), hence the coefficients b (m), bˇ (m), bˇ (m), can be calculated numerically for any desired value of m, using the explicit expressions for the scaling functions (;m,d*) (;m,d*) given in Eqs. A A5 of Appendix A. As discussed there, the numerical evaluation of these integrals for general values of m requires some care because (;m,d*) is a difference of two terms, each of which FIG.. Coefficients of 2 terms of the exponents l2 triangles, l stars, q squares for n. grows exponentially as. In this manner one arrives at the values of the 2 terms given in the second lines of Eqs In Fig. the coefficients of the 2 terms of some of these exponents are depicted for the scalar case, n. As one sees, they have a smooth relatively weak m dependence, especially for l2 l. In the special cases m2 m6, the functions (;m,d*) (;m,d*) become sufficiently simple see Eqs. A6 A8, so that the required integrations can be done analytically. This leads to b 2 5 8, bˇ , bˇ 2 8 b , bˇ 6 8, 2 9a 9b 9c 3 ln 3 2, 92a 92b bˇ 8 6 ln c 9 2 If these analytical expressions for the coefficients are inserted into the expansions 85, 86, 88 of l2, l, q with m2 m6, then Sak Grest s 3 results for those two values of m are recovered which in turn agree with Mergulhão Carneiro s 8 findings for l2 l ). As was mentioned already in the Introduction, these results for m2 m6 disagree with Mukamel s. 28 More generally, our O( 2 ) results 8 88, for all values of m,...,6,turn out to be at variance with the latter author s. The case m was also studied by Hornreich Bruce, 29 who calculated l (m) q (m) to order 2. Their results agree with Mukamel s hence diagree with ours. Upon extrapolating the series expansions 8 88 one can obtain exponent estimates for three-dimensional sys-

9 2 36 H. W. DIEHL AND M. SHPOT PRB 62 tems. Unfortunately, there exist in the literature only very few predictions of exponent values produced by other means with which we can compare our s. Utilizing hightemperature series techniques, Redner Stanley 37 found the estimate q.5.5 for the case of a uniaxial (m,d,n)(,3,) Lifshitz point. This is in conformity with the value q.59 one gets by setting.5 in the corresponding m result of Eq. 88. A more recent hightemperature series analysis by Mo Ferer 38 yielded 2 q. For the susceptibility exponent l l2 2 l2 l l, 93 the correlation exponent l, the specific-heat exponent l 2m l dm l2 9 of the (m,d,n)(,3,) Lifshitz point these authors obtained the results l.62.2, l.63., l.2.5. Utilizing these numbers to compute l via the scaling law implied by Eq. 93, l l / l, yields 39 l.2.5. This may be compared with the value l.9 one finds from Eq. 86 upon setting.5. As a further quantity for which Mo Ferer s results 38 yield an estimate that can be compared with our O( 2 ) results we consider the ratio l / l. Substituting their exponent values into l (2 l l )/2 yields 39 l.9.35 l / l From the asymptotic form 2 of (N) one reads off the scaling law G ren l l2 2 dm2 l2 l m, 95 2 which may be combined with relation 93 for l to conclude that dm2 l2 m l l l2. 96 l 22 l2 We now set mn.5 in Eqs This gives l / l2.88 l2.39. Then we insert these numbers into Eq. 96 with d3, obtaining l / l.3. There also exist Monte Carlo estimates of exponents for the case of a (m,d,n)(,3,) Lifshitz point., The more recent ones, l.9.2 l..6, due to Kaski Selke, give 39 l / l In view of the fact that the importance of anisotropic scaling its implications for finite-size effects in systems exhibiting anisotropic scale invariance 2,3 has been realized only more recently, it is not clear to us how reliable these Monte Carlo estimates may be expected to be. Note, on the other h, that the coefficients of the 2 terms of the series 8 88 are all truly small. Thus it is not unlikely that the values one gets for d3 by naive evaluation of these truncated series are fairly precise, at least for m. The 2 corrections of these exponents grow with m because of the factor (d*3) 2 ( m/2) 2. V. CONCLUDING REMARKS We have studied the critical behavior of d-dimensional systems at m-axial Lifshitz points by means of an expansion about the upper critical dimension d*m/2. Using modern field-theory techniques, we have been able to compute the correlation exponents l2 l, the wave-vector exponent q, exponents related to these via scaling laws to order 2. The resulting series expansions, given in Eqs. 8 88, correct earlier results by Mukamel 28 Hornreich Bruce; 29 for the special values m2 m6, we recovered Sak Grest s 3 findings. To clarify this long-sting controversy, it proved useful to work directly in position space to compute the Laurent expansion of the dimensionally regularized distributions associated with the Feynman diagrams. There are two other classes of difficult problems where this technique has demonstrated its potential: field theories of polymerized tethered membranes 6 critical behavior in systems with boundaries. 35,7 In the present study an additional complication had to be mastered: The free propagator at the Lifshitz point, which because of anisotropic scale invariance is a generalized homogeneous function rather than a simple power of the distance xx, involves a complicated scaling function. For powers products of simple homogeneous functions, a lot of mathematical knowledge on Laurent expansions is available. 3 Unfortunately, the amount of explicit mathematical results on Laurent expansions of powers products of generalized homogeneous functions appears to be rather scarce. Since we had no such general mathematical results at our disposal, we had to work out the Laurent expansions of the required distributions by our own tools. Difficulties of the kind we were faced with in the present work may be encountered also in studies of other types of systems with anisotropic scale invariance. Hence the techniques utilized above should be equally useful for fieldtheory analyses of such problems. ACKNOWLEDGMENTS Part of this work was done while H. W. D. was a visitor at the Physics Department of Virginia Technical University. It is a pleasure to thank Beate Schmittmann, Uwe Täuber, Royce Zia for their warm hospitality provided during this visit. M. Sh. would like to thank Kay Wiese for introducing him to the field theory of self-avoiding membranes for numerous helpful discussions. We would also like to express our gratitude to Malte Henkel Walter Selke for comments on this work their interest in it. This work has been supported by the Deutsche Forschungsgemeinschaft through the Leibniz program Sonderforschungsbereich 237 Unordnung und grosse Fluktuationen. APPENDIX A: THE SCALING FUNCTIONS AND The scaling functions () () introduced, respectively, through Eqs are given by single integrals 5 3 of the form i dq q 2 J qk q 2. A This is a stard integral, 8 which for arbitrary values of its parameters, can be expressed in terms of generalized hypergeometric functions 2 F 3. For the special values

10 PRB 62 CRITICAL BEHAVIOR AT m-axial LIFSHITZ m/2 m//2 or m//2 for which it is needed, it simplifies, giving 2 )/ ;m,d 2 2m (6m2 2m F 2 2 ; 2, 2m ; 6 2 F ;3 2, m 6 ; ;m,d 2 3m 2 m (6m2 )/ m F 2 2 ; 2, m 2 6 ; 2 2m F 2 2 ;3 2, 2m ; 6. A2 A3 At the upper critical dimension, i.e., for, this becomes ;m, m 2 ;m, m 2 (6m)/ 8 2 5m 2m F 2 ; 2, 2m ; 6 A 2 3m/ 2m/2 I m/ 2 (6m)/ 2 3m/ 2m/2 I 2 6m m/ m/ F 2 ; 3 2, 2m ; 6, A5 respectively, where the I ( ) are modified Bessel functions of the first kind. In the special cases m2 m6, these expressions reduce to simple elementary functions: One has ;2,5 2 e 2 /, ;2,5 2 ;2,5, ;6,7 2 e 2 / A6 A7 2 3, A8 ;6,7 3 2 e 2 /. A9 The reason for the latter simplifications is the following. If m2 orm6 dd*m/2 upper critical dimension, then Bessel functions K with 2 are encountered in the integral A, which are simple exponentials. 9 This entails that the required single integrations can be done analytically to obtain the results 9a 92c for the O( 2 ) coefficients. For the remaining values of m, i.e., for m,3,,5, the required integrals did not simplify to a degree that we were able to compute them analytically. However, proceeding as explained in Appendix B, they can be computed numerically. In the special cases m2 m6, the results of our numerical integrations are in complete conformity with the analytical ones. APPENDIX B: ASYMPTOTIC BEHAVIOR OF THE SCALING FUNCTIONS AND According to Eq. A, the scaling function (;m,d*) is a difference of a hypergeometric function F 2 a product of a Bessel function I m/ times a power. If one asks MATHEMATICA 5 to numerically evaluate expression A for () without taking any precautionary measures, the result becomes inaccurate whenever becomes sufficiently large. We found that such a direct, naive numerical evaluation fails for values of exceeding 9.5. This is because both functions of this difference increase exponentially as. To cope with this problem, we determined the asymptotic behavior of the scaling functions (;m,d*) (;m,d*) for. From the integral representations 3 29 of these functions one easily derives the limiting forms with ;m,d (as) ;m,d 2 m,d B ;m,d (as) ;m,d 22 m,d 8, B2

11 2 38 H. W. DIEHL AND M. SHPOT PRB 62 e m,d q i(q e ) q q 2 q d2 m 2 3dm. B (dm)5 (d)/2 At the upper critical dimension, the latter coefficient becomes m,d* 23m (6m)/ m2. B Note that for m2 the asymptotic form B is consistent with the simple exponential form A6 since (2,5). However, for other values of m, the coefficient B does not vanish. For example, (6,7)/(8 3 ), in conformity with expression A8 for the scaling function (;6,7). In order to obtain precise results for the integrals J,3 (m,d*), J,3 (m,d*), I (m,d*), on which the coefficients b (m), bˇ (m), bˇ (m) depend, we proceeded as follows. We split the required integrals as...d...d...d, choosing 9.3. In the integrals...d, we replaced the integrs by their asymptotic forms obtained upon substitution of /or by their large- approximations (as) (as) given in Eqs. B B2, respectively, then computed these integrals analytically. The integrals...d were computed numerically, using MATHEMATICA. 5 We checked that reasonable changes of have negligible effects on the results. The procedure yields very accurate numerical values of the requested integrals. The reader may convince himself of the precision by comparing the so-determined numerical values of the integrals for m2 m6 with the analytically known exact values. *Permanent address: Institute for Condensed Matter Physics, Svientsitskii str., 79 Lviv, Ukraine. address: shpot@theo-phys.uni-essen.de R.M. Hornreich, M. Luban, S. Shtrikman, Phys. Rev. Lett. 35, R.M. Hornreich, M. Luban, S. Shtrikman, Phys. Lett. 55A, R.M. Hornreich, J. Magn. Magn. Mater. 5 8, W. Selke, in Phase Transitions Critical Phenomena, edited by C. Domb J. L. Lebowitz Academic, London, 992, Vol. 5, pp Y. Shapira, C. Becerra, N.F. Oliveira, Jr., T. Chang, Phys. Rev. B 2, For a more complete set of references, see Refs Y.M. Vysochanski V.Y. Slivka, Usp. Fiz. Nauk 62, Sov. Phys. Usp. 35, E. Abraham I.E. Dzyaloshinskii, Solid State Commun. 23, C. Hartzstein, V. Zevin, M. Weger, J. Phys. France, J.H. Chen T.C. Lubensky, Phys. Rev. A, A. Aharony D. Mukamel, J. Phys. C 3, L J. Lajzerowicz J.J. Niez, J. Phys. France Lett., L M.E. Fisher W. Selke, Phys. Rev. Lett., M.E. Fisher W. Selke, Philos. Trans. R. Soc. London, Ser. A 32, I. Nasser R. Folk, Phys. Rev. B 52, I. Nasser, A. Abdel-Hady, R. Folk, Phys. Rev. B 56, C. Mergulhão, Jr. C.E.I. Carneiro, Phys. Rev. B 58, C. Mergulhão, Jr. C.E.I. Carneiro, Phys. Rev. B 59, M.M. Leite, Phys. Rev. B 6, G. Gumbs, Phys. Rev. B 33, K. Binder H.L. Frisch, Eur. Phys. J. B, H.L. Frisch, J.C. Kimball, K. Binder, J. Phys. Condens. Matter 2, B.I. Halperin P.C. Hohenberg, Rev. Mod. Phys. 9, B. Schmittmann R.K.P. Zia, Statistical Mechanics of Driven Diffusive Systems, Vol. 7 of Phase Transitions Critical Phenomena Academic, London, 995, pp J. Krug, Adv. Phys. 6, For example, in a recent investigation Ref. 27 of the question of whether anisotropic scale invariance may lead to an even larger invariance group reminiscent of conformal or Schrödinger invariance such systems were explicitly considered. 27 M. Henkel, Phys. Rev. Lett. 78, D. Mukamel, J. Phys. A, L R.M. Hornreich A.D. Bruce, J. Phys. A, J. Sak G.S. Grest, Phys. Rev. B 7, G.S. Grest J. Sak, Phys. Rev. B 7, J.F. Nicoll, G.F. Tuthill, T.S. Chang, H.E. Stanley, Phys. Lett. 58A, A.A. Inayat-Hussain M.J. Buckingham, Phys. Rev. A, I.M. Gel f G. E. Shilov, in Generalized Functions Academic, New York, 96, Vol., Chaps H.W. Diehl, in Phase Transitions Critical Phenomena, edited by C. Domb J. L. Lebowitz Academic, London, 986, Vol., pp To see this within the present calculational scheme, note that J,2 (m,d)/s m is nothing but the Fourier transform of 2 () at zero momentum q. Hence it is equal to the integral q over the square of the Fourier transform of (), which can be read off from the Fourier-integral representation of (). Upon performing the required angular integrations, one is left with a stard integral, which yields Eq S. Redner H.E. Stanley, Phys. Rev. B 6, Z. Mo M. Ferer, Phys. Rev. B 3, In computing the error bars we have taken a cautious attitude: As smallest largest values of a ratio of the form (Aa)/(B b) with A, B, a, b positive we have taken the numbers (Aa)/(Bb) (Aa)/(Bb), respectively. Their deviations from the value A/B gave us the error bars.

12 PRB 62 CRITICAL BEHAVIOR AT m-axial LIFSHITZ W. Selke, Z. Phys. 29, K. Kaski W. Selke, Phys. Rev. B 3, K. Binder J.-S. Wang, J. Stat. Phys. 55, K.-T. Leung, Phys. Rev. Lett. 66, F. David, B. Duplantier, E. Guitter, Phys. Rev. Lett. 72, F. David, B. Duplantier, E. Guitter, cond-mat/97236 unpublished. 6 K. Wiese, Habilitationsschrift, Universität-Gesamthochschule, 999; in Phase Transitions Critical Phenomena, edited by C. Domb J. L. Lebowitz Academic, London, 2, Vol H.W. Diehl, Int. J. Mod. Phys. B, ; cond-mat/963 unpublished. 8 A. Prudnikov, Y.A. Brychkov, O.I. Marichev, Integrals Series Gordon Breach, New York, 986, Vol. 2, Eq Both Sak Grest Ref. 3 as well as Mergulhão Carneiro Ref. 8 explicitly employed the simple form A6 of the scaling function in their computations. However, the latter authors Ref. 8 did not take advantage of working with the simple function A8 in the case m6, performing complicated computations in the momentum representation instead. Sak Grest Ref. 3, on the other h, did not present any details of their calculation for the case m6. 5 MATHEMATICA, version 3., a product of Wolfram Research.