VOLUME 77, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 21 OCTOBER 1996 Near-Surface Long-Range Order at the Ordinary Transition Uwe Ritschel and Peter Czerner Fachbereich Physik, Universität GH Essen, 45117 Essen, FR Germany (Received 28 February 1996) We study the spatial dependence of the order parameter m z near surface that reduces the tendency to order. Using scaling arguments and perturbative methods (e expansion), we find that for T $ Tbc a small surface magnetic field h1 gives rise to a macroscopic length scale and an anomalous short- distance increase of m z , governed by the power law m zk (with k 1 2 hord 0.21 for the d 3 Ising model). This result is related to experiments where exponents of the ordinary transition were observed in Fe3Al, while superstructure reflections revealed the existence of long-range order near the surface. [S0031-9007(96)01450-0] PACS numbers: 75.40.Cx, 68.35.Rh, 75.30.Pd, 78.70.Ck A prototypical system to study critical phenomena in Fe3Al was studied close to the DO3-B2 transition by restricted geometries is the semi-infinite Ising model, the scattering of evanescent waves generated by the total terminated by a plane surface and extending infinitely in reflection of x rays at a 110 surface. The system was the direction perpendicular to the surface (z direction) [1]. expected to belong to the universality class of the ordinary Spins located in the surface may experience interactions transition, and, indeed, the exponents measured were in different from those in the bulk, for example, due to remarkable agreement with theoretical predictions [5]. missing neighbors at a free surface or due to a strong A somewhat disturbing feature was that superstructure coupling to an adjacent medium. In the framework of reflections revealed the existence of unexpected long- continuum field theory such as the f4 model, the surface range order (LRO) near the surface, reminiscent of the influence is taken into account by additional fields such situation at the extraordinary transition. In the sequel it as the surface magnetic field h1 and the local temperature was demonstrated by Schmid [7] that in a similar situation perturbation c0 at z 0. The latter can be related to the (at the A2-B2 transition in Fe3Al) an effective ordering surface enhancement of the spin-spin coupling in lattice field h1 in the surface can arise when the stoichiometry of models [2]. the alloy is not ideal. Assuming that an h1 is also present At the bulk critical temperature Tbc, the tendency to at the DO3-B2 transition, the observed LRO can be order near the surface can be reduced c0 . 0 or in- explained, leaving unanswered the question, however, of creased c0 , 0 , or, as a third possibility, the surface can why exponents of the ordinary transition were measured be critical as well. As a result, each bulk universality despite the LRO. In the following we show that a class, in general, divides into several distinct surface uni- small h1 may generate a universal power-law growth of versality classes, called ordinary c0 ! , extraordinary the order parameter near the surface and, as a result, c0 ! 2 , and special transition c0 c sp . a LRO considerably (and, in fact, infinitely) larger than Close to the surface, within the range of bulk corre- expected from mean-field (MF) approximations, while the lation length j jtj2n, the singular behavior of ther- correlation function near the surface is still governed by modynamic quantities is markedly changed compared the exponents of the ordinary transition. to the bulk. For z ø j the magnetization behaves as Most of the theoretical studies concerning inhomoge- jtjb1 when t T 2 Tbc Tbc ! 0 from below, with nous systems concentrated on the behavior at the fixed b1 assuming characteristic values for special and ordi- points c0 6 and c0 c sp, respectively. At Tbc and nary transition, which are, in general, different from the for h1 0 for both the ordinary and the special transi- bulk exponent b. (At the extraordinary transition the sur- tion the order-parameter profiles are zero for all z $ 0. face is already ordered at Tbc.) Further, the correlation At the extraordinary transition the surface is ordered, and functions near the surface are characteristically modified. the order decays as z2b n with increasing distance from The correlation function for points within a plane paral- the surface [8], where, in the Ising case, b n 0.52 lel to the surface is given by C r r2 d221hk , where [9]. Concerning the effects of h1 it was assumed for a r jrk 2 r0kj and the anomalous dimension hk is related long time [10], and recently also shown by rigorous ar- to b1 by b1 n 2 d 2 2 1 hk [2]. Correlations in guments [11], that the case of strong h1 and c0 . 0 (the the z direction (and all other directions, except the parallel so-called normal transition) is equivalent to the extraor- one) are governed by C z, z0 jz 2 z0j2 d221h . dinary transition. The special transition was studied by Some of the theoretical predictions [2­4] were found Brezin and Leibler [12] and by Ciach and Diehl [13]. to be in excellent agreement with experiments carried It was found that at the fixed point the scaling field h1 out by Mailänder et al. [5,6]. In these experiments, gives rise to a length scale lsp h2n Dsp1 1 . For z ¿ lsp 0031-9007 96 77(17) 3645(4)$10.00 © 1996 The American Physical Society 3645 VOLUME 77, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 21 OCTOBER 1996 one finds that m z2b n as at the extraordinary transi- [2], the short-distance behavior is given by tion. In the opposite limit, z ø lsp, the magnetization behaves as m m m z, h 1z bsp 1 2b n . Since bsp 1 # b, the or- 1 h1zk with k yord 1 2 b n 1 2 hord . der still decays, governed by a somewhat smaller expo- nent compared to large distances. For the Ising model (4) bsp In the opposite limit, z ¿ lord, the magnetization ap- 1 2 b n 20.15 [9]. What do we expect if a small h proaches the bulk equilibrium value zero as z2b n. 1 is applied in the presence of a large c Equation (4) is the central result of this Letter. It states 0, i.e., close to the fixed point of the ordinary transition? In this situation the parameter c that the magnetization even at (or slightly above) Tbc in the 0 is a so-called dangerous irrelevant variable [2,14], comparable presence of a surface field h1 shows a power-law increase to the f4 coupling constant g at and above the upper reminiscent of the situation below Tbc. The short-distance critical dimension d 4, and, in general, must not be exponent k defined in (4) is zero in MF theory. Below d , naively set to its fixed point value c however, as for the Ising system in d 3, it is nonzero 0 . Setting the bulk magnetic field h 0, the remaining linear scaling and positive. Taking the literature values for the surface fields at the ordinary transition are t and h exponents from Refs. [2] and [9], one obtains k 0.21, 1 h1 c0 [2,14,15]. Hence, the behavior of the magnetization under which implies a rapid growth of LRO with increasing z. rescaling of distances should be described by The spatial variation of the magnetization discussed above strongly resembles the time dependence of the mag- m z, t, h netization in relaxational processes at the critical point. If 1 b2b nm zb21, tb1 n, h1byord 1 , (1) a system with nonconserved order parameter (model A) is where the scaling dimension of h quenched from a high-temperature initial state to the criti- 1 is given by yord 1 Dord cal point, with a small initial magnetization m i , the order 1 n d 2 hord k 2 [2]. As usual, all quantities in (1) are made dimensionless with an appropriate power parameter behaves as m m i tu [19], where the short- of the renormalization mass m, and we set m 1 time exponent u is proportional to the difference between afterwards. the scaling dimensions of initial and equilibrium magneti- Let us first discuss the profile for h zation [20]. Like the exponent k in (4), the exponent u 1 0. As men- tioned above, for t . 0 we have m 0 everywhere. For vanishes in MF theory, but becomes positive below d . t , 0, on the other hand, the magnetization approaches There is also heuristic argument for the growth of LRO its bulk value m near the surface. As stated above, h1 generates a surface b jtjb for z ! . Close to the sur- face z ø j , the magnetization increases with a power magnetization m1 h1. Regions (on macroscopic scales) law [16]. To see this from (1), we set h close to the surface will respond to this magnetization by 1 0 and fix the arbitrary rescaling parameter b by setting it z. Then the ordering as well. How strong this influence is depends magnetization takes the scaling form on two factors. First, it is proportional to the correlated area in a plane parallel to the surface in a distance z. m z, t z2b nM While in the surface, correlations are suppressed; close to t z j . (2) the surface the effective correlation length in directions Since we expect that m z ! 0 m1 [17] and know that parallel to the surface, jk, grows as z. Second, for m small h 1 jtjbord 1 , we conclude for the short-distance form of 1 (and thus small surface magnetization) it depends the scaling function M linearly on the probability that a given spin orientation t z z bord 1 n , and, in turn, the behavior of m is given m z, t jtjbord "survives" in a distance z from the surface. The latter 1 z bord 1 2b n [16]. We now turn to the case t 0 and h is governed by the perpendicular correlation function 1 fi 0. This is the situation we are actually interested in and which is C z z2 d221hord . Taking the factors together, we important for understanding the experimental results of obtain Ref. [5]. In this case, the scaling form derived from (1) is m z h1C z jd21 k h1z12hord . (5) m z, h1 z2b nMh zh1 yord 1 1 1 . (3) Qualitatively speaking, the surface, when carrying a small m First of all, we notice from (3) that the scaling field 1, induces a much larger magnetization in the adjacent layers, which are much more susceptible and respond with h1 gives rise to a length scale lord h21 yord 1 1 quite a magnetization m z ¿ m1. This effect is not observed comparable to the situation near the special transition on the MF level since there the increase of the correlated discussed above. In order to find the short-distance surface area is exactly compensated by the decay of the behavior of Mh z we have to recall that the surface is perpendicular correlations. 1 paramagnetic at the ordinary transition [10], and m1 will In order to corroborate our scaling analysis and the respond linearly to h1 [18]. Arguing again that m z ! heuristic arguments from above, we carried out a one- 0 m1, we now find that Mh z zyord 1 for z ø 1, loop calculation for the f4 model employing the e 1 and, in turn, with the scaling relation h hk 1 h 2 expansion. Expanded in powers of the coupling constant, 3646 VOLUME 77, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 21 OCTOBER 1996 the magnetization can be written in the form m propagator C 0, z, z0 appearing in (7) (for c0 ! ) takes m 0 1 gm 1 1 O g2 , where m 0 is the well known the simple form MF solution [10,21] and m 1 is the one-loop term. The 1 1 latter was calculated exactly for arbitrary c0 and h1 in C 0, z, z0 z5 5 , 2 z51 , (9) Refs. [12] and [13]. However, the improvement by means z2, z2. of the renormalization group was done at (or in the where , . denotes the smaller (larger) of z and z0. vicinity of) the special transition in these references. As Further analysis shows that the UV divergences can a consequence, the anomalous short-distance behavior at be absorbed in the standard fashion by renormalization the ordinary transition was missed. of the coupling constant Kdg0 u 1 1 3u 2e 1 O u2 The MF solution that satisfies the boundary condition and of the scaling field h1,0 h1 1 2 u 4e 1 O u2 [2]. After this, the coupling constant is set to its fixed zm 2 cmjz 0 h1 at the surface is given by s point value u 2e 3. Eventually, after exponentiation 12 1 m 0 z (6a) of logarithms, we find the asymptotic power laws g z z211e 2 for z ¿ lord, with m z, h1 h1ze 6 for z ø lord. (10) z z 1 z1 and As expected, the decay of the profile for z ¿ lord is gov- p erned by the one-loop result b n 1 2 e 2. The short- c2 g 12 1 2 2 c z21 0 1 4h1 0 distance behavior is consistent with our scaling analysis; 1 , (6b) 2 in first order e expansion k 1 2 hord which holds for general c e 6 [2]. 0 and h1. Close to the ordinary A more detailed account concerning the behavior of transition (large c0) the mean-field length scale becomes the magnetization in between the asymptotic regimes of z1 lord 12 g 1 2c0 h1. As expected from (4), there Eq. (10) will be given elsewhere. A qualitative pre- is no anomalous short-distance behavior on the MF level. view on the form of the scaling function M z The profile has its maximum value at z 0, and for h1 z2b nM z [see Eq. (3)] is shown in Fig. 1, where z ¿ lord the profile decays as z2b n with the MF value h1 the asymptotic power laws are quantitatively correct but b n 1. the crossover is described by a simple substitute for the The one-loop term m 1 is given by [13,22] scaling function. Regarding the crossover between ordi- 1 Z Z nary h m 1 z 2 dz0C 0; z, z0 m 0 z0 C p; z0, z0 , 1 0 and the extraordinary (or normal) transi- 2 tion h 0 p 1 , the following scenario should hold. While (7) at the ordinary transition m z vanishes everywhere, for R h where m 0 z is the zero-loop (MF) profile (6a) and 1 fi 0 the magnetization increases as zk up to z R p 2p 12d dd21p. The propagator C p; z, z0 is Fourier lord h21 yord 1 1 (1 yord 1 1.36 for the Ising model) and transformed with respect to the spatial coordinates paral- thereafter crosses over to the long-distance form given in lel to the surface. It can be calculated exactly [12,13], (10). When h1 becomes larger, the short-distance increase and the somewhat lengthy results will be omitted here. is steeper and lord shrinks. For h1 ! we have lord ! 0, The integrations in (7) necessary to obtain the full scaling and one finds m z2b n for all (macroscopic) distances, function M the result at the extraordinary transition. Largely analo- h are complicated and can only be carried out 1 numerically. However, it is straightforward to extract the gous results-monotonous behavior at the fixed points divergent terms, poles 1 e in dimensional regulariza- and profiles with one extremum in the crossover regime- tion, and the short-distance singularities log z, which, were found by Mikheev and Fisher for energy density of when exponentiated, give rise to power laws modified the two-dimension Ising model [23]. compared to the MF theory. Collecting these terms, m 1 is given by (7) with Z K Z C p; z, z d21 z221e dk p 2 1 3 k12e e22k e2klord z 2 1 µ 3 3 2 3 3 1 1 1 2 k k2 k2 1 finite , (8) where Kd 2 4p d 2G d 2 and "finite" stands for FIG. 1. Qualitative shape of the scaling function M z terms which are finite for e ! 0 and z ! 0. Terms of h1 z2b nMh z of the magnetization. More details are described O 1 c 1 0 are also omitted in (8). The zero-momentum in the text. 3647 VOLUME 77, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 21 OCTOBER 1996 Also, other quantities exhibit a crossover behavior simi- [6] For a review of the experimental work, see H. Dosch, in lar to the one described in the previous paragraph for m z . Critical Phenomena at Surfaces and Interfaces, edited by As an example, consider the correlation function C r, z G. Höhler and E. A. Niekisch, Springer tracts in Modern for points in a plane parallel to the surface in a distance Physics (Springer, Berlin, 1992). r from each other. The respective structure function was [7] F. Schmid, Z. Phys. B 91, 77 (1993). measured in the experiment by Mailänder et al. [5]. When [8] From the power-law decay m z2b n at the extraordi- r ¿ z the behavior of C r, z is governed by surface ex- nary transition, it appears that the magnetization diverges ponents, by the ones of the ordinary transition for z ø lord when going towards the surface. One has to bear in mind, however, that the power law is only valid for distances and by the ones of the extraordinary (or normal) transition much larger than microscopic scales. Upon approach- for z ¿ lord, with a crossover at z lord. ing the surface, the magnetization would depart from the In conclusion, we studied the effects of a small surface power law and assume a finite value in the surface. magnetic field in the vicinity of a surface that disfavors [9] C. Ruge, S. Dunkelmann, and F. Wagner, Phys. Rev. Lett. order. Our main result is that, for T * Tc, the order 69, 2465 (1992). parameter exhibits an anomalous short-distance behavior [10] A. J. Bray and M. A. Moore, J. Phys. A 10, 1927 (1977). in the form of a power-law increase m h [11] H. W. Diehl and T. W. Burkhardt, Phys. Rev. B 50, 3894 1z12hord for z & lord, implying a much larger magnetization density (1994). (long-range order) in this regime than could be expected [12] E. Brézin and S. Leibler, Phys. Rev. B 27, 594 (1983). from mean-field theory. As a consequence, as at the [13] A. Ciach and H. W. Diehl (unpublished). [14] H. W. Diehl, G. Gompper, and W. Speth, Phys. Rev. B 31, extraordinary transition, one has to expect superstructure 5841 (1985). reflections in scattering experiments that are sensitive to [15] The correct scaling field at the ordinary transition is the near-surface behavior of m z (such as one reported in actually h1 cy0, where the exponent y 1 in MF theory Ref. [5]). However, the correlation function in directions but fi1 in general. It is discussed in detail in Ref. [2] parallel to the surface (and thus the structure function) is that, in the framework of the loop expansion, one does not still governed by the exponents of the ordinary transition capture the deviation from the MF value in this exponent, in the near-surface regime z & lord. Thus, assuming that while, e.g., the z dependence of expectation values is there exists a small ordering field h reproduced correctly. For details, we refer to Refs. [2] and 1 in the system studied by Mailänder et al. [5], our scenario gives a plausible [14]. explanation for the experimental findings. The possible [16] G. Gompper, Z. Phys. B 56, 217 (1984). significance of our result for other experiments, such as [17] This is in accord with, and actually motivated by, the field-theoretical short-distance expansion [see the one recently carried out by Desai, Peach, and Franck K. Symanzik, Nucl. Phys. B190(FS3), 1 (1981); H. W. [24], remains to be explored in the future. Diehl and S. Dietrich, Z. Phys. B 42, 65 (1981)], where We thank H. W. Diehl, S. Dietrich, E. Eisenriegler, and field operators near a boundary are expressed in terms of R. Leidl for useful discussions and hints to the litera- boundary operators multiplied by c-number functions. ture. This work was supported in part by the Deutsche [18] The relation m1 h1 seems to be inconsistent with the Forschungsgemeinschaft through Sonderforschungs- definition of the surface exponent d11 via m1 h1 d11 1 and beriech 237. the result dord 11 0.6 for the Ising model [2]. One has to bear in mind, however, that dord 11 governs the leading singular behavior of m1 which is weaker here than the [1] For reviews on surface critical phenomena, see K. Binder, linear dependence on h1 [10]. in Phase Transitions and Critical Phenomena, edited by [19] H. K. Janssen, B. Schaub, and B. Schmittmann, Z. Phys. C. Domb and J. L. 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