SURFACE MAGNETIC ORDER AND EFFECTS OF THE NATURE OF THE INTERACTIONS

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SURFACE MAGNETIC ORDER AND EFFECTS OF

THE NATURE OF THE INTERACTIONS

C. Tsallis, A. Chame

To cite this version:

C. Tsallis, A. Chame.

SURFACE MAGNETIC ORDER AND EFFECTS OF THE NATURE

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SURFACE MAGNETIC ORDER AND EFFECTS OF THE NATURE OF THE

INTERACTIONS

C. Tsallis and A. Chame

Centro Brasileiro de Pesquisas Fisicas/CNPq, Rua Xavier Sigaud 150, 22290, Rio de Janeiro, RJ-Brazil Abstract. - We discuss, within a real-space renormalization-group, interesting thermal effects (surface singularity at the bulk transition; Ising-Heisenberg crossover) concerning the free surface of a semi-infinite d = 3 spin 1 / 2 anisotropic Heisenberg ferromagnet. Comparison with the Mermin-Wagner theorem and with experimental work is done.

1. Introduction

Surface magnetism is nowadays an active field of research. This is due both to its theoretical richness and t o its important applications (corrosion, catalysis, information storage). See reference [I] for a general re- view, and references [2], [3] for reviews of respectively real-space and reciprocal - space renormalization- group (RG) approaches.

The theoretical prototype for surface magnetism is the spin 112 Ising ferromagnet in simple cubic lattice with a (0, 0, 1) free surface, Js and JB respectively being the surface and bulk coupling constants. The corresponding phase diagram is indicated in figure 1: three distinct phases, namely the bulk ferromagnetic (BF; both bulk and free suffaee are magnetized), the surface ferromagnetic (SF; only the free surface is magnetized) and the paramagnetic (P; both bulk and surface are disordered) ones, join at a multicritical point (located at A

=

J s

/

JB

-

1 = Ac).

In the present work we discuss, within a real-space RG framework, two interesting effects, namely (i) the

influence of the symmetry of the magnetic interaction and its connection with the Mermin-Wagner theorem, and (ii) the singularity which the surface magnetization thermal behavior exhibits at the bulk critical point for A

>

A, (extraordinary transition). In section 2 we introduce the model and the RG formalism; in sections 3 and 4 we discuss the influence of the interaction symmetry and the surface singularity respectively. We finally conclude in section 5.

2. Model and RG formalism

We consider the spin 112 anisotropic Heisenberg dimensionless Hamiltonian

where /3

=

I

/

~ B T and (2, j ) run over all pairs of first-

neighboring sites on a semi-infinite simple cubic lattice with a (0, 0, 1) free surface; (K,,, qij) equals (Ks,qs) if both sites belong to the surface and equals (Kg, q ~ ) otherwise (Ks r J s l k ~ T and K B JBIICBT); Js, JB 2 0 and 0

5

qs, q~

5

1 (q,j = 1 recovers the standard Ising interaction and qij = 0 recovers the isotropic Heisenberg interaction); a:, a: and a: are the spin 112 Pauli matrix.

To obtain the phase diagram in the (KB, Ks, q ~ , qs) space (or equivalently in the ( ~ B T I J B , JsIJB, q ~ , qs) space) within the RG approach we have to establish the recurrence relations

/' I I

{+Ac I I Notice that the bulk affects the surface but the oppo-

0 /' I \I

I

I site is not true, since the surface is subextensive with

0 .I 2

J~'JB respect to the bulk.

Fig. 1. - Phase diagram for the spin 1/2 Ising ferromagnet We expect the Ising particular case (i.e., VB = in the semi-infinite simple cubic lattice with a ( 0 , 0, 1) free VS = 1) to be a subspace which remmains invariant

surface. under renormalization since it presents a fully devel-

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C8

-

1620 JOURNAL DE PHYSIQUE oped axial symmetry. For this to occur it suffices that

h (KBl 1) = k (KB, Ks, 1 , l ) = 1 for all (KB, Ks)

.

The recurrence equations then become

We finally expect that the isotropic Heisenberg particular case (i.e., q~ = 7 ) ~ = 0) also remains invariant

under renormalization, since nothing disturbs the per- fect isotropy of the interactions. For this to occur it suffices that h (KB, 0) = k (KB, Ksr 0,O) = 0 for all

(KB, Ks)

.

The recurrence equations then become

Both the above invariances (i.e., qij = 1 and q j = 0)

can be simultaneously satisfied if we establish the RG equations (2) following reference [4]. The method essentially consists in preserving, through renormaliz% tion, the correlations between the roots (or terminal sites) of conveniently chosen two-rooted finite graphs. In other words, . Tr e - P H is to be preserved

Internal sites

under renormalization. If the model is classical (which happens here if 77ij = 1)

,

this method provides the ex- act answer .[5] for the hierarchical lattice associated with the particular graph that has been chosen. There is no unique manner for determining the best choices for graphs: this partially relies on a kind of "culinary art"

.

Two-rooted graphs very well fitted t o the simple cubic lattice have been introduced in reference [6].

Here we shall rather follow reference [7] and adopt the simpler twe-rooted graphs, Migdal-Kadanoff-like, indicated in figure 2. By so doing we partially loose the information corresponding t o the particular site con- nections existing in the simple cubic lattice and its (0, 0, 1) free surface, but the results are expected t o remain qualitatively (even quantitativelly occasionally)

--

Fig. 2. - RG cluster transformation for the bulk (a) and its free surface (b); (e) and ( 0 ) respectively denote internal

and terminal sites.

correct: this has in fact been so for the Ising particular case (rlij = 1).

Let us now focus a problem of further complexity, namely the calculation, for all temperatures, of the bulk magnetization MB as well as of the surface one Ms. To perform this calculation we adopt the simple RG procedure introduced in reference [8] and foIIow along the lines of reference [9]. Excepting for appropri- ate scaling factors, MB and Ms respectively coincide with pg and p ~ , the dimensionless elementary magnetic dipoles respectively associated with the bulk and the surface sites (see [8, 91 for details). The RG equations for p~ and ps are established by imposing that the bulk and surface total magnetic moments (mten- sive quantities) must be preserved through renormalization. The equations have the following forms:

Together with equations (2) they close the RG procedure. For q~ = q~ = 1 they become

and, for q~ = % = 0, they become

Equations ( 6 ) (Eqs. (7)) are t o be used together with equations (3) (Eqs. (4)) to study the Ising (anisotropic Heisenberg) particular case.

3. Influence of t h e n a t u r e of t h e interactions The Mermin-Wagner theorem

[lo]

essentially states that no spontaneous magnetization at finite temperatures can exist if the three following conditions are simultaneously satisfied:

(i) the system only involves short-range interactions; (ii) the system is two dimensional;

(iii) the symmetry breakdown corresponds to interactions which are associated with a continuous group of symmetries.

The question we want to focus is whether the SF can exist if vs = 0, no matter the value of VB. - . There is a relatively widespread confusion concerning this central point. A (fallacious) argument is frequently raised

as follows: condition (i) is obviously satisfied; the surface is a two dimensional system hence condition (ii) also is satisfied; finally qs = 0, therefore condition (iii) also is satisfied; consequently the SF cannot exist and this for any value of q ~ . In fact this is not true: it is only if VB vanishes as well that the SF cannot exist.

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tonh K S

I

lanh K,

I

0 0.5 tanh KB

Fig. 3. - RG flow diagrams in t h e invariant subspaces (a) 7~ = 1, 7 s = 1 and (b) r ] ~ = 7s = 0; ( m ) , (0) and ( 0 )

respectively denote trivial (fully stable), critical (semi-stable) and multicritical (fully unstable) fixed points; dashed lines are indicative; BF, S F and P respectively denote the bulk ferromagnetic, surface ferromagnetic and paramagnetic phases.

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C8 - 1622 JOURNAL DE PHYSIQUE

theorem does not apply because condition (ii) is not satisfied, the system strictly being a d = 3 one (semi- infinite bulk). And this is of course correct. However we believe, on physical grounds, that the surface magnetization vanishes, for A

>

A,, exponentially while

entering deep into the bulk. This exponential behavior

Fig. 6.

-

Surface spontaneous magnetization MS as a func- tion of the temperature for the semi-infinite Ising ferromag- net; J s 1 . 7 ~ = 0,0.5,1 correspond t o A

<

A,; JS/JB

"

1.74 corresponds t o A = A,; JS/JB = 2, 2.5 and 3 correspond t o A

>

A,. In this last case, at the bulk critical point there is a discontinuity in the thermal derivative of Ms. The bulk spontaneous magnetization MB is also shown (dashed line) as a reference.

Fig. 5. - (vB,vs)-dependence of the location A, of the multicritical point appearing in figure 4. Notice, in the

r ) ~ = 0 plane, a slight minimum.

leads to a system which can, in practice, be considered as a oo x co x finite one, i.e., a d = 2 one, and consequently condition (ii) can, in practice, be considered as satisfied. Consistently, only remains condition (iii) as the one whose satisfaction would lead us out of the fallacy. And it is indeed so, because to satisfy (iii) all the relevants interactions have to satisfy it, which is not the case. Indeed the finite width of the active sur-

face includes much more layers than the unique free surface layer (for which 0 s = 0). Summarizing, the SF phase is expected to exist if at least one of qs and VB

is non vanishing. It must disappear only if both QS and

VB vanish since then the Mermin-Wagner theorem, in

practice, applies. In figures 3 and 4 we reproduce the RG results [7] which are consistent with the above con-

It is well known that

and

in the neighborhood of the temperatures where MB and Ms vanish. All these facts are consistently recov-

ered within the present RG approach. What we want to discuss here is a more subtle point, namely the singularity which might exist in Ms (T) at the bulk critical point

~2~

if A

>

A,. We expect, in this case,

siderations. _{The central questions we address here are what the} Let us discuss a last point' For the SF phase to _{values of}_{A - / A + ,}

z+ and z- are. The present RG disappear in the (VS, VB) + (0, 0) limit, it is not nee-

numerical [g] are consistent with %+ = x- =

essary that A, diverges: it could well remain finite and _and_A- ₊ _{i.e., a discontinuous temperature}

even so the SF phase through the collapse of _{derivative of}_{Ms (A+ and A- depend on Js/}_JB,_but the P-SF critical line on the SF-BF critical line. It is _{the ratio A - / A + roughly remains equal}_{to 4). This} necessary that Ac diverges, it does (as shown _{result is in ,-Iear} _{with Mean Field c&-ulations}

in Fig. 5). _[ll]_{since they yield}_{A - / A +}₌_1,_{i.e., a}_continuous temperature derivative of Ms.

4. Surface magnetization On experimental grounds, surface magnetism in Gd

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that Gd seems to be close to the isotropic Heisenberg model, whereas the present calculation has been done for the Ising model. The theoretical treatment of the dependence of A-/A+ on (VB, qs) would enlighten this point, but this is still t o be done. However, if even in the limit r ] ~ = qs = 0,

A - / A +

remains different from unity, then the understanding of Rau and Robert results should be searched elsewhere.

Since the bulk order acts on the surface as an exter- nal magnetic field, one could expect for A-/A+

<

1. It is however A-/A+

> 1

which actually occurs. We believe this must be due to surface-bulk correlation phenomena: a kind of feed-back effect related to the fact that the bulk is magnetically more susceptible for

T 2

~2~

than for T 5 T : ~ . Equivalently, for T 2 T : ~ the bulk acquires a relatively important magnetization induced by the surface magnetization; this bulk magnetization in turn enhances Ms.

Let us finally mention that we observe in figure 6 a slight, but surprising, non monotonicity in the Ms vs.

T curves as function of

Js/JB

for

Js/JB

<<

1.

Domb and J. L. Lebowitz (Academic Press) Vol. 8 (1983).

[2] Tsallis, C., Influence of Dilution and Nature of the Interaction on Surface and Interface Magnetism in Magnetic Properties of Low-Dimensional Sys- tems, Eds. L. M. Falicov and J. L. Moran-Lopez (Springer-Verlag) 1986.

[3] Diehl, W., Field-theoretical Approach t o Critical Behavior at Surfaces in Phase Transitions and Critical Phenomena, Eds.

C,

Domb and J. L. Lebowitz (Academic Press) Vol. 10 (1986). [4] Caride, A. O., Tsallis, C. and Zanette, S. I., Phys.

Rev. Lett. 51 (1983) 145;

Mariz, A. M., Tsallis, C. and Caride, A. O., J.

Phys. C. 18 (1985) 4189.

[5] Berker, A. N. and Ostlund, S., J. Phys. C 1 2

(1979) 4961;

Bleher, P. M. and Zalys, E., Commun. Math.

Phys. 67 (1979) 17.

161 Da Silva, L. R., Tsallis, C. and Schwachheim, G., J. Phys. A 17 (1984) 3289;

5. Conclusion Costa, U. M. S., Tsallis, C. and Sarmento, E. F.,

J. Phys. C 18 (1985) 5749;

The main results of the present RG treatment of sur- _{Da Silva, L.}_{R., Costa, U. M. S. and Tsallis, C.,} face magnetism in a semi-infinite ferromagnetic bulk _J._Phys._C₁₈_{(1985) 6199.}

can be summarized as follows. _{[7] Mariz, A. M., Costa,}_{U. M. S. and Tsallis, C.,} (i) The surface ferromagnetic (SF) phase exists as _{Europhys. Lett.}₃_{(1987) 27.}

long as either the surface or the bulk (or both) interac-

[8] Caride, A. 0 . and Tsallis, C., J. Phys. A 20

tions are Ising-like. It disappears in the simultaneous

surface and bulk isotropic Heisenberg case, and it does (1987) L665.

so through the divergence of A,. The entire behavior [9] Chame, A. and Tsallis, C., to be published. is consistent with the Mermin-Wagner theorem; [lo] Mermin, N. D. and Wagner, H., Phys. Rev. Lett.

(ii) At the bulk critical point and for A

>

A,, the 17 (1966) 1133.

temperature derivative of the surface magnetization is

[l11

L u b e n s k ~ j T. C. and Rubin, M. H., P h ~ s . Rev. discontinuous for the Ising model. This could be con- 12 (1975) 3885;

sidered as consistent with the surface tension exper- Bray, A. J. and Moore, M. A., J. Phys. A 10

iments [14] on liquid *He, but clearly disagrees with (1977) 1927.

the recent experiments [13] on Gd ( ~ o s s i b l ~ isotropic [12] Weller, D., Alvarado, S. F., Gudat, W., Schroder, Heisenberg like). In order to clarify this question, the K. and Carnpagna, M., Phys. Rev. Lett. 54

study of the influence of the nature of the magnetic (1985) 1555.

interactions on the surface magnetization temperature [13] Rau, C. and Robert, M., Phys. Rev. Lett. 58

derivative would be very welcome. (1987) 2714.

[I] Binder, K., Critical Behavior at Surfaces in Phase [14] Magerlein, 3. H. and Sanders, T. M., Phys. Rev.