Surface ferro (or antiferro) magnetism in bulk antiferro (or ferro) magnets : renormalization group analysis

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Surface ferro (or antiferro) magnetism in bulk antiferro (or ferro) magnets : renormalization group analysis

E.F. Sarmento, C. Tsallis

To cite this version:

E.F. Sarmento, C. Tsallis. Surface ferro (or antiferro) magnetism in bulk antiferro (or ferro) magnets : renormalization group analysis. Journal de Physique, 1986, 47 (7), pp.1115-1118.

�10.1051/jphys:019860047070111500�. �jpa-00210297�

Surface ferro (or antiferro) magnetism ^{in bulk antiferro} (or ferro) magnets :

renormalization _group analysis

E. F. Sarmento (*) ^{and C. Tsallis} (+)

(+) Centro Brasileiro de Pesquisas Fisicas, CNPq/CBPF, Rua Dr. Xavier Sigaud, 150, 222902014 Rio de Janeiro, RJ, ^Brasil

(*) Departamento ^de Fisica, Universidade Federal de Alagoas, 57000 2014 Maceio, AL, ^Brasil (Reçu le 8 novembre 1985, accepté sous forme définitive ^{le 10 mars} 1986)

Résumé.

Les techniques de groupe de renormalisation dans l’espace ^{réel sont} appliquées pour la première ^fois

au magnétisme de surface de systèmes magnétiques, pour ^tous les signes ^des constantes de couplage ^{de surface}

et de volume. En particulier, ^{nous nous sommes} int6ress6s au modèle de Potts à q ^{états et nous exhibons une} intéressante variation du diagramme ^de phase ^{en fonction de} la valeur de _q. Ainsi dans le cas du modèle d’Ising (q ⁼ 2), ^un phénomène remarquable apparaît ^{dans le cas} d’interactions de surface et de volume de signe opposé :

l’ordre de surface ferromagnétique (ou antiferromagnétique) peut disparaître quand ^on augmente ^la température

d’un système antiferromagnétique (ou ferromagnétique) ^{dans le} volume, puis réapparaître ^avant que le système

ne devienne paramagnétique.

Abstract

The renormalization group techniques ^are applied, for the first time, ^{to surface} magnetism ^{in bulk} magnets, for ^all signs ^of surface and bulk coupling constants. The q-state ^{Potts model is} specifically focused upon, and an interesting q-evolution ^{of the} phase diagram ^is exhibited. In particular ^the Ising ^model (q ⁼ 2) presents ^a remarkable feature in the case of competing bulk-and surface interactions : surface ferro (or antiferro) magnetism

can disappear ^while heating ^{an antiferro} (or ferro) magnet, and reappear again ^for higher temperatures, ^{before the} system ^{enters in the} paramagnetic phase.

Classification

Physics ^Abstracts

05.50 - 05.70F - 64.60C - 75. lOR

Ordering ⁱⁿ semi-infinite systems (such ^{as bulk}

materials presenting ^{a free} surface) ^{can in} principle

exhibit interesting conflicts, when the surface favours

a type ^of ordering ^which competes with that favoured

by the bulk. This is typically ^{the case of surface} (crystalline ^or amorphous) ^structure reconstruction.

This can also be the case of magnetic systems ^where the surface coupling ^constant J$ differs in sign ^from

the bulk coupling ^constant JB (see ^{the recent review} by ^Binder [1] ^{and Ref.} therein). ^{This seems to be} precisely ^{the case of Cr} (antiferromagnet ^{with a Neel} temperature ^{of 312} K). ^Its (100) free surface has been very recently investigated [2] (using angle-resolved photoelectron spectroscopy), ^{and it} presents ^a ferro- magnetic ordering up ^to 780 + ^{50 K. The} experiment

has been deviced with particular ^{care for} avoiding

undesirable surface contamination effects [3].

Magnetic competition between surface and bulk has already ^received theoretical attention through

Mean Field Techniques [4] applied ^{to the} spin 1/2 Ising model. However the problem ^{presents a theo-}

retical and experimental ^interest which deserves

alternative (and ^more sophisticated) approaches. ^In

the present ^{letter we} discuss theoretically ^the q-state

Potts model (from ^{which the} Ising model is recovered

for q ⁼ 2, ^{and bond} percolation for q ^{= 1 and ferro-} magnetic coupling constants) in semi-infinite simple

cubic lattice assuming the free surface to be a (1, 0, 0)

one. The framework is a real-space renormalization group (RG) one; although ^such treatments are

already ^available [5, 6] ^{for the} ferromagnetic ^case,

this is the first time (as ^{far as we} know) they ^are applied allowing ^{for all} signs of Js and JB. ^{We follow} along ^the

lines of reference [6] ^{and use} Migdal-Kadanoff - like

b ⁼ 3 clusters (b ^is the RG linear expansion factor)

which have been quite satisfactory for the ferroma-

gnetic ^case (Js > 0 and JB > 0).

We consider the Hamiltonian

where ( i, j > ^denotes nearest-neighbours, ^and Jij equals JS(JS 0) when both sites i and j belong ^{to the}

free surface, ^and equals JB(JB > 0) otherwise. It is

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047070111500

1116

convenient to introduce the variables (thermal ^trans-

missivity ; ^{see Ref} [7] ^{and Refs.} therein)

which, for q > 1, varies in the interval [ -l/(q -1), 1]

(r = S, B).

To construct the RG recursive relations in the

(tB, ts) ^{space, we use the} procedures described in reference [7]. The bulk and surface equations ^are

obtained through the renormalization indicated in

figures l(a) ^and 1 (b), ^which immediately yields

and

with

x(t) - { ( 1 - t 3)/[l ⁺ (q - 1) t3] } 3. ^These equations completely close the recurrence problem, ^and yield

the results presented ⁱⁿ figure ^2.

For q = ^{2 we} have indicated the RG flow in detail

(Fig. ² (a)) : ^{this can} be considered as a prototype ^as, for all q, ^{the RG} flow scheme varies smoothly ^with q.

Seven physically ^different phases ^are identified through

seven trivial (fully stable) ^fixed points (denoted by o

in Fig. ^2, excepting ^two of them which are denoted bye because, for q ⁼ 2, they ^are super-imposed ^{to semi-}

stable fixed points). ^These phases ^{are the} paramagnetic (P), ^{the bulk} ferromagnetic (BF), ^{the bulk} antiferro- magnetic (BAF), ^the surface ferromagnetic (SF ; ^{the bulk}

is magnetically disordered), ^the surface antiferromagne-

tic (SAF; ^{the bulk is} magnetically disordered), ^the

Fig. ^1.

RG recursive relation for the bulk (a) ^{and the}

surface (b) transmissivities (0 ^{and 4D} respectively ^denote

terminal and internal sites). ^The big cluster of (b) ^contains only ^{six branches as the} other three correspond ^{to the}

vacuum. The cluster renormalization transformation (exact

for the present model) consists in preserving ^{Tr e - pH where}

we trace over the internal sites.

Fig. ^2.

q-evolution of the RG flow and phase diagrams

in both (tB, ts) ^and (r, JSIJ,) spaces, where

Tic being ^{the bulk Curie} temperature. m, 4D ^{and 0} respecti- vely ^{denote trivial} (fully stable), ^critical (semi-stable) ^and

multicritical (fully unstable) ^fixed points. ^The dashed lines

are indicative.

simultaneous surface ferromagnetic ^{and bulk anti-} ferromagnetic (SF/BAF), and the simultaneous surface

antiferromagnetic ^{and bulk} ferromagnetic (SAF/BF)

ones. A great ^{amount of} physically different critical

universality ^{classes are} exhibited : these correspond

to the P-SF, P-BF, P-SAF, P-BAF, SF-BF, SAF-BAF, SF-SF/BAF, SAF-SAF/BF, SF/BAF-BAF ^and SAF/

BF-BF critical lines. Also four physically ^different multicritical universality ^{classes are} present (characte-

rized by ^four fully unstable fixed points ^denoted by ⁰

in Fig. 2) : they correspond ^{to the} P-BF-SF, ^P-BAF- SAF, P-SF-BAF-SF/BAF ^and P-SAF-BF-SAF/BF

multicritical points. ^{It is} worthy ^{to note that the}

present ^RG recovers, for q ⁼ 2, ^{the well known} (JB, Js) ^- ( - JB, - Js) symmetry (which, among

others, determines the isomorphism between ferro-

magnetic ^and antiferromagnetic Ising ^{models in}

square and simple ^cubic lattices). ^{As a} consequence of this fact, ^several universality classes which are

different for q # 2, ^{coincide two} by ^two for q ^{= 2.}

Let us now focus on the q-evolution ^{of the} (tB, ts) phase diagrams (see Fig. 2). When q increases above 2, the ts ^{= -} 1 /(q - 1) ^{axis as well as} the tB ⁼

-

1 /(g - 1) ^one approach ^the origin t. = ts ⁼ 0, ^suc- cessively forcing ^several phases ^to disappear. ^We adopt ^the following notation : ql ^{is the} particular

value of q for which the P-SAF-BF-SAF/BF ^multi-

critical point touches the ts ^{= -} 1/(q - 1) ^axis (and consequently ^the SAF/BF phase disappears), q2 is

the value for which the P-SAF critical point (at tB =o)

touches the same axis (and consequently ^{the SAF} phase disappears ^{for the} square lattice), q3 is the value for which the P-SAF-BAF multicritical point

touches the same axis (and consequently ^{the SAF} phase disappears), and q4 is the value for which the P-

SF-BAF-SF/BAF multicritical point ^touches the tB ⁼

-

1 /(q - 1) ^axis (consequently ^{the BAF} phase disap- pears). ^The following inequalities ^are satisfied : 2 ql q2 q3 q4. Baxter has _proven [8] ^that

q2 ⁼ 3. The present ^RG approximation yields

qi ⁼ 2.21, q2 ⁼ 2.25, q3 ⁼ 2.28 and q4 ^{= 2.64. One}

of the most interesting features is the ^« re-entrances » that are present ^{for the} Ising model with competing

surface and bulk coupling constants. If we have let us

say ^a ferromagnetic ^bulk (JB > 0), ^{and an antifer-} romagnetic ^surface (Js 0), ^{within an} appropriate

range of Js/JB, ^{the surface} antiferromagnetic ^order

parameter (sublattice magnetization) ^{can vanish and}

then _reappear below the bulk Curie temperature Tc, presumably presents ^{a soft} singularity ^at Tic, ^and finally ^vanish again ^{at a} Neel- like temperature ^above

Tc. ^In spite ^{of its} apparent strangeness, ^this interesting phenomenon ^{can be} intuitively understood in some sense if one takes into account that the bulk ferroma- gnet acts, ^{on the} (relatively strong) surface antiferro- magnet ^{and as} long ^as non-negligible bulk order exists, quite similarly ^{to an} external uniform magnetic ^field acting ^{on a standard} antiferromagnet (we recall that the Neel temperature ^{of the} antiferromagnetic ^surface

would be proportional ^to Js if ^{the surface were} magnetically isolated from the bulk). ^{If we denote} by

J i I J B the value of the ratio JS jJB associated with each

one of the four multicritical points, ^the present ^RG yields the results depicted ⁱⁿ figure ^3; they ^satisfac- torily ^compare with series [4] ^{and Monte Carlo} [9]

results available for q ^{= 2. For} large ^values of q (q > q*

where q* = 4 for d = ^{2 and} q* = ^{3 for d =} 3), ^{it is}

well known that the Potts ferromagnet phase ^transition

is a first order one. For such _range of q ^an enlarged

RG parameter space would be required ^{in order to}

include vacancy-like degrees ^{of freedom, known to be} responsible for the first order phase transitions of the model [10].

Let us summarize the present ^work by saying ^that

the renormalization _group analysis that has been undertaken suggests qualitatively interesting ^features

in the full phase diagram (both signs ^for JB and Js)

Fig. ^3.

q-evolution ^of J’IJ, (for I JS/JB I above I J’/JB ^I

surface magnetic ordering can occur even in the absence of bulk ordering). At q ⁼ 2, J’IJ, ^{equals 1.60} (series [4]),

1.50 (Monte ^Carlo [9]) ^{and 1.74} (present RG) for the P-BF- SF case, and equals - ^1.9 (series [4]) and - 2.28 (present RG). ^{If intended for the} simple ^cubic lattice, ^the present ^RG

cannot be retained much above q ^{= 3 as the bulk} transitions

will become of the first order; if intended for the hierarchical

lattice of figure ^{1, it is exact} for all values of _q.

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of the Potts model. Further experimental and/or

theoretical evidence would be _very welcome. In

particular, ^{it would be} interesting ^to establish, ^on (nearly) ^exact grounds, whether the peculiar phase diagram herein obtained for _q ⁼ 2, indeed _appears at q ⁼ 2, ^{or rather is to be} associated with q > 2.

Also the exact value [8] ^for q2 being ^3, the unknown exact values for q3 and q4 ^are presumably higher

than 3. Consequently they possibly belongs ^{to the}

interval of q where the transitions are of the first order type (we recall that q*(d ⁼ 3) N 3). Nevertheless

we have good hope ^{that the} q-evolution ^{of the} phase diagram ^remains qualitatively ^{the same} (e.g., ^RG’s using self-dual clusters yield ^{the critical} point

t c = l/vq + 1), which is the exact one for the square lattice for all values of _q, and not only ^for q q*(d ⁼ 2) ⁼ 4). Finally the surface magnetization

maximum (as function of temperature) very recently

detected on epitaxial ^Gd [11] could be related to the type ^of phenomena ^described herein; indeed both works present ^{the same} peculiarity, namely ^surface

order parameter which varies in a non monotonous manner with temperature.

Acknowledgments.

We acknowledge useful discussions with R. J. Celotta,

J. S. Helman and L. R. da Silva. One of us (C.T.) ^is grateful ^hor hospitality ^{received at} Universidade Federal de Alagoas where this work was initiated;

the other one (E.F.S.) acknowledges hospitality ^receiv-

ed at Centro Brasileiro de Pesquisas Fisicas where it was concluded, ^{as well as} partial support ^form CNPq/Brazil.

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