Anisotropic Heisenberg surface on semi-infinite Ising ferromagnet : renormalization group treatment

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Anisotropic Heisenberg surface on semi-infinite Ising ferromagnet : renormalization group treatment

U.M.S. Costa, A.M. Mariz, C. Tsallis

To cite this version:

U.M.S. Costa, A.M. Mariz, C. Tsallis. Anisotropic Heisenberg surface on semi-infinite Ising ferro-

magnet : renormalization group treatment. Journal de Physique Lettres, Edp sciences, 1985, 46 (18),

pp.851-859. �10.1051/jphyslet:019850046018085100�. �jpa-00232909�

L-851

Anisotropic Heisenberg ^{surface on} semi-infinite

Ising ferromagnet : renormalization _group treatment

U. M. S. Costa (1,2), ^{A. M. Mariz} (1,3) and C. Tsallis (1)

(1) Centro Brasileiro de Pesquisas Fisicas/CNPq, Rua Dr. Xavier Sigaud, ^{150 -} Urca,

22240 - Rio de Janeiro, RJ - Brazil.

(2) Departamento ^de Física, Universidade Federal de Alagoas, ^Cidade Universitária,

57000 - Maceió, Al - Brazil.

(3) Departamento ^{de Fisica Teórica e} Experimental, Universidade Federal do Rio Grande do Norte, Campus Universitário, ^{59000 -} Natal, RN - Brazil.

(Reçu ^le 24 juin 1985, accepte ^le 29 juillet 1985 )

Résumé.

Nous utilisons une transformation de renormalisation de Migdal-Kadanoff pour étudier le comportement critique ^d’un ferromagnétique d’Ising, cubique simple, cubique infini dont la surface libre (1, 0, 0) contient des interactions anisotropes (dans l’espace ^des spins) ferromagnétiques ^de Heisenberg. ^Le diagramme ^de phase présente ^trois phases (paramagnétique, ferromagnétique ^en

volume et ferromagnétique ^en surface) qui ^se rejoignent ^{en un} point multicritique. ^Les coordonnées de ce point ^{sont calculées en} fonction de l’anisotropie. ^Les diverses classes d’universalité du problème

sont mises en évidence.

Abstract

We use a Migdal-Kadanoff-like renormalization group approach ^to study the critical behaviour of a semi-infinite simple ^cubic Ising ferromagnet ^whose (1, 0, 0) free surface contains

anisotropic (in spin space) Heisenberg ferromagnetic interactions. The phase diagram presents ^three phases (namely ^the paramagnetic, ^{the bulk} ferromagnetic and the surface ferromagnetic ones) ^which join ^{on a} multicritical point The location of this point is calculated as a function of the anisotropy.

The various universality classes of the problem ^{are exhibited.}

LE JOURNAL DE PHYSIQUE - ^LETTRES

J. Physique ^{Let!. 46} (1985) L-851 - L-859 15 SEPTEMBRE 1985, :

Classification

Physics ^Abstracts

05.50

05.70F - 75.10J

82.65D

1. Introduction.

The appearance of surface effects in magnetic systems has been studied extensively during ^last

years. Both theoretical and experimental efforts have been dedicated to the subject (see ^{Ref [1]}

for a recent review) because of its various applications ^{as well as} its intrinsic richness. A quite interesting and realistic particular situation is that where the surface magnetic interactions

differ, ⁱⁿ strength ^and evenjin nature, from the bulk ones. A possible experimental implementation

of such a situation can be done by adsorbing, ^on top ^{of the} magnetic bulk, ^a layer ^{of different} magnetic ^atoms. Among the various theoretical approaches ^{that can be used to} discuss this

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019850046018085100

L-852 JOURNAL DE PHYSIQUE - ^LETTRES

type of system, ^the real-space renormalization-group (RG) ^{is a} quite convenient one as it hope- fully provides ^a good description ^{of its} criticality [2-6].

In the present paper ^we discuss ^a spin 1/2 semi-infinite simple ^{cubic lattice} Ising ferromagnet

with a (1, 0, 0) ^{free surface} (square lattice) whose interactions are of the anisotropic Heisenberg type. ^Our approach ^{is a} Migdal-Kadanoff-like ^{RG which follows} along the lines of references [5]

and [7]. ^{In section 2 we} introduce the model and formalism, in section 3 we present the results and finally ^we conclude in section 4; operational ^{details are} given ^{in the} Appendix.

2. Model and formalism.

We consider the following Hamiltonian :

where i, j ? ^{runs over all} pairs ^of first-neighbouring ^{sites on a} semi-infinite simple ^{cubic lattice}

with a (1,0,0) ^{free surface} (see Fig. 1); ^{the a’s are} the standard Pauli matrices; (Jij, ’1ij) ^equals (JBI 1) ^{in the bulk} (Jp ~ 0), ^and equals (Js, ’1) ^{on the} free surface (Js > ^0, 0 ~ ’1 ~ 1 ). The 1’/ ^{= 1} and ’1 ^{= 0 limits} respectively correspond ^{to the} Ising ^and isotropic Heisenberg ^{models. We}

introduce the following convenient variables :

where T is the temperature and kB the Boltzmann constant

The phase diagram ^{for the} purely Ising ^case (’1 ⁼ 1) presents ^three phases [5, 6, 8-12], namely

the bulk ferromagnetic (BF ; both the bulk and the surface are magnetized), ^the surface ferroma- gnetic (SF; finite surface but vanishing ^bulk magnetizations), ^{and the} paramagnetic (P ; ^no spon- taneous magnetization) ^{ones. We intend to} study the evolution of the phase diagram ^when

quantum aspects (’1 #= 1) ^are introduced in the free surface. To perform ^this analysis ^{we follow}

Fig. ^1.

Semi-infinite simple ^{cubic lattice. The dashed} (full) ^lines represent ^{the surface} (bulk) ^bonds

associated with (Js, j7) (with (Jp, 1)).

Fig. ^2.

^RG cell transformation : (a) for the bulk (Kg ~ JB/kB T ) ; (b) for the free surface (Ks ⁼ JS/kB T).

0 and. respectively represent the terminal and internal (being decimated) ^nodes.

along the lines of reference [5] ^{and construct a RG with} Migdal-Kadanoff-like ^clusters [13, 14]

(see Fig. 2) for both surface and bulk bonds. The partition function of each cluster is preserved through ^the renormalization. The clusters of figure ^{2 are} reducible in sequential series and parallel operations. ^This fact enables the quick (and exact) calculation [15] of the cluster of figure 2a, all the bonds of which are Ising (i.e., classical) interactions. We also take advantage of this fact to make easier the calculation of the cluster of figure ^{2b. In this case} however, quantum ^effects

are present and therefore the result is not exact; nevertheless it constitutes an excellent approxi-

mation [7].

For the bulk we obtain the following recursive relation :

which is equivalent ^to equation (7) ^{of reference} [5]. ^This equation ^{admits two trivial} (stable)

fixed points (at KB ^{= 0 and} KB ⁼ oo), ^{as well as a critical} (unstable) ^one (at ^a finite value of KB).

The calculation of the recurrence for the surface is more complex. ^We first solve (see Appendix)

a series _array of three bonds (each of them associated with (Ks, ’1)), ^and obtain the equivalent

parameters ^noted K3(Ks, ’1) ^and r~3(Ks, ’1). If the three bonds are of the bulk type (ie., Ising interactions), ^the equivalent parameters ^are K3 (KB, 1) ^and qs(K 3 B, 1 ) ^{= 1. We now} approach

the cluster of figure ^2b by ^six parallel ^branches [being three of the type (K3(Ks, ’1), r~3(Ks, 1]))

and three of the type (K3 (KB, ^1), 1)] ^{and obtain} [7] ^the following recurrence :

Equations (5) ^to (7) formally close the construction of the RG : the flow they ^determine (in ^the (KB, Ks, ’1) ^{space for} instance) ^{yields the phase diagram} and exhibits the relevant universality

classes.

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3. Results.

The present ^RG provides ^{the flow} diagram indicated in figures ^{3a and 3b} (in ^the (tB, ~s? r~) space), ^{where we note two} interesting ^invariant subspaces, namely il ^{= 1} (purely Ising problem) and" = tB ^{= 0} (fully bulk-disconnected isotropic Heisenberg ^free surface). ^{We also note the} following ^{facts :} (i) ^{three trivial} (fully stable) ^fixed points, namely ^at (tB, tS, 17) ⁼ (0, 0,1 ), (1,1,1)

and (0, 1, 1), characterize the three phases ^{of the} system, respectively ^the P, ^{BF and SF} ones;

(ii) ^five main semi-stable fixed points ^are present, namely ^at (tB, ts, ~) ⁼ (tB3D, 1, 1 ) (characterizing

the standard D ⁼ 3 universality class; tB3 Dcorresponds ^to the finite critical temperature ^for the D ⁼ 3 Ising model), (tB D, tl, ¹⁾ (characterizing ^{the non trivial} universality ^{class associated}

with the surface phase transition occurring simultaneously ^{with the bulk} one ; ts ^{is a finite cons-}

tant), (0, ts D, 1) (characterizing the standard D ⁼ 2 universality class; tS2 Dcorresponds ^{to the}

finite critical temperature ^{for the D = 2} Ising model), (~, ts B, 1 ) (characterizing ^the universality

class of the multicritical line where all three P, ^{BF and SF} join together) ^and (0,1,0) (charac- terizing ^the universality class of the D ⁼ 2 isotropic Heisenberg ^{model, whose critical} tempe-

rature vanishes); (iii) ^one fully ^{unstable fixed} point ^at (tB, ts, q) ⁼ (tB3 D, 1, 0) ^{which is a} special point ^on the above mentioned multicritical line, and determines by ^{itself a new} universality

class. In short the universality ^{classes of the} problem ^are Ising-dominated, ^{and the} isotropic Heisenberg ^nature prevails only ^{when no source of} Ising symmetry ^{exists at all. In} figure ^{4 we} present ^cuts of the critical surface (in ^{the (A, T)} space) ^{for typical} values of the anisotropy ^fl.

The location of the multicritical point ^{as a function} of ’1 is indicated in figure 5 ; Ae is ^{the value}

of A above which surface magnetic ^{order can subsist even} if the bulk is disordered). For" ^{= 1}

we recover the result of reference [5], i.e., ~ ~ ^{0.74, which compares} reasonably well with the series result [8] 0.6 ± 0.1, ^{the Monte Carlo one} [11] ^{0.50 ±} 0.03, and other RG result [6] ^0.569.

4. Conclusion.

We have used a simple renormalization _group approach, ^with Migdal-Kadanoff-like ^clusters,

to study surface critical effects in semi-infinite lattices of spins 1/2 which interact through ^standard

Fig. ^3.

^{RG flux} diagram and critical surface : (a) ^{in the} (tB, ts, r~) space (tr ^{=- tanh} Kr (r ⁼ B, S));

(b) ^for the" ^{= 1} subspace (purely Ising problem). 8 denotes trivial (fully stable) ^fixed points; . ^denotes

critical (semi-stable) ^fixed points; ⁰ denotes the multicritical (fully ^{unstable) fixed} point P, ^{SF and BF}

respectively denote the para-, surface ferro- and bulk ferromagnetic phases. Dashed lines are indicative.

Fig. ^4.

Fixed ~ ^cuts of the critical surface in the J 2013 T space (J ⁼ JS/JB - 1).

Fig. ^5. - 11 dependence ^of J..

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Ising ferromagnetic couplings ^{in the} bulk, ^and through anisotropic Heisenberg ferromagnetic couplings ^on the free surface. The anisotropy ’1 is assumed to vary between 0 (isotropic ^Hei- senberg ^free surface) ^{and 1} (purely Ising problem, relatively well discussed in the literature).

The present ^{results can} alternatively ^be applied ^to the hierarchical lattice determined by ^the topological recurrence in figure 2, and for such system they ^are strictly ^exact for ’1 ^{= 1} [17]

and approximate ^otherwise [7] ; ^{or be} applied ^to the semi-infinite simple cubic Bravais lattice with a (1, 0, 0) ^free surface, and for such system they are, for all values of _~, approximate (although qualitatively correct). ^The phase diagram presents ^three phases, namely ^the paramagnetic, ^the

bulk ferromagnetic and the surface ferromagnetic ^{ones. All three} phases join, ^{for a} given ^value

of _yy, on a multicritical point, whose location ^can be characterized by Ac (defined ^{as the} particular

value of d - JsIJB - 1 above which surface ordering ^{can exist even in the absence of bulk} ordering). Ac monotonically ^and continuously ^decreases while ’1 increases from 0 to 1. It attains its minimum for ’1 ⁼ 1, ^{where we} obtain J~ ~ 0.74 which _compares satisfactorily ^{with other}

values available in the literature. On the other hand, it reaches its maximum value at ’1 ⁼ 0, where it presents ^a finite ^value (Ar 2.5). ^{In other} words, ^for JS/JB high enough, ^{it is} possible

to have a surface-dominated phase transition occurring ^at temperatures above ^the bulk critical

point, ^even for ^the isotropic Heisenberg model, ^whose strictly two-dimensional version is believed to have a vanishing ^critical temperature. However, ^the paradox ^is only apparent ^{because our} system is ^not two-dimensional, ^a paramagnetic (deep) ^bulk being something quite different from

a disconnected bulk ; indeed, ^surface spins ^are correlated through both surface and bulk interac- tions. The bulk that has been assumed in this _paper is an Ising ^{one, which} clearly constitutes a

favourable situation for having bulk-assisted surface ordering. ^It is intuitive that bulk models closer and closer to the isotropic Heisenberg model will require higher ^and higher ^{values of} JS/JB for the surface ferromagnetic phase ^{to exist It is} possible that the ’1surface = 0 critical value Ac ^will diverge ^when the bulk model becomes the strictly isotropic Heisenberg ^model (’1bulk = 0)’ ^{We are} presentty working ^{on this} interesting problem, ^{in order to see, in} particular,

whether the RG approach agrees ^or disagrees ^{with a recent} random-phase-approximation

result [16] obtained for the FCC lattice which suggests the existence of a finite positive ^critical

value ’1c below which the surface ferromagnetic phase ^{cannot exist}

As intuitively expected, ^most regions of the critical surface belong ^{to the} universality ^classes

associated with the lowest symmetry ^{of the} problem, ^the Ising ^{model in our case. In} particular

all points ^{but one} (corresponding to ’1 ⁼ 0) of the multicritical line (on which all three phases join) belong ^to the bulk-surface Ising universality class; the q ^{= 0} point constitutes a non trivial

universality ^class by ^itself

Acknowledgments.

We acknowledge fruitful remarks from N. Majlis and S. Selzer.

Appendix.

We consider a series _array of three (K, ’1) bonds and four sites (see ^{Fig. 6, where (K’, ’1’) is to}

be identified with (K~, ’1~) of the main body ^{of the} paper). ^The array is characterized by ^the

dimensionless Hamiltonian

This _array is renormalized into ^a single bond whose dimensionless Hamiltonian is given by

Fig. 6.

RG transformation for a series array of three bonds.

where K~, ^K’ and ^{are to be found as functions} of (K, ’1) by imposing

where Tr denotes the trace over the internal sites of the series _array.

3,4

Let us now outline the intermediate steps of the calculation which follows along ^{the lines}

of reference [7]. ^{We first} expand both sides of equation (A. 3), ^{and obtain}

and

These two expressions replaced ^into equation (A. 3) immediately yield

By expressing J~i~ in the basis which diagonalizes ^{it we can} easily ^find K~ = Ko(a’, bi2, ~12)’

K’ = K’(a’, b i 2 , C’,2) ^and ’1’ ⁼ ’1’(a’, ~2~ ~12)’ By diagonalizing ~ 12 34 ^{we can} similarly ^find

a ⁼ a(K, ~), b12 - b12(K, ’1) and c~2 - C12(K, ~). ^All these relations together ^with equa- tion (A. 6) ^to (A. 8) provide the relations Ko = Ko(K, ~), ^{K’ = K’(K,} ’1) ^and ’1’ ⁼ ’1’(K, r~)

we were looking ^{for. We} obtain the following expressions :

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with

where the Ei’s ^{are the roots of a cubic} equation ^{and are} given by

with

and

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