A counterexample to the « maximal subgroup rule » for continuous crystalline transitions

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A counterexample to the “ maximal subgroup rule ” for continuous crystalline transitions

J.C. Tolédano, P. Tolédano

To cite this version:

J.C. Tolédano, P. Tolédano. A counterexample to the “ maximal subgroup rule ” for continuous crys- talline transitions. Journal de Physique, 1980, 41 (3), pp.189-192. �10.1051/jphys:01980004103018900�.

�jpa-00209234�

189

A counterexample ^{to the « maximal} subgroup ^{rule »}

for continuous crystalline transitions

J. C. Tolédano

Centre National d’Etudes des Télécommunications 196, rue de Paris, ⁹²²²⁰ Bagneux, ^France

and P. Tolédano

Groupe ^dc Physique théorique, ^Faculté des Sciences 33, rue St-Leu, ⁸⁰⁰⁰⁰ Amiens, France

(Reçu ^{le 16 octobre} 1979, accepté ^{le 5 novembre} 1979)

Résumé.

Nous décrivons, pour la première fois, ^un exemple théorique qui ^{contredit une} conjecture ^d’Ascher

consistant en une règle du sous-groupe maximum, applicable ^aux changements ^de symétrie qui ^se produisent

aux transitions cristallines du second ordre. Il s’agit ^d’un paramètre d’ordre à 4 dimensions qui ^{sous-tend une} repré-

sentation irréductible du groupe spatial Go ⁼ I41. ^Nous montrons, ^sur la base d’un développement ^{de Landau}

au 4e degré, que ^ce paramètre d’ordre induit des transitions vers deux groupes G1 ^et G2 tels que G2 ~ G1 ^~ Go (G2 ^non maximal). ^En revanche, ^nous n’avons trouvé aucun contre exemple ^{à la} conjecture plus précise ^{de Michel} qui exprime que la règle du sous-groupe maximum devrait être respectée relativement au groupe d’invariance

complet ^du polynôme de Landau du 4e degré. Ce groupe ^{est en} général ^un supergroupe de Go.

Abstract.

We describe for the first time a theoretical example contradicting ^Ascher’s conjecture ^{of a maximal} subgroup rule for the symmetry changes ^at continuous crystalline transitions. The example is that of a 4-dimen- sional order parameter spanning ^an irreducible representation of the space-group Go ⁼ I41. ^We show that this order parameter ^induces transitions towards groups G1 ^and G2, ^with G2 ^~ G1 ^~ G0 (G2 ^non maximal), ^{on the}

basis of a Landau free-energy ^{limited to fourth} degree ^terms. By contrast, ^we have found no counterexample

to the more precise conjecture by ^Michel stating that the maximal subgroup rule should hold with respect ^{to the} complete invariance group of the 4th degree ^Landau polynomial ^which is, ⁱⁿ general, ^a supergroup of Go.

LE JOURNAL DE PHYSIQUE

,1. Physique ⁴¹ (1980) ^{189-192 MARS} 1980,

Classitication Physics Ahslracls

64.60

In the Landau theory [1], ^the symmetry-change

which takes place ^{at a} continuous crystalline ^transi-

tion is determined by ^the symmetry properties ^of

the transition’s order parameter ^{whose n components}

span ^a n-dimensional irreducible representation ^of

the high-symmetry space-group Go. ^More precisely,

the space-group G of the low-symmetry phase ^coin-

cides with the complete ^{set of} symmetry operations belonging ^to Go ^and leaving ^{invariant the vector :}

where the (pi constitute a basis of the abstract vector

space E ^{of the} representation. Then coefficients

are particular ^{values of the order} parameter compo-

nents ’1i’ corresponding ^{to the absolute minimum}

of the Landau free-energy [1] F(’1i’ ^lox, f3k). ^{In the latter} polynomial expansion, ^{« N} (T - rj ^{is the} coefficient of the quadratic term, while ^the fik’s ^{are the coeffi-}

cients of the independent homogeneous polyno-

mials of higher degrees. For n > 2, depending ^on

the relative algebraic ^{values of the} f3k’s, ^{the absolute}

minimum of F, ^{below the} transition, ^can correspond

to various vector directions 6p; ^{E E,} generally ^asso-

ciated to distinct low-symmetry groups G;. ^To

enumerate the set of possible Gi groups compatible

with the considered order parameter, ^{one has there-} fore to locate, ^for the whole range of the f3k ^values,

all the directions 6p making ^{F an absolute minimum,}

and for each such direction to identify its invariance group G, subgroup ^of Go. ^{In this aspect, one has to} distinguish ^{between the cases} of discontinuous (first order) and continuous (second order) transitions.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004103018900

190

In the first case, the order parameter ^{does not} become vanishingly ^small, just ^{below the} transition.

Consequently, ^the enumeration of all possible groups

Gi, ^{has to take into} consideration the absolute minima of a free-energy ^F expanded ^{to an} arbitrary degree.

For such an expansion, ^{it has} been shown by ^Michel [2]

that a suitable choice of the expansion’s coefficients could _{make any} 6p ^{E E the} absolute minimum of F.

The possible low-symmetry groups ^are therefore the entire set { G } ^{of the} invariance _{groups of} all the directions bp ^{E E. This set} { G } ^{has been} effectively

determined by Janovec et al. [3] ^{as well as} by ^Michel

et al. [4] ^{for the} crystalline transitions which preserve the number of atoms in the unit cell. Also, Vinberg

et al. [5] ^have performed ^{such an} analysis ^{for the order} parameters ^{of the Pm3m} (Oh) space-group, which

comply with Lifshitz symmetry ^criterion [6].

For a continuous transition, Michel’s result cannot be used since the relevant terms of the expansion ^’are

the lowest degree ^{ones. The} algebraic discussion has to be applied, ⁱⁿ general [7], ^{to a} free-energy ^limited

to fourth degree ^{terms. On} varying ^the corresponding f3k coefficients one then obtains a few stable bp;

directions only.

Several authors [2, 4, 8, 9] ^have attempted ^{to avoid}

the minimization of F, ^{and have} conjectured ^that

the preceding bp; directions could be selected on

the basis of symmetry considerations.

Thus, ^{it has been stated} recently by ^Ascher [9],

that the directions 6p; ^{E E} corresponding ^{to the}

absolute minimum of F were the ones having ^an

invariance _group Gn’ constituting ^{a maximal} subgroup

of Go. ^{If we} consider the set { G } specified above,

the maximal subgroups ^{are defined} by ^{the two follow-} ing conditions :

In other terms, ^if G1 and G2 ^are respectively ^the

invariance groups of 5pi ^and 6P2 E E, ^with

then the direction 6P2 ^cannot correspond ^{to the}

absolute minimum of F for _any set of values of the

f3k coefficients.

Actually, ^{in the} restricted framework of conti-

nuous crystalline transitions associated to order parameter dimensions n 3, ^it has been established

by Michel and Morzrymas [4] ^{that the} low-symmetry phases ^indeed correspond ^{to maximal} subgroups.

However, ^these authors, ^{as well as} Cracknell et al. [10]

have pointed ^{out that the converse} property ^{is not}

true. For instance in the case of the 3-dimensional vector representation [6] of the m3m point group, the mm2XY group is not stable ^{below a continuous}

transition though it constitutes a maximal subgroup

of m3m.

Thus for n 3, ^{the maximal} subgroup ^{rule is} rigorously ^{satisfied as a} necessary condition presiding

over second order transitions. Lorenc et al. [11 ]

have also found this necessary condition valid for

a more complex example ^{of a} 6-dimensional order parameter, ^and they ^{have stated that it is} probably generally obeyed, though ^no rigorous proof ^exists

at present.

In the present paper, ^we describe for the first time

a theoretical example ^{of a} 4-dimensional order parameter which is in clear contradiction with the maximal subgroup ^{rule, thus} demonstrating ^the

failure of its general validity.

This example ^{is that of the} unique irreducible

representation ^{of the} I41(Cf) space-group ^at the

N-point [12] of the Brillouin-zone boundary (kli ⁱⁿ

the notations of Kovalev’s tables [13]). The group of the k vector is reduced to the identity, ^{and the}

small representation ^{is one} dimensional. The star

of kll ^{has 4 arms} and the considered representation

is a four-dimensional one. This representation complies with all the symmetry criteria of the Landau

theory [6]. ^{It is therefore} compatible ^{with the occur-}

rence of continuous transitions between strictly crys- talline phases.

We can note that since k 11 ^{has rational coordi-}

nates [13], ^{the set} of distinct matrices representing

the elements of Go (including the pure translations)

in the 4-dimensional representation space, forms a

finite _group Lo homomorphic ^of Go (Lo ^{is the image [2]}

of the considered representation). ^{Each direction} bp,

is left invariant by ^{a set} of distinct matrices forming

a subgroup ^{L of} Lo. ^The corresponding ^invariance subgroup ^{G of} Go ^{is also} uniquely ^{defined as L x K,}

where K is the kernel of the above homomorphism.

Since Lo ^is finite, ^{the set} { L } ^{of its} subgroups ^which

leave invariant all possible directions 6p ^{E E is} finite, and consequently { G } ^also comprises ^a finite number of subgroups ^of Go.

Lo ^{is of order 32. It is} generated by the three follow-

ing ^{matrices :}

This image ^is isomorphous ^{to one of the 227} point-

groups occurring ⁱⁿ 4-dimensional crystallography [14]. It has been labelled (59.1) ⁱⁿ Morzrymas [15]

tables and (32.09) ⁱⁿ Wondratscheck’s tables [14].

The various groups constituting { L } ^and { G },

as well as the corresponding ^vector directions bp

are indicated in table I. For each Gi ^E { G }, ^{there is}

a set of equivalent subgroups [2] Gi ^E { G }, ^which

are the invariance groups of directions 6p[ = go 6p;

with go ^E Go. These groups have only ^one represen- tative in the table : physically, they correspond ^to

different domains [3] ^{of the same} low-symmetry phase (mathematically they ^are conjugated subgroups

relative to different points ^{of the same orbit} [2]).

Table 1.

Physically unequivalent invariance subgroups of ^{the various} directions in bp ^{E E.}

There are six unequivalent groups in { L } ^or { ^G }, ^{two of which} only, ^{labelled V and VI are maxi-}

mal subgroups. ^{Let us} examine their stability ^below

a second order transition. As emphasized above,

we only ^{need to consider a fourth} degree expansion.

For the considered representation, ^this expansion

has the following ^form

The absolute minima of F are straightforwardly

determined for the various _ranges of the f3k ^coeffi- cients, by expressing ^the cancellation of the first derivatives of F, ^the positiveness ^{of the matrix of the}

second derivatives, ^and by comparing ^{the different}

minima which can occur for given f3k coefficients.

In agreement ^{with Landau’s} theory [1], ^{we restrict}

to the phases ^{which can} be related to Go through

a line of continuous transitions. Consequently ^we

exclude the transitions which would only ^{occur for} particular values of the f3k coefficients.

As shown by ^table II, ^we obtain, ^as possible ^low

symmetry phases, ^{the ones labelled} II, IV, ^{V and VI.}

Their respective stability ranges ^can be represented by ^a phase diagram ^{in the} (f3i, f32, 133) space. Its section by ^the f3i > 0 plane ^is reproduced ^on figure ^1.

Fig. ^1.

Section of the phase diagram ^{in the} (fli, P2, P3) space

corresponding ^{to a 0 and} PI > 0. For PI ^{0 no} continuous transition occurs. The above diagram ^{is not a real} phase diagram

in the pressure-temperature space since a, /3,, #2, (33 ^{are functions} of only ² quantities (T, P). ^{Thus the} special points A, B, C are not

physically accessible.

Table II.

Possible low’ symmetry phases ^{relative to} the considered order parameter.

192

It appears that two of the former phases, namely ^V

and VI correspond ^{to maximal} subgroups. ^{The two} others, ^{II and IV have non maximal} invariance sub- groups, thus providing ^a counterexample ^{to Ascher’s}

maximal subgroup ^rule.

From a symmetry point ^of view, ^the thermodyna- mically stable directions

do not play any special ^role among respectively,

the sets of directions (qi 0 fJ3 0) ^and (’11 ’II ’13 ?13)-

The latter directions have the same invariance _groups

as the former ones. bplj ^and bplv ^are singled ^out by

the specific symmetry properties of the considered 4th degree expansion ^{which has a} higher symmetry than Lo. ^{While the} unlimited Landau expansion ^would

have exactly ^the symmetry Lo, ^{the truncated one}

is actually ^invariant by ^a group Lo ^{of order 128,}

supergroup of Lo, ^and isomorphous ^to the 4-dimen- sional crystallographic point-group ^labelled (101.01)

in Morzrymas ^tables [15].

With respect ^to L’, ^the four stable directions ðPn, 6 piv, 6 pv ^and 6pvj, ^all correspond ^to maximal sub- groups [7].

Thus, ^it appears that in the preceding ^counter- example, ^the stability of certain non maximal sub- groups is ^{related to the fact that the} symmetry group Lo

of the physical problem ^{dealt with} (i.e. the deter- mination of the continuous symmetry changes ^{in the}

framework of Landau’s theory) ^{is a} subgroup ^{of the}

full invariance _group Lo ^{of the relevant} free-energy

terms.

In a systematic investigation recently performed [7],

we have found a few other examples of 4-dimen- sional order parameters ^also contradicting ^Ascher’s conjecture. However, ^{in all these} examples ^{the maxi-} mality principle ^is preserved ^with respect ^{to the} invariance _group Lo ^{of the truncated} free-energy expansion. ^Thus, up to now, ^no counterexample ^seems

to available to the more precise mathematical conjec-

ture made by ^Michel [2], ^and stating that the maximal

subgroup rule relative to Lo ^should always ^{be true.}

Acknowledgments.

^{We are} grateful ^{to Prof.}

L. Michel for clarifying ^{to us} many aspects ^{of the} considered problem ^{and for a careful} checking ^of

the presented counterexample.

References

[1] LANDAU, L. D. and LIFSCHITZ, ^E. M., Statistical physics (Addison Wesley, Reading, Mass.) ^1958.

[2] MICHEL, L., Colloquium ⁱⁿ the honor of Antoine Visconti.

Marseille July ^1979. Preprint CERN. Division TH.

[3] JANOVEC, V., DVO0158AK, ^{V. and} PETZELT, J., Czech. J. Phys.

B25 (1975) ^1362.

[4] MICHEL, ^{L. and} MORZRYMAS, J., Proc. VIth International

Colloquium ^on Group Theoretical Methods in Physics, Tübingen (1977).

[5] VINBERG, ^E. B., GUFAN, ^Yu. M., SAKHNENKO, V. P. and SIRO- TIN, Yu. I., Kristallografiya ¹⁹ (1974) 21 ; Sov. Phys.

Crystallogr. ¹⁹ (1974) ^10.

[6] LYUBARSKII, ^G. Ya., ^The application of group theory ⁱⁿ Physics (Pergamon, ^New York) ^1960.

[7] ^TOLÉDANO, J. C. and TOLÉDANO, P. To be published ^Phys.

Rev. B (1980).

[8] BIRMAN, ^J. L., Proc. VIth International Colloquium ^on Group

Theoretical Methods in Physics, Tübingen (1977).

[9] ASCHER, E., ^J. Phys. ^{C 10} (1977) ^1365.

[10] CRACKNELL, ^A. P., PRZYSTAWA, J. and LORENC, J., ^J. Phys.

C 9 (1976) ^1731.

[11] LORENC, J., PRZYSTAWA, ^J. and CRACKNELL, A. P., Preprint Univ. Wroclaw (Poland).

[12] ZAK, J., CACHER, A., GLÜCK, ^{H. and} GUR, Y., The irreducible representations of space-groups (Benjamin N.Y.) ^1969.

[13] KOVALEV, ^O. V., Irreducible representations of space-groups

(Gordon ^and Breach, N.Y.) ^1965.

[14] NEUBÜSER, J., WONDRATSCHEK, ^{H. and} BÜLOW, R., Acta

Crystallogr. ^A27 (1971) ^517.

[15] MORZRYMAS, J. and SOLECKI, A., Rep. ^Math. Physics ⁷ (1975)

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