WEAKLY PERIODIC STRUCTURES AND EXAMPLE

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EXAMPLE

S. Aubry

To cite this version:

S. Aubry. WEAKLY PERIODIC STRUCTURES AND EXAMPLE. Journal de Physique Colloques,

1989, 50 (C3), pp.C3-97-C3-106. �10.1051/jphyscol:1989315�. �jpa-00229456�

JOURNAL DE PHYSIQUE

Colloque C3, supplément au n°3, Tome 50, mars 1989 C3-97

WEAKLY PERIODIC STRUCTURES AND EXAMPLE

S. AUBRY

Laboratoire Léon Brillouin (CEA-CNRS). CEN-Saclay, F-91191 Gif-sur-Yvette Cedex, France

Résumé: Les états fondamentaux de modèles de structure non quantique, dont l'Hamiltonien est invariant par translation, sont faiblement périodiques quelque soit la dimension finie de l'espace. En d'autres termes, étant donné une précision e, pour chaque partie finie de la structure de l'état fondamental (bloc), il existe une distance R telle que le même bloc est retrouvé à la précision e, dans toute boule de rayon R. Cette propriété reste vraie pour les états fondamentaux des modèles de pseudo-spins invariants par translation sur un réseau périodique ( avec e=OJ et dans ce cas, la réciproque peut être prouvée. Pour toute structure faiblement périodique C de pseudo-spins distribués sur un réseau périodique, il existe un Hamiltonien invariant par translation dont l'ensemble des états fondamentaux est entièrement engendré par la structure C, toutes ses translatées et leurs limites. Cette propriété est utilisée pour démontrer l'existence d'Hamiltoniens invariants par translation dont l'état fondamental qui est unique (mis à part les glissements déphasés) est une structure récemment étudiée et qualifiée d'intermédiaire entre quasi-périodique et aléatoire. Son facteur de structure ne possède pas de pics de Dirac mais seulement des "quasi-pics" avec des lois d'échelle et un spectre présumé singulier continu. Ceci est le premier exemple connu d'un état fondamental, qui est ni périodique, ni quasi-périodique sans pour autant être aléatoire.

Abstract: Models for non-quantum structures with translaiionally invariant Hamiltonian in any finite dimension space, have ground-states which are weakly periodic. In other words, for any finite piece of the ground-state structure (block) and some given accuracy e, then there exists a length R, such that the same block is found again (within this accuracy z) in any ball with radius R. This property is also true for the ground-state of any translaiionally invariant pseudo spin models on a lattice (but with e=0) where the reciprocal property can also be proven. For any weakly periodic structure C of pseudospins on a periodic lattice, there exists a translationally invariant Hamiltonian which has this configuration C, all the translated configurations ofC and all their limits as equivalent ground-states and there exists no other ground-state for this Hamiltonian. This property is used for proving the existence of a translationally invariant hamitonianfor which the unique ground- state (apart phase shifts) is a recently studied structure intermediate between quasiperiodic and random. Its structure factor has no Dirac peaks but has " quasi-peaks" with scaling properties and is presumably a singular continuous spectrum. This is the first known example of ground-state which is neither periodic nor quasi- periodic although not random.

1 -INTRODUCTION: The problem which we are going to discuss, concerns general properties of the ground-states of translationally invariant Hamiltonians. These ground-states are believed to be the most frequently, periodic structures (crystals). It is now well-known that there exists ground-states which are incommensurate structures. This situation may occur when there exists competing interactions between the particles of the structure. Typical examples are PEEERLS chain (quasi-one dimensional conductors with electron-phonon coupling) where the electron concentration per atom is an irrational number. Then, the structure looses its periodicity and develops incommensurate Charge Density Waves which have been observed in many real compounds!!]. More recently, the discovery of quasi-crystalst^] provided new physical examples of quasi- periodic structures. At least from the theoretical point of view, the quasi-periodic Penrose tilings which are the models used for describing these quasi-crystals can be viewed as ground-states of translationnally invariant Hamiltonians (Such Hamiltonians for the rigid tiles can be obtained for example as the sum of local energy terms which are minimum only when the matching rules are fulfilledt?>3]).

Some years ago, it has been proven under rather general hypothesis^] that any translationally invariant Hamiltonian has a ground-state with a minimal kind of order called "weak periodicity". However, attempts for finding non-trivial examples were unsuccesful till now. In this paper, we answer positively to the theoretical question: Is it possible to find translationally invariant Hamiltonians where the non-degenerate classical ground-state is neither a periodic crystal nor a incommensurate structure, nor a quasicrystal ? This question has a more precise mathematical formulation through the Fourier spectrum of the ground-state structure. In other words: Is it possible that this Fourier spectrum be different from a sum of Dirac peaks (usually called BRAGG peaks for crystals and also by extension for quasi-crystals) ? The structure studied in refJS] will be proven to be the non-degenerate ground-state (apart a trivial phase degeneracy) of a certain translationally invariant Hamiltonian.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989315

Models for non-quantum real sauctures at finite temperature generally consist into finding the absolute minima of some free energy determined on the basis of microscopic arguments. The temperature is often taken into account in a phenomenological approach by temperature dependant parameters in the model. At zero degree K, this method becomes exact for finding the classical ground-state. For example, let us consider a mixture o f different elements A,B,C ...( or rigid molecules) for which the pair potential interactions v ~ ~ ( r ) , v ~ ~ ( r ) , etc.. are known as a function of the relative distance r of the atoms of the pairs AA, AB, etc..

A B C

and let us assume that the total energy @ ( { ~ i l , { ~ j ],{uk 1,

...

) of this system is the sum of all possible pair potenrials in the system

A B

where ui denotes the set of coordinates of the atom i of the element A, ui denotes the coordinates of the atom j of the element B, etc ... Usually these pairs potentials have a strongly refhive core potential (eg.

V ~ b ( r ) diverges for I r I

+

0 ) and are sometime modelised by a hard core potential V A B ( ~ ) which is infinite for I r I < ~ A B and zero elsewhere ( ~ A B being the minimum distance between atom A and B). The ground- states should depend on the concentrations of each of the elements A,B,C.. etc, but separations into phases with different compositions may also occur.

Finding the ground-state (and in particular the most packed configuration for mixture of unequal hard spheres) is generally an unsolved problem as well analytically and numerically. The numerical problern originates from the fact that in most interesting cases, there exists infinitely many metastable states which are generally chaotic but have more energy than the unknown ground-state. (For a finite system, the number of metastable states ows exponentially with the size!). Although in some exccptional cases (eg. the Frenkel- Kostorowa m o d e l ~ l ) , the properties of the ground-state can be found analytically. there is no general method to pick out the true ground-state among this infinite set of metastable states. In spite of that, it is possible to use some of the ideas developped in ref.181) for proving that the ground-states have a minimum property called

"weak periodicity".

2-WEAK PERIODICITY: Let us describe now the hypothesis which are needed for proving this

A B C

property of weak periodicity for the ground-state. A configuration is described as a set { {ui ] , ( u , 1, ( u k ) .... ⁾ of coordinates in the space ~d where the atoms move. The dimension d can be arbitrary, but in isual physical situations d=3. It is also convenient to represent equivalently a configuration by the atomic densities for each element A,B,C

...

(2) P ~ ( r ) = z 6 ( r - u i A )

1

pg(r),

...

which are sums of Dirac functions with amplitude 1 located at the positions u f , u ~ , . . o f the atoms A,B,C

....

The topology of the configuration space used in the next, is the topology of measures (then thc vague topology of measures corresponds to the weak topology in the configuration space )' .We now restrict the space of all possible configurations to a space of configurations fulfilling a DELAUNEY conditionl61 :

A B C

Definition 1: A configuration { ( u i ),{uj ],{uk 1,

...I

fulfills the DELAUNEY condition with parameters O<b<B,when

1- for any open ball with radius b, there exists gt most one atom inside this ball.

2-for any closed ball with radius B, there exists at least one atom inside this hall.

The spuce of conjigurarions which fulfill this condition is called C (b,B)

(Most) physical configurations for condensed matter fulfill some DELAUNEY condition with two pzramcters b and B [except perhaps in models dealing with the formation of aggregates with fracta!-like structures!). The first condition is equivalent to a hard core condition (the density of matter cannot be locally infinite!). The second condition means the absence of infinitely large empty space in the considered structure.

From now-on, we assume that these conditions are both fulfilled. A straightforward consequence of definition 1 is that the space C (b,B) is closed for the weak topology. In addition, it can be proven:

k o ~ o s i t i o n 1 : The space C (b,R) of'configurarionr fulfillitzg rhis DDELAUNEY condrriotz is cc~moocr (/i)r tlze weak topology ) und does not contain fhe vucuum.

( I ) ( ~ e t us recall that by definition the sequence of measure pi(r) is said to converge to a measure p(r) if for any continuous function @(r) with compact support, one has: Lim

jpi(r)

$(r) d r = jp(r)$(r) d r

i + m

(In addition, Michel B A U E R [ ~ I ] has shown that there exists a metric distance which defines the weak topology restricted to the subset of configurations C (b,B) fulfilling a DELAUNEY condition). Clearly, the vacuum which is the configuration characterized by p~(r)=O,pg(r)=O,etc..(no atoms in the configuration) does not belong to C (b,B). However, note that C (b,oo) is a!so cornpact but contains the vacuum. A consequence of this result is that for any set of measures TZi (r),p$ (r),

...I

representing configurations in C (b,B), there exists a subsequence which converges for the vague topology to a set of measures

? % = { p ~ ( r ) , p g ( r ) , ...

1

ⁱⁿ C (b,B) for the vague topology. We consider now, an energy form

A B C

@({ui ),{uj ] , ( u k ), ...) defined on C (b,B) such that

Hvuothesis 1: The form of the energy is invariant by anyfinite translation by R

A B C A B C

(3) O ( ( U ~ ) , ( u j 1,Iuk I,

...

) = O ( ( u i + R ) , l u j + R J , l u k +Rl,...)

Note that the translation invariance (2) does not imply that the energy form is the sum of pair potentials as for example (1).

A B C

Hvnothesis 2: For each given atom (i,A) and for any configuration ( ( u i ],{u, 1, ( u k ) ,...) i n C (b,R), the energy variation

A B C A B C

(4) O({ui + 61,(uj l , { u k 1,

...

^{) -} @({ui l,{uj 1 , l ~ k 1, ...) = V(6, { u i l ) is a continuous function of the displacement 6 of this urorn.

This condition implies for example the absence of long range pair potential (decreasing as l/r in 3 dimension) because the energy variation (4) could be the sum of a divergent series and then be not defined. For an infinite system, the concept of ground-state needs to be clearly defined because in principle the energy fornl (3) is divergent and cannot be minimized in a strict sense (Only, the energy per atom or per unit volun~e is defined). As well as for the Frenkel-Kontorowa model r4], we have to distinguish the concept of "minimum energy configuration" from the concept of "ground-state configurations". We take the same definition as in ref.r81:

.

A B C

Definition 2: A "minimum energy configuration" is a confi~uration ( ( u i ) ,{uj ) , ( u k ) ,.

..

⁾ of atoms such that the displacement of anv finile number of atoms over finite distances necessarily produces a positive variarion ofthe energy (4). We call this set of configurarion Q as in ref. [8].

(We have to consider only finite displacements for a finite number of atoms in order that the energy variation be defined). Another definition for Q which is equivalent, is

The set Q of "minimum energy configuration" is the set of all possible limits of minimum energy cotfigurations infinite boxes when the volume V of these boxes and the numbers of atoms NA, NB,Nc.. go to infinity.

Clearly, there are many possible limits according to the densities of the elements A,B,C ...

respectively which can be fixed arbitrarily and the possible boundary conditions. This set is not empty. A straighforward proof which generalises a result already obtained in ref (81, allows one to assert:

Proposition 2: The set Q of "minimum energy configuration" is closed for the weak topology.

Equivalently, the limit of any convergent sequence C i of minimum energy configurations, is also a minimum energy configuration. Similarly to the work on the FK model of ref.181, we will use in the next, the fact that Q is invariant under the action of the translation group l%d. For continuing our proof in the general case, we need another hypothesis for the considered model.

Hvpothesis 3: There exists a minimum energy configurarion in Q whichfillfills some DELAUNEY cotzdirions with parameters Oib<B.

This condition which is physically intuitive, is very weak. For many models with physically reasonable pair interactions (e.g. Morse potential), the proof of this hypothesis is easy in finite systems with b and B independant of the size of the system. This property remains preserved when taking the limit of configurzttions at infinite size.

A A

Definition 3: Let us call Q , the subset of confiurations in Q (Q =, Q ) which fulfills sonie DELAUNEY condition.

Let us now explain why a minimum energy configuration defined as above is not necessarily a ground-state in the usual physical sense. It is well-known that there exist physical situations where elements cannot be mixed (for example A and B). Then, if the concentrations in finite boxes of both elements A and B are fixed to some non-zero value, the limit configuration (which is a minimum energy configuration) exhibits a phase separation between elements A and B with a phase boundary. This configuration is not usually considered as ground-state although it is a minimum energy configuration for some boundary conditions. However, far away from the phase boundary, either in phase A or in phase B, the configuration goes to a ground-state. In the case of an homogeneous phase which could be for example a crystal, a minimum energy configuration may still contain topological defects such as dislocation lines or grain boundaries. For defining properly the physical concept of ground-state, one has to extend the mathematical process (used in ref@]) for eliminating all possible localized defects from the minimum energy configur?tion Since the energy form @ ( ( u ~ ] , { u ~ } , { u ~ ] ,

_ ^-

...) is invariant by translation, if

A B C

C =( { u i ) ,(ui ] ,{uk ), ...} is a minimum energy configuration in Q , the set of translated configurations

'4 B C

T R ( C ) ={ ( u i +R},{uj +R],{uk +R],...]also belongs to Q. The "accumulation configurations" which by definition are limits of converging sequences of translated configurations also belongs to Q because Q is closed.

Pro~osition 3: Suppose that configuration C fulfills a DELAUNEY condition with parameters O<b<B, then all the configurations TR(C ) and their limits fulfill the same DELAUNEY condition.

Let us call

&

the (closed) set of all configurations TR(C ) and of their limits ( C is the closed orbit A

of C by the translation group in EXd). The orbit

3

of a configuration C' in the orbit

&

of C (C' E

6

),

A A A A A

is obviously included in C ( C 2 C' ). If the orbits of C and C' are identical ( C = C' ), C belongs toC' which means that the configuration A C can be reciprocally obtained as a limit of translations TR(C') of C'. Then, the two configurations C and C' can be considered as physically equivalent. For example, if C is an incommensurate configuration, C' is another incommensurate configuration with a different phase.

A A h A

When C 3 C' (C # C' ), C and C' cannot be considered as physically equivalent but C' can be said to be"more perfect" than C. Indeed, in a physical interpretation, the fact that the minimum energy

A A

configuration C' has an orbitC' smctly included in the orbitC of the minimum energy configuration C, means that some topological defects of C (but not necessarily all of them) have been swept away when taking the limit of translations of C. (see for example the FK model181 for a better understanding of this point). The natural question which comes now is: Is it possible to sweep away all kinds of defects of the structure? If this is possible, one gets a "perfect" minimum energy configuration which now can be called "ground-state". If C is

A A

such a "ground-state", for any limit configuration C'of translations TR(C ) of C, one has C = C' . Then, by dcfinition:

Definition 4: A ground-state C is a minimum energy configuration which has a minimal orbit

2

with respect to the translation group IXd (i.e the orbit

8'

of any minimum energy configuration C' E

2

^is

A A

identical to C , or TR(C' ) is dense everywhere in C ).

For proving the existence of ground-states, we have to check that the relation "more perfect than"

fulfills the axioms of an order relation in Q and that it has the essential property:

Proposition 4: The order relation in Q "more perfect than" is inductive. A

This property means that for any family of configurations C i ( i ~ I, I is an arbitrary set of indices)

A A A A

which is totally ordered, (for any pair C i and Cj with i~ I and j~ I), one has either Ci 2 Cj or Cj ² Ci ),

A A

there exists a configuration C i n 4 such that for any i~ I , Ci 2 C

.

For finding C, one has to take it into the intersection

n ti

using the theorem that the intersection of any totally ordered family of compact sets is never

I € I

empty). The condition for applying the ZORN lemma[9] is then fulfilled. This lemma asserts:

Theorem 1 : For any minimum energy configuration C' in Q , A there exists a minimal configuration

A A

C (called ground-state) such that C' 2 C

.

A A

(A configuration C is minimal when there exists no, configuration C" such that C 3 C".). In physical term, there always exists a defect free configuration C corresponding to a minimum energy configuration C' but this one is not necessarily unique. For example, if C' contains a phase boundary between two homogeneous phases A or B, configuration C may be either the phase A or the phase B.

C being an arbitrary configuration, B an arbitrary open ball in Kid and E an arbitrary positive number, a base of open sets O(C ,B,E) for the weak topology of the configufition space can be determined by defining O(C ,B,E) as the set of configuration C'such that inside the open ball B

1) C and C' have the same number of atoms (of each kind)

2) Inside ball B, the atoms of C and C' can be associated in pairs (of the same kind) at a distance smaller than E.

Then the property of minimality of configuration C is easily proven to be equivalent to the property of weak periodicity defined as

Definition 5: A configuration C is said to be weakly periodic, iffor any open ball B in Kid and any positive E, there exists apositive number R(B,E) such that in any ball in IRd with radius R(B,E) there exists a translation vector, such that TR(C ) belongs to the neighbourhood O(C ,B,E) of configuration C .

In simplest terms, weak periodicity means that within any given accuracy, any finite block of the structure is found again in the whole structure at a bounded distance R of any point. Then we can also say that a weakly periodic structure has "a local order at all scale". In general, for nonperiodic groundstates, it is physically intuitive that the smaller is E and the bigger is the open ball B, larger will be R(B,E).

It is interesting to note that if instead of the vague topology of measures for the configuration space, one considers the uniform topology of functions (absolutely continuous measures), the property of weak periodicity becomes the definition of almost periodic functions of Harald B O H R [ ~ ~ ] ) . Then, the FOURIER spectrum of the structure is essentially a countable sum of DIRAC peaks. (The difference we make between quasiperiodicity and almost periodicity is that the DIRAC peaks of a quasiperiodic structure are distributed on the integer combination of a finite number of wavevectors, instead of being distributed on an arbitrary countable set of wave vectors).

Although periodicity or quasi-periodicity implies weak periodicity, the reverse is not true. However, weak periodicity is clearly not randomness. The above theory is essentially based on the Hamiltonian invariance with respect to a translation group Kid. The point group of rotations and symemes O(d) which is compact, does not need to be considered. However, it is useful to note that the point group of symetry G for aweakly periodic configuration C can be defined as the subgroup of O(d) which lets globally invariant the orbitd'. This group is not necessarily a crystallographic group as shown for example by the Penrose lattices which have symmetry of order five.

The same theory can be built for a lattice Hamiltonian H((ui)) where ui is for example a set of local and bounded coordinates on a square periodic lattice at site i, when this Hamiltonian is invariant under the action of a discrete group Zd defined by the translations {ui}+(ui+,) for n E Zd. Then, the reader can readily find that the ground-states of H((ui)) are weakly periodic with a definition similar as above.

It is also possible to consider a Hamiltonian H({ui)) with non bounded coordinates {ui) which is only invariant under the action of a discrete group 7Z.d instead of Kid (this is the situation when the atoms are submitted to an external periodic potential). A ty ical and non trivial example in one dimension (extendable in d dimensions) is the Frenkel Kontorowa modellhl, the Hamiltonian of which is invariant by the translation (ui)+{ui

+

2n). In that case, the o r b i t 6 of the ground-state can be mapped as an invariant set in the standard rnap[8]. There are commensurate (periodic) ground-states for which

6

is mapped as periodic cycles and incommensurate ground-states which depending on the model parameters are mapped either as 1-circle (KAM tori) or as Cantor sets (CAM). There also exists minimum energy configurations which are the discommensurations of the commensurate ground-states.

We now present an example of weakly periodic structure which is neither periodic, quasi-periodic nor random. For finding this example, it is now convenient to work with pseudospin models.

3-PSEUDOSPIN MODELS: It is often possible to replace a model with continuous variables by a model with discrete variable. For simplicity, we consider a mode1 with all the atoms identical. Then, we divide the space into small cubic boxes (smaller than the constant b in the DELAUNEY condition). The pseudospin oi = 1 is associated to the box i, when an atom is present in the box, this pseudospin oi is 0 when the box is empty. (If there is several kinds of atoms, several pseudospins

df

have to be associated to the box i). For each pseudospin configuration, the energy of the whole system can be minimized with an atom in each of the boxes i

where o i = 1. Then one obtains a pseudospin Hamiltonian H({oi]) on a cubic lattice which we assume to be expandable into a sum of pair, triplet,

...

interactions

Because of the invariance of the energy by translation, one has for any vector n

A similar theory as above can be built for such a Hamiltonian (5-a). A minimum energy configuration is a configuration such that by flipping any finite number of pseudospins, the energy variation is positive. Note that the definition of a minimum energy configuration is not equivalent to those of the initial model because flipping a pseudospin is equivalent to add or to withdraw an atom. The boundary conditions of the initial model are dropped and we look to the absolute minimum of the Hamiltonian with free boundary conditions where the density of atoms is not fixed. However, adding a chemical potential allows one to change _- this density of atoms which is equivalent to add a "magnetic field

I:

H o i to the pseudospin hamiltonian.

1

This pseudospin model is in fact simpler and no DELAUNEY conditions or similar ones has to be considered. The translation group which lets invariant the Hamiltonian form (5-a) is now discrete, and the topology which has to be considered on the pseudospins configurations is the discrete topology. Then one obtains the theorem:

Theorem 2: Any pseudospin Hamiltonian on a cubic lattice which is translationally invariant has weakly periodic minimum energy configurations (called ground-states).

A weakly periodic spin configuration can be equivalently defined by:

For any arbitrary finite block B of spins in a weakly periodic configuration of pseudospins on a cubic lattice, there exists an integer N such that any cube with size N contains a block identical to this block B.

In general, for a non periodic ground state, the bigger is the block, the larger is N. In the case of pseudospin models, weak periodicity is a necessary and sufficient condition for a configuration to be the ground-state of some translationally invariant harniltonian as proven by this theorem:

Theorem 3: For any weakly periodic configuration of pseudospins C = { o i ) on a cubic lattice, there exists a well defned Hamiltonian for which the set of ground-states is identical to the closed orbit of this configuration C = (oil under the translation group ~ d .

These ground-states are physically equivalent configurations. For example, for an incommensurate structure, this set of ground-states is generated by the phase variation of the modulation between zero and 27t.

This Hamiltonian is obviously not unique, but the degeneracy of its ground-state is the smallest uossible in order that it contains the weakly periodic configuration C.

We can also prove that for a non weakly periodic configuration C", there exists a Hamiltonian which accepts this configuration as a minimum energy configuration, but this Hamiltonian should also accepts as minimum energy configuration all the configurations

of^ .

Taking for example C" as a random configuration of pseudospins,

C""

would be the whole configuration space. Since any configuration has to minimize the Hamiltonian energy, this Hamiltonian has to be a trivial constant which clearly is a non interesting case.

The proof of theorem 3 is in fact rather simple. For defining a Hamiltonian, let us consider the full set of all possible configurations (which contains 2Nd elements) of a cubic pseudospin block BN with a given size N. For each pseudospin configuration of a block BAN)=(si), we define a variable tg, = 0 if this block BAN)appears in the configuration C = { o i ) , (which means that there exists n such that ( ~ i + ~ ] = ( s i ) for

i E BN ) and t ~ , = 1 if does not appear. Then, we define an energy term (6-a) HB, ((oil)=

n

^{[ o i si +} ( 1 - o i ) (1-si)

I

~ € B N

which is zero if the block configurations (oil and {si) are different and one if they are identical. We now define for example a pseudospin Hamiltonian as:

where K is some positive constant smaller than 1, in order than the energy per site in (6-b) be given by an absolutely convergent series. The sum in (6-b) is done on the block size N, for all possible blocks of size N and on the site variable n so that this form is invariant by any translation. (Note that instead of cubic blocks, we could also consider all the arbitrary finite blocks of pseudospins. In order to have less terms in the sum (6- b), we could also define tg, = 1 only for the "primary " blocks defined as pseudospin block configurations which do not appear in C but such that any sublock in Ba appears in C and tga = 0, for all pseudospin block configurations which either appear in C or which contains a primary block. By definition, the energy of any configuration is positive or zero. For the initial cbnfiguration C = (oi), it is clearly zero because for all the pseudospin blocks in C, tg,=O. It is also zero for all the translations of C and their limits. Consequently, C = (Oi) and all the configurations of its closed orbit

6

are ground-states of H({oi)).

Let us now consider C' =

(4')

a minimum energy configuration of this Hamiltonian. The definition of a minimum energy configuration allows C' to contain "forbidden blocks" i.e. which does appear in C. If C' does not contains any such blocks, it is clear that C' belongs to the orbit C

.

If it does, let us consider such a (primary: spin block B,. It is clear that Hamiltonian(6-b) yields a non-zero positive energy for this block. It can be proven that the average number of such blocks per unit volume must be zero. Suppose that this property does not hold and then consider a cubic block BN in which the block density of sublock Ba is finite and larger than some number q>0. The size N of this block can be infinitely large. Replacing the whole block BN by a cubic block which appears in C', the energy variation of C' is the sum of two terns. The first one is negative and smaller than a number proportional to q, to the volume ~d of BN and to the energy Ea of block Ba

.

The second term is positive and smaller t+,an a number proportional to the surface area 2dNd of the block. But since C' is a minimum energy configuration, the density of all "forbidden" blocks in C' must be zero. From this result and from the definition of the groundstates as minimum energy configurations with minimal orbit, it

A A

comes out that the ground-states

C"

of Harniltonian (6-b) such thatC ' 2 C " are limits of translations of C and thus belongs to the orbit& of C. This result proves that the set of ground-states of H is identical to the orbitC ^A

.

4-EXAMPLE: A MODEL WITH A NON-ANALYTIC OUASTPERIODIC BOND MODULATION : According to the definition of weak periodicity, it is clear that periodic and quasiperiodic structures are special cases of weakly periodic structures. The question which comes out now is: Do weakly periodic ground-states which are neither periodic nor quasiperiodic exists at least on some theoretical models? It is well known that periodicity and quasi-periodicity are characterized by a discrete Fourier spectrum as for example in crystals, incommensurate structures and quasicrystals. Therefore a weakly periodic structure which is not periodic or quasi-periodic must have a continuous part in its spectrum. However, since no estimation of the upper distance R(B) for a block B can be given as a function of the block size without making any assumption about the Hamiltonian, no general results about the Fourier spectrum of a weakly periodic structure (neither about its entropy). We just provide an example of a weakly periodic structure where this spectmm has scaling properties[5] with exponents which can be analytically calculated and which suggest that it is a singular continuous spectrum' (However, in spite of very strong arguments, a rigorous mathematical proof of this property is still missing). Physically for some given resolution, a singular continuous measure exhibits peaks but with a better resolution, each of the peaks are observed to split into several smaller peaks which are very close. Increasing again the resolution, these smaller peaks splits again into new smaller peaks and so on. In the limit of an infinitely narrow resolution, there exists no Dirac peak (with finite intensity) but only a "singular continuous measure" with generally multifractal properties. Our example consists into a generalization of a FIBONACCI sequence defined by the sequence of atomic positions:

(7-a) Ui+l - Ui = 12

+

( 11 - 12) Oi

where 11 or 12 (tiles) is the distance between consecutive atoms and o i = 0 or 1 is a quasiperiodic sequence of pseudospins Oi = X( i

< ⁺

^{a ) =} 0 or 1 where ~ ( x ) is a I-periodic function defined in the first period as:

(7-b) ~ ( x ) = 1 for O I x < A and ~ ( x ) = 0 for A S x < 1

5

is an irrational number, A is some arbitrary number between 0 and 1 and a is an arbitrary phase.

In the special case

a-

¹

(8) < = A = - 2

(UA singular continuous spectrum is a positive measure with n o Dirac functions and for which their exists a support with zero Lebesgue measure. (A set S is a support for a measure d p if S has full measure with respect to dp. Note that with this definition, the support of d p is not necessarily closed and that it is not unique).

one obtains by definition (7-a), the well-known Fibonacci latticeL2] which is the onedimensional analoguous of a PENROSE lattice. Note that replacing in (7-a), ~ ( x ) by a smooth continuous function, it is readily proven that the structure (ui] becomes quasiperiodic. The properties which we describe now explicitely depends on the fact that ~ ( x ) is a discontinuous periodic function (as for example the "non-analytic" hull function in the FK model)).

Nevertheless, the sequence of pseudospins { o i ]is quasiperiodic, and then it is also weakly periodic. According to theorem 3 above, there exists a translationally invariant pseudospin Hamiltonian for which the whole set of ground-states is given by (7) and 0la<l.'Note that when i

5 +

a = 0 or A, there is two possible determinations of Ci (as for non-analytic incommensurate groundstates in the FK modell41).

Subtituting o i by the atomic positions given by (7-a) and adding V(ui+l-ui) to this Hamiltonian where for i

example V(x)= ~ ( x - l ~ ) ~ ( x - l ~ ) ~ is a double potential with equal minima at x= 11 and x=12 and taking K=+oo, proves the existence of a translationally invariant Harniltonian for which the ground-states are given by (7-a).

The Fourier spectrum of this atomic chain ("static structure factor") is defined as 1 N

(9) S(q) = lim S N ( ~ ) with S N ( ~ )

= R

I e x p ( i q u n ) 1 2

N+- n= 1

The average distance between consecutive atoms a = < ui+l- ui>i = A 11

+

(1- A ) 12 determins an average lattice u$O) = i 1 + uo and suggests to write ui as ui= ujO)+ Vi = i a

+

uo

+

vi where vi is the fluctuation. In fact, (when

6

is irrational),the KESTEN theorem[12] asserts that this fluctuation or equivalently

N

(10-a)

C

~ ( i r + a ) - N A i= 1

is bounded if and only if A is an integer multiple modulo 1 of

5

(10-b) A = p 5-m

where p and m positive or negative integer. Morover, we proved thad5] when this condition is fulfiIld

(1 1) u i = u O + [ f ( i a + a t ) - f ( a l ) ]

where a'=- a and f(x) is a piecewise linear function (hull function). In addition, g(x)=f(x)-x which

5

is the hull function of the modulation vi is periodic with period -. 1 Consequently, when (10-b) is fulfilled, the

5

structure is quasiperiodic and S(q) given by (9) is a sum of Dirac peaks located at Q,,, = m Q1

+

ⁿ Q2 2 7 ~ 2.n

6

where m and are integers and Q1

=a

^, _{4 2} =

7 -

When the Kesten condition (10-b) is not fulfilled, it is clear that no function g(x) with bounded variation can describe the modulation vi. A combined analytic and numerical study suggests that the sequence (ui) is not quasi-periodic but is only weaklyperiodic. For doing this study, it is shownt51 that the sequence of pseudospins ( o i ) given by (7-b) or equivalently the sequence of tiles 11 and 12 can be constructed by using a sequence of "inflation" procedures analogous to those used for Fibonacci sequences or Penrose lattices. In general, these inflation procedures are not all the same but their sequence is determined unambiguously both by the continued expansion of

6

and by a special expansion of A (which has been called <-expansion)[5]. This expansion extends the usual expansion on base 2 on a base of numbers generated by the best approximation to zero by the integer multiples modulo 1 of

5.

Instead of two tiles as in the Fibonacci case, this inflation procedure involves three tiles (two of them being equal at the first step). (In the case, where the Kesten condition is fulfilled, two tiles only are left after a finite number of iterations.). It is especially convenient to do the analysis when both the expansions of

6

and of A are periodic in order to have a periodic sequence of inflation procedures. For having one of the simplest cases which do not fulfill the Kesten condition, we choose a continued fraction expansion for

5

with constant coefficients equal to 1, which yields

(12-a)

5

^{= -}

7 -

^{= - 1 ( r is the} golden mean) and for A the

"6

-expansionM

where for p integer sp is a series with period 3 (for i22) defined by q p + 2 = q p (trying shortest periods yields a value of A fulfilling the Kesten condition). Fi=Fi-l+Fi-2 is the usual Fibonacci sequence with the initial condition Fo=O,Fl=l. Then, the sequence of inflation procedures also has period 3. The product of three consecutive inflation procedures yields the symbolic transformation:

(13-a) An = {cn-1 An-1 cn-1

1

(13-b) B n = (An-1 Cn-1 Cn-1 An-1 Cn-11 (13-C) Cn = (‘411-1 Bn-1 Cn-1 An-1 Cn-1

1

which by iteration from the initial step with A0 = 11 , BO = 11 , CO = 12, yields three sequences An, Bn and Cn of blocks of tiles 11 and 12 such that for any n, the initial sequence (7-a) with condition (12-b) is also a sequence of blocks An, Bn and Cn

.

It is useful to consider the characteristic matrix M of this inflation rule (13). the coIumns of which counts the number of symbols An-~,Bn-1,Cn-l in An,Bn,Cn respectively

(14)

G = [ : i]

because a theorem due to BOMBIERI and T A Y L O R [ ~ ~ ] asserts that when the characteristic matrix of an inflation rule has only one eigenvalue with modulus larger than or equal to 1, the sequence generated by this inflation procedure has a discrete specuum S(q). In our situation, the three eigenvalues of M are 23,-1 and - 2-3 so that we see that we are in a marginal case where this theorem does not apply. Defining the amplitude

N - .

factor of a block W={un)(n=l,..N) as AW

=x

e x p ( i q un), the amplitude factors A n ( q ) = n= 1

[A~,(q),A~,(q).Ac,(q)l of the finite blocks An,Bn,Cn can be calculated recursively from An-1(q) by the

- -

linear relation An(q) =Fn(q) An-1(q) where the 3*3 matrix Fn(q) generally depends on n. This matrix is determined from the inflation rules (13) and from the length LA,,LB,,Lc, of the blocks which are calculated

-

n

-

by diagonalization of the characieristic matrix

a.

From the behavior of this matrix product

I1

^{R(q) for}

i= 1

large n, one obtains the behavior of M~,(q)l

,..

and thus of 1 the static sttucture factor (9) of the L ~ n

chain per unit length. There exists a dense set of values for q defmed by

(15) 2 x 1 .

q =

- - (I+

k 2) j and k integer a 4

-

- -

for which Fn(q) becomes independant of n and equal

OF,(^)

for iarge n. Then, it is proven that S N ( ~ ) behave unusually as a power law of the size N of the block with an explicitely calculable exponent 0<y51. This one depends on q and is generally smaller than 1. When there is a Dirac peak, this exponent should

11 . .

be 1. In the situation where

-

is mational, it is never equal to 1 for any value of q given by (15) which implies 12

the absence of Dirac peaks in S(q) (except for q=O) on this set of q values. Since we can prove that a necessary

-

and kfficient condition for the sequence of matrices Fn(q) to converge toward a periodic cycle is that q be defined by (15) with j and k rational, these calculations can be extended in principle for all these values of q (but then the exponent y becomes harder to be calculated analytically). There also exists a set of values of q (with zero Lebesgue measure) for which this matrix product has no limits.

A complementary analysis of S(q) can be done by using the series of best approximations to A by integer multiples of mod 1 obtained by truncations of explicitely analytically calculable as sums of Dirac peaks and it is numerically found that the intensity of each Dirac peak decreases as the order of the approximation increases while new smaller peaks grows and so on. The nice scaling properties which are numerically observed

1

in the vicinity of "quasipeaks" located at the q values determined by (15) where the exponent

a

is larger than , confirms the absence of Dirac peaks in the limit A =

y

1

.

(see refs.[5] for more details).

In conclusion, it has been shown that it is possible that the groundstate of a translationally invariant

Hamiltonian be perfectly ordered without any Dirac peaks. In the example we have shown the Fourier spectrurr has multifractal properties("Devil'spectrumm'). In general, proving that this situation could occur for a giver frustrated Hamiltonian is mathematically very difficult but we took advantage here of the existence of inflatior rules for constructing the groundstate sfructure. In spite the model considered here is very simple, it might have a physical relevance for example in alloys where there exists an incommensurate modulation of composition anc where the atomic distances should be also modulated. Let us now emphasize that a poor experimental resolution should confuse the "quasipeaks" of S(q) with Dirac peaks and the weakly periodic structure presented here with an incommensurate one. However, such a phenomena could be suspected in the case where incommensurate structures exhibits unusually intense high order harmonics and peak width anomalies. Refined high resolution measurements in good samples should be necessary for a possible observation of "Devil's spectrum".

Aknowledgement: The author is indebdted for useful discussions with Michel BAUER from SPT Saclay concerning the mathematical fundation of this theory and with Christophe OGUEY from Ecole Polytechnique concerning the possible relevance of this work to the theory of quasicrystals.

[ l ] see for example Pierre MONCEAU (ed.) Electronic Properties of Inorganic quasi-onedimensional Materials D.Riedel Publishing Cie. (1985)

Claire SCHLENKER (ed.) Low Dimensional Electronic Properties of Molybdenium Bronzes and Oxides D.Riedel Publishing Cie.in press

[2] Paul J. STEINHARDT and Stellan OSTLUND (ed.) "The Physics of Quasicrystals" World Scientific Publishing Co. (1988)

[3] M.DUNEAU and A.KATZ Phys.Rev.Lett. 54, 2688 (1985) and A.KATZ and M.DUNEAU J.Phys.(Paris) 47,181 (1986)

[4] S.AUBRY J.Phys.(Paris) 44, 147-162 (1983)

[5] S. AUBRY, C. GODRECHE and J.M. LUCK Europhys. Lett. 4, 639 (1987) and Journal of Statistical Physics 51, 1033 (1988) C.GODRECHE,J.M.LUCK and F.VALLET J.Phys.AZ0 4483 (1987) [6] DELAUNEY 1928 1sv.Akad.Nauk.SSSR otdel.Fiz. Mat. Nauk. 79-110,147-172, DELAUNEY et a1

1976 Soviet Math. 17 319, and 1976 Doklady Akad. Nauk SSSR 227 [8] S.AUBRY and P.Y. LE DAERON, Physica D8 381-422 (1983)

[9] see for example Walter RUDIN Real and Complex Analysis MacGrawHill Pub.Cie (1979) or Irvine KAPLANSKY Infinite Abelian Group (Univ.of Michigan Press, Ann Harbor)1954 p.6

[lo] see e.g. F. RIESZ and B.NAGY L e ~ o n s d'Analyse Fonctionnelle, (Gauthiers-Villars Paris ) 1965 [ l 11 M.BAUER private communication

r121 H.KESTEN. Acta Arith. 12. 193 (1966-67)

[13j E.BOMBIERI and J.E. TAYLOR'J. phys.'(paris) Coll. C3 (1986) and Contemp. Math. 64,241 (1987)