Phase Defects and Order Parameter Space for Penrose Tilings

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Phase Defects and Order Parameter Space for Penrose Tilings

T.Sh. Misirpashaev

To cite this version:

T.Sh. Misirpashaev. Phase Defects and Order Parameter Space for Penrose Tilings. Journal de

Physique I, EDP Sciences, 1995, 5 (3), pp.399-407. �10.1051/jp1:1995134�. �jpa-00247064�

Classification Pllysics Abstracts

61.70R

Phase Defects and Order Parameter Space for Penrose Tilings

T.Sh.

Misirpashaev(")

Landau Institute for Theoretical Physics, 2 Kosygin St-t Moscow 117334, Russia

(Received

22 June 1994, revised 8 November 1994, accepted 30 November

1994)

Abstract. A _new invariant classifying phase defects of Penrose tilings is coustructed. This invariant takes values in trie group closely related to trie fundamental _group of _a certain topo- logical _space, which is trie image of trie tiling itself under identifications dictated by matching rules. This _space plays trie rote of trie order parameter _space ^for pentagonal quasicrystals. Trie

invariant enables _us to discriminate between mismatches of dioEerent directions.

l. Introduction

It is known that

studying

_any

regular

ordered medium _one cannot avoid the

question

of its structural defects. Defects _are

inevitably

created under the _most careful

fabricating techniques,

often

being

_necessary ta

explain

trie most fundamental properties of materials

(e.g., plasticity

is

explained

^{with trie aid of}

dislocations). Sometimes, they

facilitate the

growth

of _new

phases.

Finally, they give

use to the _most

striking properties

of ordered _rnatter

(such

_as

phenornena

connected with the existence of vortices in

liquid phases

of He and

superconductors).

Arnong

all

possible

violations of _a

perfect

structure those whose

stability

is dictated

by

_some

topological

_{reasons are} of the main interest. The

topological theory

of defects is

especially

well elaborated for media with broken continuous syrnrnetry

[1-3].

In this _case the state of the mediurn at each

point

is charactenzed

by

_a value of the order parameter that

belongs

to the relevant order parameter _space P

(also

called the manifold of

degenerate states).

^The

problem

of

finding topological

defects _{m a}

physical

volume V then reduces to _a

homotopy

classification of continuous

mappings

from

Vi

D ta P where D is _a point ^for

pointlike defects,

a

line for linear

defects,

etc. For contractible volumes V the mappmgs are dassified

by

elements of the appropnate

homotopy

_group

7rk(P)

or

by conjugation

classes if k

= 1, and

7ri(P)

is non-commutative. In

particular, pointlike

^{defects of} _a 2-dimensional matter are dassified

by conjugation

classes in

7ri(P).

One _cari also try to _{use a} similar

approach

for ordered media that _are

essentially

discrete,

namely crystals

and

quasicrystals. Consider,

for

example,

a flat

2-periodic

lattice.

Taking

the quotient _space of the

plane by

the _group of translations of the lattice _one obtains _a torus.

Elements of the fundamental _group of the torus

7ri(T~)

₌ Z~

consistently dassify complete

(")e-mail:

missir@landau.ac.ru.

@ Les Editions de Physique 1995

400 JOURNAL DE PHYSIQUE I N°3

Fig. l. Matching rules for Penrose tiling: decorated thick and thin tiles.

dislocations of the lattice in terms of components ^of the

Burgers

vector. That is

why

one

sometimes _says that the torus is the relevant order parameter _space,

though important partial

dislocations _are not

induded,

neither _can

complicated

rearrangements of atomic

planes

ob- served in

experiments

be described in these terms. Even _more serious difliculties arise in the

case of

quasicrystals

_{[4, Si,} in which apart from dislocations _a

variety

of defects of

matching

rules called mismatches becomes

possible.

An attempt to build the

general approach

to the latter based _on the

homotopy theory

_was done in [4, Si. ^Hereafter we present a concrete treat- ment

specific

for the _case of _a

pentagonal quasicrystal.

We

disregard

^the atomic decoration of unit

cells, studying only

their

packing

arrangements, 1.e.,

quasiperiodic tilings.

^Associated

with ideal

pentagonal quasicrystals

_are Penrose

tilings

^of the

plane

_[6], formed

by

two types of rhombi. In

general, tilings corresponding

to non-ideal

quasicrystals

_may contain distorted

rhombi, overlaps,

and non-covered _regions, characteristic for elastic

(or phonon)

defects [7).

Another dass of defects

(phase defects)

arises because of non-trivial

matching

rules of Penrose

tilings.

Phase defects _on which _we focus _our attention in this paper do not invoke _any dis- tortion of the

tiling.

^From now on the word

"tiling"

will

everywhere

mean a

tiling

of _a whole

plane

or its finite part

by

thick and thin rhombi of the Penrose

tiling.

2. Mismatches of Penrose

Tilings

It follows from the

algebraic theory

of Penrose

tilings

_[6] ^{that for} an infinite

tiling

^of _a ^whole

plane

the two

following

statements are

equivalent.

l The

tiling belongs

to the local

isomorphism

dass of Penrose

tiling,

which is the dass _~t

= o

in terms of references [6, 8].

2)

A decoration of all

edges

of the

tiling

with

simple

and double _{arrows exists} such that each thick and thin rhombi _are decorated _as shown _m

Figure

1.

One _can, therefore, define _an ideal

tiling

_as a decorated

tiling

with

matching

_arrows. While

studying

structural defects _one

always

_assumes that violations of the structure _are not too

drastic,

1-e- that the ideal structure

persists

^almost

everywhere

except for separate ^{and localized} defective

regions.

We thus restrict ourselves _to the _case of

tilings

that _can be decorated

with _arrows in such _a way that

rùatching

rules _are satisfied almost

everywhere

except ^for

small isolated dusters of mismatches. A

primitive

defect in this

approach

is _a

single

^isolated

mismatch,

that is _an isolated

edge

that cannot be

consistently

decorated

(Fig. 2).

In many

practically interesting

_cases

only

this kind of defects _appears. In

particular,

there is _a suitable way of

obtaining

defective

tilings,

which consists _in

applying

the

cut-and-projection

method [9], the band

orland

the multidimensional

integer

lattice

being

deformed.

(Only

deformations with

no net

displacement along

the direction

(i,

_{i, i, i,}

i)

should be considered in order for the

tiling

to stay ⁱⁿ the _same local

isomorphism dass.)

^One can show

[io-12]

that the

only

defects

arising

for small _curvature deformations of the band _are isolated mismatches. In particular, this is the

case of approximants [13] that _can be considered _as defective

quasicrystals

with _one mismatch per approximant unit cell [10]. ^{Isolated mismatches} are aise the

only

defects which _appear

Fig. 2. A piece of _a tiling containing _a surgie isolated mismatch. Bolded edge should _{carry a} simple

arrow but orientations of this _arrow dictated by two adjacent tiles _are contradictory.

outside the _core of _a

dislocation,

with _a

density decreasing

_as

1/r.

Note that each isolated mismatch is situated _{on an}

edge connecting

two vertices

prohib-

ited for Penrose

tiling. (By

_{vertex we mean} the

configuration

of rhombi

sharing

the vertex

point.

Disregarding

the _arrow decoration there _are 7 different

permitted

vertices [6]. ^Isolated

prohibited

vertices _are

possible

but

they

_are net pnmary

defects,

_as

they

came

along

with

non-isolated mismatches and _appear

only

for

large

curvature deformations of the band. We also _note that each isolated mismatch is localized _on such _an

edge

that _two

adjacent

^rhombi

demand different orientations of two

simple

_arrows

(not

_two double _{arrows or one}

simple

and

one double

arrow).

This fact which _con be verified

directly

^will be

given

below _a

topological explanation.

Isolated mismatches _can be created in pairs with the aid of _a

simple

rearrangement ^of a

previously

ideal _region of _a

tiling

and _can be moved

by

limited distances

along

"worms". This pnmary

dynamics

of

primary

defects

probably plays

_an

important

role in the

quasicrystal

kinetics

[loi.

3.

Topological

Invariants for Mismatches

TO show that _a

given

defect is stable _one should attribute _a

topological charge

to it. TO this end

one can encirde the defect

by

_a

sufliciently large loop passing entirely through

non-deformed

regions.

In order to be _sure that the defect _carnet

dissipate involving only changes

in the _very

vicinity

of its _{core one} should be able to detect its presence

knowing only

the local arrangement in the

vicinity

of the

loop.

This is done

by assigning

_an element of _a certain

dassifying

_group

G _to each such

loop

_so that elements

assigned

to

homotopic loops

_are m the _same

conjugation

dass.

Dissipation

of such _a

defect,

then, would involve rearrangements on a whole hne

joining

the defect with another defect _or with _a

boundary.

The energy cost of such

rearrangements

guarantees ^the relative

stability

of the defect. In the _case of

tilings

it is natural to consider

loops

which _are broken

lines,

segments

being

the

edges

of the

tiling.

Such

loops

will be called

tiling loops. Choosing

a

tiling loop

that

belongs

to

perfect

_{regions, one can use arrows} to

construct invariants.

The

simplest possibility

_[14] is to orient the

loop

and take the

algebraic

_sum of all

edges

belonging

to it, orientation of each

edge being

_given

by

its _arrow. An

edge

contributes +i to the invanant if it is onented

along

the

loop

and -i otherwise. This commutative invariant

402 JOURNAL DE PHYSIQUE I N°3

b A

A

e a

Fig. 3. A group generator a, b,_c,d,_e

(A,

B, C,D,

E)

is attributed to each edge _m trie corresponding direction, carrying simple

(double)

_arrow. There

are 10 dioEerent relations of "commutativity" of tiles.

Ail 10 rhombi giving dioEerent relations _are present ^{in this} figure, they form _a decagon occurring in Penrose tilings. Instead of this decagon _any other configuration of trie _same 10 rhombi coula bave been chosen. Vertices of type ¹ _are bolded.

a takes values in the group of

integers

^{Z and} assures the

topological

nature of mismatches that have _a

= +2, ^while it does _not enable _us to

distinguish

between mismatches of diiferent directions.

Nothing

_new is achieved if _{one counts}

separately simple

^and double _arrows, which

gives

_an invariant with values in _a commutative group Z~.

Principally

_new invariants _cari be

constructed with values in non-commutative groups.

To define the _most

general

invariant _we introduce io ₌ 5_x2 group

generators

_a, b, _c,

d,

_e,

A, B, C,

D, E

corresponding

to 5

possible

orientations of _an

edge carrying

a double _{or a}

single

_ar-

row. In order for invariant to be

independent

of the

loop

ail rhombi of the

tiling

must be

"commutative", which

gives

io relations ab~~

=

BA~~,

^ad~~

=

D~~A,

bc~~

=

CB~~,

db~~

=

B~~D,

cd~~

=

DC~~,

be~~

=

E~~B, (i)

de~~

=

ED~~,

ec~~

=

C~~E,

ea~~

=

AE~~,

ca~~

= A~~C

(Fig. 3).

We have thus defined _a non-commutative group G of values of the invariant. To each defect of _a

tiling

_a certain

conjugation

dass of G

corresponds iii.

Let _~t be _an oriented

tiling loop passing through

ideal regions of the

tiling.

The element

g(~t)

of G attributed _{to ~t is} defined _{as a} formol

product

of generators ^attributed to segments ^{of the}

loop

in their sequence, the generator

being replaced by

its inverse if its orientation given

by

the

arrow on the

corresponding edge

diifers from the orientation of the

loop (Fig. 4).

This element

is determined _{up to a}

conjugation

because

loops

have _no fixed

origin. Obviously,

absence of defects mside the

loop

entails that the invariant g(~t) = 1. We do _not know whether the inverse is true, that is if this invariant is fine

enough

to detect _any

phase

defect and

distinguish

between any pair of diiferent defects. On the other

hard,

the _group G

already

looks quite

complicated

and the invariant hard _to calculate. Some information _can be

obtained, however, by studying quotient

_groups of G. Let F be _a normal

subgroup

of G and H

= G

IF

the

corresponding

quotient _group. Then _to

get

an invanant with values in H it is suflicient to

replace

the dass of

conjugation

in G of the element g(~t)

by

the dass of

conjugation

in H of its _coset

g(~t)F.

One thus obtains _a "reduced" version of the invanant in which _some information is lost but which

may occur easier to treat. The

simplest

_way of

constructing

quotient groups is to introduce

additional relations. Each _group, for

example,

_can be "commuted"

by adding

to the

defining

relations those of

commutativity

of each pair of generators. One _{can see} that

commuting

the

a

A

d

Fig. 4. Definition of trie general ^invariant. Take _an arbitrary tiling loop that does not contain any mismatch. Choose arbitrarily _an initial vertex and fix _an orientation. Then the corresponding invariant

is the conjugation class in G of _a product of generators ^attributed to edges _m their sequence, orientation being taken into account. E.g., the element attributed to the bolded loop oriented counterclockwise

with the initial point Q equals ^ae~~ Dbe~~c~~bd~~cae~~

Dbe~~c~~bd~~c,

_{which is in} the _same class with A~.

group G _one obtains Z~ and the

corresponding

reduced invanant coincides with that introduced above.

Putting

further _a

= A _one obtains the invariant _a.

We will

apply

this scheme to construct 5

simple

invariants that enable to

distinguish

isolated mismatches of diiferent directions. It _was mentioned above that isolated mismatches _are

always

mismatches of

simple

_arrows. Take the defective

edge itself, passed

from _one end _to another and

back,

_{as a}

loop defining

the invariant. One thus obtains the

following

elements

A~~, B~~, C~~, D~~,

^E~~ of the _group G

corresponding

to isolated mismatches _m 5 diiferent directions.

To get the invariants _we impose additional relations

a=A, b=B, c=C, d=D,

e=E

(2)

and _one of the rive

following

sets of relations

(3a-e)

E=B=E~~=B~~,

C=D=1

,

(3a) A=C=A~~=C~~,

_D=E=1

,

(3b) B=D=B~~=D~~,

E=A=1

,

(3c) C=E=C~~=E~~,

A=B=1

,

(3d)

D ₌ A

=

D~~

=

A~~,

B ₌ C =1

(3e)

Consider,

_e.g., the

resulting

_group GA defined

by (1), (2)

and

(3a).

It is

engendered by

two

independent

generators

A,

B with relations

B~

_{=1, BAB}

=

A~~

This group is

isornorphic

to the semidirect

product

^of 22 and Z and _can be realized _as the group of

symmetries

of the set

(x

_E

Z)

of

integer

points on a

line,

B _{: x ~+ -x,} A _{x ~+ x} +1.

It follows that the order of A is infinite

and,

in

particular,

A~

#

1.

Consequently,

the invariant

404 JOURNAL DE PHYSIQUE I N°3

a

y

^c

B

Fig.

enerators. Old can be

expressed

in terms of new nes,

e.g., = e~~ B~~ _and C a~~ _C~~e.

Ail other relations _of ⁽⁴⁾ obtain by cyclic

in GA assigned to

a

mismatch

A~~ is non-trivial,

D~~, E~~

are

trivial. this onsideration

with groups GB ; . .

,GE

_defined

(2)

and

(3b),... ,

(3e)

one proves that all ismatches

are

diiferent

land annot nnihilate).

An

important

ormations

of

ideal ^{nfinite tilings}

[6].

_{To deflate}

a

_tiling one cuts tiles into

then pplies a scale ransformation

estoring the initial

size

of

_tiles, which is also possible

for

finite

and

even

defective tilings.

A

finite tiling with a single mismatch ^comes after

deflation

another tiling _with a

single

^{srnatch in}

the

sarne onsequently, the deflation dimin-

ishes

the

densityofdefects by the _{factor r~.} In

the

algebraic language, deflation

an ndomorphism

a

-+ à,

b

-+ Î,. of the _group

G, where

à

=

 =

b =

a~~C~~, ^É

=

e~~B~~d,

é =

b-i

_j~-i

à

-

_ > _ >

Geometrically, à, generators

_on new edges and arrows

(Fig. 5).

One

can _see that

this ^domorphism

the

invanants with values ⁱⁿ groups GA;.,GE,

hich is _in

agreement

with the

statement

_that

deflation does not change _solated

mismatches.

4. Order

Parameter

Space

We are now

going

a _{purely algebraic}

way.

We

will

_appeal

to the

eneraltheory of cellular _{complexesils].}

the followmg

_intuitive escription of

what

a _cellular complex

C

is will

be

uflicient.

It is

built

inductively

from

_cells of dimensionality.

First,

take ^everal points, _which

are disks

(0-cells).

They form _the

o-skeleton sko C

of the complex C

Then take several

segments,

-dimensional disks

(1-cells), and omehow ttach theirends

to

0-cells _chosen in the ^previous step.

This

gives us the _1-skeleton ski C

that take everal -dimensional disks _(2-cells), and attach their _boundary circumferences to the

1-skeleton.

Actually,

at this stage a restriction for

possible

attachments

arises,

this restriction will be described below. We thus get ^{the 2-skeleton} sk~ C.

Continuing

this

procedure

_up to cells of the

highest dimensionality

_n

(or infinitely

for infinite-dimensional

complexes)

_we get the whole cellular

n-complex.

The

topology

of the

resulting

_space

depends

_upon the quantities of cells of each dimension k < _n and upon the

mappings

_we have used to attach boundaries of

ÎÎcells

_to

(k -1)-skeletons.

The above-mentioned restriction for

possible

attachments _is the

following.

An attachment of _{any new} k-cell is _some

mapping

F from _its

boundary ^ôD~

₌ ^S~~~

to skk-i C

(called attaching mapping).

Consider the image of this

mapping F(ôD~

in

skk-i

C.

Let D~ be _any 1-cell in

skk-i C,

< k -1. We demand that either _D~

(

^ôD~ is

entirely

inside the

image F(ôD~)

_or has empty intersection with the

image.

Consider

Figure

3 _{as a} cellular

2-complex assuming

that

edges

marked

by

the _same letters

are identified

along

_arrows. One _can

easily

_see that there remain 2 non-identical vertices

(0- cells),

10 non-identical

edges (1-cells),

and 10 non-identical rhombi

(2-cells).

Denote P the

resulting topological

_space. Instead of the

decagon

in

Figure

3 _one could have considered the entire decorated Penrose

tiling

_{as a}

complex

and then

identify

tiles of the _same orientation

along

_arrows. The

resulting complex

would be the _same. We stress the fact that vertices of Penrose

tilings

fall into two types:

1)

vertices in which at least _one

edge

with _a double _arrow

ends;

2)

ail other

vertices,

1.e. vertices in which

edges carrying simple

_{arrows can} start and end while

edges

with double _{arrows can}

only

start.

Edges

with

simple

_arrows connect vertices of type 2 and

edges

with double _{arrows go} from type 2 to type ^{1 vertices. All the vertices of the} same type _are identified in P.

Suppose.that

there is _an isolated mismatch in the

tiling.

The _presence of _a

single

mismatch does not break the division of vertices into two types, ^{therefore this} is _a mismatch of _a

simple

_{arrow as was}

stated in Section 2.

The way we have defined the group G almost coincides with the standard

algorithm

for

a cellular calculation of _7ri

lis].

This

algonthm

is based _on the theorem which _states that

7ri(C)

=

7ri(sk2 C)

for _any cellular

complex C, permitting

to reduce the question to that for

2-complexes.

We _assume that C is connected and the fixed

point

O in C with respect to which

we define

7ri(C)

coincides with _one of 0-cells. The fundamental _{group is} found in terms of

generators

^{and relations} in three steps.

l)

For each 1-cell choose _an orientation and attribute _a formol generator a~.

2)

For each 2-cell consider the

corresponding attaching mapping

S~ _-

ski C, decompose

its

image

into the _sequence of oriented 1-cells ii12 i~

(taking

into account that _some of1- cells _may enter more than _once if the image contains

multiple points),

and add the relation a~~o~~ a~~ = 1. Note that the restriction _on

possible

attachments is important ^here.

3)

For any o-cell diiferent from the fixed

point

O choose _a

path connecting

it with O and

consisting

of onented 1-cells

ji,j2, ,jp,

and add the relation aj~aj~ aj~ = 1.

The

only

distinction between groups

7ri(P)

and G arises _on the third step of calculation of

7ri(P)

from the fact that the cellular

complex defining

P has 2 diiferent o-cells. It follows that in order to calculate

7ri(P)

_one has to contract _one of the

edges joining

vertices of diiferent

types, _e.g.,

just join

the relation _a

= to the relations

(1).

Note that _we have obtained the _space P from the Penrose

tiling nearly

in the _same way

as that we followed _to get a torus from _a

periodic lattice, namely by

_some identifications of

equivalent

areas. This

analogy together

with _an intimate connection between

7ri(P)

and

the group G

dassifying

defects suggests to call P the order parameter space for

pentagonal

quasicrystals.

^{The essential difference between}

^periodic

^and

quasipenodic

_cases consists in the fact that the torus _is the

quotient

_space of _a

plane

under the action of the _group of translations

Z~,

while P and the group G do _not admit such _an interpretation. This _means that _{we are}

406 JOURNAL DE PHYSIQUE I N°3

dealing

with _some

generalization

of the standard situation in which _a discrete _{group acts on}

a

homogeneous

_space. ^{In the standard} case the _space is

partitioned

into

disjoint

fundamental

regions,

_so that each of them contains _no

equivalent points,

and _any two con be identified

by

the action of _some element of the discrete group. We _{see no}

straightforward analogy

^for

the

quasipenodic

_case. ^{A related}

complication

is that _our

quotient

_space P is

self-intersecting singular

^surface.

Indeed,

_a

simple inspection

shows

(see Fig. 3)

that each

edge

of P is shared

by

4

faces,

instead of 2 _as it should be for

regular

^surfaces.

Unfortunately,

_we do not possess any

general theory

of

quasipenodic

_group actions, and do not know whether such _a

theory

exists.

Note that _our candidate P for the order parameter _space in the _case of

pentagonal

symmetry does not coincide with rhombic icosahedron with identified

opposite

faces

(RI), proposed

in

references [4, Si. As _a matter of fact, _7ri

(RI)

= Z~ is commutative, while _we believe that mis- matches should be described

by

non-commutative invariants. The

advantage

of the

approach adopted

in [4,_Si consists in its

generality,

the drawback

being

^{the absence} of clear connection with concrete

phase

defects. Our treatment has been

empirical,

since _we have started from

Penrose

matching

rules which _are themselves

empirical,

but has

permitted

_us to

analyze

mis- matches. The

problem

consists in

finding

_a

geometrical

derivation of Penrose

matching

rule.

We believe that the notion of the order pararneter space P will be

helpful

to achieve such _a derivation.

5. Conclusions

We have constructed _{a new}

geornetrical

invariant for rnismatches of Penrose

tilings.

^The

invariant takes values in _a non-commutative group G which

plays

the role of the fundamental group of the order parameter _space introduced for

ordinary crystals

and continuous media with broken symmetry. ^This _group is

dosely

related to the fundamental _group of _a certain space, which _may be

considered, therefore,

_as the relevant order parameter _space. To _any

quotient

group of G _a reduced _version of the invariant

corresponds.

We have shown

using

these

simpler

invariants that mismatches of different directions _are

topologically

different. We do _not know if this invariant permits a

complete description

of ail

phase

defects of Penrose

tilings.

The relation with another

approach

to the

topological defects,

based _on

homology

classes

[16],

^also

remains to be elucidated.

Acknowledgments

I thank M. Kléman and O. Radulescu for fruitful

discussions, ànd

_P.

Kalugin

and L. Levitov for

comprehensive

consultations. This work _was

begun during

the stay of the author in the

Laboratoire de

Minéralogie-Cristallographie,

Universités Paris 6-7. I wish _to thank the staff of the

laboratory

and

especially

J.-F. Petroff for kind

hospitality.

The work _was

partially supported through

the ENS

(Paris)

Landau Institute for Theoretical

Physics (Moscow)

agreement.

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JOURNAL DEPHYSIQUEI T 3,N°3, MARCH 1993 ~