RENORMALISATION ON THE PENROSE LATTICE

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RENORMALISATION ON THE PENROSE LATTICE

C. Godreche, Henri Orland

To cite this version:

C. Godreche, Henri Orland. RENORMALISATION ON THE PENROSE LATTICE. Journal de

Physique Colloques, 1986, 47 (C3), pp.C3-197-C3-203. �10.1051/jphyscol:1986320�. �jpa-00225731�

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JOURNAL DE PHYSIQUE

C o l l o q u e C3, s u p p l é m e n t au n o 7 , T o m e 47, j u i l l e t 1 9 8 6

RENORMALISATION O N THE PENROSE L A T T I C E

C. GODRECHE' and H . ORLAND" '

'service de Physique du Solide et de Résonance Magnétique, CEN-Saclay, F-91191 Gif-sur-Yvette Cedex, France

n t

Service de Physique Théorique, CEN-Saclay, F-91191 Gif-sur-Yvette Cedex, France

Introduction and summary

The discovery of quasicrystals has started an intense theoretical activity.

Much of this activity i s devoted to the study of the structure and

cristallography of these new phases. Other studies concern their stability, their elementary excitations ...

We investigate here the magnetic structure of quasi-crystals : we want t o study the critical behaviour of the lsing mode1 w i t h Hamiltonian

x = - L .

_{o j >}

^J..s.s.

_IJ _i_J ( 1 )

where <i,j> are nearest neighbour sites on a Penrose tiling. The variables Si=* 1 are k i n g spins. The Penrose tiling w i l l be described here using Robinson tiles [1,2].

In this talk we w i l l mainly explain how, by taking advantage of the self similarity of the Penrose tiling, it i s possible t o match a real space renormalisation method to the deflation (composition) of the tiling. This w i l l be illustrated here by the cumulant method [3] applied t o the dual of the Penrose tiling. We w i l l then give a brief account of the results. The remainder of this section i s a summary of the talk.

There are two kinds of Robinson triangles called P and

Q

(Fig. 11. One may associate t o these two triangles two kinds of atoms w i t h spins located at the center of the triangles, i.e. on the dual lattice. The magnetic

interactions between the atoms are described by coupling constants J,,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986320

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JOURNAL DE PHYSIQUE

Figure 1. Robinson triangles and the process of composition. Angles are in units of n/5.

r = (45

+ 1 )/2

which depend on the environments of the triangles. One has thus different types of couplings between atoms P and 0, P and P, 0 and 0.

The renormalisation method consists i n renormalising i n two steps, using

&Q types of blocks. The cumulant method w i l l be performed to f i r s t order only. We f ind ferromagnetic and antiferromagnetic phase transitions corresponding t o the sarne f ixed point. Another renorrnalisation scheme, namely the Migdal-Kadanoff approximation, applied on the Penrose tiling itself, and an extensive study of the antiferromagnetic phases w i l l be described w i t h f u l l details elsewhere [4].

Penrose t i l i n g w i th Robinson triangles

The Penrose tiling i s usually described with darts and kites or w i t h two rhombs. If one cuts those polygons i n a particular way one obtains the two

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triangles P and O of Fig. 1. They w i l l be referred t o as A-tiles. The tiling with A-tiles i s subject t o the following two matching rules.

1. Each edge of a triangle must be abutted by the edge of another triangle i n such a way that the colors of the vertices match.

2. In the case of a monochromatic edge fie. an edge joining 2 vertices of the same color), the smaller angle of one triangle must abut the smaller angle of the other. This i s equivalent to Say that each monochromatic edge i s oriented. The monochromatic edges of adjacent tiles must have the same orientation.

The A-tiles can be composed t o form new tiles (Fig. 1 ). Composing P and

Q

gives the triangle

rQ'

which may be obtained from Q by expanding by the factor

r

and reversing the colors. Thus there are two basic cases of tiling w i t h triangles, A and 8. In case A, there i s a large triangle P and a small triangle Q. In case

B,

the same triangle P has become the small triangle, and the large triangle i s

rQ'.

Similarly the composition of

rQ'

and P form the triangle

rP'.

Consequently there i s a third case of tiling denoted TA' consisting of triangles rP' and

rQ'.

This i s isomorphic t o case A, since both triangles are expanded by the factor

r

and have the colors reversed

(Robinson

[Il).

We may proceed further : by composition, the TA'-tiles lead t o TB'-tiles, the TE-tiles t o t 2 ~ - t i l e s . Let us summarise the successive steps (Fig.2) :

In the parentheses we wrote i n the f i r s t place the larger t i l e that appears i n the tiling. The important remark for what follows i s that at each step i n the composition process, the links or edges that are deleted are al1 cohtained in a particular polygon (Fig. 2 and 3). In the f i r s t and third steps the compositions of t iles take place i n a rhomb. In the second and fourth steps they take place i n a kite.

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JOURNAL DE PHYSIQUE

T B ' ( T ~ O , T P ' ) T ~ A ( T ~ P , T ~ Q )

(dl le)

Figure 2. From the A- tiling to the I'A-tiling. Hatched polygons are rhombs in a) and

CI,

kites in b) and dl. (This figure i s a modified version of a figure of Ref. [2])

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Figure 3. The process of composition (deflation).

The dual lattice. Renormalisation in two steps

The method we use consists i n renormalising the Hamiltonian in two steps.

The f irst step corresponds to the passage from the A- til ing to the B-tiling. The second step to the passage from the 8-tiling to the TA'-tiling. In the f i r s t step the block chosen i s the rhomb, i n the second step the block i s the kite. Since the resulting lattice i s isomorphic to the initial one with inverted colors on the vertices, we do not consider the two next steps leading to the r2~-tiling.

Furthermore, i n order to be able to implement a block spin renormalisation method- such as the cumulant method -one has to define block spins, and separate the bonds into intra-block bonds and inter-block bonds. This separation i s easily done i f we locate the spins on the dual lattice instead of the direct one. Thus one has to consider the environments around each

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C3-202 JOURNAL DE PHYSIQUE

triangle at each step (Fig. 4). These environments define 8 coupling constants KI-% at the f i r s t step and 5 coupling constants LI-LS at the second step (those coupling constants are related t o the Ji, of Eq.( 1) by including the temperature i n their definition : Ka= Jij /kT).

The f i r s t renormalisation leads t o the relation

L2, L3> L4s L5a

$,

^K7s $1

=

Wl(KI, ^K23

%>

^K4>^K5s

&* 5,

^K8)

The second one t o the relation

(KI, K',, K',, K',, K',, K',, K',, K',)

=

W2(Ll, L2, L,, L4, L5,

$,

K7,

K8)

Finally one has

(K.,, K',, K',, K',, K,, K',, K',)

=

R,. RI (K,, KZ,

%,

K4, )<J,

s, 5, G)

After those t w o steps the resulting lattice i s scaled by

r.

The IR, and IR2 renormalisation transformations are given at f i r s t order w i t h the cumulant method by the following equations.

Def ining x

=

I /( 1 +exp(-2)<J)) and y

=

1 /( 1 +exp(-2L2)) the renomalisation equations read :

1 ) At step R,

LI

=

x2 (%+K7) L,

=

x

K,

L,=x 2

K,

L4

=

^X²KI L,=xK,

2 ) At step IR2

K',

=

L, K',=yL,

K',

=

L, K',

=

L,

K',=yL,

4 =

y* (L,+%)

K',=y 2 LI kg

=

K7

These equations admit a unique fixed point : x*

=

.82661, y*

=

.88215. One finds a ferromagnetic transition : the isotropie critical point is given by K,

=

K,= .959970. When certain couplings are negative, one finds anti ferromagnet i c ordering a t low temperature, corresponding t o the same ferromagnetic fixed point. The complete phase diagram 1s difficult to

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Figure 4. Different types of coupling constants associated to different environments, for the A-tiling (P and

Q

triangles) and for the B-tiling

(rQ'

and P triangles).

describe since the space of coupling constants i s of dimension eight. This i s nevertheless done i n Ref.[4} for the case of the Penrose tiling itself.

This case i s more tractable since one has four coupling constants to describe the system, and one finds an infinite number of antiferromagnetic phases.

Ref

erences

[ l ] R. Robinson, University of California preprint ( 1 975) unpublished (21 B. Grünbaum and G.

C.

Shepard, "Tilinas and Patterns", unpublished [3] Th. Niemei jer and J. M. J. van Leeuwen, in Phase Transitions and Critical Phenornena, ed. by C. Domb and M. S. Green, vol. 6 ( 1 976)

[4] C. Godrèche, J. W. Luck

and

H. Orland, i n preparation