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Electronic properties of the octagonal tiling : a new renormalization-group calculation
J. Zhong, R. Mosseri
To cite this version:
J. Zhong, R. Mosseri. Electronic properties of the octagonal tiling : a new renormalization-group calculation. Journal de Physique I, EDP Sciences, 1994, 4 (10), pp.1513-1525. �10.1051/jp1:1994204�.
�jpa-00247009�
Cla,,ification Phi.ii<.i A/>.iii.a<.i.1
71.20 61.41)M 71.25
Electronic properties of the octagonal tiling : a new renormalization-group calculation
J. X. Zhong (' 2) and R. Mosseri (')
(' Groupe de Phy;ique de, ,olide,. Univer,lié Pari, 7 et 6. Tour 23. ? place Ju,,jeu. 75251 Pari, Cedex 05, France
(2) Laboratory of Modem Physics~ Xiangtan University, 411105 Hunan. P. R, china
(Re( PliPd 2/ Apii/ /994, ie<.en.ed in final foitii 2?./trie /994, a< <.opte(/ 30./wie /994)
Abstract. we present a new renormalization-group calculation of the octagonal tiling excitation spectrum. Two new self-;imilar octagonal tilings are introduced, a triangle-Lite and a triangle-dan tiling, which are entangled with the standard rhombus-~quare tiling. The decimation procedure
involved in the renormalization group calculation transforms each tiling enta a different one and goe; back tu the (rescaled) original tiling after three steps. The local and average density of state;
are calculated. In the case of a Laplacian-like tight-binding Hamiltonian, on the standard rhombus- square octagonal tiling, Dur results compare reasonably well to numerically denved spectra and
improve some previou; renormalization group calculations. in the pure hopping ca;e, the
renormalization i~ ~hown tu be inaccurate by comparing to numencal density of states obtained by
a continued fraction method.
1. Introduction.
There ha, been much intere,t in ~ua,iperiodic sy,teins since the discovery oi ~uasicry,tal; (1[.
Considerable effort has been devoted to the under,tanding oi electronic behavior in ,uch
,tructure,, intennediate between periodic ,ind random. The Lind oi long range order
(~ua,iperiodicity) that is nervertheles, present haï not yet allowed for ~i,impliiied irameworL ]'Le, for instance, the one that Bloch theorem and 1-space calculations provide in the presence oi translational symmetry. However, some ~uasiperiodic tilings present inflation-deil~ition
,ymtnetnes that permit real-space renormalization-group (RSRG) method, to be ~ipplied.
For ID ~uasiperiodic lattices. several skilliul ~inii poweriul RSRG approache, hâve heen
developed in order to inve;tigate the electronic propertie,. The mo,t iniluenti~il method i, the dynamical-map method introduced by Kohmoto. Kadanoii and Tang [?[, ~ind independently by Ostlund ci ai. [3[, which associates the energy spectum oi an infinite ID ~ua,iperiodic
structure to a trace map and a map invanant ai periodic systems (approxiniants) which are
,uccessively constructed according to well known inflation rules. Another widely used RSRG method is the real-space decimation methods developed by several groups [4-7[. Ail these method;, which are exact in one dimension, have provided a very detailed knowledge oi the spectral propertie;~ in tenu, oi the nature oi the ,pectrum and of the eigeniunctions (for a
recent review, see [?4[)
In compati,on with the 1-dimen;tonal ~y~tems, much iewer worL ha~ been devoted tu the electronic propertie, of ?D ~uasiperioiiic tilmg; ix-14[, >nô, owing tu their complexity, mo,t of'it ha, been purely numerical. Detailed investigation, ~, (1[ ,howed that the rhombu, tiling, (ter-ft)lLi ,ymmelric Penro,e and octogonal tiling,) put-e hopping,pectrum hâve a central peaL which 1, a iielta iunction a;,ociated with a,pecial Lind ni localizeLl,tate~ 1-e-, the coniined
,tate. In the Penro,e c~i,e, two finite gaps ,eparate this peak tram trie re,t oi the ,pectrum~
whi le there 1; no numencal eviLlence for gap, ,urrounding thi; central peak in octogonal tiling;.
In addition tu these numerical calculation,. RSRG approache, [13, 14[ hâve been propo,ed
for these ?D ,tructures. The problem in using RSRG decimation method, in ?D came, tram
trie diiiiculty in getting a compact renormalization group. The latter 1, obtained by neglecting
,orne interaction, upon renormalization, which leads thereiore to aproximated ;pectra. The
tir,t RSRG ,cheme for ?D ~ua;iperiodic lattice wa, proposed by Sire and Bellissard [13j f'or trie ~uasiperiodic octogonal tiling (paved by rhombus anLi ,~uare as b~i,ic tiles) on the basis of trie Schur f'oriiula f'or calculaling [lie ienormalized Haiiilloman on [lie decimated structure.
These author, calculate the ,upport oi the spectrum for a tight binding Hamiltonian which
include, a hopping terni between neighbors and a diagonal term proportional to the site
coordination number. It is po;sible to weight the diagonal and off-diagonal contribution, and
f'or instance interpolate between a
« molecular
» level ,ituation and a Laplacian-like mortel.
Another RSRG decim~ition approach was then developed, which u,es a more elaborate
decimation and calculate the local density of state, jLDOS) of Penrose tiling [14j.
In thi; paper we iollow the latter tuethod and calculate the octogonal tiling density of st~ites.
While in the previou; calculation on this tiling 13 [, the decimation proceeded in one ;tep irom the tiling to its deilated scaled version~ here it takes three succe,sive steps with only one Lind oi,ites being elimin~ited at e~ich step. The two intermediate steps correspond to eight-iota ,ymmetric ~u~isipenodic point sets. By consistently deiining edge, in these point set,, they give n,e to two new tilings, the Triangle-Dart (TD) and Triangle-Kite (TK) octogonal tilings.
The standard octogonal tiling i, hereaiter called the Rhombus-S~uare (RS) tiling. Dur result
improve upon the previous one; in that the juil expected ,pectrum is recovered in the
« molecular
» level limit (while one branch was missing near E
=
7 in Ref. il3])~ and also because the total bond width is also better approached in the Laplacian-like case. In addition
the DOS and LDOS are calcul~ited and the pure hopping ca,e 1, con,idered. This paper i;
organized a; follow,. In section ?, the three octogonal tiling, are presented and their mutual
tr~in,ionuations analy£ed. The RSRG approach is described in ,ection 3. In ,ection4, trie
calculated spectra are given, along with a di,cu,,ion.
2, Self-similar entangled octagonal sets.
The well-known RS octagonal tiling [15, 16[ ~ind its acceptance zone are presented in
figure, la and 16- According to the accept~ince zone description~ the RS sites con be cla,siiied
into seven types corresponding to different local environments. The seven types of sites, i e.,
c, ;, p, ~, m, r ~ind e, which ,ire displ~iyed in figure lc have the ireqqency ii~ = w,
ii, =
? w ~, n~ = w ~,
ii~~ = w ~, R,,, =
? w~, n,
=
w~ ~ind n~ =
w~ with
w = , ? 1.
The RS octogonal tiling presents a well-Lnown self-simil~ir property. Removing c~ ,, and
~ types of,ites and joining the rem~iining sites with new edge,, one obtains a new RS octogonal lattice with a re,c~iled iactor
= + , 2, This is exactly the decimation th~it w~is
clone in one step in reierence 13j. Here, we iirst remove type c sites and add new edge, on the
long diagonal oi the original RS rhombi. This immediately gives a new lattice made oi two diiierent tries, ,i triangle and ~i kite (Fig. 2~i). Thi, new tiling i; ,tilt eight-iota symmetric (it, acceptance zone has an eight-iota symmetric ,tar shape) and is called the TK tiling. It; ;even types of,ite, are illu,trated in figure ?b.
s
e p c
s
a) b)
c
~ 2
m
c)
Fig. l. Trie Rhonibu,-,qiiare iiling a) a piece Dl ihe iiling where 3-fold and 4-ifild cooriiinaicii ,ite, ,ire eiupha;ized (.) 3-f'old ,lie,. (OI 4-fold ;tie;) b) the nccepi;ince domaii with trie ,ub-demain,
a,,ociated with each type of ,lie, c) the ,even type, of,ite, with their local envirfimuent.
We now eliminate ;ites ai type; 6 and 7 in the TK tiling. Note that tho,e ,ite, correspond to the original « s » sites oi the RS tiling. Alter proper edge addition, a new tiling is deiined
which is composed oi ba;ic triangles and darts. It i-s cJlled the TD octogonal tiling (Fig. ~a) and
its eight types oi sites are di,played in figure 3b. It is ea,y to see that ,iiter removing the
type 8 sites in the TD tiling, une finally obtain, the RS tiling with
a iescaled jacter
À. The type 8 sites in the TD tiling correspond to the type ~ ;ites Dl'the original RS tilin~>. SO
une c~in see th~it f'or an RS octogonal tiling, ii we remove c type~ s type~ and ~ typel;ites
,uccessively~ we obtain ~i TK~ TD ~ind fin~illy ~i rescaled RS tiling. In the t'ollowing we ;hall
a)
b)
2
~7
6
Fig. ?. The Triangle-Kite tiling a) a piece of the tiling, b) the ,even type, of ,ites with their local environment.
,ilso denote TK. TD, and RS the decimations which transform the RS. TK, and TD point,ets
into the TK, TD, and RS point sets, respectively. These three transformations will be used to derive a (approximate) compact renorm~iliz~ition group to study the electronic properties oi these octogonal tilings. Note th~1tlij>oiiafii fi>llow.iii,q point for these intermedi~ite tilings, the hopping terms oi the Hamiltonians are not restricted to be along the edges that deiine the tiles.
More complex interactions are generated upon renormaliz~ition, and will be given below.
a)
b)
6
~
7 ~
~
Fig. 3, The Triangle-Dart tiling. ai i piece of the iiling : bj ihe eighi type, ut ,ire, wiih their local environment.
3. Renormalization group for octogonal tilings.
We consider the iollowing general tight-binding Hatuiltonian H
= LE>[') (il +iv,j l'l (i1 (')
whre ~, i; the site energy and v,~ ii the hopping integr~il between sites and j. The Green function G(H) i, deiined a;
(Z-H)G(H)=/ with Z=E+i~. (2)
The ,ites ai the original lattice are c~illed oi type or ? corre;ponding respectively to tho,e
,ites being kept and tho,e being decimated. Alter decimation, one ha;
(z-Hli)Gii(Hli)=' (3)
where H(j is the renormalized Hatuiltoni~in l'or the new set composed oi only type site;, which 1, given by
~Îl
~ ~ll + ~12(~ ~22) ~21 ~~)
In order to c~ilculate the DOS, we u,e the gener~iting iunction iormali;m introduced by
Tremblay and Southern [17[, and proved to be u;eiul for,eli-similar and ~ua,iperiodic
structures Il 8-20]. Let us deiine the generating iunction by
,z ,
F (H
=
2 In 1-x,fl dii~ exp [1U~ (Z H U (5)
where U~ (iij fi~,
ii~
and ii~ are continuous variables deiined on the site,. Succe,,ive decimation gives
K,,,~,
F (H
=
jj In (det (Z HÎÎ')) + constant (6)
1-,
in which 1 denote, that the carre,ponding variables belong to1-th decimation. Now, the DOS p (E) con be deduced irom the derivative oi the generating iunction F (H)
p (El
=
Im
lim jj G,,
=
lim ~ ~~ j( ~~~~. (7)
~ N-~ ,~j N-~ ~
For ID ~uasipenodic structures, suitable choice of HÎÎ' often make (Z-HÎÎ')~' ~ind
det (Z H((') ex~ictly solvable and hence give the LDOS ~ind DOS. However,
in the 2D case~
even though one chooses an H((' corresponding to solvable ;mail clusters, the successive decimations make the treatment of H((' as difficult as that of the original Hamiltonian because of the increasing number of the parameters in the Hamiltonian of the renormalized structures.
Some approximations hâve to be made in order to get a compact renormaliz~ition group~ such that the number of parameters remains constant under decimation. Usually H(('
is ~ipproxim~ited
by a local Hamiltonian constructed on ;mail clusters whose mutu~il interactions are
ignored 13 [.
In the octagonal case, tve use the three ~ibove transformations TK, TD and RS in order to
generate the renorni~ilization. TO obtain the RSRG transformations equ~itions, let us fir;t
con;truct the Hamiltonian for the different structures. We assume that different types of sites hâve diiierent site energie,. For the RS octagonal lattice~ the seven kinds of site energies are
(~.~, ~.,. ~~, ~,~, ~,,, ~,, ~~) corresponding to seven types oi ;ites, c, s, p, ~, tu, r, and e. The
,ite energie, oi the TK and TD tilings taLe the values (~j. ~j, ~~. ~~. ~.,. ~~,.
~~) ,ind (ej, ci, ei, e~, e~, e~,, e~, ex), respectively. The three type, oi hopping integrals
i,, i,,, 1,, in the three tilings, are illustr~ited in figure 4. In Dur RSRG, we only con,ider the contribution oi the neare,t-neighbor decimated sites to the renorni~ilized site energie;.
AccorLiing to e~uation (4j, we obtain the iollowing e~uation, for the TD tran,ioriuation Ii we deiine
c
= i/(z- ~~i i>j = Ii(] +21,c), pi =t(c
Iii = pj/(1 -2p~c), />~
=
2plc/(1 2 jj~c)
~~j~~
/
~j~fil
tl
b)
v4
v8
v2 v3
vi
'~~
~
~ ~
~~
~ki
~~~
~k5
~ ~ ~
k2
~
~f
~k k6
k9
~ ~Ù
k8
Fig. 4. Definition of the Haiiiltonian hopping parameter, ; a) rhombu,-,avare iiling ; b) triangle-kite tiling, c) triangle-dort tiling.
then one gets,
~
? p(C
~j ~~.+4/jC +?pj p~C +
2(p~ + p
~ C
~~ ~~~~~~ ~
' ~l~i~c ~i = ~, +
~'~~
~~/~~'
~~~ ~P' ~~ ~~ ? iC
~
?t~C (8j
' '
~ ~~
<, ~ +
' ~~~$~'
~
211C
~~= ~,+ijc +
-/sC
Î'j=i~,+?iitjc, i~j=t~,+tji~c~ 1>~=t~, 1,~=t~
us = i, + ij i, C i'~, = 11 + /~ i, C 1>~ = i
~ ~
v~ = i, and
D, = C ~( - ? p~ p~ [
D~. =
C ~(C ' ? ~)~[ (C ' ? p~ - p~ - p( ]
F H~s) = F
(H~~) - R~ N In D,,, - fi~ N In D~ - n~ N In
N is the
number oi site; m RS octogonal lattice. In ~uation, (~). D,,,, D, and D~
determinant, with the local
lusters oi c-type ,ites around sites of type m,
is.
Now we periorm the TK tr~insiormation irom TK to TD, i e., decimate ,ite, of type,
in
TK iling
sites to the site, in the original RS tiling). Deiining
ej=>j+81)R~ e~=~j+6n)R+?>)S, e~=~j+41>)(R+S)~
e~ = ~j + ~ v)R + 6 v),i
e~= ~~+? nÎR+?(n~+ "~)S. e(,= ~i+ "~R+? i'~S+? ~~,
~
'~Î
~'Î~
p, = ~;+~ v/~~
~> = ~j+4
/~' "
7
i'x
lj
=
~ nj i>~R. Ii
= nj n~R l~
= 21,j 1,~,,i~ l~
= 2 vj v~S~
1;
= v~ v~R + i>j v, S
j,2R vi
j, RI
l~, = >,~ + ~
~ l~
= 1,~ + v; »~,,i, lx
= 1>~, l,~ = ~ Î
~ jù~
-1~ R i>jR-
and
J~ >i~
F (H~D)
=
F (HTK) m N In D)~ 2
~ N In DÎ~ + constant (1 1)
À
~~th D1°
=
R 2 1,j, DID
~ s
where 1,
=
2 , ? is the fre~uency oi rhombi in the RS tiling, ~ind D(~ Dΰ
corne tram the
contributions oi two-site molecular cluster (6-type oi site) and single site (7-type ,ite) (see Fig. 3)~ re,pectively.
Finally when the RS transformation is applied to the Tk tiling, a new RS octogonal tiling ii obtained with the iollowing renonualized Hamiltonian parameters
~/ = e~ 4 1( W. ~,
= e~, + ~ Ii W,
~.j = e~ + 31( W,
F~~ = e, + (1( + ? Iii W
~,,, = ej 4 21( w, ~j
= ej + il W, ~.j
= ej
Ii
=
i Ii
= ii + i~ ix w. /~ = i, + i~ i~ w
~~~
ii = 1~ + 2 ii i~ w. Ii i~,. Ii
=
in
wheie W Z ex. The generating iunction f'or the TK lattice i, gin,en by
n~
F (H~~
=
F (H(s)
~ N In W. (13)
Now we hâve denved a RG tram an RS octogonal lattice to a new RS octogonal 1,ittice. Ii one
,tort, tram TK or TD octogonal lattice, the ahove RSRG ti,in,formations con also give the
corresponding renormalized parameters for these TK or TD octogonal tilings. Since ail these
tiling, have one special site, with periect eight-iold point ,ymmetry which ,urvives the whole ,e~uence of decimation;, trie local density oi,tate; ±it this site can be calcul±ited iroii trie RG e~uation, (8)~ (lù) and (1?). Using the recur,ion relation, (~). Il )~ Il 3) for the generating iunction, one con calculate the (average) density of,tates oi the octogonal tiling.
4. Results and discussion.
4.1 LAPLACIAN-LIKE HAMILTONIANS. We focu; the di,cu;,ion on the excitation spectruiu
of the RS tiling. Figure 5 shows the support of the spectrum for the Laplacian-like Hamiltonian generated by taking ~,
= =,, ij = ii = t~ = t~ = i, i~ = i~, =
ù~ ù
~ t ~ in which =,
=
3, 4, 5, 6, 7, 8 is the coordination number of the seven types of sites c, s, p, ~, m, i e, respectively.
The ca,e t
= corresponds ta the spectrum of the discrete Laplacian on that tiling. We con see
that the present calculation improve, the previous one[13[ in both liiiits, t =0 and
t
t ) l
i
0 ,
3 4 s 6 7 8
Energy
Fig. 5. Spectrum of the Laplacian-like Hamiltonian with 0 <1w1
t
0
3 4 5 6 7 8
Energy
Fig. 6. Spectruii of the H;imiltonian related to that in figure 5 >nô di,cu,,ed in trie te~t.
é 8.OO
Î Î 7.OO
6.OO ,e( 5.OO
14.OO
$( 2.OO
100É
g g
ui à oi c4 à à à c4 1 $ ÎÎ
nergy
à) à
1
.é Î ui
g
~$U~ D Q - - 4 D
UJ
n«gy
b)
Fig.
ii ver;ige : b) local DOS iJn ihe,lie wiih
8-fold,jmmeirj,.
