Table Of ContentFractal dimensions of the wavefunctions and local spectral measures on the Fibonacci
chain.
Nicolas Macé,1 Anuradha Jagannathan,1 and Frédéric Piéchon1
1Laboratoire de physique des Solides, Université Paris-Saclay, 91400 Orsay, France
(Dated: August 4, 2016)
We present a theoretical framework for understanding the wavefunctions and spectrum of an ex-
tensively studied paradigm for quasiperiodic systems, namely the Fibonacci chain. Our analytical
results, which are obtained in the limit of strong modulation of the hopping amplitudes, are in
good agreement with published numerical data. In the perturbative limit, we show a new symme-
6
try of wavefunctions under permutation of site and energy indices. We compute the wavefunction
1
renormalization factors and from them deduce analytical expressions for the fractal exponents cor-
0
responding to individual wavefunctions, as well as their global averages. The multifractality of
2
wavefunctionsisseentoappearatnext-to-leadingorderinρ. Exponentsforthelocalspectralden-
g sity are given, in extremely good accord with numerical calculations. Interestingly, our analytical
u resultsforexponentsareobservedtodescribethesystemratherwellevenforvaluesofρwelloutside
A the domain of applicability of perturbation theory.
3
Introduction As distinct from periodic crystals on the obtain explicit expressions for the fractal exponents cor-
]
l one hand, where electronic states are typically extended, responding to individual wavefunctions. We show, using
l
a and disordered systems on the other hand, where states the conumbering scheme, that wavefunctions are sym-
h are typically localized for low dimension and/or strong metric under exchange of site and energy indices in the
- enoughdisorder,electronicstatesinquasicrystalsarebe- perturbative limit. We show that the multifractal prop-
s
e lieved to have an intermediate “critical” character. The ertyofwavefunctionsappearsatnext-to-leadingorderin
m study of tight binding models on the Fibonacci chain, ρ. We give expressions for their globally averaged val-
. a one dimensional paradigm for quasicrystalline struc- ues. We compute next the generalized exponents of the
t
a tures, is particularly important, as a first step towards globalandthelocalspectralmeasures. Ourresultsagree
m understanding the physics of these systems. These mod- very well with numerical computations on approximants
els have been extensively investigated theoretically, as in of the Fibonacci chain.
-
d [1–6]. There have also been many experimental stud- In Sec.I we introduce the model, and some basic defi-
n
ies of electronic properties of this model. To cite some nitions and notations. In Sec.II we review the real space
o
recent works, in [7] investigated transport due to topo- renormalization group used to calculate the wavefunc-
c
[ logically protected edge states in a Fibonacci photonic tions in perturbation theory. In Sec.III we present the
waveguide. In [8], the density of states of a Fibonacci main steps of the calculation of fractal exponents, show-
2
tight-binding model was studied by direct observation of ing in detail how to obtain results to leading order. In
v
polariton modes in a one-dimensional cavity, and shown the following section Sec.IV, we present the higher or-
2
tohavethefractalstructure,log-periodicoscillationsand der corrections, and we obtain theoretical predictions
3
5 gaps labeled as predicted by theory [4]. which we then compare with numerical data. In Sec.V
0 we discuss some of the implications of these results. We
0 While the spectral properties in the model are now presentaninequalityrelatingthedifferentfamiliesofex-
1. reasonably well understood, wavefunctions are less well ponents which, we speculate, might be more generally
0 characterized. More generally, despite the belief, sup- valid in other 1D models. Summary and conclusions are
6 portedbynumericalevidence,thattherearecriticalwave presented in Sec.VI. The Appendix shows details of the
1 functions in quasicrystals, there are no analytical calcu- higher order calculation in perturbation theory.
: lations for the fractal properties of all wavefunctions, to
v
i our knowledge. In view of the importance of the struc-
X
tureofeigenstatestounderstand, forexample, dynamics I. MODEL AND DEFINITIONS
r ortransport,obtainingatheoreticaldescriptionofstates
a
in this 1D quasicrystal is necessary. In this paper, we
Thetight-bindingHamiltonianthatwewillconsiderin
present a detailed calculation of the properties of all the
this work is the pure hopping model given by
wavefunctionsintheoff-diagonalFibonaccitight-binding
model,inwhichthehoppingtermshavetwopossibleam- (cid:88)
H = t (i i+1 + i+1 i), (1)
plitudes, ordered according to the Fibonacci sequence. i | (cid:105)(cid:104) | | (cid:105)(cid:104) |
i
In the strong modulation regime where the ratio of hop-
ping amplitudes ρ 1, one can write an approximate where the hopping amplitude between sites i and i +
(cid:28)
renormalization group transformation for this model [9– 1, t , can take the value t (strong) or t (weak). The
i s s
11] and obtain recursion relations for multifractal expo- ratio of the number of hopping amplitudes of each type
nents of the spectrum. Returning to this approach, we is N(t )/N(t )=ω where ω =2/(1+√5) is the inverse
w s
2
of the golden ratio. This ratio is irrational, showing that degeneracy. We now consider separately the case of the
the chain cannot be periodic. t varies along the chain atomic and of the molecular energy levels.
i
according to the rule Atomic levels At first order, each atomic energy level,
localizedonanatomicsitecouplestotheatomiclevelslo-
(cid:40)
t = tw when imodω−1 ≥ω, (2) calizedonthetwoneighboringatoms(figure(2)). Inper-
i t otherwise. turbation theory, the effective bond coupling two neigh-
s
boring atoms takes on only two possible values (as illus-
TheFibonaccichaincanalternativelybeconstructedre- trated on figure (2)), a strong and a weak one, arranged
cursively by the inflation method. We start with the again according to the Fibonacci sequence. More pre-
finite chain C =t , to which we apply the inflation rule cisely, upon replacement of the couplings between atoms
0 s
by renormalized couplings, one passes from the chain C
(cid:40) n
r d=ef ttw →ttwts (3) ttoionthethcehaaitnomCinc−d3.eflWateiocna.ll tThhisegreeonmoremtraicliazletdracnosufporlimngas-
s w
→ are linked to the old ones by a multiplicative factor z
to obtain a series of longer and longer chains : C1 = [11]. One finds t(cid:48)s =zts, t(cid:48)w =ztw, with z =ρ2.
r(C ) = t , C = r(C ) = t t , ... C = rn(C ). The
0 w 2 1 w s n 0
Fibonaccichainisthendefinedasthesemi-infinitechain tw ts tw ts tw tw ts tw
C .
∞Innumericalcalculations,wewillreplacetheFibonacci
chain C by a periodic system, whose elementary cell is t t
the finit∞e chain C (the so-called nth periodic approxi- 0w 0s
n
mant). In the above formula, this amounts to replacing
Figure 2: Illustration of the atomic deflation rule: here the
ωbyarationalapproximant,ω =F /F ,whereF
isthenth Fibonaccinumber(stnartingn−fr2omnF−1=F =1)n. fifth approximant is transformed to the second.
1 2
Molecular levels. In a similar fashion, each state local-
F =8
5 izedonadiatomicmolecule,iscoupledtotheneighboring
molecules through only two possible effective couplings
Figure 1: The periodically repeated block of the fifth
between two neighboring molecules. Upon replacement
approximant to the Fibonacci chain. Weak couplings t
w of the couplings between molecules by renormalized cou-
are represented by a single line, and strong couplings t
s plings, one passes from the chain C to the chain C
by a double line. n n 2
(see figure (3) for an example). We call this geometric−al
transformation the molecular deflation. The renormal-
Atomsandmolecules. Inthestronglymodulatedlimit,
ized couplings are linked to the old ones by a multiplica-
one has a natural classification of sites into “molecule-
tivefactorz. Onefindst =zt ,t =zt ,withz =ρ/2.
”type (m) and “atom-”type sites (a) depending on their (cid:48)s(cid:48) s (cid:48)w(cid:48) w
Inaddition,themolecularrenormalizationintroduceson-
local environment. Atom-type sites have weak bonds on
site potentials t on the deflated chain, shifting the
s
the left and on the right, so they are weakly coupled to ±
whole energy spectrum by +t (bonding molecular lev-
s
therestofthechain. Molecule-sitesarelinkedbyastrong
els), or t (antibonding molecular levels).
s
bond to another molecule-site, and have a weak bond on −
eitherside. Inthelimitt 0,thesepairsformisolated
w t t t t t t t t
→ w s w s w w s w
diatomicmolecules,whiletheremainingsitescorrespond
to isolated atoms.
t t t t
0w 0s 0w 0s
II. THE PERTURBATIVE t t t
s s s
RENORMALIZATION SCHEME ± ± ±
Figure 3: Illustration of the molecular deflation rule: the
We now recall the main ideas behind the perturbative fifth approximant is transformed to the third.
renormalizationschemeintroducedbyNiuandNori[10],
and independently by Kalugin, Kitaev and Levitov [9]. Tosummarize,theHamiltonianofthenthapproximant
The Hamiltonian depends on the single parameter ρ = decouples into the direct sum of three Hamiltonians:
t /t , with ρ 1 in the strong modulation limit.
wWshen ρ =(cid:28)0, the atoms and the molecules decou- Hn =(zHn 2 ts) (zHn 3) (zHn 2+ts) + (ρ4)
(cid:124) −(cid:123)(cid:122)− (cid:125)⊕ (cid:124) (cid:123)(cid:122)− (cid:125) ⊕ (cid:124) −(cid:123)(cid:122) (cid:125) O
ple. The spectrum consists of three degenerate levels:
bondinglevels atomiclevels antibondinglevels
E = t , corresponding to molecular bonding and anti- (4)
s
±
bonding states, and E =0, for the isolated atomic state. In the limit n , the chain becomes quasiperiodic
→ ∞
When ρ = 0, the states in each of the three degener- and its wavefunctions and its spectrum become nontriv-
(cid:54)
ate levels weakly couples to each other, thus lifting the ial: they exhibit multifractality [1–3, 12].
3
A. Renormalization paths, equivalence between
energy labels and conumbers.
Renormalization paths of the energy bands.
Eq. (4) tells us that the spectrum of the Hamiltonian
H is the union of three energy clusters – the antibond-
n
ingmolecularcluster,theatomicclusterandthebonding
molecularcluster–eachofwhichisascaledversionofthe
spectrum of a smaller approximant. Molecular clusters
are separated from the atomic cluster by a gap of width
∆ t (1 z). Eachofthesemainclusterscanbedecom-
s
∼ −
posed into three sub-clusters, and so on. The spectrum
has therefore a recursive, Cantor set-like description, as
shown by fig. (4).
Energy spectrum
n-3
Figure 5: Every energy band can be labeled by its
n-2 renormalization path. As an example, here are shown the
renormalization paths of the energy bands labeled by 0+
(red) and +−− (blue).
n-1
z z z
n
Figure 4: Spectrum of the approximant H (n=8)
n
constructed geometrically from the spectra of H and
n−2 m
H (relation (4)). z and z are the two scaling factors.
n−3
m m m
One can assign to the energy bands belonging to the
bonding, atomic or antibonding clusters respectively the mm mm
labels+,0or . Tothis,onecanappendanother+,0or
−
according to the sub-cluster type of each energy. Re-
−
peating this procedure recursively, one obtains for each
energybandauniquesequenceofletterscalleditsrenor-
am am ammmmmmm mmmmmmam
malizationpath [3,11]. Figure(5)showstherenormaliza-
tion paths of two particular energy bands. Let us note
that although the renormalization path labeling of the Figure 6: Every site can be labelled by its renormalization
path, as shown here for the sites of non-zero amplitude, at
energy bands has been derived in the perturbative limit,
first order, of the wavefunctions labelled by 0+ (red) and
it continues to hold for 0<ρ<1 since no gap closes.
+−− (blue).
Renormalization paths of the sites. One assigns
to every atomic (resp molecular) site of the chain C a
n
label“a” (resp“m”). Becausethesetofatomsandtheset
of molecules of C are mapped to the chains C and only on atomic/molecular sites. This is again true at ev-
n n 3
C respectively, one can repeat the labeling pro−cedure erystepoftherenormalizationprocess,sothatbyrecur-
n 2
rec−ursively. Thus, to each site is associated a sequence sion,agiveneigenfunctionhasnonzeroamplitudeatfirst
of letters, that we call the renormalization path of the order only on sites whose renormalization path matches
site. Note that because we have not distinguished be- the one of the energy level associated energy band (pro-
tweenbondingandantibondingstatesonmolecules,sev- videdthatwemaketheidentification mand0 a).
±↔ ↔
eral molecular states can have the same renormalization In order to further understand the symmetry between
path. the renormalization paths for sites and energies, we now
Symmetry between renormalization paths for make use of the well known cut an project method. Let
sites and energies. In the perturbative limit, us recall very briefly that the cut and project method
because of (4), an eigenfunction associated to an considers the sites on the chain C as projections along
n
atomic/molecular energy band has nonzero amplitude an axis E of selected sites on a square lattice (see fig.
(cid:107)
4
13
12
molecules 11
10
9
8
atoms 7
6
5
4
molecules 3
2
1
1 9 4 12 7 2 10 5 13 8 3 11 6
Figure 7: Example of the cut and project method showing sites along the horizontal physical axis, and their conumber along
the perpendicular space (vertical) axis. Conumbering naturally orderes sites according to their local environment: atomic
sites (in blue) are clustered around the center of the windows, while molecular sites (in red) are grouped at the sides.
(7)). We can also consider the projection of the selected
sitesalongtheE axis,orthogonaltoE . Theprojection
of the sites on E⊥ forms a regular latti(cid:107)ce whose density
increases with th⊥e approximant size. In E , as seen on |ψ|²
20
fig. (7), the sites regroup in 3 clusters : a c⊥entral atomic 1.0
x
cluster surrounded by two molecular clusters. Keeping
e
onlysitesbelongingtotheatomic/molecularclustersex- d 0.8
n
actly amounts to performing an atomic/molecular dec- i 40
imation. Since the deflated chains are again Fibonacci y 0.6
g
chains, the 3 clusters are made of 3 molecular-atomic- r
e
molecular subclusters, and so on. Thus, in E the sites n 60 0.4
areorderedexactlyinthesamewayastheene⊥rgybands. e
This hints that looking at the sites in E is of interest. 0.2
We therefore choose to label the sites acc⊥ording to their
80
projection on E (from bottom to top on fig. (7)). Let 0
us call i the ind⊥ex of each site according to the order
in which it appears in real space. Then, for the nth ap- 20 40 60 80
proximant, the conumbers c are given by c(i) = F i conumber index
n 1
mod F . This relabeling of the sites was first introduc−ed
n
by R. Mosseri [6, 13], and was called conumbering. Be-
cause of the symmetry between the ordering of the sites
in E and the ordering of the energy bands, the conum-
ber l⊥abels play the same role as the energy labels. These |ψ|²=1 ρ²λ λ
symmetry will prove itself useful in our analysis of the
wavefunctions.
As an illustration of the relevance of conumbering, we λ ρ²λ
show the local density of states as a function of the en-
ergy labels and the conumber labels. This plot is invari-
ant under the exchange of the position/energy axes, in
the ρ 1 limit (figure (8)), for the reason explained
(cid:28)
previously. Crucially, this symmetry is revealed only if
one uses conumbering for the sites, and remains hidden
otherwise.
B. The gap labelling theorem in the perturbative
limit
Figure 8: Upper figure: intensity plot of the numerically
computed LDoS. x-axis: conumber index, y-axis: energy
To conclude this section devoted to the spectral prop-
index. Color reprents the presence probability (ie the
erties of the Fibonacci chain, we show how we can inter-
LDoS). Lower figure: the first few steps of the geometrical
pret the gap labelling theorem using the insight we gain
construction of the LDoS according to our perturbation
from the renormalization scheme. theory.
In the gaps, the integrated density of states (IDOS)
5
N(E ), takes values in a set specified by the gap la- gaps. From the recursive construction of the gap labels
gap
beling theorem. In the case of the Fibonacci chain, the (7), it is clear that the stable gaps are precisely the iter-
theorem states that atesthroughrenormalizationofthetwomaingaps,while
the transient gaps are the iterates through renormaliza-
tion of the E =0 gap.
N(E )=ωq mod 1=ωq+p (5)
gap
where p and q are integers. q is the label of the gap, and
p(q) is such that the IDOS satisfies 0 N 1.
For the nth approximant, the gap la≤bellin≤g becomes
1
N (E )=ω q+p (6)
n gap n
with ω = F /F , and q [1,F ). As we have seen,
n n 1 n n
in the strong m−odulation lim∈it the spectrum has a hier- 1 1
archical, ternary tree structure, and therefore the gaps
have this structure as well. To demonstrate the gap
structure, let us call G the set of IDOS of the gaps
n 2 1 1 2
of the nth approximant. We then have the following re-
cursion relations for the set of gap values in the bond-
ing/atomic/antibonding clusters:
3 2 1 4 1 2 3
F
G−n = Fn−n2Gn−2 ≡d−(Gn−2) (7)
F F
G0n = Fn−n3Gn−3+ Fn−n2 ≡d0(Gn−3) (8) 5 3 2 6 1 4 4 1 6 2 3 5
F F
G+n = Fn−n2Gn−2+ Fn−n1 ≡d+(Gn−2) (9) FBilguuer:est9a:bTlehgeagpas,prleadb:eltsr,aqn,sioefntthgeapfisr.stAfellwthaeppstraobxliemgaanptss.
where we have defined the three mappings d , d , d are iterates of the 2 main gaps, while all the transient gaps
0 +
− are the iterates of the E =0 gap.
corresponding to the three different clusters. These rela-
tions have a geometrical interpretation that can be best
seen if we replace the gap N (E ) = ω q +p by the
n gap n
vector g = (p,q). Then the above relations are replaced
by the following affine transformations, depending on
LetusstressthatalthoughthevalueoftheIDOSinside
whether g is in the bonding, atomic or antibonding clus-
agivenstable gapvarieswiththesizeoftheapproximant
ter:
(but converges in the quasiperiodic limit), its gap label
d (g)=M 2g (10) (p,q) is independant of the size of the approximant, and
−
− equal to the value we would have found for the infinite
d0(g)=M−3g+g1 (11) quasiperiodicsystem. Incontrast,thegaplabelofatran-
d (g)=M 2g+g (12) sient gap varies with the size of the approximant. Also
+ − 2
note that whereas the width of the transient gaps goes
where g1 = (1, 1) and g2 = (0,1) are the labels of the to zero in the quasiperiodic limit, their fraction stays fi-
−
two main gaps (corresponding to q = 1, see figure (9)), nite. Itconvergesto(4+3ω)/(18+11ω) 0.24. Amore
and M is the substitution matrix ± detaileddiscussionoftheroleofgaplabe(cid:39)lingforapprox-
(cid:20) (cid:21) imants is under preparation and will be given elsewhere
1 1
M = (13) [14].
1 0
that generates the Fibonacci sequence by acting repeti- To summarize this short section, we have seen that
tively of the letters A and B. the renormalization group picture gives us a natural in-
The recursive gap labeling procedure we just men- terpretation of the gap labelling theorem in terms of in-
tioned labels each of the F 1 gaps of the nth approx- flation/deflationtransformations. Ithasbeenshownthat
n
−
imant. However we know that some of these gaps are thesegaplabelscanbephysicallyinterpretedintermsof
goingtodisappear inthequasiperiodiclimit, whilesome the topological properties of edge states of a finite chain
are going to persist. Following [11], we call the former [7, 15]. We have also seen that the gap labelling, often
transient gaps and the latter stable gaps. The gap at mentionedinthecontextofquasiperiodicsystems, natu-
E = 0 that appears for even F is an example of tran- rallyextendstoperiodicapproximants. Theonlypriceto
n
sient gap. The two main gaps that separate the molecu- payforthisextensionistheintroductionofthetransient
larclustersfromtheatomicclusterareexamplesofstable and stable gaps.
6
III. EXPRESSIONS FOR FRACTAL B. Fractal dimensions of the wavefunctions
DIMENSIONS TO LEADING ORDER
The fractal dimensions Dψ(E) of the wavefunction as-
q
Weturntothequestionofthefractaldimensionsofthe sociated to the energy E are defined by
spectrum and of the wavefunctions in the limit n .
Ffroarctcaolmdpimleetnensieosns,swofetfihrestspdeecstcrruibme[t1h1e,d12er].ivation →of t∞he χnq(E)=(cid:88)i |ψin(E)|2q n→∼∞(cid:18)F1n(cid:19)(q−1)Dqψ(E) (19)
χ (E)istheinverseparticipationratio,andtheexponent
A. Fractal dimensions of the spectrum 2
Dψ(E) provides information as to the degree of localiza-
2
tion of the state. The value Dψ(E) = 1 indicates that
The fractal dimensions of the global DoS can be de- 2
terminedusingthethermodynamicalformalism[16]. We thestateE isextended,whileD2ψ(E)=0characterizesa
define the partition function localizedstate. Intermediatevalues0<Dψ(E)<1area
2
signatureofacriticalstate,whosemultifractalproperties
Γ (q,τ)=(cid:88) (1/Fn)q (14) can be probed by varying q.
n (∆ (E))τ At leading order in ρ the renormalization of the eigen-
n
E states is simple, and well understood [19]. For a wave-
function in the atomic cluster, we have ψn(E) =
awceorshgsneoytrcreliieba∆vtueetnldis(o,Etnoas)ntoidhsfettuaheskeinneebngrogtenyoqdulbienaevgteit,olhanlenatw(bi4ebi)dlo,ltenwhddeoiEnfog.tbhtaSeaneipdennaaretarogtmiynigbcatenhnde- t||ψψhiienn(cid:48)−(oE3r()iEg|i=(cid:48)n)a|.|lψFcin(cid:48)ho−ra2ia(nEwa(cid:48)a)n|vd/e√fou2nn.ctEthioeandniednflEatht(cid:48)eeadmreoontleheceru|eelsnapireerccgltiueivs|steeolyrn.,
Therefore, weobtainimmediatelytheleadingorderfrac-
(cid:18)F (cid:19)q (cid:18)F (cid:19)q tal dimensions of the wavefunction associated to the en-
Γn =2 n−2 z−τΓn 2+ n−3 z−τΓn 3 (15) ergy E:
Fn − Fn −
log2
Taking the quasiperiodic limit, one obtains an implicit Dψ (E)= x(E) + (ρ2), (20)
equation for the spectral fractal dimensions D : q,0 − logω O
q
where
2ω2qz−(q−1)Dq +ω3qz−(q−1)Dq =1. (16)
n (E)
x(E)= lim m (21)
We can solve it at first order in ρ, obtaining
n n
→∞
Dq = 1 1 qlog(cid:2)ω−q(cid:0)√lo1g+ρω−q−1(cid:1)(cid:3) +O(cid:18)(log1ρ)2(cid:19) wiziatthionnmp(aEth) tohfeEn,umi.eb.erx(oEf)+i/s−thleettfrearsctiinonthoefrRenGorsmteapls-
− (17) spent in molecular clusters. x is a non trivial function of
theenergy(figure(10)),whosestructureisreminiscentof
Forthecaseq =0,werecall,oneobtainstheHausdorff
theoneofthelocaldensityofstates(8). Inthequasiperi-
dimension D , which helps to characterize the nature of
0
the spectrum. D = 0 for a pure-point spectrum, while odic limit, x varies continuously between 0 and 1/2, and
0
D = 1 indicates that the spectrum has an absolutely Dψ (E) is a continuous function of x. The distribution
0 q,0
continuous component. An intermediate value 0<D < of x is given by
0
1 is the signature of a fractal spectrum. Multifractality
(n +n )!
corresponds to situations where Dq varies with q, which Ω(x(n0,nm))=2nm 0 m (22)
is the case here. n0!nm!
Here, we find
WhereΩ(x)isthenumberoccurrencesofagivenvalueof
log(√2 1) (cid:18) 1 (cid:19) x. It coincides with the distribution of the widths of the
D = − + (18) energy bands [11]. Therefore, in the quasiperiodic limit,
0 logρ O (logρ)2
the distribution of x is given by f, the Legendre trans-
form of the fractal dimensions of the spectrum: P(x)
in agreement with the result of Damanik & Gorodetski ∼
[17], using trace-map-based methods. For ρ > 0, one Fnf(x)−1. This distribution is sharply peaked around the
sees that 0 < D0 < 1, and therefore we recover the most probable value, xmp = 2(3ω − 1)/5 (cid:39) 0.3416....
well-establishedresultthatthespectrumoftheFibonacci States with this value of x are statistically the most sig-
Hamiltonian is fractal for nonzero ρ, however small. nificant.
These results compare well with numerical data only Sincethefractaldimensionsonlydependonx,weper-
for extremely small values of ρ. Note however that im- form the change of variables Dqψ,0(E) → Dqψ,0(x). Since
proved multifractal analysis that goes beyond the first 0 x 1/2, we have 0<Dψ(x)<1: the wavefunctions
≤ ≤ q
order was carried out [18]. are critical, as we expect for a quasiperiodic system.
7
ψ 2 =λψ 2
| i| | i0|
0.4
λ λ λ
0.3
) ψ 2
E | i0|
(
x 0.2
ψ 2 =λψ 2
| i| | i0|
0.1
λ λ λ
0.0 λψ 2
0 50 100 150 200 | i0|
E
Figure 11: SchematizationoftheRGprocedure. Topfigure:
Figure 10: The parameter x as a function of the energy a wavefunction of the atomic cluster, bottom figure: a
labels (or equivalently of the sites in conumbering), for the wavefunction of the molecular cluster.
approximant of 233 sites. Lines are drawn to guide the eye.
A. Renormalization group for the wavefunctions
The parameter x determines completely the fractal At leading order in ρ, we know that the wavefunction
properties of the wavefunctions. For example, x =0 for amplitudesonthenth approximantarerelatedbyatriv-
a
the level in the center having the renormalization path ialmultiplicativefactortotheonesonasmallerapproxi-
00... (cf figure (10)). The corresponding eigenstate has a mant. Athigherorder,westillrelatewavefunctioncoeffi-
zero fractal dimension, and is thus completely localized. cientsonlargeapproximantstowavefunctionsonsmaller
Ontheotherhand,themaximalvalueofx= 1 isreached ones through multiplicative factors. We call these multi-
2
for the levels E =E ,E = t /(1 z) at the edges plicative factors λ if the wavefunction is of atomic type,
min max s
± −
ofthespectrum, forwhichtherenormalizationpathsare and λ is it is of molecular type (figure (11)).
++... and ... The corresponding eigenstates are
themostext−en−de−d. Theyoccupyafraction(1/Fn)−2lloogg2ω (cid:40)|ψi(n)(E)|2 =λ|ψi((cid:48)n−3)(E(cid:48))|2 if E is atomic (23)
of the sites. |ψi(n)(E)|2 =λ|ψi((cid:48)n−2)(E(cid:48))|2 if E is molecular
To conclude, we note that, to leading order in ρ, the
λ and λ are renormalization group parameters, and
fractal dimensions of the wavefunction do not depend on
theyplayintherenormalizationofthewavefunctionsthe
q. Thus the wavefunctions are not multifractal at this
role z and z plays in the renormalization of the energy
order in ρ and multifractality appears only at the next-
bands.
to-leading order, as discussed in the next section. This
We find (details of the calculations are given in the
first-orderdescriptionofthewavefunctionshasbeencom-
Appendix):
pared to numerical results in [20], where the agreement
was found not to be very good. We argue that this is 2
λ(ρ)= (24)
because the wavefunctions becomes rapidly multifractal (cid:112)
(1+ρ2)2+ (1+ρ2)4+4ρ4
as ρ is increased. Fortunately it is possible to calculate
1
higher order corrections, and thereby vastly improve our λ(ρ)= (25)
(cid:112)
theoretical predictions concerning the wavefunctions, as 1+ρ2γ(ρ)+ 1+(ρ2γ(ρ))2
shown below.
withγ(ρ)=1/(1+ρ2). Atleadingorderinρ, werecover
λ(0)=1/2, λ(0)=1 as expected. At next order,
1
λ(ρ)= + (ρ2) (26)
1+ρ2 O
1
IV. HIGHER ORDER RENORMALIZATION λ(ρ)= + (ρ2) (27)
GROUP AND MULTIFRACTALITY 2+ρ2 O
Although our calculations are done in the strong modu-
At higher order the picture of molecular and atomic lation limit, in the periodic limit ρ 1, we obtain the
→
eigenstates and energies remains relevant, but it is now exact expression for the renormalization factors:
possible for an atomic eigenstate to have nonzero ampli-
λ(ρ) ω3 (28)
tude on molecular sites, and vice-versa. In this section −ρ−−→1
→
we explain our ansatz for the wavefunctions. In the next λ(ρ) ω2 (29)
section,wewillapplyittothecomputationofthefractal −ρ−−→1
→
dimensions of the wavefunctions. .
8
B. Local wavefunction dimensions from 0 or 1, x is not enough to describe the renormaliza-
tion parameter λ. One actually needs to know the whole
renomalization path of the states, whereas our formula
For q 0 and when ρ 1, we can write
≥ (cid:28) takes into account only a single parameter x. The the-
(cid:40)(cid:0)λ(ρ)q/λ(ρq)(cid:1)χn 3(E ) if E is atomic, oretical expression gives λ accurately in only two cases:
χn(E) q− (cid:48) the states that are always bonding or always antibond-
q (cid:39) (λ(ρ)q/λ(ρq))χnq−2(E(cid:48)) if E is molecular. ing (ie the states at the edge of the spectrum). The
(30) states with alternating bonding and antibonding charac-
Iteratingthisrelation,weunderstandthatthefractaldi- ter have the largest deviation from the theoretical value,
mensions depends only on the renormalization path of in the perturbative limit. The figure shows, as well, that
the energy E we started with. Actually, it only depends the agreement with numerics is extremely good for the
on the parameter x (21). Solving the recurrence we ob- λrenormalizationfactor,correspondingtothewavefunc-
tain an explicit expression for the fractal dimensions in tion at E = 0. The small discrepancy between the nu-
the quasiperiodic limit: merics and the analytical predictions is only due to nu-
merical finite-size effects. Indeed, the fractal dimensions
(cid:34)(cid:18)λ(ρ)q(cid:19)x(cid:18)λ(ρ)q(cid:19)(1−2x)/3(cid:35) of this E = 0 wavefunction have been determined ex-
(q 1)Dψ(x)=log /logω.
− q λ(ρq) λ(ρq) actly [3], using trace map methods. Interestingly, these
exact fractal dimensions coincide with our perturbative
(31)
predictions for λ (32), for all values of ρ, meaning that
It is easy to check that we recover the first-order expres-
our perturbative expression for λ is in fact exact.
sion for the fractal dimensions (20) if we take ρ=0.
0.8
1.0
0.6
0.8
)
x
(
0.6 λ(ρ) ψ 0.4
2 )
D E
λ(ρ) (
0.4 0.2 D2
0.2 E
0.0 0.2 0.4 0.6 0.8 1.0 0.0
ρ 0.0 0.1 0.2 0.3 0.4 0.5
x
Figure 12: Numerical results and theoretical predictions for
the renormalization factors λ(ρ) and λ(ρ). Dots: Numerical Figure 13: Numerical results and theoretical predictions for
results (n=19, 4181 sites). Solid lines: theoretical
the fractal dimensions of the wavefunctions. Dots:
predictions (32).
Numerical results ( n=19, 4181 sites). Dashed line:
theoretical prediction at leading order (eq. (20)), solid line:
From this we can express the renormalization factors theoretical prediction including multifractal corrections (eq.
in terms of the fractal dimensions for q , setting (31)). Inset: The fractal dimension D2ψ(E) for every energy.
x=0 for λ, and x=1/2 for λ: → ∞ In accordance with theoretical predictions, the fractal
dimensions organize in lines, each line corresponding to a
given value of x(E).
λ(ρ)=ω3Dψ(0)
∞
λ(ρ)=ω2Dψ(12) (32) We now check our theoretical predictions against nu-
∞
merical results. The inset of figure (13) shows how the
Thus, computing numerically the fractal dimensions for fractal dimension for a given q (here we chose q = 2)
q large gives us a numerical estimation of the renormal- depend on the chosen state of energy E. We observe
ization factors, that we can compare to the theoretical thatthevalueofthefractaldimensionorganizesinlines.
predictions. We expect the agreement to be good in the AlongeachlineDψ(E)isconstantuptosmallvariations
q
strong modulation ρ 1 and in the weak modulation andcorrespondstoagivenvaluex(E)=x. Thatis,upto
(cid:28)
ρ 1 regimes, because we know that in these limits the small corrections that should vanish in the limit ρ 0,
∼ →
renormalization factors are exact. In fact, we see that the fractal dimension of a given wavefunction does not
the agreement with numerics is excellent for all values of depend on the energy E, but only on x(E). This is in
ρ (fig .(12)). agreement with the theoretical predictions (31). Figure
It can be seen in fig (12) that the values of λ calcu- (13) shows, for example, the x dependence of the fractal
lated numerically have a spread, which is not described dimension for q = 2. Calculated numerical results are
bythetheoreticalformula. Thisisbecause,whenρisfar seen to be in very good agreement with the theoretical
9
predictions. As the figure shows, the multifractal prop- Inthissectionwecalculatescalingpropertiesofthewave-
erties of the wavefunctions – which were not captured at functions after averaging over all states. We define an
leadingorderinρ–arerelevant, evenforthesmallvalue averaged fractal dimension Dψ by
q
coupling ratio ρ = 0.1. For x = 0 the first order contri-
bution to the fractal dimensions vanishes, so that only 1 (cid:88) (cid:18) 1 (cid:19)(q−1)Dψq
the multifractal correction term remains. χq = χq(E) (33)
(cid:104) (cid:105) F ∼ F
n n
E
ρ=0.15
This quantity, studied in disordered systems at the An-
D ψ(x)
q derson localization transition [21], has been computed
1.0 x=0.473684 numerically by [20] for the Fibonacci model. Within our
x=0.368421 perturbation theory, we obtain an implicit equation for
0.8 x=0.315789 the averaged fractal dimensions:
x=0.210526
0.6
x=0.157895 λ(ρ)q λ(ρ)q
0.4 x=0.0526316 2ω2 ω−2(q−1)Dψq +ω3 ω−3(q−1)Dψq =1 (34)
λ(ρq) λ(ρq)
x=0.
0.2
This equation is perturbative, valid to order ρ2q. The
q
1 2 3 4 derivationcanbefoundintheAppendix,whichalsotakes
into account higher orders in ρ, resulting in a lengthier
Figure 14: The fractal dimensions Dqψ(x) of the expression. Notethesimilarityofstructurebetween(34)
wavefunctions for the different values of x accessible and the implicit equation obtained for the spectral di-
numerically. Dots are numerically computed data points,
mensions (eq (16), see also [11]).
solid lines are the theoretical predictions (eq. (31)).
The resulting theoretical predictions are compared
withnumericalresultsonafinitesizesysteminfig. (15).
To conclude this section, we show the q dependance of The agreement is excellent for all positive values of q
the fractal dimensions for fixed values of x (fig. (14)). compared to the lowest order theory used in [20].
The agreement with the theoretical predictions is excel- For larger ρ, (34) can be corrected to include higher
lent for all positive values of q. This demonstrates that order terms. The resulting theoretical prediction (see
ourtheoreticalanalysisindeedcapturestheqdependance Appendix) agrees with the numerical computations even
of the fractal dimensions. Since it is the multifractality for large ρ as shown in fig. (15) for the choice ρ = 0.5.
of the wavefunctions that is responsible for the nontriv- The reason for this unexpected robustness of our pertur-
ial q dependance of the fractal dimensions, we conclude bativetheoryoutsideitsdomainofvalidityisunclear. It
again that the multifractal corrections are relevant even supportstheideathattherenormalizationgrouppicture
at small coupling. stemming from the geometrical inflation/deflation prop-
erty of the Fibonacci chain contains all the fundamental
physics determining its electronic properties.
C. The spectrally averaged fractal dimensions of
wavefunctions
D. The local spectral dimensions and their average
1.0
0.4
0.8
ψ 0.6 0.3
Dq ρ=1/10
0.4 ρ=1/2 μq 0.2
0.2 D ρ=1/10
0.1
0.0
0 5 10 15 20
q 0.0
0 5 10 15 20
q
Figure 15: The averaged fractal dimensions of the
wavefunctions Dψq as a function of the multifractal Figure 16: The averaged local spectral dimensions Dµ as a
q
parameter q, for ρ=0.1,0.5. Dots: numerical results, solid
function of the multifractal parameter q, for ρ=0.1. Dots:
line: theoretical predictions.
numerical results, solid line: theoretical predictions (eq.
(39)).
In the previous section we have defined the fractal di-
mensionsofindividualwavefunctions,andhaveseenthat In this section we consider the local density of states,
they were associated with the energy level parameter x. and the associated fractal dimensions. For a finite-size
10
system, the local density of states (LDoS) at site i is generalization of this relationappears to hold also forall
q > 0, namely Dµ = Dψ D . This relation is
dµ (E)= 1 (cid:88)Fn δ(E E )ψ (E )2dE (35) satisfied numericaqlly for qval1u+e(sq−o1f)Dρψ<q 0.2. For larger
i F − a | i a | values of ρ, it transforms into the ineq∼uality:
n
a=1
TalhlesigtelosboafltdheensLiDtyooSf.sTtahteesloicsaolbdteaninsietdy boyf stthaetessumdefionveesr Dqµ ≥DψqD1+(q−1)Dψq (40)
µ , the local spectral weight at site i associated to the
i
band a of width ∆n: B. An upper bound for the diffusion moments
a
(cid:90)
µ (∆n)= dµ (E) (36) Ketzmerick et al [22] obtained a lower bound for the
i a i
E banda exponent σq, describing the moments of the spreading of
∈
a wavepacket: σ Dµ/Dψ. We propose a new upper
The local spectral weight at site i sums up all the in- q ≥ 2 2
bound
formationaboutthespectralandwavefunctionproperties
of the Hamiltonian. To probe the multifractality of the Dµ
σ q (41)
local density of states one defines the partition function (1−q)Dψq ≤ Dqψ
Γ (q,τ;i)=(cid:88)(µi(∆na))q (37) thatwederivefromtheinequality(40),usingtherelation
n (∆n)τ σ =D [19]. Toourknowledge, thisisthefirstupper
a a bqound 1th−aqt is proposed for the diffusion exponents. It
using which one can compute the local spectral fractal is interesting in that it involves the spectral properties
dimensions for individual sites. One can also define a (throughDµ), andthewavefunctionproperties(through
site-averaged gamma function through: Dψ). Work in progress on the study of dynamical corre-
lationsontheFibonaccichainwillbereportedelsewhere.
1 (cid:88)
Γ (q,τ) = Γn(q,τ;i) (38)
n
(cid:104) (cid:105) F
n
i
VI. SUMMARY AND CONCLUSIONS
Thesite-averagedlocalspectraldimensionsDµ,obeythe
q
implicit equation: Inthispaperweprovideatheoreticaldescriptionofthe
spectrum and the wavefunctions of the Fibonacci pure-
2ω2λ(ρ)qz(1−q)Dqµ +ω3λ(ρ)qz(1−q)Dqµ =1 (39) hopping model in the strong modulation limit, using a
λ(ρq) λ(ρq) perturbative renormalization group scheme. The pertur-
bative approach allows to discuss the structure and la-
We compare these theoretical predictions with numeri- bellingofgaps,propertiesoftopologicaloriginandthere-
cal data in fig. (16)) for ρ = 0.1 and q > 1, finding an fore valid for all values of the coupling ratio. We show
excellent agreement between the two. We note, finally, howusingtheconumberingbasisallowsonetocharacter-
that the theoretical prediction for the Hausdorff dimen- ize wavefunctions conveniently according to their renor-
sion Dµ given by the equation (39) agrees also well with malization path. We show that the system has an ap-
0
the numerical result. proximatesymmetry,intheperturbativelimit,underthe
exchange of site and energy indices. The leading order
expressions for exponents are observed to agree with nu-
V. RELATIONS BETWEEN EXPONENTS merical calculations only for the smallest values of the
coupling ratio ρ. We obtain the analytical description of
Comparing the relations (16), (34) and (39) for the the spectrum and wavefunctions of the Fibonacci chain
globalspectral, the average wavefunction andlocal spec- atnext-to-leadingorderinρ. Theseexpressionsshowex-
traldimensionsrespectively,oneseesthattheybeargreat plicitly how the multifractality of wavefunctions appears
similarity. This suggests that there might be a relation forlargervaluesofρ. Theextendedtheoryisshowntobe
between the three families of exponents. We present be- in very good agreement with numerical results for values
lowtwopossibleinequalities, tobeinvestigatedinfuture ofρinawiderangefromsmallvaluesallthewayuptoρof
work. order unity. Exponents for the local and global spectral
measures for individual states are calculated. Averaged
exponents are defined as well, and compared with nu-
A. A relation between fractal dimensions merical data. New inequalities relating these exponents,
andthediffusionexponent,areproposedandnumerically
Consider the case q = 0. In this case, the local and tested.
global spectral dimensions coincide, and the wavefunc- Acknowledgments We would like to thank J.M. Luck
tion dimension is 1, so that we have Dµ = Dψ D . A (IPhT, Saclay) and M. Duneau for helpful discussions.
0 0 0
