Phonon modes in the Frenkel-Kontorova chain: exponential localization and the number theory properties of frequency bands

0.34 MB·English

by  O. V. Zhirov

Phonon modes in the Frenkel-Kontorova
hain:
exponential lo
alization and the number theory properties of frequen
y bands
∗
O.V. Zhirov
Budker Institute of Nu
lear Physi
s, 630090 Novosibirsk, Russia
†
5
G. Casati
0
International Center for the Study of Dynami
al Systems, Università degli Studi dell'Insubria and
0
Istituto Nazionale per la Fisi
a della Materia, Unità di Como, Via Valleggio 11, 22100 Como, Italy and
2
Istituto Nazionale di Fisi
a Nu
leare, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy

n
‡
a
D.L. Shepelyansky
J
Laboratoire de Physique Quantique, UMR 5626 du CNRS,
0
Université Paul Sabatier, 31062 Toulouse, Fran
e
1
(Dated: 2nd February2008)

] Westudynumeri
allyphononmodesofthe
lassi
alone-dimensionalFrenkel-Kontorova
hain,in
n
the regime of pinned phase
hara
terized by the phonon gap and devil's stair
ase, as well as by a
n
largenumberofstates(
on(cid:28)gurationalex
itations),whi
henergysplittingfromthegroundstateis
-
s exponentiallysmall. Wedemonstrate,thesestates behavelike disorder media: their phononmodes
di areexponentiallylo
alized, in
ontrasttothephononmodesinthegroundstate,wherephononsare
. prelo
alized only.
at We demonstrate also, the phonon frequen
y spe
trum of the ground state has an hierar
hi
al
m stru
ture,a dire
t manifestation of hierar
hi
al spatial stru
ture, foundfor theground state of the
FK
hain in our re
entwork.
-
d
PACSnumbers: PACSnumbers: 05.45.Mt
n
o
c
I. INTRODUCTION a
ting with elasti
for
es, pla
ed in periodi
potential,
[
whi
h period di(cid:27)ers from a mean interparti
le distan
e.

1 Thegroundstate(GS)ofthismodelisde(cid:28)nedasastati
,
v The most trivial disorder originates in media due to
equilibrium
on(cid:28)guration of the
hain, that
orresponds
8 randomstati
impurities(see,e.g. [1℄). However,another
to the absolute minimum of the
hain potential energy.
8 butveryinterestingpossibilitiesareglasses,whi
hhavea
The ground state is unique and has some spe
ial order
1
huge number of (meta)stable degenerated states. Origi-
1 of parti
les in the
hain, that was dis
overed by Aubry
nallyglassysystemhasahomogeneousHamiltonianwith
0 [9, 15℄ more than twenty years ago. The positions of
no intrinsi
random parameter, and disorder o

ursin it
5 atoms in the
hain are des
ribed by an area preserving
0 dynami
ally. Re
ently [2℄ we have demonstrated that a map, whi
hiswellknownin the (cid:28)eldofdynami
al
haos
/ popularFrenkel-Kontorovamodel[3℄presentsanexample
t astheChirikovstandardmap[20℄. Theratioofthemean
a ofglassysystem,whi
hhasalotofstati
states,knownas
interparti
le distan
e to a period of the external poten-
m
on(cid:28)gurational ex
itations of the
lassi
al ground state,
tial in the FK model determines the rotation number of
- withenergysplittingextremely(exponentially)small. As the invariant
urves of the map, while the amplitude of
d it wasshownin [4℄, this model has a nontrivialquantum
n dynami
s,thequantumphasetransition: if quantumpa- theperiodi
KpotentiKal<givKescthevalueofthedimensionless
o parameter . For the KAM
urves are smooth
rameter ex
eeds some
riti
al value, the "pinned" glassy
c and the spe
trum of longwavephonon ex
itations in the
v: phase turns into "sliding" phonon gas.
hain is
hara
terizedby a linear dispersion law starting
i The Frenkel-Kontorova model (FK) [3℄ is widely used from zero frequen
y. In this regime the
hain
an freely
X
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14℄ in the solid state physi
s slide along the external (cid:28)eld (the (cid:16)sliding(cid:17) phase). On
K >K
r to get insight on generi
properties of non
ommesurate the
ontrary, for c the KAM
urvesare destroyed
a
systems. Itsgroundstate,whi
hisratherquasiperiodi
al and repla
ed by an invariant Cantor set, whi
h is
alled
[2, 9, 15℄ than periodi
al, attra
ts also an attention [9,
antorus. Inthis regimethephononspe
trum hasagap,
10, 16, 17, 18℄ as some interplay [17℄ between order and and the
hain is pinned ((cid:16)pinned(cid:17) phase). Later, on the
disorder [19℄. example of Ising spin model to whi
h the FK model
an
be approximately redu
ed [21℄ it has been shown [22℄
This model des
ribes a
hain of atoms/parti
les inter-
thattheGShassomewellde(cid:28)ned hierar
hi
alstru
ture,
whi
hparti
ulardetailesaredeterminedbynumberprop-
erties of the ratio of the mean interparti
le distan
e to
∗
†Ele
troni
address: zhirovinp.nsk.su the period of the external (cid:28)eld. Re
ently our numeri
al
‡Ele
troni
address: giulio.
asatiuninsubria.it study[2℄oftheoriginalFKmodelinthepinnedphasehas
Ele
troni
address: dimairsam
.ups-tlse.fr
2
shown, that theGS hasindeed an hierar
hi
alstru
ture, stru
tureofthephononfrequen
yspe
trumintheGS.It
but in some important detailes di(cid:27)erent from predi
ted is well known, that this spe
trum is splitted into bands
in [22℄. [10, 26℄ but, to our knowledge, up to now there is no
Inshort,weputourattentiontothestrikingfa
t,that
learexplanationfororiginofitssplittingintoparti
ular
in the pinned phase of the FK
hain there are some par- bands and subbands. We show, that this splitting is a
ti
les, whi
h positions exponentially
lose to bottoms of dire
t
onsequen
e of parti
ular spatial stru
ture of the
wells of the external potential:
orresponding external
haininitsGS.Wehavefound,thatthisstru
tureisalso
for
e a
ting to su
h parti
le is extremely
lose to zero. hierar
hi
allyordered,withde(cid:28)niteresemblan
eanddis-
Obviously, in stati
equilibrium ea
h of these parti
les tin
tionswithrespe
ttoaspatialstru
tureofunderlying

an be
onsidered as some dummy (cid:16)glue(cid:17) that only
ou- GS.
ples two adja
ent parts of the
hain, whi
h ends has al- Lo
alisation properties of phonons in in
ommensurate
most identi
al (exponentially
lose) tension for
es. one-dimentional
hains are intesively studied in re
ent
Anotherimportantobservationisthatsmalldeviations works [17, 18, 23, 25℄, with strong indi
ations [23, 25℄,
ofglueparti
lesfromtheirwellbottomsaregrouppedinto that phonon modes in the GS of FK
hain are not lo-
well de(cid:28)ned hierar
hi
ally ordered s
ales. Now, if one
alized, and even at edges of frequen
y bands they are

utthe
hainintofragmentsviaglueparti
les,whi
hbe- ratherprelo
alized, thanlo
alized. Westudyalsophonon
long to level with least deviation from well bottoms, one properties as in GS, as in CES of the
hain. Our results
getsseveralfragmentsoftwosizes,ortwospe
iesofsome
on(cid:28)rm,thatin the GSphononmodesareonlyprelo
al-
(cid:16)bri
ks(cid:17)[30℄. Thenonemayrepeatthepro
edure,
utting ized. However,thesituationappearstobequitedi(cid:27)erent
(cid:16)bri
ks(cid:17) via (cid:16)glue(cid:17) parti
les that belong to next s
ale of for CES. Even for∆CEUS, w1h0i−
h12energy splitting (in natu-
deviations and getting twonew spe
iesof smallerbri
ks, ral problems
ale) ≤ therearephonon modes,
andsofar. Atthelaststeponegetstwospe
iesofsmall- whi
h are lo
alized exponentially. Moreover, for CES
∆U
est posAsi(b0l)e bri
ks with no glue parti
le inside: smaller with higher splitting we see, that there are entire
bri
ks whi
h
onsBis(t0)of 2 parti
les inside a single bands of exponentially lo
alized phonon modes.
well, and larger bri
k whi
h
onsist of 4 parti
les
(two pairs in two adja
ent wells). In this wayA(oi)n,eBg(ie)ts
an hierar
hi
allyorderedset ofbri
k spe
ies II. THE MODEL.
with very simple
omposition rules [2℄: (cid:8) (cid:9)
The Hamiltonian of the FK model is
A(i+1) = B(i)gA(i)gB(i),
B(i+1) = B(i)gA(i)gB(i)gA(i)gB(i), (1) H = s Pi2 + (xi−xi−1)2 Kcosx .
(2) 2 2 − i (3)
i=1
g X
where symbol denotes of an insertion of glue parti-

le, whi
h (cid:16)glues(cid:17) two adja
ent bri
ksA. (Ti)he di(cid:27)Be(rie)n
e of The (cid:28)rst term in the Hamiltonianmis=a k1ineti
energy,
tensionfor
esatboundariesofbri
ks and isex- where we put masses of parti
les , the se
ond
ponentially small and de
reaserapidly with a number of termdes
ribesinterparti
leintera
tionwithelasti
ity
o-
the hierar
y level. In prin
iple, these rules are su(cid:30)
ient e(cid:30)
ient put to unity, while the third term
orresponds
to
onstru
taGSforaFK
hainofanylength, if thero- to parti
le intera
tion with external periodi
al(cid:28)eld with
K s
tation number paraνm=ete(√r o5f th1e)/
2hain approximates the r
oupling
onstant . All parti
lesaredistributed over
mean golden value − . period/wells of the external potential, whi
h period,
2π
Besides the GS there exists(cid:16)
on(cid:28)gurationalex
itation without any loss of generality is taken equal to . The
ν = r/s
states(cid:17) (CES), presentedbystati
equilibrium
on(cid:28)gura- ratio gives [9℄ the rotational number of
orre-
tions
orresponding to lo
al (rather than absolute) min- sponding standard map [20℄.
P P
0 s
ima of the
hainpotential, with energy very
loseto GS. We assume periodi
al boundary
onditions: ≡ ,
x x L L=2πr
0 s
Within the pi
ture just outlined above CES
orrespond ≡ − ,where isthelengthofthe
hain. In
todi(cid:27)erentpermutationsofbri
ks[2℄,thereforethenum- oursubsequentanalysiswetake(assometypi
alexample
r/s=377/610
ber ofthem
anbe
ombinatori
allyhuge. Atany a

es- ofFK
hain)the
hainwith asanapprox-
ν¯ = (√5 1)/2
siblesmalltemperaturetheir
ontributions
andominate imation of the golden mean value − , and
K = 2 K (ν =
c
over the
ontribution of GS. parameter well above the
riti
al value
ν¯)=0.971...
In this paper we address to phonon ex
itations of the [27℄. The te
hnique to obtain GS and CES

hain, small vibrations around stati
GS and CES
on- is des
ribed in our previous work [2℄. Let us stress, that
(cid:28)gurations of the
hain. These ex
itations are relevant validity of the GS
an be
he
ked either by dire
t analy-
as for heat transport properties [18, 23℄ of the
hain, as sis [2℄ of its spatial stru
ture, or proved by monotonous
forsomequantume(cid:27)e
ts[24℄,espe
iallyinthequasi
las- behavior [15℄ of its hall fun
tion. Note, that the GS is
si
al limit. As in the previous paper, we
on
entarte on almost degenerate with enor∆mUou=s nUumberUof CES:10th−e8r0e
ECS GS
the
aseof pinned phase ofthe
hain,whi
h
orresponds arehundredsofstateswith ∆U 10−−12 ≤ 109,
to a nonzero phonon gap. We start with analysis of the while the number of ECS with ≤ ex
eeds !
3
E(cid:27)e
tive Hamiltonian for phonon modes
an be ob- that, at last, areexpanded into basi
bri
ksofzerolevel:
tained by expansion up to se
ond order terms of the
originalFK
hainHamiltonian(3)arounda
hosenstati
12=(4g2g4), 20=(4g2g4g2g4).

on(cid:28)guration of the
hain: (10)
s Π2 1 s ∂U
Hph = i + ψ ψ , In this way it is easy to
al
ulate, that the
hain
an be
i k
2 2 ∂xi∂xk (4)
ut into pie
es either as
i=1 i,k=1
X X
144 g+55 (2)+89 (4),
ψ =x x¯ × × × (11)
i i i
where − aresmall deviations of parti
lesfrom
x¯ Π
i i
their equilibrium stati
positions , and are
orre- or
sponding parti
le momenta. The elasti
ity matrix 34 g+13 (12)+21 (20),
∂2U × × × (12)
R =Kcos(x¯ )δ δ δ .
ik i ik i,k+1 i+1,k
≡ ∂x ∂x − − (5)
i k or
8 g+3 54+5 88,
Solving the eigenvalue problem × × × (13)
R ω2I ψ =0, 2 g+(232)+(376)
− (6) or × ,see(7). Mutualstati
disturban
e
(cid:0) (cid:1) ofbri
ksissmallandde
reasesexponentiallywithalevel
numeri
ally, we get both the spe
trum of phonon fre- of hierar
y, therefore we have at any level of hierar
y a
quen
ies and
orresponding ve
tors of phonon modes. sequen
ein someorderof twospe
iesof almostidenti
al
stru
tures (bri
ks).
Nowletusmakesomeimportantremark. Inourstudy
III. PHONON FREQUENCY SPECTRUM in [2℄ of stati
stru
ture of the
hain, the glue parti
les
playasomepassiveroleonly,whi
hresultsintheirspe
-
Itiswellknownforalongtime[10,26℄,thatfrequen
y i(cid:28)
ation as some dummy (cid:16)glue(cid:17). However, in dynami
al
spe
trum of phonons in the Frenkel-Kontorova
hain is problem of motion parti
les in the
hain, that we now
splitted into several bands. Main features of the spe
- address to, the role of glue parti
les be
omes more im-
trum for a
hain in the GS look as very universal: (i) portant, sin
e they have masses. A
tually, in this
ase
the number of main bands are independent of the pa- one should
onsider as repeating stru
tures elementary
K
rameter as well as of a length of the
hain; (ii) it was
ells, whi
h we get if atta
h to ea
h bri
k (e. g. at the
alsonoti
edin[17℄(withoutanyexplanation,either)that left side) one glue parti
le[31℄.
boundaries of the bands
orrespond to Fibona

i num- At the basi
level of hierar
y one have 55 elementary
bers; (iii) repla
ement in interparti
le intera
tions elas-
ells of 3 parti
les and 89αe(l0e)m=en(tga2r)y
ellsβo(0f)5=p(agr4ti)
les,
ti
for
es by the Lenard-Jonespotential does not
hange letthembe
alledas
ells and ,re-
qualitativelyapatternofsplittingphononspe
trum into spe
tively. Ea
helementary
ell,beingisolated(and,e.g.
band [17℄. periodi
alαly(0)
losed) has its ownβe(i0g)enfrequen
ies: three
Now we
an explain these and more features as dire
t for a
ell and (cid:28)ve for a
ell . Then, if these
ells

onsequen
eofparti
ularspatialstru
ture [2℄ of the GS, in the
hain be really isolated, then, see Fig.1 we get
whi
h main relevant detailes are summarized in Intro- three bands ea
h of 55αd(0e)generated states (solid lines)
du
tion. The key point of our explanation is that the whi
h belongs to
ells , and (cid:28)ve bands ea
h of 89
GS
onsistsoflargenumberofalmostidenti
alelements. degenerated states (dotted lines).
For
larity sake, we
onsider a parti
ular
ase of the FK Due to
ells intera
tions all degenerated frequen
ies
r/s=377/610

hain with , but all our arguments
an be are really splitted into (cid:28)nite width bands, see Fig.1.
easily generalizied for a
hain of di(cid:27)erent size. The
loser inspe
tion of frequen
y spe
trum shows, that
Applying the pi
ture of CS spatial stru
ture outlined seven largest breaks
ut the exa
t spe
trum in bands
n = (1,89) (90,144) (145,233) (234,322) (323,377)
in Introdu
tion to our parti
ular
ase, we
an see, that , , , , ,
(378,467) (468,523) (524,610)
our CS
an be presented as a
omposition of two bri
ks , , sothebandwidthshave
of 3d level of hierar
y: the same order 89, 55, 89, 89, 55, 89, 55, 89, exa
tly as
610=(g232g376), thesequen
eofsolidanddottedhorizontallinesinFiαg(.10)!
(7) Noβw(,0)let us remind, that the sequen
e of
ells
and in our GS is not random but they belong to
with subsequentexpansion,in a

ordan
ewith
omposi-
tion rules (1),(2), into bri
ks of 2nd level: 3α4(1)
e=lls(go1f2t)he=n(egx4tg2hgie4r)ar
y level, seeβ((11)2)=: (1g320
)el=ls
and 21
ells
(g4g2g4g2g4)
232=(88g54g88), 376=(88g54g88g54g88),
. Again,ifthis
ellswouldbede
oupled,we
(8)

ould see bands, that
onsist of 13 and 21 degenerated
whi
h, in turn, are expanded into bri
ks of 1st level: states. Indeed,inthemiddlepartoftheFig.1,wherethe

entral band is shown with greater resolution, we see a
54=(20g12g20), 88=(20g12g20g12g20),
(9) sequen
e of bands, that
ontain 21,13, 21, (21+13), 21,
4
Figure 1: Frequen
y spe
trum for FK model in the ground Figure2: Thesame,asinFig.1,buthereafourthbandandits
state. Intheleftbox:thetotalspe
trumof
hain,inthemid- fourthsubbandareshownwithgreaterresolution. Inright(cid:28)g-
dle and right boxes the
entral band and its
entral subband urebox openand
losed
ir
les
orrespond to phonon modes
α β
areshownwith greater resolution. Solidanddasαh(e0d) horizβo(n0)- of the
ells and , respe
tively.
tal lines show eigenfrequen
ies for isolated
ells and
respe
tively. ee ee
it must
ontain 13 states.
We
an repeat the whole pro
edure ones more; then
13,21 number ofstates, whi
h is similarto that we have we
ome to next level e(cid:27)e
tive Frenkel-Kontorova
hain
β
at the main level of hierar
y![32℄ However, at higher res- with 13 e(cid:27)e
tive parti
les distributed among 8 poten-
α
olution (the right box in Fig.1) the spe
trum in
entral tial barriers , and to next generation
ells, one of the
band be
omes stru
tureless. α eβ
kind and two of the kind . Numeri
al data presented
e
This is not surprising, sin
e the (cid:16)
ells(cid:17) we have intro-
du
ed are not weakly
oupled obje
ts with respe
t to in Figee.2
on(cid:28)rm our pi
tureeein all the detailes. Here we

onsiderthefourthbandofspe
tra,whi
hiswellresolved
phonon modes. In fa
t, what
ells provides,is that some
from other bands, see, the left bβo(0x)of Fig.2. It
ontains

hainfragmentshave(cid:28)xedperiods;thesefragmentsform
89 states, the number of
ells . In the middle box
lo
ally their band stru
ture. If bands of di(cid:27)erent frag-
we take againthe fourth band whi
h in turn
ontains13
mentsoverlap,theirlevelsare
olle
tivisedintoone
om- β
states,thenumberof
ells . Atlast,intherightboxwe
mon band. The
entral band is the
ase, howeverin this α
show by open
ir
les 3 states of the
ell and by
losed

ase band levels are less sensitive to small variations in- e
2 5 β
trodu
ed by extra regularities of next levels of hierar
y ones × states of two
ells . ee
of spatial
hain stru
ture. Note, that this new kind ofeehierar
y is quite di(cid:27)erent
However, a quite new interesting phenomenonβo(0
)
urs, from that we found in [2℄ with respe
t to spatial stru
-
if one
onsider band, whi
h belongs to aα
(0e)ll but ture, sin
e the transformation rules between levels of hi-
doesnotoverlapwithanybandofthe
ell . Phonons erar
y are more
ompli
ated.
with frequen
ies insidαe(0t)his band will be damped along In
on
lusionofthis se
tionit shouldbe stressed,that
the
ells of the kind , these
ells will play a role of all the universal features of the global band stru
ture
βso(m0)e(cid:16)potentialbarriers(cid:17),whi
hde
ouple
ellsofthekβin(0d) mentioned at the beginning aregoverenedby nearestor-
ea
h froαm(0)other. Our
hain
ontains 89
ells der in the
hain. In
ontrast, the (cid:28)ne stru
ture depends
and 55
ells , whi
h is like to 89 parti
les separated
ru
ially on the farorder.. The latter is destroyed in the
by 55 barriers, or distributed among 55 wells. Now we CES, therefore in CES the (cid:28)ne stru
ture is washed out,
haveβg(o0t)a new e(cid:27)e
tive Frenkel-Kontorova
hain, where and frequen
y spe
tra be
ome smoother, see Fig.3. In

αe(l0l)s play role some parti
les with potential barriers parti
ular,weseealsoin Fig.3howtwolowestbandsare
among them. For a
larity sake, let us tak=e mα(o0r)e merged into
ommon one.
graphi
alnotation=sfβor(0t)hesee(cid:27)e
tive(cid:16)barrier(cid:17) ∧ ,
and (cid:16)parti
le(cid:17) • . Then, se
onary bri
ks, whi
h
A =
o

ur in this e(cid:27)e
tive FK
hain, are ∧ • •∧ and IV. PHONON LOCALIZATION
B =
∧••∧••∧, and
orresponding elementary
ells
e

an be obtained adding to bri
ks one (cid:16)glue(cid:17) parti
le at
e α = β = The ordering of bri
ks, whi
h persists in the GS, is
the left: • ∧ • • ∧, • ∧ • • ∧ • •∧, having graduallydestroyedwith
on(cid:28)gurationalex
itationofthe
again threee and (cid:28)ve eigenαfreequen
ies, reβspe
tively. Our
hain. The lowest ex
itations are equivalent destru
tion
e(cid:27)e
tive
hainhas8
ells and 13
ells . Now, ifin the oflargestbri
ksduetopermutationsofbri
ksofpre
eed-
frequen
y spe
trum of new e(cid:27)e
tive FK
hain we take a inglevelofhierar
y: thesmallerbri
ksaredestroyed,the
e β e α
band, whi
h belongs to the
ell but not to the
ell , higheristheex
itationenergy[2℄. Thenumberofalmost
e e
5
Figure 3: Comparison of the ground state (left box) and ex-

ited state (rigth box)frequen
yspe
tra.
Figure4: ComparisonofPR(upperplot)andMGV(bottom
n
plot) versus phonon mode number for the ground state of
degenerateCES,originatingfromdi(cid:27)erentbri
kspermu- the
hain.
tationis
ombunatori
allylarge,thereforesomearbitrary

hosenCEShasratherrandomsequen
eofbri
ks,andis
similar to disordered media, e.g. spin glass system. Iψn partℓi
−u1l/a2r,exfopr( aitypii
a/lℓe)xpℓonentially lo
alized state
i 0
From the very beginning it is
lear, that perfe
t ran- | | ∼ −| − W| , ℓi−s1aexlpo(
alsiz/a2tℓi)on length,
and value of the MGV: ∼ − , i.e. be-
domizationof bri
ksordermay result in Andersonlo
al-

omes exponentially small. Moreover, one
an get from
izationofphononmodes. Anewinterestingpointis,that
the value of MGV an estimate of the lo
alization length
FK model does not require any external disorder: ran-
as
domization of bri
ks order in FK
hain o

urs dynami-
Wℓ

ally. Another interesting feature, whi
h
hara
terizes a ℓ= s/2ln s/2ln( W/2lnW).
degree of
haotization of bri
ks in CES, is that all the − s ≃− − (16)
(cid:18) (cid:19)
examples of exponential lo
alization are obtained from
single arbitrary taken CES, without averaging over any Noteanimportantdi(cid:27)eren
ebetweentheestimateofthe
en
emble of nearest CES. lo
alization length as inverseparti
ipation ratio and our
A typi
al quantity used traditionally in studies of lo- estimate(16). Theformerestimatesasizeofthedomain

alization phenomena (see, in parti
ular [17, 23℄) is the where the eigenstate is lo
alized, and is sensitive to par-
parti
ipation ratio (PR), de(cid:28)ned as ti
ular short range dynami
s for formation of the given
s −1 state. Onthe
ontrary,thelatterisrelatedtotherateof
1
R= ψ4 , the exponentialfallo(cid:27) at the tails of the eigenstates,and
s i=1 i! (14)
hara
terizesproperties of the disordered media.
X
Nowletusturntolo
alizationpropertiesofphononsin
ψ ψ2 =1
where isanormalized( i i )ve
torofthephonon theFK
hain. Asinthepreviousse
tion,we
on
entrate
eigenstate. Its value
orPrespond to a
hain fra
tion o
- on the nνum=erri/
sal=st3u7d7y/6o1f0the
hain with thKe r=ot2ation

upied by the lo
alized state, but whether the state is number , and parameter . In
lo
alized exponentially
an be un
lear. Typi
al feature Fig.4 we present our
omparison of PRRand MWGV for
of exponentially lo
alized state is that
omponents out- GS. One
an see, that bWoth quantities and look
side the
enter of lo
alization are exponentially small. very similar. Note, that for GS is well distin
t from
Meanwhile, this
omponents do not
ontribute in PR at zero (dotted line). This means, that all phononRstates
all. Ware not exponentially lo
alized; the smallness of and
To our opinion a better
hara
teristi
s to indi
ate an for some phonon modes means only that this modes
exponential lo
alization
an be the generalized mean ge- areprelo
alizedonly,whi
hagreewithearlierstudies[23,
ometri
al value (MGV), de(cid:28)ned as 25℄.
s s In orderto see, how the exponential lo
alization man-
1
W =s ( ψ2)1/s =sexp( ln(ψ2)), ifest itself in PR and MGV, let us address to CES. We
· s (15)
iY=1 Xi=1 startwi∆thUa=typUiC
aElSCEUS,GwSh=i
h10is−s1t2illinenergyvery
lose
to GS: − . Despite to,thaten-
s
where, a normWalization fa
tor provides that for ex- ergy splitting is small, this CES belong to 5th10b9and in
tenWded states be order of unity. The reason in favor theen∆eUrgy.s1tr0u−
1t2ureofCES[2℄,andRitisoWneof CESs
to is that it essentially better probes the exponen- with . InFig.5weplot and forthesame
tially small
omponents (tails) of the phonon eigenstate. CES.ItisseenthatthebehaviourofPRremainsinmain
6
ω
Figure 7: MGV versus the frequen
y of the phonon. It is
seen, that phonon are exponentially lo
alized ot the edges of
ferquen
y bands.
Figure5: ComparisonofPR(upperplot)andMGV(bottom
n
p∆loUt)=v1er0s−u1s2phonon mode number for a typi
al CES with
.
n
Figure 8: The MGV versus phonon mode number in two
Figure 6: Phonon eigenvie
tor
omponent distribution versus CES.
the
omponent number . Examples of non-lo
alized and lo-

alized modes.
We see, that lo
alized modes are lo
ated at edges of the
frequen
y spe
trum bands.
detailsthesame,whileintheMGVplottherearepoinWts, Now, how this pi
ture lo
alization looks for CES with
wherethe
urvetou
hesazero line. Thismeans,that highe∆rUspl=itt1in0g−1f2romGS?Weexpe
t[2℄,thatatthesplit-
at these points is exponentially small, i.e. there is an ting the largest robust elementary
ells[33℄
13 21
exponential lo
alization of these modes. Note, that the are and , that take part in mutual permutations
PR plot at the same points has no
lear indi
ations that only, whi∆leUla=rg1e0r−
9ells10a−re8 destroye2d1. In the range of
these modes are lo
alized exponentially. splitting ÷ the
ell
andisso
iate[2℄,
(n=305) 13
Sometypi
alexamplesofnonlo
alizedmode that in
rease a number of smaller
ells and de
rease
(n=378)
and lo
alized one for this CES is presented in a∆U
or=ela1t0io−n5 len1g0t−h4of the di1so3rder. Next, at splitting
Fig.6. We see, that the latter is perfe
tly lo
alizedexpo- ÷ the
ell
an disso
iate too, and
ℓ 12 5
nentially, with a lo
alization length ∼ , whi
h
har- permutations of
ells
ome into play, that de
rease a
a
terizesa
orrelationlengthofdisorder. For
omparison
orrelation length of the disorder even more. In appar-
weshowalsothesamemodeintheGSina(cid:16)prelo
alized(cid:17) ent agreementwith our expe
tations de
reaseof the dis-
state. ∆U = 10−12 order s
ale results in a total exponential lo
alization of
In fa
t, CES with
orresponds an early substantial fra
tion of phonon modes, espe
ially in high
lo
alization of phonon modes: only small fra
tion of the frequen
y regoin, as seen from Fig.8.
is exponentially lo
alized, as seen from Fig.5. To get In Fig.9 we plot fo the same two CES our estimate
insight, what modes are lo
alized at the (cid:28)rst turn, in for lo
alization length (16). Two dotted lines show lev-
ℓ = 5 13
Fig.7 we plot MGV as a fun
tion of phonon frequen
y. els and , whi
h
orrespond to expe
ted sizes of
7
show a
lear tenden
y to be lo
alized too, see Fig.10.
V. DISCUSSION AND CONCLUSIONS.
In this paper we have studied properties of phonon
modes in the Frenkel-Kontorova
hain, taken in the
regime of pinned phase. Spatial "bri
k" stru
ture of the
groundstateofthe
hainfoundin[2℄,appearstobevery
useful for understanding the (cid:28)ne stru
ture of the
hain
phonon spe
trum. A
tually, similar analysis
an also be
performedforele
troni
spe
trum,studiedinre
entwork
[29℄.
However,it isobvious,thatin arealphysi
als
alethe
groundstateishighlydegeneratedwithahugenumberof
ℓ
the stati

on(cid:28)gurationalex
itationsstates (CES) of the
Fnigure 9: Lo
alization length versus phonon mℓ=od5e num1b3er
. Horizontal dotted lines
orrespond to levels and .
hain. A
tually this means that the true groundstate of
the
hain is pra
ti
ally ina

essible. CES has properties
quitedi(cid:27)erentfromthatofthegroundstate: theirspatial
stru
tureisrather
haoti
[2℄. Asaresult,they
an
ause
the Anderson-like (exponential) lo
alization of phonon
modes similar to that seen in disordered media. This
means, that results of previous studies of phonon[17, 18,
23, 25℄ and ele
tron properties[29℄ should be revized or
extended to more realisti
states of the
hain.
Con(cid:28)gurational ex
itation states (CES)
an provide a
possibility to study a gradual transition from the order
to disorder. It is important, that the number of CES
grows with energy splitting from the GS very fast. Even
at very small splitting the number of ECS is huge. It
is
urious that ea
h parti
ular CES has intrinsi

haos,
whi
h reminds the situation in
lassi
al spin glass [28℄,
but in
ontrast to the latter, the
haos in ECS arises
Figure 10: Pro(cid:28)les of phonon eigenstates: eigenve
tor
om-
dynami
ally, without any external noise.
ponent distribution versus
omponent number. The upper ~
The quantization of FK model in small limit is in

orrespon∆dsUto=C6S·,1w0h−i8lenex2t.t4w·o10p−re4sentCESwithsplitting
fromGS and respe
tively. essen
ethequantizationofitsphononmodes[24℄. There-
fore lo
alization of phonon modes means lo
alization of
quantum states. Note also, that phonon quantization
largest ro∆bUust e1le0m−e4ntary10
−el8ls, whi
h survive at energy problem is very
lose to ele
tron quantization problem
splitting ∼ and ,respe
tively. Wesee,that [29℄, where transition from lo
alization to delo
alized
minimal lo
alization length follows the size of maximal state is interesting as a insulator-metal transition .
robust stru
ture in the
hain. This work was supported in part by EC RTN net-
Sin
e the
hainisnothomogeneous,lo
alizationprop- work
ontra
tHPRN-CT-2000-0156. One ofus (O.V.Z.)
erties are not homogeneous too. As seen from Figs. 8,9, thanksCariplofundation,INFNandRFBRgrantNo.04-
the lo
alizationis maximalat the high frequen
y part of 02-16570 for (cid:28)nan
ial support. Support from the PA
phonon spe
trum, while low frequen
y part seems non- INFN (cid:16)Quantum transport and
lassi
al
haos(cid:17) is grate-
lo
alized. In fa
t, at least longwave modes of phonons fully a
knowledged.
[1℄ J.M. Ziman, Models of disorder (Cambridge University [4℄ O.V. Zhirov, G. Casati and D.L. Shepelyansky, Phys.
Press, Cambridge,1979). Rev. E67, 056209 (2003).
[2℄ O.V. Zhirov, G. Casati and D.L.Shepelyansky, Phys. [5℄ F. Nabarro, Theory of Crystall Dislo
ations (Clarendon,
Rev. E 65, 026220 (2002). Oxford, 1967).
[3℄ Y.I. Frenkel and T.K. Kotorova, Zh.Eksp.Teor.Fiz.8, [6℄ V.L. Pokrovsky and A.L. Talapov, Theory of In-
1340 (1938).
ommesurate Crystals, Soviet S
ienti(cid:28)
al Reviews Sup-
8
plementSeries Physi
s, vol.1 (Harwood, London, 1984). B 22, 2988(1980).
[7℄ S.C. Ying, Phys.Rev. B3, 4160 (1971). [22℄ F. Vallet, R. S
hilling and S. Aubry,Europhys. Lett., 2,
[8℄ L.Pietronio,W.R.S
hneiderandS.Strässler,Phys.Rev. 815 (1986); J. Phys.C 21, 67 (1988).
B 24, 2187(1981). [23℄ P.Tong,B.Li,andB.Hu,Phys.Rev.B59,8639(1999).
[9℄ S.Aubry,inSolitons and Condensed Matter Physi
s,ed. [24℄ B.Hu,B.Li,andH.Zhao,Europhys.Lett.53342(2001).
byBishopA.R.andS
hneiderT.,(Springer,N.Y.,1978). [25℄ J.A. Ketoja and I.I. Satija, Physi
a D 104, 239 (1997).
[10℄ S. Aubry,J. Phys.(Fran
e), 44, 147 (1983). [26℄ M. Peyrardand S. Aubry,J. Phys. C 16, 1593 (1983).
[11℄ L.M. Floria and J.J. Mazo, Adv.Phys., 45, 505 (1996). [27℄ J.M. Greene, J. Math. Phys., 20, 6 (1979);
[12℄ M.WeiserandF.J.Elmer,Phys.Rev.B53,7539(1996); R.S. Ma
Kay, Physi
a D 7, 283 (1983).
Z. Phys. B104, 55 (1997). [28℄ M. Mézard, G. Parizi, and M.A. Virasoro,Spin Glass
[13℄ O.M. Braun, T. Dauxois, M.V. Paliy, and M. Peyrard, Theory and Beyond, (WoldS
i., Singapore, 1997).
Phys.Rev.Lett.78,1295(1997); Phys.Rev.E,55,3598 [29℄ P. Tong, B. Li and B. Hu, Phys. Rev. Lett. 88, 046804
(1997). (2002).
[14℄ L. Consoli, H.J.F. Knops, and A. Fasolino, Phys. Rev. [30℄ Thenumberofbri
kspe
iesdependsonthenumberprop-
Lett. 85, 302 (2000). erties of the ratio of a mean interparti
le distan
e to a
[15℄ S. Aubry, Physi
a D, 7, 240 (1983); S. Aubry and P.Y. period of the external (cid:28)eld, the
hain (cid:16)rotation number(cid:17)
ν
LeDaeron, Physi
a D 8, 381 (1983). .Themostsimplepi
turequotedhereassumesthatthis
[16℄ S.AubryandG. Abramovi
i,Physi
a D43, 199 (1990). rotational numberapproximates the golden meanvalue.
[17℄ E. Burkov, B.E.C. Koltenba
h, and L.W. Bru
h, Phys. [31℄ Note,thatat anylevelof hierar
yanumberofgluepar-
Rev. B 53, 14179 (1996). ti
les is equal to thetotal numberof bri
ks.
[18℄ B.Hu,B.Li,andH.Zhao,Phys.Rev.E61,3828(2000). [32℄ Here two
entral subbands of widths 21 and 13 are
[19℄ Ya. G. Sinai,Introdu
tion to Ergodi
Theory, (Prin
eton merged dueto their overlapping.
UniversityPress, 1977). [33℄ Letusremind(seepreviousse
tion),thatanelementary
[20℄ B.V. Chirikov, Phys.Rep.52, 263 (1979).
ell is a bri
kwith one glue parti
le added.
[21℄ H.U. Beyeler, L. Pietronero and S. Strässler, Phys. Rev.