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