Dynamical two electron states in a Hubbard-Davydov model

Transcription

1 EPJ mauscrip No. (will be isered by he edior) Dyamical wo elecro saes i a Hubbard-Davydov model Firs auhor L. Cruzeiro, J.C. Eilbeck, J.L. Marí, ad F.M. Russell CCMAR ad FCT, Uiversidade do Algarve, Campus de Gambelas, 8 Faro, Porugal. Deparme of Mahemaics, Herio-Wa Uiversiy, Ediburgh EH AS, U.K. May, Absrac. We sudy a model i which a Hubbard Hamiloia is coupled o he dispersive phoos i a classical oliear laice. Our calculaios are resriced o he case where we have oly wo quasiparicles of opposie spis, ad we ivesigae he dyamics whe he secod quasi-paricle is added o a sae correspodig o a miimal eergy sigle quasi-paricle sae. Depedig o he parameer values, we fid a umber of ieresig regimes. I may of hese, discree breahers (DBs) play a promie role wih a localized laice mode coupled o he quasiparicles. Simulaios wih a purely harmoic laice show much weaker localizaio effecs. Our resuls suppor he possibiliy ha DBs are impora i HTSC. PACS. 7..-k 3..Pw 3..Ry Iroducio. I spie of he may sudies [ 3,,] made sice i was firs discovered [], high emperaure supercoduciviy (HTSC) remais a challege. The aure of he carriers ad he mechaism behid pair formaio are sill uclear. Accordig o may researchers, HTSC ca be explaied by a purely elecroic model, such as ha described by he J or he Hubbard Hamiloias, for which charge ad/or spi ieracios are paramou. This view is esseially based o he absece of isoope effecs see i some experimes [7] ad he appare d-symmery of he supercoducig wavefucio. However, accumulaig experimeal evidece exiss for elecro-laice effecs i high emperaure supercoducors [8,, ], ad heories based o elecro-phoo ieracios have also bee proposed [ ]. Here we follow he idea ha boh elecroic correlaios ad elecro phoo ieracios are impora [] ad sudy a model i which a Hubbard Hamiloia is coupled o dispersive phoos. Our mai aim is o explore oe exra igredie, which has geerally bee igored uil ow, he imporace of he aharmoic characer of laice vibraios. Whils our ulimae aim is

2 Firs auhor L. Cruzeiro e al.: Dyamical wo elecro saes i a Hubbard-Davydov model o udersad HTSC, here we propose a specific mechaism for pair formaio ha ivolves he ieracio of polaros hrough a oliear laice mode, which will have applicaios i oher areas. We sudy he sabiliy of such a pair as a fucio of he elecro-elecro (or hole-hole) ieracio. displaceme (see below). Boh egaive ad posiive values of γ will be cosidered, correspodig o he aracive ad repulsive Hubbard models, respecively. As i he Davydov model for eergy rasfer i proeis [3], Ĥ qp-ph, he Hamiloia for he ieracio of he quasiparicle wih he laice icludes he couplig o The Hubbard-Davydov Hamiloia. acousic (or Debye) phoos: Ĥ qp-ph = χ,σ [ (u+ u ) ( ĉ σĉ σ )] (3) The Hamiloia Ĥ we use has hree pars: Ĥ = Ĥqp + Ĥqp-ph + H ph () where Ĥqp is he Hamiloia for a quasiparicle wih spi, Ĥ qp-ph describes he ieracio of he quasiparicle wih he laice ad H ph is he laice (phoo) Hamiloia. The Hamiloia for he quasiparicle is he D Hubbard Hamiloia: where χ is a parameer which describes he sregh of he quasiparicle-laice ieracio. May previous publicaios have icluded elecro-phoo ieracios i he framework of he model of Holsei [], i which oly shor-rage ieracios are are cosidered. As has bee poied ou elsewhere [], whe he elecro screeig is poor, such as i cupraes, elecro-phoo ieracios are log rage, which ca be described by acousic phoos. The phoo Hamiloia is as follows: Ĥqp = ɛ (ĉ ) σ ĉ σ + γ ĉ ĉ ĉ ĉ (),σ (ĉ σ ĉ σ + ĉ σĉ+σ),σ where he sums are over he sies, goig from o N, (N is he oal umber of laice sies) ad σ refers o he spi ad ca be up or dow. ĉ σ is he creaio operaor H ph = H ph co + Hos ph [ H ph co N ( ) = κa a 7 a + u = u ( ) ] a a + u u H ph os N ( = κ u + ) u + N p M = = () for a quasiparicle of spi σ a sie. ɛ is he self-eergy of he quasiparicle, he rasfer erm for he quasiparicle o move bewee eighbourig sies. We depar from he usual oaio i ha he o-sie quasipariclequasiparicle couplig is here desigaed as γ (ad o U) o avoid cofusio wih he variables {u } used for laice where u is he displaceme from equilibrium posiio of sie, p is he momeum of sie, a is he equilibrium disace bewee sies, κ is he elasiciy of he oliear laice ad κ is a similar cosa for he o-sie poeial. Here, he couplig ieracios bewee sies are described by a Leard-Joes poeial, H ph co, a poeial commoly

3 Firs auhor L. Cruzeiro e al.: Dyamical wo elecro saes i a Hubbard-Davydov model 3 used o describe ieracios bewee aoms. I a high emperaure cuprae, his poeial describes he ieracios of he copper ad oxyge aoms i oe Cu-O layer. The o-sie poeial H os ph is as used i may breaher sudies []. I ca be cosidered o represe he effec, i a mea field approach, of he res of he crysal o he oe dimesioal chai whose saes are sudied explicily. I a cuprae, his models he effec of he eighbourig layers o he Cu-O layer. Our Hamiloia (-) icludes wo sources of oliear effecs. The firs comes from he irisic olieariy of he Leard-Joes poeial, H ph co ad he o-sie poeial, H ph os. The secod source of olieariy is exrisic ad comes from he ieracio of he quasiparicle wih he laice (cf. Eq. 3). The former is he source of olieariy i he sudies of discree breahers [] ad he laer is he cause of localizaio i polaro heory. he fully quaum sysem. A. K, he wo approximaios lead o virually he same value [7]. We would like o emphasize ha he approximaio we cosider here is o a adiabaic approximaio. I a adiabaic approximaio he kieic eergy of he phoos is egleced wih respec o he kieic eergy of he quaum paricle. We do o do ha here, as our dyamical equaios, eqs () ad (7) below, iclude he ime derivaive of he momea of he laice sies. Wha we do is o cosider ha he dyamics of he laice ca be reaed classically. Boh he semiclassical (or quaum/classical, as we prefer o call i o differeiae from oher use i he lieraure) approach we apply here ad he adiabaic approximaio lead o similar resuls whe we cosider he groud saes of he sysem (because he correspodig soluios have zero kieic eergy), bu hey are differe whe we deal wih dyamics, as we do i his work. We adop a mixed quaum-classical approach i which he laice is reaed classically, while he quasiparicle is reaed quaum mechaically. Accordigly, he displacemes u ad momea p are real variables. The quasiparicle variables are operaors, a disicio which is marked by he has above he operaors. The imporace of quaum effecs of he laice ca be assessed by cosiderig he full quaum model a fiie emperaure, which has already bee doe for he Davydov Hamiloia. I was foud ha, a.7 K, he laice displaceme correlaed wih he posiio of he quaum paricle i exac semiclassical Moe Carlo simulaios differed by % from he correspodig variable i exac simulaios i Ulimaely, he eed for a full quaum reame comes from compariso wih experimeal resuls. Isoopic effecs ca oly be described i a fully quaum framework. Our mai aim here is o explore he imporace of aharmoiciy i he laice for he dyamics of paired saes, somehig which is much more complicaed o do wihi a fully quaum formalism. Thus, as a firs approximaio, we resric ourselves o he mixed quaum-classical regime ad sudy he behaviour of a pair of quasiparicles, coupled o a oliear laice.

4 Firs auhor L. Cruzeiro e al.: Dyamical wo elecro saes i a Hubbard-Davydov model Wih hese assumpios, he exac wo quasiparicle wavefucio for he Hamiloia (-) is: ψ() =,m=,n φ m ({u }, {p }, ) ĉ ĉ m () where φ m is he probabiliy ampliude for a quasiparicle wih spi up o be a sie ad a quasiparicle wih spi dow o be a sie m. The probabiliy ampliude is depede o he laice displacemes ad momea i a way ha is o specified a priori ad is deermied by he equaios of moio. Similarly o oher sysems 3 Dyamical saes. We cosider he case i which he quasiparicle desiy is low ad he sarig poi is ha of a isolaed quasiparicle ieracig wih he laice. We wish o fid if he addiio of a secod quasiparicle wih opposie spi o ha sae ca lead o pairig of he wo quasiparicles, ad how he relaive sabiliy of he paired sae depeds o he quasiparicle-quasiparicle ieracio γ. We sar from he sae of a sigle quasiparicle. The wavefucio is [8], he equaios of moio for probabiliy ampliudes φ m are derived by iserig he wavefucio () i he ψ σ = φ ĉ σ (8) Schrödiger equaio for he Hamiloia (-), ad he equaios for he displacemes ad momea are derived from he Hamilo equaios for he classical fucioal E = ψ Ĥ ψ. They are: ı h dφ jl d = (φ j l + φ j+l + φ jl + φ jl+ ) + γφ jl δ jl + Miimum eergy saes for his oe quasiparicle ca be foud by umerical miimizaio of he eergy fucioal E = ψ Ĥ ψ wih respec o he probabiliy ampliude for a sigle quasiparicle i sie, φ, ad o he displacemes u [9]. Two kids of miimum eergy saes are + χ (u j+ u j + u l+ u l ) φ jl () dp j d = H ph (7) u j χ ( ϕ j ϕ j+ + ϕ j ϕ j+ ) where ϕ j, he probabiliy for he quasiparicle wih spi up o be i sie j ad ϕ j, he probabiliy for he quasiparicle wih spi dow o be i he same sie. These are give by: ϕ j = ψ ĉ j ĉj ψ = ϕ j = ψ ĉ j ĉj ψ = N φ jl, l= N φ lj l= foud. For sufficiely large quasiparicle-laice ieracio χ, he quasiparicle saes are localized ad here is a associaed laice disorio. We call his he sigle paricle polaro, or simply polaro. Below a hreshold value for χ, he saes are delocalized, as i he usual Bloch saes, ad he laice is udisored. We have cosidered a value of χ ad oher parameers such ha he iiial oe quasiparicle polaro sae is eiher oo weak or oo sable whe compared wih delocalized, Bloch saes for he same values. While i is impora o fid he behaviour of he wo quasiparicle saes cosidered here for differe values of he parameers, our choice esures ha he resuls here are o he cosequece of exreme values.

5 Firs auhor L. Cruzeiro e al.: Dyamical wo elecro saes i a Hubbard-Davydov model The dyamical saes we sudy are he perurbaios of he sigle polaro sae, iduced by he presece of a secod quasiparicle wih opposie spi. Because he umber of variables φ m ha characerize he wavefucio () icreases wih he square of he laice size, i order o be able o iegrae he equaios of moio for a sufficiely log ime, he size of he laice was kep relaively shor, i. e. he umber of sies is N =. The aim is o ivesigae he ifluece of he sregh ad sig of he quasiparicle-quasiparicle ieracio γ o he dyamics of he paired quasiparicle saes. The parameers of he simulaios i he figures are he same, excep for he quasiparicle-quasiparicle ieracio γ. I Fig. we se γ/ = i a aracive Hubbard model. The addiio of a secod quasiparicle leads ϕ u o a localized sae for he pair, wih a very sligh peak p oscillaio, ha is hardly visible i he figure. (The probabiliy for he secod quasiparicle is he same as ha show ad is o displayed). The laice, however, ses io a breaher-like oscillaio [], i.e., a localized exciaio wih a ieral oscillaio. Ideed, a he sie of he iiial laice disorio, oscillaios are clearly visible i he laice displacemes ad momea. A srikig observaio is ha he amou of radiaio geeraed is very small, ad mos of he eergy of he laice is associaed wih he breaher. Fig., which displays aoher ps period of he dyamics a a laer ime, demosraes he sabiliy of his soluio. Fig.. Time depedece for (a) he probabiliy for oe quasiparicle o be i sie, ( = N, N = ), (b) he laice displaceme ad (c) he momeum of sie. Time is i picosecods. The parameers are = J, χ = pn, κ = N/m, κ = κ, a =.Å ad γ = J. A Hubbard Hamiloia wih a much weaker aracio, correspodig o a raio of γ/ =., is cosidered

6 Firs auhor L. Cruzeiro e al.: Dyamical wo elecro saes i a Hubbard-Davydov model ϕ.8... ϕ.8... u u p p Fig.. Same as Fig., a a laer ime. Fig. 3. Same as Fig., bu wih γ = J. i Fig. 3, where he las picosecods of a picosecod simulaio are displayed. A modulaio of he peak of he probabiliy disribuio is ow clearly see, which has he same frequecy as he mai modulaio of he laice breaher. The modulaio of he quasiparicle prob- abiliy is associaed wih a periodic chage of shape i which a lower peak wih a sligh ail appears. Eve a his comparaively much weaker ieracio, he amou of radiaio is very small ad mos of he laice eergy is

7 Firs auhor L. Cruzeiro e al.: Dyamical wo elecro saes i a Hubbard-Davydov model 7 i he breaher. The frequecy of he mai modulaio of he breaher is as for γ/ =. I Fig. a repulsive Hubbard Hamiloia is cosidered, wih γ/ = +.. The modulaios ad he associaed ails of he probabiliy disribuio for he quasiparicle are ow more proouced, bu heir mai frequecy is uchaged. Alhough here is a sligh icrease i he radiaio i he laice, he sabiliy of he breaher ad of he quasiparicle soluio is appare. ϕ.8... u I Fig. he repulsive ieracio is icreased o γ/ = +. The modulaios i he probabiliy disribuio for he quasiparicles lead o greaer periodic chages of shape, sill wih he same frequecy as for he oher values of γ. The radiaio i he laice is ow more visible, bu he breaher remais sable I Figures ad 7, a large repulsive value, correspodig o γ/ = is ake. This leads o a chage i he probabiliy disribuio for he quasiparicles, from a sigle sie peak io a wo sie peak, wih periodic oscillaios which make oe probabiliy a oe sie larger ha he p oher. The laice variables show ha, cocurrely wih he appearace of he breaher, a cosiderable amou of radiaio is geeraed. Also oiceable is he fac ha he frequecy of he modulaios has chaged. Fig. 7 shows ha he ew quasiparicle probabiliy disribuio is sable, as well as he laice breaher, eve if he oise which resuls from successive passes of he radiaio hrough he periodic boudaries, cosiues a sigifica par of he Fig.. Same as Fig., bu wih γ = + J. laice eergy. I Figures 8 ad 9, a repulsive ieracio correspodig o γ/ = is used. Fig. 8 shows ha a drasic rasformaio akes place i which he iiial disribuio chages io a wo peak disribuio. Oe of he peaks is locaed where he iiial laice disorio was ad he secod peak is as far away from i as i ca be i his laice.

8 8 Firs auhor L. Cruzeiro e al.: Dyamical wo elecro saes i a Hubbard-Davydov model ϕ.8... ϕ.8... u u p p Fig.. Same as Fig., bu wih γ = + J. Fig.. Same as Fig., bu wih γ = + J. Also, while he peak ha is locaed a he origial laice disorio sie remais umodulaed i ime, as well as is associaed laice disorio, he secod peak oscillaes wih approximaely he same frequecy as ha i Figs. ad 7. The momea i Fig. 8 show clearly ha he secod peak has a associaed laice breaher, while he firs peak is associaed wih a disorio ha is esseially saic. Afer some ime, because of he repeaed reflecio of he radiaio from he boudaries, his picure is o so clear. Boh peaks show oscillaios i he displacemes

9 Firs auhor L. Cruzeiro e al.: Dyamical wo elecro saes i a Hubbard-Davydov model 9 ϕ.8... ϕ.8... u u p p Fig. 7. Same as Fig., bu a a laer ime. Fig. 8. Same as Fig., bu wih γ = + J. Dyamical saes i he fully harmoic ad he momea of he laice are raher oisy. However, approximaio Fig. 9 does illusrae he sabiliy of he wo peak soluio, eve i he presece of such relaively large amou of oise. The early heory of pair formaio via ieracio wih phoos assumed ha he laice moio was harmoic. I

10 Firs auhor L. Cruzeiro e al.: Dyamical wo elecro saes i a Hubbard-Davydov model bu ow he phoo Hamiloia is give by: ϕ.8... H harm ph H co-harm ph H os-harm ph = H co-harm ph + H os-harm ph = N κ (u u ), = = κ N = ( u ) + M N = (9) The phoo Hamiloia (9) ca be obaied from () by cosiderig he limi of small displacemes, i which oly he liear erms remai. I his case, he oly oliear erm lef for he oal Hamiloia is ha which p u describes he quasiparicle-laice ieracio. I should be poied ou ha, if we disregard he correlaio erm i (), he equaios of moio for his sysem are hose sudied by a umber of auhors [,] for a sigle sigle polaro, ad for ay umber of polaros by Alexadrov []. p Fig. shows ha whe he effecive ieracio is such ha γ/ =, he addiio of a exra elecro o he miimum eergy sigle polaro leads o a sae i which boh elecros are i he same sie wih a srog laice deformaio of breaher ype associaed wih heir presece. The ime evoluio of he momea, however, shows ha here is o breaher formaio, oly phoos which ravel alog he laice. Because of he periodic Fig. 9. Same as Fig., bu a a laer ime. boudary codiios, hese phoos eveually come back ad afer hey have crossed each oher may imes he is ieresig o see how he dyamics of he wo elecro saes would be i his case, ad his secio is devoed o ha quesio. The firs wo erms (), (3) i he Hamiloia we cosider i his secio are he same as before, laice becomes very oisy. The laice deformaio associaed wih he wo elecros oscillaes periodically because of he ierferece of hese phoos, bu does o move. Also, he sae of he wo elecros remais localized o oe sie all he ime.

11 Firs auhor L. Cruzeiro e al.: Dyamical wo elecro saes i a Hubbard-Davydov model ϕ.8... ϕ.8... u u p p Fig.. Same as Fig., bu wih γ = J ad for he harmoic laice 9. Fig.. Same as Fig., bu wih γ = + J ad for he harmoic laice 9. Whe he elecro-elecro ieracio is repulsive ad Similar dyamics akes place for γ/ =., excep ha very sligh oscillaios i he probabiliy disribuio for he elecro saes also akes place (o show). such ha γ/ = +, he phoo emissio leads o flucuaios i he elecro probabiliy disribuio ha are clearly visible i Fig.. The dyamics is similar o ha

12 Firs auhor L. Cruzeiro e al.: Dyamical wo elecro saes i a Hubbard-Davydov model of fig., wih phoos propagaig alog he laice ad causig oscillaios i he oherwise cosa disorio iduced by he wo elecros. Agai, he momea show ha here is o breaher formaio ad all he dyamics of he laice is due o he phoo propagaio ad ierferece. For a repulsive ieracio for which γ/ = +, he wo elecros spli up ad he probabiliy disribuio shows wo peaks, boh of which have a associaed laice deformaio wih he breaher profile (see Fig. ). Phoos are geeraed from each of hese locaios ad ϕ u heir ierferece eveually leads o a oisy laice. The. wo peaks i he probabiliy disribuio for he elecros oscillae i a less regular fashio ha i he aharmoic laice, bu remai sable hroughou he simulaio. I should be oiced ha for his harmoic approximaio p also, he laice displacemes iduced by he elecros/holes. are o small. Hece, a accurae represeaio of he dy-. amics should iclude he oliear erms i he laice Hamiloia, as was doe i he previous secio Discussio Our aim was o ivesigae he relaive sabiliy of a correlaed pair of quaum quasiparicles wih opposie spis wih respec o heir ucorrelaed saes. The sarig Fig.. Same as Fig., bu wih γ = + J ad for he harmoic laice 9. poi was hus he sae of a sigle quasiparicle polaro ad we sudied he dyamic saes which arise whe a secod quasiparicle is added o he firs sae. The Hamiloia used icludes several physical igredies. O he oe had, i coais wo sources of olieariy, oe irisic o he laice ad aoher which arises from he quasiparicle laice ieracio. Such oliear laices have bee show o possess geeric soluios kow as discree breahers (DBs) []. The sudy of sysems i which o-

13 Firs auhor L. Cruzeiro e al.: Dyamical wo elecro saes i a Hubbard-Davydov model 3 liear laices are coupled o oe quaum quasiparicle, o he oher had, is jus begiig [,3]. To our kowledge, his is he firs ime ha he couplig of wo quaum quasiparicles o a oliear laice has bee cosidered. Ideed, a secod igredie is he iclusio of quasiparicle-quasiparicle ieracios, i addiio o he quasiparicle-laice ieracios foud i he polaro model. The quasiparicle-quasiparicle ieracios ca represe Coulomb ieracios, ad/or spispi ieracios, ad be eiher aracive or repulsive. The dyamical simulaios idicae ha for hese exeded sysems, DBs are geeric soluios also ad ca be geeraed by he presece of a secod quasiparicle. These laice breahers ca i ur sabilise localized, paired, quasiparicle saes, for a large rage of γ values. Widows of γ were foud for which similar soluios are obaied. Thus, for a raio of γ/ bewee ad + (Figs. -), DBs are foud i he laice ad i he quasiparicle, wih he same mai modulaio frequecies. For larger values of γ/, wo differe soluios were foud (see Figs. -9). I oe soluio he quasiparicles disribuio is spli io equal values i wo eighbourig sies ad i he secod a wo peak disribuio, wih he peaks as far apar as possible i he laice used, is observed. This Hamiloia icludes he wo mai physical causes for quasiparicle pairig ha have bee cosidered i HTSC ad allows for ierpolaio bewee hem, by varyig he sregh of he releva parameers. Accordig o our resuls, a greaer imporace of quasiparicle-laice ieracios i pair formaio should arise i sysems for which he dyamics of he laice is fas eough compared o he quasiparicle dyamics, so ha he laice relaxes whe he wo quasiparicles mee. Coversely, a correspodig greaer imporace of quasiparicle-quasiparicle ieracios should be associaed wih sysems i which he laice dyamics is much slower ha he quasiparicle dyamics. A implici assumpio i his sudy is ha he oliear characer of he laice plays a impora role i HTSC. Alhough he laice disorios are weak i coveioal supercoducors, ad hus he laice dyamics ca be approximaely described by a liear sysem, we argue ha i HTSC hese disorios are such ha he laice eers a oliear regime. This may be why he soud velociy decreases by a few pars per millio i coveioal supercoducors, whereas i a high Tc maerial here is a icrease which is wo or hree orders of magiude larger ha i he former case. Our simulaios wih he harmoic laice show ha he perceage of eergy rasferred o ravellig phoos is much larger ha for he aharmoic laice. The breaher-like soluios foud i he dyamical simulaios are a sigaure of he oliear dyamics of he laice. The possibiliy ha breahers are associaed wih HTSC has bee suggesed elsewhere [, ]. Our sudy idicaes ha DBs are geeric exciaios i sysems govered by he Hamiloia used here. Moreover, wihi a cerai rage of he parameers, he saes i which wo quasiparicles are paired ad coupled o a DB are eergeically more favourable ha hose of ucorrelaed quasi-

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