Novel results from an algebraic approach to molecular bending dynamics

No el esul s om an algeb aic app oach o molecula bending dynamics F ancisco P´e ez-Be nal1, Osi is ´ Al a ez-Bajo1, Jos´e M A ias2, Miguel Ca ajal1, Jos´e E Ga c´ıa-Ramos1, Danielle La ese3, and Ped o P´e ez-Fe n´andez2 1Depa amen o de F´ısica Aplicada, Fac. de CC. Expe imen ales, Uni e sidad de Huel a, Huel a 21071 (SPAIN) 2Depa amen o de F´ısica A ´omica, Molecula y Nuclea , Facul ad de F´ısica, Uni e sidad de Se illa, Apdo. 1065, Se illa E-41080 (SPAIN) 3Depa men o Chemis y, Yale Uni e si y, P.O. Box 208107, New Ha en, CT 06520-8107 (USA) E-mail: [email p o ec ed] Abs ac . We p esen a b ie e iew o esea ch opics o cu en in e es ha depend on an algeb aic app oach o molecula bending dynamics. This app oach is based on a u(3) spec um gene a ing algeb a. In pa icula , we b ie ly p esen esul s on h ee opics: he calcula ion o ini e-size analy ical co ec ions o mean ield esul s, he applica ion o he model o he la ge- ampli ude ib a ional bending mode o he NCNCS molecule, and he analysis o he in luence o quad a ic Casimi ope a o s on exci ed s a e quan um phase ansi ions. 1. In oduc ion The modeling o n-dimensional sys ems using a u(n+ 1) Lie algeb a as he sys em’s spec um gene a ing algeb a (SGA) has p o ed success ul in an ample se o physical si ua ions [1]. This app oach has caused a majo impac in nuclea s uc u e s udies, whe e he in e ac ing boson model (IBM), based on a u(6) SGA and o mula ed by Iachello and A ima in he se en ies, has become a s anda d heo e ical ool [2]. Following he ail o he IBM, he Vib on Model was bo n in he eigh ies, ea ing molecules as bosonic sys ems whe e bosons ep esen ib a ional and o a ional exci a ions ( ib ons) [3]. The SGA in his case is a u(4) Lie algeb a, due o he h ee-dimensional na u e o he p oblem. The Vib on Model, hough use ul, p o ed qui e cumbe some when applied o molecula species composed by mo e han ou a oms. In he cases ha do no equi e a simul aneous ea men o molecula o a ions and ib a ions, a possible al e na i e app oach is he one- dimensional (1D) limi o he ib on model, based on a u(2) SGA [4]. In his limi o a ional deg ees o eedom a e neglec ed and each local ib a ional deg ee o eedom has an associa ed u(2) Lie algeb a. The e o e his model conside s anha monici y om he ou se , due o an isomo phism be ween he u(2) Lie algeb a and he 1D Mo se po en ial [5]. In addi ion, his app oach na u ally embeds disc e e symme y conside a ions [6]. The bending ib a ional dynamics o linea and quasi-linea molecules imply he in e play o ib a ional and o a ional deg ees o eedom. In his case an app op ia e model is he wo- dimensional (2D) limi o he ib on model, wi h a u(3) Lie algeb a as he sys em’s SGA [7]. GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049 Published unde licence by IOP Publishing L d 1 The 2D limi o he ib on model has success ully modeled spec al e m ene gies and in ensi ies o a ious molecula species, om igidly-linea o igidly-ben molecules, including quasi-linea , and non- igid in e media e si ua ions [8, 9, 10]. A ho ough desc ip ion o bo h quan al and classical aspec s o his model can be ound in Re . [11]. In ecen yea s he s udy and cha ac e iza ion o quan um phase ansi ions (QPTs) in algeb aic models has a ac ed much a en ion, a line o esea ch ha can be aced back o he seminal wo k o Gilmo e [12]. These ansi ions, also called shape phase ansi ions and g ound s a e ansi ions, a e ze o- empe a u e phase ansi ions be ween di e en geome ic limi s o he sys em’s g ound s a e upon a ia ion o one o se e al con ol pa ame e s in he Hamil onian. This esea ch line has been ecen ly bols e ed up wi h he disco e y o p ecu so s o QPTs in nuclea sys ems. Re e ence [13] is a excellen and ecen e iew on his opic. Mo e ecen ly, he s udy o exci ed s a e quan um phase ansi ions (ESQPTs), ansi ions ha ake place upon he a ia ion o he sys em’s exci a ion ene gy ins ead o he Hamil onian pa ame e s, has been p oposed [14]. In his case he ansi ion happens o some pa icula exci ed s a es o he sys em. This is especially ele an in he ield o molecula spec oscopy, whe e no el spec oscopic echniques allow he access o highly-exci ed s a es. In ac , quan um monod omy e ec s, whose appea ance in se e al molecula sys ems has been expe imen ally con i med [15, 16], can be ega ded as an ESQPT om he algeb aic model pe spec i e [11]. We inish his sec ion wi h a b ie ou line o he p esen wo k. Sec ion 2 is de o ed o a sho p esen a ion o he bosonic algeb aic app oach o 2D sys ems and i s classical limi . Analy ical co ec ions o he classical limi o he model, going beyond i s mean ield limi a e p esen ed in sec ion 3. An applica ion o a ealis ic case can be ound in sec ion 4, whe e we i ib a ional bending ene gies o a la ge ampli ude bending mode o NCNCS. Sec ion 5 con ains some in e es ing esul s conce ning he in luence o quad a ic Casimi ope a o s on he model ESQPT. In sec ion 6 we inish by gi ing some concluding ema ks. 2. Algeb aic app oach o 2D sys ems In his sec ion we p esen , in an ab idged o m, he 2D limi o he ib on model and i s classical limi . 2.1. Basic esul s The building b icks o he u(3) Lie algeb a used o modeling 2D sys ems a e h ee bosons o wo di e en ypes. One o hem is a scala boson, σ†, and he o he wo a e Ca esian bosons, {τ† x, τ† y}. The ope a o s ollow he usual bosonic commu a ion ela ions: [σ, σ†] = 1 and hτi, τ† ji=δi,j wi h i, j =x, y. All o he commu a o s a e ze o. Fo he sake o con enience we in oduce ci cula bosons [11] τ† ±=∓τ† x±iτ† y √2, τ±=∓τx∓iτy √2.(1) The nine u(3) gene a o s a e buil wi h he possible bilinea p oduc s o σ†and τ† x,τ† yc ea ion and annihila ion ope a o s. They a e con en ionally exp essed as [7]: ˆn=τ† +τ++τ† −τ−,ˆns=σ†σ ˆ l=τ† +τ+−τ† −τ− ˆ D+=√2(τ† +σ−σ†τ−),ˆ D−=√2(−τ† −σ+σ†τ+) ˆ Q+=√2τ† +τ−,ˆ Q−=√2τ† −τ+ ˆ R+=√2(τ† +σ+σ†τ−),ˆ R−=√2(τ† −σ+σ†τ+). (2) GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049 2 The ope a o ˆ lis he 2D angula momen um, as one can easily see once i is exp essed in e ms o Ca esian boson ope a o s. The e a e wo possible dynamical symme ies ha conse e 2D angula momen um, s a ing in he u(3) SGA and ending in he so(2) symme y algeb a. u(3) ⊃u(2) ⊃so(2) Chain I ,(3) u(3) ⊃so(3) ⊃so(2) Chain II .(4) Nume ical calcula ions in he p esen a icle ha e been ca ied ou using he basis associa ed wi h dynamical symme y I. A de ailed discussion o bo h dynamical symme ies can be ound in [11]. Ano he ing edien o he algeb aic app oach is he Casimi (o in a ian ) ope a o s associa ed o each subalgeb a chain [5]. The i s and second o de Casimi ope a o s o he subalgeb as in chains (3) and (4) a e ˆ C1[u(2)] = ˆn , ˆ C2[u(2)] = ˆn(ˆn+ 1) , ˆ C2[so(3)] = ˆ W2= ( ˆ D+ˆ D−+ˆ D−ˆ D+)/2 + ˆ l2, ˆ C1[so(2)] = ˆ l , ˆ C2[so(2)] = ˆ l2. (5) The so(3) Casimi ope a o ˆ W2can be eplaced by he pai ing ope a o ˆ P=N(N+1)−ˆ W2, whe e he o al numbe ope a o , ˆ N= ˆns+ ˆn, has been eplaced by i s alue, N, due o he ac ha we conside sys ems wi h a ixed numbe o bosons. The mos gene al o a ion and pa i y-in a ian , one and wo-body Hamil onian can be w i en as a linea combina ion o he ou possible i s and second o de Casimi ope a o s (5) ˆ H=E0+ǫˆ C1[u(2)] + αˆ C2[u(2)] + βˆ C2[so(2)] + Aˆ C2[so(3)] .(6) Ma ix elemen s in he wo possible bases o he Casimi ope a o s (5) can be ound in Re . [11]. 2.2. Mean ield limi o he model The mean ield limi o he model, also called he he modynamical limi , he la ge Nlimi , o he classical limi o he model, can be ound ollowing an algo i hm es ablished by Gilmo e [17] known as he cohe en (o in insic) s a e app oach. In making use o he cohe en s a e app oach, we use p ojec i e cohe en s a es ha de ine an in insic g ound s a e |[N]; , θi=1 √N!b† cN|0i,(7) whe e and θa e pola coo dina es associa ed o Ca esian coo dina es xand y, and b† cis he boson condensa e b† c=1 √1 + 2σ†+xτ† x+yτ† y.(8) The cohe en s a e (7) is he numbe p ojec ed gene alized cohe en s a e o u(3) [2]. The cohe en s a e app oach p o ides an ene gy unc ional ha is an app oxima ion o he g ound s a e ene gy o a gi en Hamil onian. This app oxima ion becomes exac o N→ ∞. The exis ence o a second o de QPT be ween he wo possible dynamical symme ies can be easily p o en using he a o emen ioned cohe en s a e app oach [11]. In o de o do so, i is con enien o de ine a simpli ied model Hamil onian, wi h one ope a o o each chain and p ope ly no malized ˆ H=ε(1 −ξ)ˆn+ξ N−1ˆ P.(9) GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049 3 The pa ame e εis an ene gy scale and ξis a con ol pa ame e ha akes alues be ween ze o (u(2) dynamical symme y) and one (so(3) dynamical symme y). The ene gy pe boson o he essen ial Hamil onian (9) is [9, 11] Eξ( )≡1 Nh[N]; , θ|ˆ H|[N]; , θi h[N]; , θ|[N]; , θi=ǫ (1 −ξ) 2 1 + 2+ξ 1− 2 1 + 2!2 .(10) The ex emiza ion o he ene gy unc ional (10) e eals ha he e a e wo di e en geome ic limi s: symme ic (o linea ) and de o med (o ben ). In he symme ic case he ene gy minimum is a e= 0, while in he second case e=p(5ξ−1)/(3ξ+ 1) 6= 0. The symme ic phase akes place o con ol pa ame e alues ξ≤ξc= 0.2, and he de o med phase o ξ > ξc= 0.2. When e alua ed a = e, he ene gy unc ional is Eξ( e) = (ξ0≤ξ≤ξc −9ξ2+10ξ−1 16ξξc< ξ ≤1.(11) By e alua ing he de i a i es o Eξ( e) wi h espec o ξ, one inds ha , acco ding o Eh en es ’s classi ica ion scheme, he u(2) −so(3) phase ansi ion is o second o de [11]. Ano he obse able o in e es is he expec a ion alue o he u(2) numbe ope a o in he sys em g ound s a e h[N]; , θ|ˆn|[N]; , θi=N 2 e 1 + 2 e =(0 0 ≤ξ≤ξc 5ξ−1 8ξξc< ξ ≤1.(12) This ope a o can ac as a classical o de pa ame e o he ansi ion be ween he symme ic and de o med phases [11]. 3. Beyond mean ield analy ical co ec ions We e alua e analy ical co ec ions o he mean ield limi o he algeb aic app oach o go beyond he esul s p esen ed in he p e ious sec ion. The mean ield o classical limi is only exac in he la ge Nlimi , and he calcula ed co ec ions allow us o calcula e he N0co ec ions o he g ound s a e ene gy (11) and he numbe o τbosons (12). In o de o compu e hese co ec ions we pe o m a Hols ein-P imako expansion and a shi ans o ma ion, ollowed by a Bogoliubo ans o ma ion, ollowing Re . [18]. This e e ence is a gene al wo k, aimed a bosonic wo-le el Hamil onians wi h SGA u(2L+ 2) wi h L= 1,2,.... In he symme ic egion Dusuel and collabo a o s go a s ep u he han us, making use o he con inuous uni a y ans o ma ions app oach (CUTS) [19]. The Hols ein-P imako expansion implies he de ini ion o a new pai o Ca esian bosons, b† i(bi), i=x, y, wi h he usual bosonic commu a ion ela ions, [bi, b† j] = δij. The inclusion o his boson emo es he dependence on he scala σ†boson o he Hamil onian τ† iτj=b† ibj, τ† iσ=√N b† iq1−ˆnb/N =σ†τi†,(13) ˆnσ=σ†σ=N−ˆnb, whe e i, j =x, y, and ˆnb=b† xbx+b† yby. A hi d se o bosons, c† i(ci), wi h i=x, y is de ined ia a shi ans o ma ion b† i=√Nλδix +c† i;i=x, y . (14) GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049 4 The pa ame e λ, known as he shi pa ame e , is ze o in he sphe ical phase and nonze o in he de o med phase. Once he model Hamil onian (9) is ans o med, we ob ain an expansion in powe s o N, wi h e ms ˆ H=ˆ H1+ˆ H1/2+ˆ H0+O(1/√N),(15) whe e ˆ Hiinco po a es he e ms wi h a Nidependence. The i s e m is ˆ H1=Nhξ+ (1 −5ξ)λ2−4ξλ4i.(16) Se ing λ= /√1 + 2 his e m is no hing mo e han he mean ield esul (10). The nex o de in he expansion o he model Hamil onian, ˆ H1/2, is ze o once he equilib ium alues o (o λ) a e subs i u ed. This is due o he ela ion ˆ H1/2=2 √N dˆ H1 dλ . The e o e, he i s ini e-size co ec ion o he mean ield limi is he ˆ H0 e m, ha is quad a ic in he cboson ope a o s H0=λ2(4λ2−1)ξ+ (1 −3ξ+ 14λ2ξ)c† xcx+ (1 −3ξ+ 4λ2ξ)c† ycy + (5λ2−1)ξc† xc† x+cxcx+ (2λ2−1)ξc† yc† y+cycy.(17) The Hamil onian (17) can be diagonalized wi h a Bogoliubo ans o ma ion c† i=uia† i+ iai, ci=uiai+ ia† i,(18) ha should be independen ly pe o med in he symme ic and de o med phases [18, 20]. The inal esul p o ides he co ec ion o he g ound s a e ene gy pe pa icle in he symme ic and de o med egions Esym 0=ξ+N−1h3ξ−1 + Ξsym(ξ)1/2i,(19) Ede 0=−9ξ2+ 10ξ−1 16ξ+N−1"1−6ξ−27ξ2+ 8ξΞde (ξ)1/2 16ξ#,(20) whe e Ξsym = 5(ξc−ξ)(1 −ξ) and Ξde (ξ) = 5(ξ−ξc)(1 + 3ξ). In he de o med phase he y coo dina e con ibu ion is a spu ious Golds one boson, associa ed wi h a g ound s a e o a ion. Compa ed o he mean ield limi , he deduced ini e-size co ec ions g ea ly imp o e he ag eemen wi h nume ical esul s. This can be clea ly seen in Fig. 1 whe e he mean ield and beyond mean ield (BMF) co ec ions o he g ound s a e ene gy a e compa ed o a nume ical calcula ion o N= 20. The ini e-size co ec ion can be calcula ed o o he obse ables. We include esul s o he expec a ion alue in he g ound s a e o ˆn(5). In his case we can make use o he Hellman- Feynman heo em, de ining a new con ol pa ame e xsuch ha ξ= 1/(1 + x), we ob ain o he symme ic and de o med phases hˆni N=d dx [(1 + x)Esym 0(x)] = 1 N 1−3ξ−Ξsym(ξ)1/2 Ξsym(ξ)1/2+O(N−2),(21) hˆni N=d dx h(1 + x)Ede 0(x)i=5ξ−1 8ξ+1 N 4ξ(ξ−1) + (1 −3ξ)Ξde (ξ)1/2 8ξΞde (ξ)1/2+O(N−2).(22) We compa e his esul , he mean ield limi and he nume ical calcula ion o N= 20 in Fig. 2. As in he p e ious case, he ob ained co ec ion no ably imp o es he mean ield esul , in pa icula in he icini y o he c i ical con ol pa ame e . GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049 5 0.0 0.2 0.4 0.6 0.8 1.0 Con ol Pa ame e (ξ) 0.00 0.05 0.10 0.15 0.20 0.25 G ound S a e Ene gy pe Pa icle (ε0=E0/N) Mean Field BMF Symme ic N = 20 BMF De o med N = 20 Nume ical, N = 20 Figu e 1. (Colo online) G ound s a e ene gy pe pa icle (E0) in a bi a y uni s as a unc ion o he con ol pa ame e ξ. 0.0 0.2 0.4 0.6 0.8 1.0 Con ol Pa ame e (ξ) 0.0 0.1 0.2 0.3 0.4 0.5 No malized O de Pa ame e (〈n(ξ)〉/N) Mean Field BMF Symme ic N = 20 BMF De o med N = 20 Nume ical N = 20 Figu e 2. (Colo online) Expec a ion alue o he numbe ope a o ˆnin he g ound s a e o Hamil onian (9) as a unc ion o he con ol pa ame e ξ. The esul s p esen ed a e encompassed in an open line o esea ch [21]. We a e cu en ly wo king on compu ing he BMF co ec ion o o he obse ables, on analy ically de i ing ini e- size scaling exponen s o he u(2) −so(3) second o de phase ansi ion, and on es ablishing a connec ion be ween Re . [18] esul s and ou s. 4. Applica ion o he ν7bending mode o cyanogen iso hiocyana e The bending dynamics o se e al molecula species [7, 9, 10, 22, 23, 24, 25] ha e been modeled using he wo-dimensional limi o he ib on model. We a e cu en ly in e es ed in he analysis o he spec a o non- igid molecules, as hese p obably p o ide examples o ESQPT. As he exci a ion ene gy inc eases, he non- igid molecule’s exci ed s a es ha e o o e come a po en ial hump in he o igin, and change om a ben -like o a linea -like cha ac e . This ac was known long ago [26], bu nowadays such sys ems ha e ocused an inc easing deg ee o a en ion due o quan um monod omy e ec s and i s implica ions [16]. We ha e pe o med a i ing o he bending ene gy spec um o cyanogen iso hiocyana e (NCNCS), a non- igid molecule whose la ge ampli ude bending spec um displays monod omy e ec s and is expe imen ally accessible [27]. We i Hamil onian (6) o he NCNCS e m alues ob ained wi h he Gene al Semi igid Bende Hamil onian model in Re . [27]. The inal ms is 2.2 cm−1, in a i o 70 e m ene gies. The op imized pa ame e s a e gi en in Table 1. The calcula ions we e pe o med using he FORTRAN code ia u3 [28]. Table 1. Op imized pa ame e s o he one- and wo-body Hamil onian (6) and hei associa ed unce ain ies ob ained in he i o ν7 e m alues o NCNCS [27]. All pa ame e s, excep ing N, a e in uni s o cm−1. The i oo mean squa e de ia ion is ms = 2.2 cm−1. N ǫ α β A 70 203.6(19) -2.58(3) 1.496(10) 0.813(6) The ob ained esul s a e p omising, especially i he simplici y o he model is conside ed. GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049 6 The esul s in Re . [27] and ou i a e compa ed in a quan um monod omy plo in Fig. 3 whe e he good ag eemen be ween bo h app oaches is ema kable. We ep oduce he change in slope in he o igin ha cha ac e izes he c i ical monod omy poin . -8 -6 -4 -2 0 2 4 6 8 Vib a ional Angula Momen um (K) 0 200 400 600 800 1000 Ene gy (cm-1) This wo k PRL 95 243002 Figu e 3. (Colo online) Quan- um monod omy plo o calcula ed ν7mode bending ib a ional le els o NCNCS. The ib a ional angula mo- men um lis labelled K ollowing he usual spec oscopic no a ion. We a e cu en ly wo king in he i o he ib a ional bending spec um o wa e , whe e quan um monod omy e ec s ha e been expe imen ally epo ed [15]. 5. In luence o he quad a ic u(2) Casimi ope a o in ESQPT As shown in Table 1, he second o de Casimi ope a o o he u(2) subalgeb a plays an impo an ole when i ing expe imen al da a. This also happens in o he cases [9, 10] and has sugges ed us o explo e he ole o his anha monic e m in he model, in pa icula i s e ec s on g ound s a e QPT and ESQPT [29]. This can be accomplished wi h a new model Hamil onian, con enien ly scaled, ha inco po a es a second con ol pa ame e ˆ H=ε(1 −ξ)ˆn+α N−1ˆn(ˆn+ 1) + ξ N−1ˆ P.(23) The cohe en s a e app oach, when applied o his case, gi es as a esul ha he g ound s a e ansi ion is mos ly unpe u bed by he addi ion o he new e m. The ene gy unc ional E0(ξ, α) in he mean ield limi becomes E0(ξ, α; ) = εξ+ (1 −3ξ) 2+ (1 + α) 4 (1 + 2)2.(24) The minimiza ion o he ene gy unc ional (24) p o ides he equilib ium alues e= 0 ,q5ξ−1 3ξ+2α+1 ha implies a lowe bound α > −(1+3ξ)/2 i ξ > 0.2. The c i ical alue o he con ol pa ame e is, once mo e, ξc= 0.2. Fo ξ≤ξc he ene gy unc ional minimum lies a he o igin, while o alues o ξ > ξca new minimum appea s and he minimum a he o igin becomes a maximum. When e alua ed o = eand ε= 1, he ene gy unc ional is E0(ξ, α; e) = (ξ0≤ξ≤ξc −9ξ2+10ξ+4αξ−1 16ξ+4αξc< ξ ≤1,(25) which has a discon inuous second o de de i a i e in ξ=ξc. Al hough he second o de g ound s a e ansi ion is basically una ec ed by he new con ol pa ame e , he e a e conspicuous e ec s in he sys em’s ESQPT o nega i e α alues. This GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049 7 is specially no iceable once he exci a ion ene gy diag am o he Hamil onian (23) is plo ed o di e en α alues. Ins ead o one sepa a ix be ween he con igu a ion wi h linea and ben cha ac e , like in he α= 0 case [11], he e a e wo sepa a ices. In Figs. 4 and 5 we plo he exci a ion ene gy diag am o ze o angula momen um (l= 0) s a es o and α= 0 and −0.4, espec i ely. The numbe o bosons in bo h cases is N= 100 and he ene gy is plo ed as a unc ion o he con ol pa ame e ξ. The colo code in he cu es is assigned in acco dance o he magni ude o he maximum squa ed-componen o he eigen ec o in he wo bases o choice. I he eigen ec o is close o dynamical symme y (3), he maximal componen will be close o one in he u(2) basis and he poin will be plo ed in ed colo . I he maximal componen is in he so(3) basis he mapped colo is blue. This way o ep esen ing he da a clea ly ma ks he s a es ha co espond o he sepa a ix in he monod omy plo , in he high le el densi y egion ha sepa a es he linea - and ben -like s a es. The appea ance o wo sepa a ices in Fig. 5 is ma kedly clea . Figu e 4. (Colo online) Exci a ion ene gy diag am in a bi a y uni s (ε= 1) o ze o angula momen um eigens a es o Hamil onian (23) as a unc ion o he con ol pa ame e ξ o α= 0. The numbe o bosons is N= 100 and ene gies a e no malized by N. The colo code indica es he s a e p oximi y o he u(2) o so(3) dynamical symme ies (see ex ). Figu e 5. (Colo online) Exci a ion ene gy diag am in a bi a y uni s (ε= 1) o ze o angula momen um eigens a es o Hamil onian (23) as a unc ion o he con ol pa ame e ξ o α=−0.4. The numbe o bosons is N= 100 and ene gies a e no malized by N. The colo code indica es he s a e p oximi y o he u(2) o so(3) dynamical symme ies (see ex ). The posi ion o he wo sepa a ices in Fig. 5 depends on he alues o he ene gy unc ional (25) maximum a he o igin and i s asymp o ic alue [29]. The alue a he o igin is only a unc ion o ξ, while he asymp o ic alue depends only on α. In pa icula , he e is a c i ical alue o α,αc=ξ−1, whe e he wo sepa a ices c oss and i is associa ed wi h equal alues o he ene gy unc ional a ze o and o la ge alues. In he case o ESQPT he e is no a clea de ini ion o an o de pa ame e [14]. The expec a ion alue o he numbe ope a o ˆnin he di e en eigens a es p o ides a possible app oxima ion. This obse able changes ab up ly in he c i ical exci a ion ene gy bu i does no go o ze o in any o he phases [14]. We plo in Fig. 6 he expec ed alue o ˆn o he eigens a es o Hamil onian (9) as a unc ion o he no malized s a e exci a ion ene gy. We ha e GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049 8 ixed he numbe o bosons (N= 800) and he con ol pa ame e ξ(ξ= 0.6). Fo α= 0 ( ed cu e in Fig. 6) we eco e he esul s published in [14], whe e he c i ical ene gy is ma ked by he wa ped minimum in he plo ed unc ion. Fo 0 > α > αc, in addi ion o he displacemen o la ge ene gies o he ini ial minimum, a maximum appea s o la ge exci a ion ene gies (o ange cu e in Fig. 6). As α ends o αcbo h ex emes app oach. We ound he su p ising esul ha o α=αc he expec a ion alue o ˆnbecomes cons an o he ull ene gy ange excep a sudden a ia ion a ound he c i ical ene gy ha co esponds o he c ossing poin o he wo sepa a ices (g een cu e in Fig. 6). 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Exci a ion Ene gy (E/N) 0.2 0.4 0.6 0.8 1.0 O de Pa am. [〈n(E)〉/Ν] N = 800, ξ = 0.6 α = 0.0 α = -0.25 α = -0.4 Figu e 6. (Colo online) Expec a- ion alue o he numbe ope a o ˆn in he eigens a es o model Hamil- onian (23) o N= 800 and ξ= 0.6 as a unc ion o he exci a ion en- e gy o he s a e no malized by N o h ee di e en alues o α. 6. Concluding ema ks F om ou poin o iew, he wo dimensional limi o he ib on model is a simple algeb aic model ha s ill encompasses enough complexi y o be applied o in e es ing eal physical sys ems. The calcula ion o ini e-size co ec ions o he mean ield limi g ea ly deepens he knowledge o he model. These co ec ions a e a undamen al help o cha ac e ize he QPTs be ween he di e en geome ic limi s o a model and o analy ically ex ac he scaling exponen s o he model, as i will be shown o he model unde s udy in a o hcoming publica ion [21]. We ha e applied he model o he bending dynamics o a non- igid molecule. Despi e he simplici y o he app oach, i is capable o coping wi h complica ed si ua ions, as i has been shown in he case o NCNCS, whe e quan um monod omy e ec s appea . We a e especially in e es ed in he desc ip ion wi h his model o wa e ’s ib a ional bending le els. Being he simples bosonic wo-le el model wi h a non- i ial angula momen um, his limi o he ib on model is a e y con enien aid in he s udy o ESQPTs and hei implica ions. The expe imen al access o bending exci a ion ene gies abo e he monod omy c i ical ene gy makes his ea u e especially a ac i e. The ob ainmen o expe imen al da a o such s a es is no possible in o he ields, as nuclea spec oscopy, which ha e been he adi ional es g ound o g ound s a e ansi ions. Acknowledgmen s This wo k was suppo ed in pa by he Spanish Jun a de Andaluc´ıa unde p ojec s P07- FQM-02962, P07-FQM-03014, and P07-FQM-02894 and by he Spanish MICINN and he Eu opean egional de elopmen und (FEDER) unde p ojec s FIS2008-04189 and CPAN- Ingenio (CSD2007-00042). PPF acknowledge he Spanish MEC o a FPU g an . The au ho s hank Jo ge Dukelsky, F ancesco Iachello, Pie e an Isacke , and Ami am Le ia an o aluable commen s and sugges ions. GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049 9