Finite-size scaling exponents in the interacting boson model

RAPID COMMUNICATIONS PHYSICAL REVIEW C 72, 011301(R) (2005) Fini e-size scaling exponen s in he in e ac ing boson model S´ ebas ien Dusuel,1Julien Vidal,2Jos´ eM.A ias, 3Jo ge Dukelsky,4and Jos´ e En ique Ga c´ ıa-Ramos5 1Ins i u ¨ u Theo e ische Physik, Uni e si ¨ a zu K¨ oln, Z¨ ulpiche S . 77, D-50937 K¨ oln, Ge many 2Labo a oi e de Physique Th´ eo ique de la Ma i` e e Condens´ ee, CNRS UMR 7600, Uni e si ´ e Pie e e Ma ie Cu ie, 4 Place Jussieu, F-75252 Pa is Cedex 05, F ance 3Depa amen o de F´ ısica A ´ omica, Molecula y Nuclea , Facul ad de F´ ısica, Uni e sidad de Se illa, Apa ado 1065, E-41080 Se illa, Spain 4Ins i u o de Es uc u a de la Ma e ia, CSIC, Se ano 123, E-28006 Mad id, Spain 5Depa amen o de F´ ısica Aplicada, Uni e sidad de Huel a, E-21071 Huel a, Spain (Recei ed 10 Ma ch 2005; published 14 July 2005) We in es iga e he ini e-size scaling exponen s o he c i ical poin a he shape-phase ansi ion om U(5) (sphe ical) o O(6) (de o med γ-uns able) dynamical symme ies o he in e ac ing boson model, making use o he Hols ein-P imako boson expansion and he con inuous uni a y ans o ma ion echnique. We compu e exac ly he leading-o de co ec ion o he g ound-s a e ene gy, he gap, he expec a ion alue o he d-boson numbe in he g ound s a e and he E2 ansi ion p obabili y om he g ound s a e o he i s exci ed s a e and de e mine he co esponding ini e-size scaling exponen s. DOI: 10.1103/PhysRe C.72.011301 PACS numbe (s): 21.60.Fw, 05.10.Cc, 21.10.Re, 75.40.Cx The in e es in he s udy o quan um phase ansi ions (QPT) has kep g owing in he las yea s in di e en b anches o quan um many-body physics, anging om mac oscopic sys ems such as quan um magne s, high-Tcsupe conduc o s [1] o dilu e Bose and Fe mi gases [2] o mesoscopic sys ems such as a omic nuclei o molecules [3]. Al hough, s ic ly speaking, QPT occu s only in mac oscopic sys ems, he e is a enewed in e es in s udying s uc u al changes in ini e- size sys ems whe e p ecu so s o he ansi ion a e al eady obse ed [4]. The unde s anding o he modi ica ions on he cha ac e is ics o he QPT induced by ini e-size e ec s is o c ucial impo ance o ex end he concep o phase ansi ions o ini e sys ems. In he p esen s udy, we analyze hese ini e-size co ec ions in he in e ac ing boson model (IBM) o nuclei [5], bu he same echnique can be applied o o he boson sys ems, o ins ance, o he molecula ib on model [6] o o a mul ile el boson model o Bose-Eins ein condensa es whe e simila QPT ake place [7]. The IBM is a wo-le el boson model ha includes an angula momen um L=0 boson (scala sboson) and i e angula momen um L=2 bosons (quad upole d-bosons) sepa a ed by an ene gy gap. The sand dbosons ep esen s- and d-wa e idealized Coope nucleon pai s. The algeb aic s uc u e o his model is go e ned by he U(6) g oup and he model has h ee dynamical symme ies in which he Hamil onian, w i en in e ms o he in a ian (Casimi ) ope a o s o a nes ed chain o subg oups o U(6), is analy ically sol able. The dynamical symme ies a e named by he i s subg oup in he chain: U(5), SU(3), and O(6). The classical o he modynamic limi o he model was in es iga ed by using an in insic s a e o malism ha in oduces he shape a iables βand γ[8–10]. Wi hin his geome ic pic u e he U(5), SU(3), and O(6) dynamical symme ies co espond o sphe ical, axially de o med, and de o med γ-uns able shapes, espec i ely. T ansi ion be ween wo o hese dynamical symme y limi s a e desc ibed in e ms o a Hamil onian wi h a con ol pa ame e ha mixes he Casimi ope a o s o he wo dynamical symme ies. As a unc ion o he con ol pa ame e , he sys em c osses smoo hly a egion o s uc u al changes in he g ound-s a e wa e unc ion o ini e numbe No bosons. In he la ge Nlimi , he smoo h c osso e u ns in o a sha p QPT be ween wo well-de ined shape phases [9,11–14]. In pa icula , he ansi ion om U(5) o O(6) has been in ensi ely s udied in ecen yea s because i has a unique second-o de QPT [13–15] associa ed wi h a iple poin in he IBM pa ame e space [14]. Fu he mo e, i was ea ly ecognized ha he IBM Hamil onian along his ansi ion was ully in eg able [16] and exac ly sol able [17,18]. Un o una ely, i is di icul o use he exac solu ion o compu e ini e-size co ec ions analy ically. Thus, we ollow a di e en ou e ha is based on he con inuous uni a y ans o ma ions (CUTs) [19–21]. Wi hin his amewo k, we compu e he i s co ec ion beyond he s anda d andom phase app oxima ion (RPA) [22], which al eady con ains he key ing edien s o analyze he c i ical poin . As al eady obse ed in a simila con ex , [23–25], his 1/N expansion becomes, a his o de , singula when app oaching he c i ical egion so ha one ge s non i ial scaling exponen s o he physical obse ables (g ound-s a e ene gy, gap, occupa ion numbe , ansi ion a es). In a second s ep, we ake ad an age o he exac sol abili y o he model o ob ain nume ical esul s o la ge numbe o bosons ha allows us o check ou analy ical p edic ions. Le us conside he U(5)-O(6) ansi ional Hamil onian H=xnd+1−x 4(N−1)(P† d−P† s)(Pd−Ps),(1) whe e ndµd† µdµ(wi h µ=−2,−1,0,1,2) is he d-boson numbe ope a o , P† s=s†2,P† d=µ(−1)µd† µd† −µ, and xis he con ol pa ame e ha mixes he U(5) linea Casimi 0556-2813/2005/72(1)/011301(4)/$23.00 011301-1 ©2005 The Ame ican Physical Socie y RAPID COMMUNICATIONS DUSUEL, VIDAL, ARIAS, DUKELSKY, AND GARC´ IA-RAMOS PHYSICAL REVIEW C 72, 011301(R) (2005) ope a o (x=1) wi h he O(6) quad a ic Casimi ope a o (x=0). The sys em unde goes a QPT a xc=1/2, be ween a U(5) (sphe ical) phase o 1/2⩽x⩽1 and a O(6) (de o med γ-uns able) phase o 0 ⩽x⩽1/2, when N→∞[13–15]. In he ollowing, we es ic ou analysis o he sphe ical (symme ic) phase ha allows us o in es iga e he c i ical poin mo e simply han om he de o med phase. The Hols ein- P imako boson expansion o one-body boson ope a o s is especially well-sui ed o pe o m a 1/N expansion o he boson Hamil onian (1). In he p esen case, i eads [26,27] d† µdν=b† µbν,(2a) s†s=N− µ d† µdµ=N− µ b† µbµ=N−nb,(2b) d† µs=N1/2b† µ(1 −nb/N)1/2=(s†dµ)†.(2c) Keeping e ms o o de (1/N)0in he Hamil onian exp essed in e ms o he new b’s yields a quad a ic Hamil onian ha can be diagonalized ia a Bogoliubo ans o ma ion. One hen eco e s RPA esul s [22,28]. A he nex o de (1/N)1, he Hamil onian is qua ic and diagonalizing i clea ly equi es a mo e sophis ica ed me hod. To achie e his goal, we used he CUTs echnique [19–21]. Fo an in oduc ion o his me hod, we e e he eade o Re s. [23–25] whe e CUTs we e applied in a simila con ex . One in oduces a unning Hamil onian H(l)=E0(l)+(l)nb+V(l):n2 b:+W(l)P† bPb +(l)(P† b+Pb)+(l)(P† bnb+nbPb),(3) which is ela ed o he ini ial Hamil onian H(0) h ough a uni a y ans o ma ion, namely H(l)=U†(l)H(0)U(l). This ans o ma ion Uis chosen such ha H(∞) commu es wi h nb. In Eq. (3), : O: deno es he no mal o de ed o m o he ope a o O, and he no a ions o he b’s a e he same as o he d’s. The e olu ion o he unning Hamil onian is ob ained om he low equa ion ∂lH(l)=[η(l),H(l)], whe e η(l)=∂lU†(l)U(l) is he an i-He mi ian gene a o o he uni a y ans o ma ion. Fo he p oblem a hand, we conside he so-called quasipa icle conse ing gene a o [29], η(l)=(l)(P† b−Pb)+(l)(P† bnb−nbPb),(4) designed o ensu e H(∞) commu es wi h nb[i.e. (∞)= (∞)=0]. The low equa ions can be sol ed exac ly, o de by o de in 1/N, and he coe icien s o he inal Hamil onian a e ound o be as ollows: E0(∞)=N(1 −x) 4+5 21 2(1 −3x)+(x)1/2(5) +5x(1 −x) N25x−9 16(x)−1 (x)1/2, (∞)=(x)1/2+x(1 −x) N9x−1 4(x)−2 (x)1/2,(6) V(∞)=x2(1 −x) 4N(x),(7) W(∞)=x(1 −x)(3x−1) 8N(x),(8) 0+ 2 x exci a ion ene gies 10.90.80.70.60.5 2 1.5 1 0.5 0 4+ 1,2+ 2 x exci a ion ene gies 10.90.80.70.60.5 2 1.5 1 0.5 0 x exci a ion ene gies 10.90.80.70.60.5 2 1.5 1 0.5 0 x exci a ion ene gies 10.90.80.70.60.5 2 1.5 1 0.5 0 x exci a ion ene gies 10.90.80.70.60.5 2 1.5 1 0.5 0 2+ 1 x exci a ion ene gies 10.90.80.70.60.5 2 1.5 1 0.5 0 o de (1/N )1 x exci a ion ene gies 10.90.80.70.60.5 2 1.5 1 0.5 0 o de (1/N )0 x exci a ion ene gies 10.90.80.70.60.5 2 1.5 1 0.5 0 FIG. 1. (Colo online) Compa ison be ween analy ical esul s (solid and do ed lines) and he nume ical esul s (ci cle, iangle and squa e symbols) o he i s exci a ion ene gies, wi h N=40. whe e (x)=x(2x−1). One can hen s aigh o wa dly analyze he low-ene gy spec um. The g ound s a e o H(∞)is hes a e|0wi h ze o b bosons, whose ene gy is E0(∞). The i s exci ed s a e is i e old degene a e and co esponds o one quad upole boson b† µ|0, whose exci a ion ene gy is (∞). These a e he i e componen s o he i s 2+exci ed s a e. Fo he wo-boson s a es, hings a e a bi mo e complica ed because o he W e m, which is no diagonal in he basis o s a es {b† µb† ν|0} wi h µ, ν =−2,−1,0,1,2. I is, howe e , easy o see ha P† bPbhas a non i ial ac ion only in he subspace {b† 2b† −2|0,b † 1b† −1|0,1 √2b† 0 2|0}. The co esponding 3×3 ma ix has eigen alues 0 ( wice) and 10. One hus inds ha he e a e 14 degene a e s a es wi h exci a ion ene gy 2[(∞)+V(∞)], and one 0+s a e ha is gi en by 1 √10 P† b|0wi h ene gy 2[(∞)+V(∞)+5W(∞)]. Le us emphasize ha he degene acy is li ed a o de (1/N)1,an e ec missed a he RPA o de . The 14 degene a e s a es a e he nine componen s o he i s exci ed 4+s a e and he i e componen s o he second exci ed 2+s a e. These 4+ 1and 2+ 2 s a es a e degene a e along he whole ansi ion line because o he common O(5) s uc u e. No e also ha , a ixed N, he 0+ 2s a e degene a es wi h he 4+ 1and 2+ 2in he U(5) limi . This low-ene gy spec um is depic ed in Fig. 1 o N=40. The ag eemen be ween nume ics and analy ical esul s is p e y good and has been checked o imp o e when Nge s bigge , as long as one is su icien ly a away om he c i ical poin . Indeed, as can be seen in Eqs. (5)–(8), he 1/N o de co ec ions di e ge a x=1/2. This singula beha io al eady ound in o he models [23–25] is a signa u e o he nonin ege scaling exponen s [30] ha we discuss below. The main s eng h o he CUTs is o allow he compu a ion o expec a ion alues o obse ables as well as ansi ion am- pli udes. Thus, one has o pe o m he uni a y ans o ma ion o he obse ables in which one is in e es ed. In he p esen case, all obse ables can be deduced om he knowledge o he low o he ope a o b† µ(l)=U†(l)b† µU(l). Fo example, he a e age numbe o dbosons in he g ound s a e o he Hamil onian His ound as nd=0|µb† µ(∞)bµ(∞)|0. This quan i y can also be compu ed using he Hellmann-Feynman heo em, 011301-2 RAPID COMMUNICATIONS FINITE-SIZE SCALING EXPONENTS IN THE . . . PHYSICAL REVIEW C 72, 011301(R) (2005) nume ics x B(E2) 10.90.80.70.60.5 600 400 200 0 o de (1/N )1 x B(E2) 10.90.80.70.60.5 600 400 200 0 o de (1/N )0 x B(E2) 10.90.80.70.60.5 600 400 200 0 FIG. 2. (Colo online) Compa ison be ween analy ical esul s (solid do ed lines) and he nume ical esul s (ci cles) o he B(E2) ansi ion p obabili y, wi h N=40. which yields nd=∂/∂y[(1 +y)E0] wi h y=x/(1 −x). One hen ge s he ollowing nd=5 23x−1 2(x)1/2−1 +5x(1 −x)2 16N−7x (x)2+8 (x)3/2.(9) Howe e , his heo em canno be applied o compu e nondiago- nal ma ix elemen s such as ansi ion ampli udes. To illus a e he powe o he CUTs o such a ask, we ocus on he B(E2) ansi ion p obabili y be ween he g ound s a e and he i s exci ed s a e ha is de ined as B(E2) =5|2,0|Q(2) 0|0,0|2in he s anda d |J,Mbasis, wi h Q(2) 0=s†d0+d† 0s. The low equa ions o b† µ(l) can s ill be exac ly in eg a ed ou o de by o de in 1/N and leads o he ollowing: B(E2) =N5x (x)1/2 +5x2−27x2−20x+5 4(x)2+4x−1 (x)3/2.(10) The compa ison be ween analy ical and he nume ical B(E2) ansi ion p obabili ies o a sys em o N=40 bosons, is shown in Fig. 2. As o he exci a ion ene gies, he e a e di e gences in he B(E2) alues a he c i ical poin , al hough hey now appea e en a he RPA o de [28]. Howe e , a ini e N alues no di e gence should appea in he physical magni udes o hei de i a i es wi h espec o he con ol pa ame e x,e ena he c i ical poin . This ob ious ema k allows us o de e mine he non i ial scaling exponen s. Such an analysis was p oposed in Re s. [23–25], and we now b ie ly ecall how i wo ks. The 1/N expansion o any physical quan i y has wo con ibu ions, he egula ( eg) and singula (sing) espec i ely, when xapp oaches he c i ical alue xc=1/2: N(x)= eg N(x)+sing N(x).(11) A close analysis o he singula pa in he icini y o he c i ical poin xcshows ha he singula pa scales as ollows: sing N(x)≃(x)ξ NnF[N(x)3/2],(12) TABLE I. Scaling exponen s o he g ound-s a e ene gy E0, he gap , he numbe o dbosons in he g ound-s a e ndGS, and he B(E2) ansi ion p obabili y. ξ n−(n+2ξ/3) E01/2 0 −1/3 1/2 0 −1/3 nd−1/2 0 1/3 B(E2) −1/2 1 4/3 whe e Fis a unc ion depending on he scaling a iable N(x)3/2only. To compensa e he singula i y coming om (x)ξ(o i s de i a i e), one hus mus ha e F(x)∼x−2ξ/3 so ha sing N(xc)∼N−(n+2ξ/3). In Table I he compu ed scaling exponen s o he low-ene gy physical quan i ies s udied a e summa ized. To check hese esul s, i is impo an o analyze he la ge Nbeha io o N. The e o e, we ha e nume ically sol ed he p oblem by diagonalizing he boson Hamil onian (1) up o N=1000. De ails o his calcula ion will be gi en in a o hcoming publica ion [31]. As shown in Fig. 3, an excellen ag eemen is ound be ween he exponen s p edic ed analy ically and he nume ical esul s. Le us unde line ha he scaling exponen o he g ound- s a e ene gy has been ecen ly ob ained by Rowe e al. [30] by using he collec i e model associa ed o he IBM Hamil onian [32]. This mapping on o a qua ic po en ial also explains why we ound he same ini e-size scaling exponen o he g ound- s a e ene gy and he gap (1/3) in o he simila models [23–25]. Howe e , such an app oach does no allow o simply compu e he ini e Nco ec ions and may no be sui able o ob ain he beha io o obse ables such as B(E2). The CUTs me hod is hus, in his con ex , a e y use ul ool. In he p esen wo k, we ha e exac ly compu ed ini e-size co ec ions beyond he RPA in he symme ic phase o he IBM model. We ha e shown ha he spec al p ope ies a he c i ical poin in he U(5)-O(6) ansi ion ha e well-de ined asymp o ic limi s and we ha e calcula ed he N-dependen scale ac o s. A na u al ex ension o his wo k would be o in es iga e he N1/3 log10(N) 32.521.51 1.2 0.8 0.4 0 -0.4 -0.8 -1.2 N−1/3 log10(N) 32.521.51 1.2 0.8 0.4 0 -0.4 -0.8 -1.2 B(E2) 5N log10(N) 32.521.51 1.2 0.8 0.4 0 -0.4 -0.8 -1.2 nd log10(N) 32.521.51 1.2 0.8 0.4 0 -0.4 -0.8 -1.2 ∆ log10(N) 32.521.51 1.2 0.8 0.4 0 -0.4 -0.8 -1.2 E0 log10(N) 32.521.51 1.2 0.8 0.4 0 -0.4 -0.8 -1.2 FIG. 3. (Colo online) Plo o he singula pa s o E0,,nd, and B(E2)/(5N) a he c i ical poin xc=1/2, in a log10 -log 10 scale. 011301-3 RAPID COMMUNICATIONS DUSUEL, VIDAL, ARIAS, DUKELSKY, AND GARC´ IA-RAMOS PHYSICAL REVIEW C 72, 011301(R) (2005) b oken phase (x<1/2) bu he p esence o Golds one modes in he low-ene gy spec um (a he RPA le el) makes i mo e in ol ed [31]. The s udy o he scaling p ope ies a he c i ical poin o a QPT o a ini e-Npa icle model is o p ime in e es in se e al mesoscopic sys ems such as nuclei, molecules, and o he physical sys ems. The p esen esul s p o ide a ool o ackle such a s udy and o cha ac e ize he app oach o he sys em o he c i ical egions as he numbe o pa icles goes o in ini y. S. Dusuel g a e ully acknowledges inancial suppo o he DFG in SP1073. This wo k has been pa ially suppo ed by he Spanish DGI unde p ojec s BFM2002-03315, BFM2003- 05316-C02-02, and BFM2003-05316. [1] M. Voj a, Rep. P og. Phys. 66, 2069 (2003). [2] M. W. J. Romans, R. A. Duine, S. Sachde , and H. T. C. S oo , Phys. Re . Le . 93, 020405 (2004). [3] F. Iachello and N. V. Zam i , Phys. Re . Le . 92, 212501 (2004). [4] R. F. Cas en and N. V. Zam i , Phys. Re . Le . 85, 3584 (2000). [5] F. Iachello and A. A ima, The In e ac ing Boson Model (Camb idge Uni e si y P ess, Camb idge, 1987). [6] F. Iachello and R. D. Le ine, Algeb aic Theo y o Molecules (Ox o d Uni e si y P ess, Ox o d, 1995). [7] J. Dukelsky and P. Schuck, Phys. Re . Le . 86, 4207 (2001). [8] J. N. Ginocchio and M. W. Ki son, Nucl. Phys. A350,31 (1980). [9] A. E. L. Diepe ink, O. Schol en, and F. Iachello, Phys. Re . Le . 44, 1747 (1980). [10] A. Boh and B. R. Mo elson, Phys. Sc . 22, 468 (1980). [11] D. H. Feng, R. Gilmo e, and S. R. Deans, Phys. Re . C 23, 1254 (1981). [12] A. F ank, Phys. Re . C 39, 652 (1989). [13] E. L´ opez-Mo eno and O. Cas a˜ nos, Phys. Re . C 54, 2374 (1996). [14] J. Jolie, P. Cejna , R. F. Cas en, S. Heinze, A. Linnemann, and V. We ne , Phys. Re . Le . 89, 182502 (2002). [15] J. M. A ias, J. Dukelsky, and J. E. Ga c´ ıa-Ramos, Phys. Re . Le . 91, 162502 (2003). [16] Y. Alhassid and N. Whelan, Phys. Re . Le . 67, 816 (1991). [17] F. Pan and J. P. D aaye , Nucl. Phys. A636, 156 (1997). [18] J. Dukelsky and S. Pi el, Phys. Re . Le . 86, 4791 (2001). [19] F. Wegne , Ann. Phys. 3, 77 (1994). [20] S. D. Głazek and K. G. Wilson, Phys. Re . D 48, 5863 (1993). [21] S. D. Głazek and K. G. Wilson, Phys. Re . D 49, 4214 (1994). [22] J. Dukelsky, G. G. Dussel, R. P. J. Pe azzo, S. L. Reich, and H. M. So ia, Nucl. Phys. A425, 93 (1984). [23] S. Dusuel and J. Vidal, Phys. Re . Le . 93, 237204 (2004). [24] S. Dusuel and J. Vidal, Phys. Re . A 71, 060304 (2005). [25] S. Dusuel and J. Vidal, Phys. Re . B 71, 224420 (2005). [26] T. Hols ein and H. P imako , Phys. Re . 58, 1098 (1940). [27] A. Klein and E. R. Ma shalek, Re . Mod. Phys. 63, 375 (1991). [28] D. J. Rowe, Nucl. Phys. A745, 47 (2004). [29] C. Kne e and G. S. Uh ig, Eu . Phys. J. B 13, 209 (2000). [30] D. J. Rowe, P. S. Tu ne , and G. Rosens eel, Phys. Re . Le . 93, 232502 (2004). [31] S. Dusuel e al., in p epa a ion. [32] J. M. A ias, C. E. Alonso, A. Vi u i, J. E. Ga c´ ıa-Ramos, J. Dukelsky, and A. F ank, Phys. Re . C 68, 041302(R) (2003). 011301-4