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
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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
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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.
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