PHYSICAL REVIEW C 75, 014301 (2007)
Two-le el in e ac ing boson models beyond he mean field
Jos´
eM.A ias,
1Jo ge Dukelsky,2Jos´
e En ique Ga c´
ıa-Ramos,3and Julien Vidal4
1Depa 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
2Ins i u o de Es uc u a de la Ma e ia, CSIC, Se ano 123, E-28006 Mad id, Spain
3Depa amen o de F´
ısica Aplicada, Uni e sidad de Huel a, E-21071 Huel a, Spain
4Labo 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
(Recei ed 31 Augus 2006; published 2 Janua y 2007)
The phase diag am o wo-le el boson Hamil onians, including he in e ac ing boson model (IBM), is s udied
beyond he s anda d mean ield app oxima ion using he Hols ein-P imako mapping. The limi a ions o he
usual in insic s a e (mean ield) o malism conce ning ini e-size e ec s a e poin ed ou . The analy ic esul s
a e compa ed o nume ics ob ained om exac diagonaliza ions. Exci a ion ene gies and occupa ion numbe s a e
s udied in di e en model space egions (Cas en iangle o IBM) and especially a he c i ical poin s.
DOI: 10.1103/PhysRe C.75.014301 PACS numbe (s): 21.60.Fw, 73.43.Nq
I. INTRODUCTION
The concep s o phase ansi ion and c i ical poin s ha e
been de ined, s ic ly speaking, o mac oscopic sys ems.
Howe e , i has been ecen ly sugges ed ha p ecu so s o
phase ansi ions can be obse ed in ini e-size mesoscopic
sys ems [1]. In nuclea physics, he di e en nuclea shapes
and he phase ansi ions be ween hem a e con enien ly
s udied wi hin he in e ac ing boson model (IBM) [2]. This
was ecognized soon a e he in oduc ion o he model
[3–7] bu has been s udied mo e ho oughly in he las ew
yea s [8–18] a e he in oduc ion o he concep o c i ical
poin symme ies [19–21]. Since he IBM was o mula ed
om he beginning in e ms o c ea ion and annihila ion
boson ope a o s, i s geome ic in e p e a ion in e ms o shape
a iables is usually done by in oducing a boson condensa e
wi h wo shape pa ame e s, βand γ(o de pa ame e s) [3,22].
The pa ame e βis ela ed o he axial de o ma ion o he
sys em, while γmeasu es he de ia ion om axial symme y.
The equilib ium shape o he sys em is ob ained by minimizing
he expec a ion alue o he Hamil onian in he in insic s a e.
Shape phase ansi ions a e s udied heo e ically using one o
mo e con ol pa ame e s in he Hamil onian. These con ol
pa ame e s d i e he sys em in di e en phases cha ac e ized
by o de pa ame e s and allows one o s udy in a simple way
phase ansi ions and c i ical poin s in nuclea physics.
The phase diag am o he IBM has been s udied using
se e al app oaches [8–12,14,16–18], and i is well known ha
he dynamical symme y associa ed wi h U(5) co esponds o
a sphe ical shape (β=0), he dynamical symme y SU(3) is
associa ed wi h an axially de o med shape (γ=0,π/3,β =
0), and he dynamical symme y O(6) is ela ed o a γ-uns able
de o med shape (β= 0 and γindependen ). These symme y
limi s a e usually ep esen ed as he e ices o a iangle
(Cas en iangle) [23]. Phase ansi ions be ween hese shapes
ha e been widely s udied, and i is known ha he phase
ansi ion om U(5) o O(6) is second o de , while any o he
ansi ion wi hin he Cas en iangle om a sphe ical o a
de o med shape is i s o de [9,12]. These s udies ha e been
pe o med, as men ioned abo e, by using he in insic s a e
o malism. Howe e , his app oxima e me hod is known o
be only co ec a leading o de in a 1/N expansion, whe e
Nis he numbe o bosons. In his pape , we p esen a
me hod ha goes beyond his o de and compu es ini e-size
co ec ions o se e al spec oscopic obse ables. We s ess ha
1/N co ec ions ob ained wi h he in insic s a e o malism (o
Ha ee-Bose me hod) a e in gene al inco ec , and hey gi e
no in o ma ion on he p ope ini e-size co ec ions.
The pape is o ganized as ollows. Fi s he model
Hamil onian is in oduced in Sec. II. In Sec. III, he Hols ein-
P imako mapping [24] is pe o med leading o a boson
Hamil onian in which we e ain e ms in o de s N,N1/2,
and N0. Then a Bogoliubo ans o ma ion is pe o med o
diagonalize he Hamil onian and o s udy bo h he symme ic
(sphe ical) and he b oken (de o med) phases. All his is done
in gene al o wo-le el boson models in which he lowes
le el is a scala sboson while he uppe le el is an a bi a y
Lboson. The IBM co esponds o he pa icula case L=2
(dµbosons). We also p esen esul s o he case L=0asan
illus a ion o he gene al me hod. In Sec. IV, we compa e
he analy ical esul s wi h exac nume ical diagonaliza ions
o di e en pa hs along he Cas en iangle. Finally, Sec. V
p esen s he summa y and conclusions.
II. THE MODEL
As al eady no ed in Re . [14], he expe imen al explo a ion
o he shape ansi ion and c i ical poin s in nuclei is di icul
because o he lack o a con inuous con ol pa ame e .
Howe e , in heo e ical s udies, his limi a ion is o e come
by using a Hamil onian w i en in e ms o one o mo e
con ol pa ame e s ha can a y con inuously. In his wo k,
we conside a wo-le el boson model in which he lowes
le el is cha ac e ized by a ze o angula momen um (sboson),
while he uppe le el has an a bi a y angula momen um L.
The Hamil onian p oposed is a gene aliza ion o he IBM
0556-2813/2007/75(1)/014301(10) 014301-1 ©2007 The Ame ican Physical Socie y
ARIAS, DUKELSKY, GARC´
IA-RAMOS, AND VIDAL PHYSICAL REVIEW C 75, 014301 (2007)
consis en -Q o malism (CQF) [25], which depends on wo
con ol pa ame e s xand χ,
H=xn
L−1−x
NQχ·Qχ,(1)
whe e nL=µL†
µLµis he ope a o o he numbe o
bosons in he uppe le el, Nis he o al numbe o bosons,
he symbol ·s ands o he scala p oduc de ined as a·b=
+L
µ=−L(−1)µaµb−µ, and Qχis a mul ipole ope a o w i en
as
Qχ
µ=(s†˜
L+L†s)(L)
µ+χ[L†×˜
L](L)
µ,(2)
whe e ˜
Lµ=(−1)µL−µ.Fo L=2(dbosons), he
Hamil onian (1) is he well-known CQF Hamil onian o IBM.
Though i is no he mos gene al IBM Hamil onian, i cap u es
he mos impo an low-ene gy p ope ies o a wide ange o
nuclei [26–28]. In pa icula , i is gene al enough o desc ibe
di e en nuclea phases and quan um phase ansi ions, and i
has been used o ha pu pose a he mean ield le el [9,10,12].
The Hamil onian (1) comp ises di e en models depending
on he alue o L. Fo ins ance, o L=1 he Hamil onian
is app op ia e o s udying he phase diag am o he ib on
model [29] o in e es in molecula physics.
III. MEAN FIELD AND BEYOND
The usual way o ge ing he phase diag am o he model (1)
is o in oduce shape a iables. This can be done by conside ing
he in insic s a e o malism, also called he Ha ee-Bose
app oxima ion [3,5,22,30]. In his app oach, he g ound s a e
is a a ia ional s a e buil ou o a condensa e o “d essed”
bosons, which a e independen bosons mo ing in he a e age
nuclea ield. Fo L=2, hese bosons a e de ined as
†
c=1
1+β2s†+βcos γd
†
0+1
√2βsin γ(d†
2+d†
−2)
,
(3)
and he Nboson condensa e is
|c= 1
√N!(†
c)N|0.(4)
The a ia ional a iables βand γa e he o de pa ame e s
o he sys em, and hei equilib ium alues a e ixed by
minimizing he expec a ion alue o he ene gy. The exp ession
o his ene gy can be ound in many e e ences [3,22,30,31]
and can be w i en schema ically as
E(N,β,γ,x,χ)=NF(1)(N,β,γ,x,χ)
+(N−1)F(2)(N,β,γ,x,χ),(5)
whe e F(1)(N,β,γ,x,χ) is he ma ix elemen o he one-
body ope a o s di ided by N, and F(2)(N,β,γ,x,χ)is he
ma ix elemen o he wo-body ope a o s di ided by N−1.
No e ha he e is no N2dependence in he wo-body ope a o
because o he de ini ion o he Hamil onian. Ac ually, he
only ele an con ibu ion is he leading one (o de N), since
he nex one (N0, o ins ance) a e incomple e as explained
below.
Fo he s anda d IBM Hamil onian (L=2), wi h an
a ac i e quad upole in e ac ion, he nucleus always becomes
axially de o med, ei he p ola e (γ=0) o χ<0 o obla e
(γ=π/3) o χ>0. As a consequence, he pa ame e γ
can be inco po a ed in he alue o β.β>0 co esponds o
γ=0, while nega i e βimplies γ=π/3. In he case χ=0,
he nucleus becomes γuns able; i.e., he ene gy is independen
o γ.
In his amewo k, one-phonon exci a ions abo e he
g ound s a e a e cons uc ed by di ec ly eplacing in he g ound
s a e (4) a condensa e boson by an exci ed boson, his p oce-
du e is known as he Tamm-Danco app oxima ion (TDA)
me hod, o by including g ound s a e luc ua ions, which is
he andom phase app oxima ion (RPA) me hod [5,31,32]. Fo
L=2, he e a e i e exci ed phonons ha a e cha ac e ized
by hei angula momen um p ojec ion Kand can be labeled
as βexci a ion wi h K=0,γ exci a ions wi h K=±2, and
inally wo K=±1 exci a ions. No e ha no all he exci ed
phonons a e always physical, some o hem become spu ious
Golds one bosons associa ed wi h b oken symme ies. This is
he case o axially de o med nuclei; he K=±1 exci a ions
a e spu ious Golds one bosons because he s a e cons uc ed
wi h his exci a ion co esponds o an O(3) o a ion o he
whole sys em. In he case o γ-uns able nuclei, he K=
±2 exci a ions also become Golds one bosons and a e ela ed
o O(5) o a ions o he g ound s a e. In he case o L=0,
only a K=0 exci a ion exis s, and i is di ec ly ela ed, as we
will see, wi h he βband o he IBM [33].
The mean ield desc ip ion o he g ound s a e ene gy jus
men ioned is only alid a o de N. The i s quan um co ec-
ions can be ob ained wi hin he RPA o malism. Al e na i ely,
he Hols ein-P imako expansion [24] o e s a simple and
na u al expansion in powe s o 1/N. The ad an ages o his
ans o ma ion a e ha i is He mi ian, p ese es he boson
commu a ion ela ion, and p o ides a co ec expansion in
powe s o N. Fu he mo e, i s leading o de coincides wi h
he mean ield con ibu ion.
The Hols ein-P imako expansion elimina es he sboson
ans o ming he bilinea boson ope a o s in he ollowing way:
L†
µLν=b†
µbν,(6)
L†
µs=N1/2b†
µ(1 −nb/N)1/2=(s†Lµ)†,(7)
s†s=N−nb,(8)
whe e he bbosons sa is y [bµ,b
†
ν]=δµ,ν. The mapping
ul ills he commu a ion ela ions a each o de in Nin he
Taylo expansion o he squa e oo .
We nex in oduce he cbosons h ough a shi ans o ma-
ion
b†
µ=√Nλ∗
µ+c†
µ,(9)
whe e he λµa e complex numbe s ha o m a (2L+1)-
dimensional ec o . This shi allows o a mac oscopic
occupa ion numbe nb. Thus, i allows one o conside a
he same ime he sphe ical phase, se ing λµ=0 o all µ,
and he de o med phase, λµ= 0. In his la e si ua ion, we
shall only conside he case λ0= 0 wi hou loss o gene ali y.
014301-2
TWO-LEVEL INTERACTING BOSON MODELS BEYOND THE ... PHYSICAL REVIEW C 75, 014301 (2007)
The Hamil onian hen eads
H=N1λ2
05x−4−4(x−1)λ2
0+(x−1)χα(L)
0,0λ0
×41−λ2
01/2+χα(L)
0,0λ0+N1/2λ0(c†
0+c0)
×5x−4−8λ2
0(x−1) +2(x−1)χα(L)
0,0λ0
×−4λ2
0+3
1−λ2
01/2+χα(L)
0,0λ0
+N03x−2−6λ2
0(x−1)nc+(x−1)(2L+1)
−(2L+3)λ2
0+1−λ2
0(P†
c+Pc)
−4λ2
0c†2
0+2c†
0c0+c2
0+2χ(x−1)
×λ0(1 −λ2
0)1/2+L

µ=−L
α(L)
0,µ +2c†
µcµ(−1)µα(L)
µ,−µ
+α(L)
0,µ+(−1)µα(L)
µ,0(c†
µc†
−µ+cµc−µ
−λ3
0α(L)
0,0
21−λ2
01/22+3c†2
0+2c†
0c0+c2
0+2nc
−λ5
0α(L)
0,0
41−λ2
03/21+c†2
0+2c†
0c0+c2
0
+χ2(x−1)λ2
01++L

µ=−L
2c†
µcµ(−1)µα(L)
0,0α(L)
µ,−µ
+α(L)
0,µ
2+(−1)µα(L)
0,µ
2(c†
µc†
−µ+cµc−µ)
+O(1/√N),(10)
=H1+H1/2+H0+O(1/√N),(11)
whe e α(L)
µ,ν =L, µ;Lν|L, µ +νand P†
c=c†·c†=(Pc)†.
The e m o o de N(H1) is exac ly he mean ield ene gy.
Se ing λ=β/1+β2,one ge s
E(N,β,x,y)=Nβ2
(1 +β2)2[5x−4+xβ2
+βy(x−1)(4 +βy)],(12)
whe e y=χα(L)
0,0. In he case o L=2, Eq. (12) educes o
he IBM g ound s a e ene gy. No e ha H1only depends on L
h ough he Clebsch-Go dan coe icien L, µ;Lν|L, µ +ν,
al hough his dependence can be abso bed in he pa ame e y.
H1p o ides he mean ield ene gy and he e o e he
equilib ium alues o he o de pa ame e . Figu e 1depic s he
phase diag am co esponding o H1. Fo gi en pa ame e s x
and χ(y), he i s s ep consis s in minimizing H1wi h espec
o β(λ), ge ing he equilib ium alue β0(λ0). The s udy o
hese minima has been shown in se e al publica ions [4,7], bu
o comple eness we summa ize he e he main ea u es:
(i) β=0 is always a s a iona y poin . Fo x<4/5,β =
0 is a maximum, while o x>4/5,i becomes a
minimum. In he case o x=4/5,β =0 is an in lec ion
S
D
O(6)
SU(3)U(5)
φ
ρ
an ispinodalc i icalspinodal
FIG. 1. Quali a i e phase diag am o he Hamil onian (1)and
L=2. Inse s show ypical ene gy su aces s de o ma ion pa ame e
βin each o he phases and a he phase bo de s. Con ol a iables
a e de ined as ρ=1−xand φ=2π
3√7(√7
2+χ).
poin . x=4/5 is he poin a which a minimum a β=0
s a s o de elop and de ines he an ispinodal line.
(ii) Fo χ= 0(y= 0), he e exis s a egion, whe e wo
minima, one sphe ical and one de o med, coexis . This
egion is de ined by he poin a which he β=0
minimum appea s (an ispinodal poin ) and he poin
a which he β= 0 minimum appea s (spinodal poin ).
The spinodal line is de ined by he implici equa ion
3x
3x−4=A
B1−1+B
A3
2,(13)
whe e A=[4 −3x+2(x−1) y2]2and B=
36 y2(x−1)2. The SU(3) case, χ=−
√7/2, p o ides
x≃0.820361.
(iii) In he coexis ence egion, he c i ical poin is de ined
as he si ua ion in which bo h minima (sphe ical and
de o med) a e degene a e. A he c i ical poin , he
wo degene a ed minima a e a β0=0 and β0=
α(L)
0,0χ/2(β0=y/2) and hei ene gy is equal o ze o.
The c i ical poin line can be calcula ed o be
xc=4+y2
5+y2=4+χ2L, 0; L0|L, 02
5+χ2L, 0; L0|L, 02.(14)
In he case o L=2,
xc=4+2
7χ2
5+2
7χ2,(15)
being in he SU(3) limi (χ=−
√7/2),x
c=9/11.
(i ) Acco ding o he p e ious analysis, a i s -o de phase
ansi ion appea s o χ= 0(y= 0); while o χ=
0(y=0), an isola ed poin o second-o de phase an-
si ion occu s a x=4/5. In his las case, an ispinodal,
spinodal, and c i ical poin s collapse in o a single
poin .
014301-3
ARIAS, DUKELSKY, GARC´
IA-RAMOS, AND VIDAL PHYSICAL REVIEW C 75, 014301 (2007)
The subs i u ion o β0(λ0) in he Hamil onian (11) implies
ha he e m o o de N1/2 anishes because i is p opo ional
o he de i a i e o H1wi h espec o λ. Mo e p ecisely,
one has ha ∂H1/∂λ =2H1/2. The i s quan um co ec ion
comes om he N0 e m which is a simple quad a ic o m in
he c-boson ope a o s. I can hus be diagonalized h ough a
Bogoliubo ans o ma ion. This ans o ma ion depends on
he phase, sphe ical o de o med, and in he nex subsec ions
bo h will be ea ed sepa a ely.
A. Bogoliubo ans o ma ion in he sphe ical phase
In he sphe ical phase, β=0(λµ=0 o all µ) and x>
4/5. In his case, he Hamil onian (11) eads as
H=(3x−2)nc+(x−1)[(2L+1) +(P†
c+Pc)] +O(1/N),
(16)
which is s aigh o wa dly diagonalized ia a Bogoliubo
ans o ma ion
c†
µ=uµξ†
µ+ µ˜
ξµ,(17)
˜
cµ=uµ˜
ξµ+ µξ†
µ,
whe e he coe icien s e i y u2
µ− 2
µ=1, wi h uµ=u−µand
µ= −µ. The phases o he coe icien s a e chosen so as o
minimize he mean ield ene gy, leading o
H=2L+1
2[−x+(x)1/2]+nξ(x)1/2+O(1/N),(18)
whe e we ha e in oduced (x)=x(5x−4),and nξis he
numbe ope a o o ξbosons. No e ha in he sphe ical phase,
he mean ield ene gy is equal o ze o. In his phase, which is
only de ined o 4/5⩽x⩽1, he spec um is, a his o de ,
independen o χ(y) and has a i ial dependence on L.As
shown in Re . [33] o L=0, one has o diagonalize Ha nex
o de (1/N) o see he ole played by his pa ame e .
In his phase he e exis s a (2L+1) imes degene a ed
phonon (5 in he IBM case), ξ. The Hamil onian is comple ely
ha monic, and he e o e he wo phonon exci a ion ene gy is
exac ly wice he one phonon exci a ion ene gy.
Ano he obse able o in e es ha can be calcula ed easily
is he numbe o Lbosons in each s a e. Fo he calcula ion
o such an obse able, he Hellmann-Feynman heo em can
be used. I es ablishes ha he de i a i e o he eigen alue
o a gi en ope a o , e.g., he Hamil onian, is equal o he
expec a ion alue o he de i a i e o his ope a o wi h he
co esponding eigen unc ion. This leads o
nL= ∂
∂θ [(1 +θ)H],(19)
whe e θ=x
1−x. In his case, he con ibu ion om he mean
ield is ze o, and he i s non anishing con ibu ion comes
om he e m p opo ional o N0in he ene gy. The e o e,
nLgs =2L+1
23x−2
(x)−1+O(1/N),(20)
nLpξ =nLgs +p3x−2
(x)+O(1/N),(21)
whe e nLgs s ands o he expec a ion alue o he numbe
o Lbosons in he g ound s a e, pis he numbe o exci ed
ξbosons and nLpξ s ands o he expec a ion alue o he
numbe o Lbosons in he s a e wi h pexci ed ξbosons. The
N0co ec ion is singula a x=4/5 as al eady no ed in simila
models [34,35].
No e ha he e we ha e chosen β0as an o de pa ame e ,
bu we could ha e aken nLgs equi alen ly. Indeed, in he
he modynamic limi , his quan i y is only non anishing in he
de o med phase, as we shall now see.
B. Bogoliubo ans o ma ion in he de o med phase
In he de o med phase, whe e β0= 0(λ0= 0), he si ua ion
is mo e complica ed and s ongly depends on L.In he
ollowing, we will discuss sepa a ely he wo cases L=
0,2,bu we emphasize ha he o m (11) o he expanded
Hamil onian allows he s udy o a bi a y L.
1. The case L =0
The case L=0 has ecen ly a ac ed much a en ion
because a he mean ield le el, i ep oduces exac ly he
IBM phase diag am (al hough, o cou se, i does no include
K=2 exci a ions). In Re . [33], we compu ed he ini e-size
co ec ions up o 1/N o de in he sphe ical phase. He e, we
will now ea he de o med phase a o de (1/N)0.A his
o de , o L=0, he Hamil onian is easily diagonalized ia a
Bogoliubo ans o ma ion o e he cscala boson, and one
ge s
H=E(x,y,β0)+1
21+β2
0−x+(7x−8)β2
0
+2(x−1)yβ0−2−yβ0+2β2
0+(x,y,β0)1/2
2
+nξ(x,y,β0)1/2+O(1/N),(22)
whe e E(x,y,β0) is gi en by Eq. (12),
(x,y,β0)=x−(3x−4)β2
0+2(x−1)yβ02+yβ0−β2
05x−4−(19x−20)β2
0+2(x−1)yβ06+3yβ0−7β2
0−β4
0
1+β2
02
(23)
014301-4
TWO-LEVEL INTERACTING BOSON MODELS BEYOND THE ... PHYSICAL REVIEW C 75, 014301 (2007)
and y=χ. In his case, one has a single phonon exci a ion
wi h K=0. Fo β0=0, one eco e s exp ession (18) se ing
L=0.
Rega ding he expec a ion alue o he numbe o
Lbosons, i can be calcula ed as be o e h ough he Hellmann-
Feynman heo em [see Eq. (19)]. No e ha in he de o med
phase, a con ibu ion p opo ional o Ncomes om he mean
ield ene gy. Mo e p ecisely, one has
nLgs =Nβ2
0
1+β2
0+(1 −x)2∂
∂x 1
21+β2
0(1 −x)
×−x+(7x−8)β2
0+2(x−1)yβ0
×−2−yβ0+2β2
0+(x,y,β0)1/2
2(1 −x),(24)
nLpξ =nLgs +p(1 −x)2∂
∂x (x,y,β0)1/2
1−x,(25)
whe e we used he same no a ion as in Eqs. (20) and (21), and
pis he numbe o exci ed ξbosons.
2. The case L =2
In his sec ion, we will ocus on he IBM case, i.e.,
L=2. Fo a bi a y L= 0, he Hamil onian (11)mus be
diagonalized sepa a ely o each alue o µ. Indeed, one has
H0=C++2

µ=−2
Hµ,(26)
whe e Cis a cons an and Hµ=H−µ. As can be seen in
Eq. (11), Hµdepends no only on µbu also on he angula
momen um L ia he Clebsch-Go dan coe icien s α(L)
µ,ν.
We diagonalize sepa a ely he modes µ=0,µ=±1,
and µ=±2 which co espond o he βphonon (K=0),
a Golds one phonon (K=1 wo- old degene a e), and he
γphonon (K=2 wo- old degene a e), espec i ely. A e
he diagonaliza ion ia a Bogoliubo ans o ma ion, he ull
diagonal Hamil onian in he de o med phase eads
H=E(x,y,β0)+1
2(1 +β2
0)−5x+(19x−24)β2
0
+12(x−1)yβ3
0++2

µ=−2
µ(x,y,β0)1/2
2
+nξµ1/2
µ(x,y,β0)+O(1/N),(27)
wi h
0(x,y,β0)
=x−(3x−4)β2
0+2(x−1)yβ02+yβ0−β2
05x−4−(19x−20)β2
0+2(x−1)yβ06+3yβ0−7β2
0−β4
0
1+β2
02,(28)
±1(x,y,β0)=x−(3x−4)β2
0+(x−1)yβ02+yβ0−2β2
05x−4−(3x−4)β2
0+2(x−1)yβ03+yβ0−β2
0
1+β2
02,
(29)
±2(x,y,β0)=x−(3x−4)β2
0−2(x−1)yβ02+yβ0+β2
05x−4−(3x−4)β2
0+2(x−1)yβ0−6+yβ0−β2
0
1+β2
02,
(30)
whe e y=−
√2/7χ.Fo β0=0, he symme y be ween
modes is es o ed [µ(x,y,0) =(x)] and one eco e s he
exp ession (18) wi h L=2.
Fo β0= 0, he phonon exci a ions depend on µ.
The exci a ion o µ=0 bosons, which co esponds o
βbosons, is he same as in he L=0 case, namely,
014301-5

ARIAS, DUKELSKY, GARC´
IA-RAMOS, AND VIDAL PHYSICAL REVIEW C 75, 014301 (2007)
0(x,y,β0)=(x,y,β0). In addi ion, he exci a ion ene gy
o µ=±1 modes anishes, since o β0= 0, one has
±1(x,y,β0)∝∂E(x,y,β)
∂β |β0=0. This is in ag eemen wi h
he ac ha he µ=±1 exci a ion co esponds o a o a ion
o he g ound s a e, i.e., o a Golds one phonon. Finally, he
µ=±2 exci a ion co esponds o a γexci a ion, which is
wo- old degene a e.
Fo he calcula ion o he expec a ion alue o he numbe
o dbosons, we p oceed as in he L=0 case and ob ain
nLgs =Nβ2
0
1+β2
0+(1 −x)2∂
∂x 

1
21+β2
0(1 −x)
×−5x+(19x−24)β2
0+12(x−1)yβ3
0
+
µ=+2

µ=−2
µ(x,y,β0)1/2
2(1 −x)
(31)
nLpξµ=nLgs +p(1 −x)2∂
∂x µ(x,y,β0)1/2
1−x,(32)
whe e, again, he same no a ion as in Eqs. (20) and (21)is
used, and pco esponds o he numbe o ξµexci ed bosons
(βo γbosons in he L=2 case).
IV. NUMERICAL RESULTS
In his sec ion, we compa e he analy ical esul s ob ained
in p e ious sec ions wi h nume ical calcula ions. No e ha o
cla i y only he i s 0+and 2+s a es a e plo ed as membe s
o he di e en bands.
A. The case L=0
In his case, we pe o m he nume ical calcula ions using
he echnique p esen ed in Re s. [33,36]. I allows us o easily
deal wi h a la ge numbe o bosons, up o a ew housands. One
can each such a numbe o bosons because o he unde lying
O(5) symme y which allows he use o a senio i y scheme
ha educes conside ably he dimension o he ma ices o be
diagonalized.
In Fig. 2we compa e he analy ical wi h he nume ical
exci a ion ene gies o a la ge numbe o bosons, N=5000
(all he ollowing calcula ions o L=0 a e pe o med o
N=5000), and χ=−
√7/2. No e ha single and wo phonon
exci a ions a e equally well desc ibed. The le pa o he
igu e co esponds o he de o med phase; he igh pa o
he sphe ical one. In his case, a i s -o de phase ansi ion
appea s, and a he c i ical poin xc=9/11, he g ound s a e
and he i s exci ed s a e a e degene a ed, one co esponding
o he sphe ical and he o he o he de o med g ound s a e.
In Fig. 3we epea he same compa ison o he case χ=0.
In his case, a second-o de phase ansi ion appea s. The
ene gy o he i s exci ed s a e becomes ze o in he de o med
phase and in he sphe ical phase a he c i ical poin , xc=4/5.
A his poin , ega ding he analy ical calcula ions, he de-
o med βexci a ion ans o ms in o he sphe ical one phonon
exci a ion. Howe e , conce ning he nume ical esul s, he 0+
2
0.78 0.8 0.82 0.84
x
0
1
2
Exci a ion ene gy
0+
2
0+
3
1 ph
2 ph
FIG. 2. (Colo online) Beha io o one and wo phonon exci a ion
ene gies, in a bi a y uni s, o L=0 as a unc ion o xnea he
c i ical poin o χ=−
√7/2. Lines a e analy ical esul s; do s,
nume ical calcula ions.
s a e, iden i ied wi h he βband in he de o med sec o , ans-
o ms in o he wo phonon exci a ion in he sphe ical sec o .
No e ha al hough in he sphe ical phase he N0co ec ion
is independen o χ(y), he e is a no iceable di e ence be ween
Figs. 2and 3because o each x alue, only he phase ha
co esponds o he lowes mean ield ene gy is plo ed. The
sphe ical phase only becomes he mos s able om x>9/11
on o χ=−
√7/2; while in he case χ=0,i is om x>4/5
on. No e also ha in he de o med phase o χ=0, he e
appea degene a e double s o le els because o he ex a pa i y
symme y in he Hamil onian in his case. Thus, he βband is
connec ed o wo and h ee phonon exci a ion in he sphe ical
phase, while he ββ band is ela ed o he ou and i e (no
shown in Fig. 3) phonon exci a ion in he sphe ical phase.
Fo he numbe o bosons, we compa e he analy ical
o mulas wi h he nume ical esul s o he case o χ=−
√7/2
0.79 0.795 0.8 0.805 0.81
x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Exci a ion ene gy
0+
1 ph
2 ph
FIG. 3. (Colo online) Same as Fig. 2,bu o χ=0. Exci ed
phonon in he de o med egion (x<4/5) is equi alen o he β
exci a ion in IBM. The lowes 0+co esponds o 0+
2.
014301-6
TWO-LEVEL INTERACTING BOSON MODELS BEYOND THE ... PHYSICAL REVIEW C 75, 014301 (2007)
-0.001
0
0.001
<n0>/ N - <n0 >m / N
Num.
Analy .
0 0.2 0.4 0.6 0.8 1
x
-0.001
0
Num.
Analy .
FIG. 4. Va ia ion o (nL=0gs −nL=0m
gs )/N as a unc ion o x
nea he c i ical poin o L=0. Full line is o analy ical and ci cles
o nume ical esul s. Uppe igu e co esponds o χ=0 and lowe
o χ=−
√7/2.
and χ=0inFig.4. In pa icula , we a e in e es ed in he
s udy o he N0co ec ions o he g ound s a e; he e o e,
we sub ac he mean ield con ibu ion, n0m
gs /N, om bo h
analy ical and nume ical esul s. As expec ed, we obse e how
he N0co ec ion imp o es he desc ip ion o n0gs, especially
nea he c i ical poin .
B. The case L=2
Fo L=2, he nume ical calcula ions ha e been ca ied
ou wi h an IBM code [37] which has been modi ied o allow
calcula ions up o N=100 bosons. All nume ical calcula ions
o IBM p esen ed below a e pe o med o N=100.
Fo L=2, he case χ=0 educes o he L=0 si ua ion
al eady discussed. In pa icula , he analy ical g ound s a e
ene gy is he same in bo h cases, al hough he N0co ec ion
di e s; he e exis s only one kind o exci a ion: he β, while he
µ=±1 and γexci a ions become spu ious Golds one bosons.
The βexci a ion ene gy is equi alen o Eq. (23). On he exac
diagonaliza ion side, he Hamil onian (1) can be ew i en in
0 0.2 0.4 0.6 0.8 1
x
0
2
4
6
8
10
12
Exci a ion ene gy
β
γ
ββ
γγ
βγ
1 ph
2 ph
FIG. 5. (Colo online) Exci a ion ene gies (analy ical), in a bi-
a y uni s, o one and wo phonon s a es as a unc ion o x o
L=2andχ=−
√7/2.
0.4 0.45 0.5 0.55 0.6
x
2
3
4
5
6
7
8
Exci a ion ene gy
β
γ
ββ
γγ
βγ
0+
2+
FIG. 6. (Colo online) Exci a ion ene gies (analy ical and nume -
ical), in a bi a y uni s, o one and wo phonon s a es as a unc ion o x
o L=2andχ=−
√7/2 in he de o med phase. Lines co espond
o analy ical, and do s o nume ical esul s. The lowes 0+s a e
co esponds o 0+
2,and he 2+s a es a e, espec i ely, 2+
3(lowes )
and 2+
5(highes ).
e ms o he gene a o s o an SU(1,1) algeb a [33,36]in he
same way as ha in he L=0 case. Consequen ly, in his
sec ion we will only conside L=2 wi h χ= 0. Any χ alue
can be analyzed, bu , as an illus a ion, he e we will p esen
esul s o he case χ=−
√7/2 ha gi es he U(5)-SU(3) leg
in he Cas en iangle.
Fi s , we plo he analy ical esul s co esponding o one
and wo phonon exci a ions (Fig. 5). In he de o med phase,
he bosons a e βand γexci a ions, while in he sphe ical phase
hey a e sphe ical ha monic phonons. A he c i ical poin , he
βand he ββ bands ans o m in o one and wo phonon bands,
0.75 0.8 0.85
x
0
1
2
3
Exci a ion ene gy
β
γ
ββ
γγ
βγ
1 ph
2 ph
0+
2+
FIG. 7. (Colo online) Exci a ion ene gies (analy ical and nume -
ical), in a bi a y uni s, o one and wo phonon s a es as a unc ion o
x o L=2andχ=−
√7/2 in he egion a ound he c i ical poin .
Do s co espond o nume ical esul s. The lowes 0+s a e co esponds
o 0+
2and he lowes 2+s a e co esponds o 2+
1.
014301-7
ARIAS, DUKELSKY, GARC´
IA-RAMOS, AND VIDAL PHYSICAL REVIEW C 75, 014301 (2007)
0.85 0.9 0.95 1
x
0
1
2
Exci a ion ene gy
1 ph
2 ph
0+
2
2+
1
FIG. 8. (Colo online) Exci a ion ene gies o one and wo phonon
s a es, in a bi a y uni s, as a unc ion o x o L=2andχ=−
√7/2.
Lines co espond o analy ical and do s o nume ical esul s.
espec i ely. Howe e , he γ,βγ, and γγ bands appa en ly
disappea when en e ing he sphe ical phase. Indeed, ha
disappea ance happens because βand γexci a ions become
degene a e o β=0. The sphe ical phonon exci a ion is a i e
degene a e exci a ion whe e he de o med βand γexci a ions
collapse oge he wi h he Golds one boson wi h p ojec ion
±1 (which is a ze o ene gy in he de o med phase).
In o de o compa e analy ical and nume ical esul s, we
will spli he analysis in o h ee di e en egions: de o med
phase (Fig. 6), c i ical egion (Fig. 7), and sphe ical phase
(Fig. 8). The ha monic cha ac e o he esul s is obse ed in
all hese plo s.
1. De o med phase
In he de o med phase (Fig. 6), one and wo phonon
exci a ions a e clea ly sepa a ed in ene gy. No e ha he exci-
a ion ene gy o he γband is highe han he co esponding
one o he βband, al hough o x=0 [SU(3) limi ] hey
a e degene a ed. Also no e ha he γγ exci a ion ca ies
he angula momen um p ojec ions K=0,±4, which in his
app oach a e degene a ed.
The co espondence be ween nume ical and analy ical
s a es is as ollows: he βband is iden i ied wi h 0+
2,γ wi h
0.4 0.45 0.5 0.55 0.6
x
0.52
0.54
0.56
0.58
0.6
0.62
<nd>/N
0+
2
2+
3
β
γ
FIG. 9. (Colo online) nd/N as a unc ion o x o L=2and
χ=−
√7/2, in he de o med phase o one phonon s a es. Lines
co espond o analy ical and do s o nume ical esul s.
0.7956 0.7958 0.796 0.7962
x
1.18
1.19
1.2
Exci a ion ene gy
2+
3
2+
4
FIG. 10. (Colo online) Exci a ion ene gies (nume ical) o 2+
3and
2+
4s a es a he egion o closes app oach (see ex ) o L=2and
χ=−
√7/2.
2+
3,ββ wi h 0+
3,βγ wi h 2+
5,and γγ wi h 0+
4. No e ha he
s a e 2+
2belongs o he βband, while 2+
4is wi h he ββ band.
The o e all ag eemen be ween analy ical and nume ical
esul s is sa is ac o y and imp o es he desc ip ion gi en in
Re . [31] o single and double phonon exci a ions, al hough
in he p esen app oach, no mixing appea s among he di e en
kinds o exci a ions.
The a e age numbe o dbosons in he de o med phase,
no malized o he o al numbe o bosons, is depic ed in
Fig. 9 o one phonon s a es ( he esul s o wo phonon s a es
a e no p esen ed o cla i y). I can be obse ed ha a smoo h
dec ease o ndoccu s when xinc eases, as is expec ed when
app oaching he sphe ical phase.
2. C i ical a ea
The compa ison a ound he c i ical a ea (Fig. 7) becomes
complica ed because one and wo phonon s a es ha e compa-
able ene gies and he e appea s an in e change o cha ac e
be ween s a es. Fo example, a he c i ical poin , he ββ is a
lowe ene gy han he γexci a ion.
S a ing a x=0.75, he co espondence be ween analy -
ical and nume ical s a es is simila o he one gi en in he
p eceding sec ion; bu al eady a x=0.8, di e en s a es
in e change hei cha ac e . The co espondence be ween
s a es is p esen ed in Table I. Clea ly, an in e change o cha -
TABLE I. Co espondence be ween analy ical and nume ical
s a es o h ee alues o x:x=0.75 de o med phase, x=9/11
c i ical poin , and x=0.85 sphe ical phase. Only 0+and 2+s a es
a e indica ed explici ly.
x=0.75 x=9/11 x=0.85
β0+
2,2+
20+
2,2+
2
γ2+
32+
4
ββ 0+
3,2+
40+
3,2+
3
βγ 2+
52+
6
γγ 0+
50+
7
1 phonon 2+
1
2 phonons 0+
2,2+
2
014301-8
TWO-LEVEL INTERACTING BOSON MODELS BEYOND THE ... PHYSICAL REVIEW C 75, 014301 (2007)
0.7956 0.7958 0.796 0.7962
x
0.18
0.19
0.2
0.21
0.22
0.23
<nd>/ N
2+
3
2+
4
FIG. 11. (Colo online) nd/N (nume ical) o 2+
3and 2+
4s a es a
he egion o closes app oach (see ex ) o L=2andχ=−
√7/2.
ac e exis s among he s a es co esponding o he γ,ββ,βγ,
and γγ bands.
An in e es ing ques ion ha a ises is whe he he in e -
change o cha ac e is due ei he o le el c ossing o o le el
epulsion. We ha e o ake in o accoun ha he ansi ion
be ween SU(3) and U(5) is no an in eg able pa h [12]; i.e., a
comple e se o mu ually commu ing He mi ian ope a o s does
no exis . This implies ha c ossings a e o bidden and only
epulsion is allowed. In pa icula , in he he modynamic limi ,
he epulsion becomes an ic ossing, i.e., in ini e epulsion.
In Fig. 10, we show a closeup o one appa en c ossing in
Fig. 7be ween 2+s a es in he egion a ound x=0.796; i is
clea ly seen ha he le els indeed epel each o he as expec ed.
To illus a e his esul and show how he wo in ol ed
le els in e change hei cha ac e , we p esen in Fig. 11
he expec a ion alue o he dboson numbe in bo h s a es. I
is clea ly obse ed ha he s a es in e change hei p ope ies
a he poin o closes app oach.
The a e age numbe o dbosons, no malized o he o al
numbe o bosons, in he egion a ound he c i ical poin
is depic ed in Fig. 12 o he g ound s a e (le panel) and
o he βband ( igh panel). One impo an ea u e is he
discon inui y appea ing a xc=9/11 due o he exis ence
o a i s -o de phase ansi ion. In he e olu ion o he β
band, a kink appea s in he nume ical esul s a he c i ical
poin . This beha io a he c i ical poin has been al eady
obse ed o o he obse ables such as isome shi s [1],
0.775 0.8 0.825 0.85
x
02
+
β
2 ph
0.75 0.775 0.8 0.825
x
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
<nd> / N
0+
1
gs analy .
FIG. 12. (Colo online) nd/N as a unc ion o x, o L=2
and χ=−
√7/2, a he c i ical a ea o one phonon s a es. Lines
co espond o analy ical and do s o nume ical esul s.
0.85 0.9 0.95 1
x
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
<nd>/N
02
+
21
+
1 ph
2 ph
FIG. 13. (Colo online) nd/N as a unc ion o x o L=2and
χ=−
√7/2, in he sphe ical phase o one and wo phonon s a es.
Lines co espond o analy ical and do s o nume ical esul s.
de i a i es o he a ios o 4+
1/2+
1exci a ion ene gies [38], o
B(E2; 4+
1→2+
1)/B(E2; 2+
1→0+
1)[15]. Also no e ha he
0+
2s a e ans o ms in o a wo phonon band when passing o
he sphe ical phase.
3. Sphe ical phase
The las egion o in e es is he sphe ical phase (Fig. 8).
He e, he e exis s a i e degene a ed phonon exci a ion. The
co espondence be ween he analy ical and he nume ical
esul s is clea : one phonon exci a ion co esponds o he s a e
2+
1; wo phonon exci a ion o he s a e 0+
2(also o 2+
2and 4+
1
s a es).
The a e age numbe o dbosons, no malized o he o al
numbe o bosons, in he sphe ical egion is depic ed in
Fig. 13 o he one and wo phonon s a es. The main
disc epancies be ween nume ical and analy ical esul s, as
expec ed, appea close o he c i ical poin . No e ha he
s uc u e o he s a es is e y simple and ha al eady o
x=0.9, he numbe o dbosons is ixed o 1 and 2 o one
and wo phonon s a es, espec i ely.
V. SUMMARY AND CONCLUSIONS
In his pape , we ha e s udied wo-le el boson models
cha ac e ized by a lowes scala sboson and an exci ed Lboson
h ough a Hols ein-P imako ans o ma ion ha allows us o
ea explici ly o de by o de an Nexpansion. This ea men
shows ha only he leading N e m o he g ound s a e ene gy
is co ec in a mean ield (o Ha ee-Bose) app oach. We s ess
ha he equilib ium nuclea shape co esponding o an IBM
Hamil onian should be ob ained only when conside ing he
leading N e m o he g ound s a e ene gy.
Depending on he alue o L, models o in e es in di e en
ields can be ob ained. Thus, L=0 is ela ed o he Lipkin
model i s in oduced in nuclea physics and hen used in
many ields, L=1 is he ib on model o in e es in molecula
physics, L=2 is he in e ac ing boson model o nuclea
s uc u e, e c. We ha e p esen ed a me hod o going accu a ely
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