PHYSICAL REVIEW C 72, 037301 (2005)
β4po en ial a he U(5)–O(6) c i ical poin o he in e ac ing boson model
Jos´
e En ique Ga c´
ıa-Ramos,1,∗Jo ge Dukelsky,2,†and Jos´
eM.A ias
3,‡
1Depa amen o de F´
ısica Aplicada, Uni e sidad de Huel a, E-21071 Huel a, 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 A ´
omica, Molecula y Nuclea , Facul ad de F´
ısica,
Uni e sidad de Se illa, Apa ado 1065, E-41080 Se illa, Spain
(Recei ed 1 July 2005; published 8 Sep embe 2005)
Exac nume ical esul s o he in e ac ing boson model Hamil onian along he in eg able line om U(5) o
O(6) a e ob ained by diagonaliza ion wi hin boson senio i y subspaces. The ma ix Hamil onian educes o a
block idiagonal o m ha can be diagonalized o la ge numbe o bosons. We p esen esul s o he low-ene gy
spec um and he ansi ion p obabili ies o sys ems up o 10,000 bosons, which con i m ha a he c i ical poin
he sys em is equally well desc ibed by he Boh Hamil onian wi h a β4po en ial.
DOI: 10.1103/PhysRe C.72.037301 PACS numbe (s): 21.60.Fw, 21.10.Re
The main goal o his B ie Repo is o epo on new esul s
ha comple e a p e ious s udy [1] on he ela ions be ween
he c i ical poin in he ansi ion om U(5) o O(6) limi s
o he in e ac ing boson model (IBM) [2] and he ecen ly
p oposed E(5) c i ical poin symme y [3]. In Re . [1] wo
di e en boson Hamil onians pe o ming he ansi ion om
U(5) o O(6) we e used o show ha a he c i ical poin
(i) hey p o ide di e en spec a and ansi ions o small
numbe o bosons (ii) hey con e ge o he same spec um
o la ge N, and (iii) bo h con e ge o he spec um p o ided
by he Boh Hamil onian [4] wi h a β4po en ial a he han
o he one p o ided by a squa e-well po en ial as in he E(5)
model. Mos o he la ge-Nanalysis was based on he solu ion
o he Richa dson equa ions [5,6], which allow us o ob ain
ene gy eigen alues, bu he o m o he eigens a es is no well
sui ed o calcula e ansi ion p obabili ies. The e o e, o s udy
ansi ion a es we had o eso o cu en IBM codes ha
es ic ed ou calcula ions o sys ems up o N=40 [1]. Fo
hese small-N alues, he ansi ion a es show a endency o
app oach he β4po en ial esul s bu hey a e no conclusi e. In
his B ie Repo we p esen an al e na i e o he Richa dson
equa ions o ob aining he low-ene gy eigen alues and, on
he same oo ing, he ansi ion p obabili ies in he U(5)–O(6)
ansi ional egion o la ge N alues.
The wo Hamil onians desc ibing he U(5)–O(6) ansi ion
s udied in Re . [1] a e
ˆ
HI=xˆ
nd+1−x
N−1ˆ
P†ˆ
P(1)
and
ˆ
HII =xˆ
nd−1−x
N
ˆ
Qχ=0·ˆ
Qχ=0,(2)
∗Elec onic add ess: [email p o ec ed]
†Elec onic add ess: dukelsk[email p o ec ed]
‡Elec onic add ess: [email p o ec ed]
whe e
ˆ
nd=
m
d†
mdm,(3)
ˆ
P†=1
2(d†·d†−s†·s†)=1
2(P†
d−P†
s),(4)
ˆ
Qχ=0=(s†×˜
d+d†×˜
s)(2),(5)
and ·s ands o he scala p oduc . We ha e in oduced in
(4) he boson-pai c ea ion ope a o s P†
d=d†·d†and P†
s=
s†·s† ha will be used la e on.
The mean- ield analysis o he quan um phase diag am
o he IBM is usually pe o med wi hin he in insic s a e
o malism [7,8] whe e, a e sepa a ing he h ee Eule angles,
he ial wa e unc ion is a boson condensa e depending on
he wo geome ical a iables βand γ. Along he U(5)–O(6)
ansi ion he ene gy su ace is γindependen and he in insic
g ound-s a e ene gy o a gi en alue o he con ol pa ame e x
co esponds o he alue o he de o ma ion pa ame e β, which
minimizes he ene gy su ace. The phase ansi ion along his
line is hen signaled by he condi ion
[d2E(N,β)/dβ2]β=0=0,(6)
which ixes he c i ical alue o he con ol pa ame e x.Fo
he Hamil onian (1) he c i ical xis xI
c=0.5, independen o
he numbe o bosons N, whe eas o he Hamil onian (2) i is
xII
c=(4N−8)/(5N−8). In he la ge Nlimi xII
c→4/5.
To s udy he eigens a es o he Hamil onians (1) and (2) we
in oduce he s- and d-boson pai algeb a [9]
K+
s=1
2s†·s†=1
2P†
s=(K−
s)†,
(7)
K0
s=1
2s†s+1
2=1
2ˆ
ns+1
4,
K+
d=1
2d†·d†=1
2P†
d=(K−
d)†,
(8)
K0
d=1
2
md†
mdm+1
2=1
2ˆ
nd+5
4.
0556-2813/2005/72(3)/037301(4)/$23.00 037301-1 ©2005 The Ame ican Physical Socie y
BRIEF REPORTS PHYSICAL REVIEW C 72, 037301 (2005)
0+
2+
4,2
++
+++ +
6,4,3,0
+++ +
6,4,3,0
0+
0+
2+
2+
4,2
++
4,2
++
ν =1, ν =3
s
d
ν =0, ν =2
s
d
ν =1, ν =1
s
d
ν =0, ν =0
s
d
τ=0
τ=1
τ=3
τ=2
ξ=1
ξ=2
ξ=3
L+
ξ,τ
FIG. 1. Schema ic spec um ob ained by diagonaliza ion wi hin
boson senio i y subspaces as explained in he ex and i s co espon-
dence wi h he one o Re s. [3] and [1].
Fo each  alue, 0 o 2, he h ee ope a o s {K+
,K−
,K0
}
sa is y he su(1,1) commu a o algeb a
K0
,K±
=±δK±
,[K+
,K−
]=−2δK0
.(9)
A comple e se o eigens a es o a gene al IBM U(5)–
O(6) ansi ional Hamil onian can be w i en in e ms o he
aising ope a o K+
ac ing on a subspace o unpai ed bosons
cha ac e ized by he senio i y quan um numbe ν,
|˜
nν=1
C˜
n
,ν
(K+
)˜
n|ν,(10)
whe e νs=0,1; νd=0,1,2,...; and |νis a no malized
s a e. The alue o νgi es he numbe o bosons o ype 
no coupled in pai s o ze o angula momen um. The label
˜
n e e s o boson pai s coupled o ze o angula momen um.
The e o e, he o al numbe o bosons is N=2˜
ns+2˜
nd+
νs+νd.Using hesu(1,1) algeb a i is s aigh o wa d o
ob ain he no maliza ion cons an s
C˜
n
,ν=ν|(K−
)˜
n(K+
)˜
n|ν= ˜
n!(2˜
n+2+2ν−1)!!
2˜
n(2+2ν−1)!! .
Now we p oceed o cons uc he comple e se o s a es as
|˜
ns˜
nd,ν
sνd=1
C˜
ns
s,νsC˜
nd
d,νd
(K+
s)˜
ns(K+
d)˜
nd|νsνd.(11)
The basis (11), al hough lacking in o ma ion on angula
momen um, is especially use ul o diagonalizing he Hamil o-
nians (1) and (2). To show his, we ew i e he Hamil onians (1)
and (2) in e ms o he gene a o s o he wo su(1,1) algeb as,
ˆ
HI=xˆ
nd+1−x
(N−1)(K+
sK−
s+K+
dK−
d
−K+
sK−
d−K+
dK−
s),(12)
ˆ
HII =xˆ
nd−1−x
N(4K+
sK−
d+4K+
dK−
s
+5ˆ
ns+ˆ
nd+2ˆ
nsˆ
nd).(13)
The ma ix elemen s o he ele an ope a o s o bo h
Hamil onians in he basis (11) a e
˜
ns˜
nd,ν
sνd|ˆ
ns|˜
ns˜
nd,ν
sνd=2˜
ns+νs,
˜
ns˜
nd,ν
sνd|ˆ
nd|˜
ns˜
nd,ν
sνd=2˜
nd+νd,
˜
ns˜
nd,ν
sνd|K+
sK−
s|˜
ns˜
nd,ν
sνd=˜
ns˜
ns+νs−1
2,(14)
˜
ns˜
nd,ν
sνd|K+
dK−
d|˜
ns˜
nd,ν
sνd=˜
nd˜
nd+νd+3
2,
(˜
ns−1)(˜
nd+1),ν
sνd|K+
dK−
s|˜
ns˜
nd,ν
sνd
=1
2√˜
ns(˜
nd+1)(
2˜
ns+2νs−1)(
2˜
nd+2νd+5).
The Hamil onians (1) and (2) do no mix s a es wi h di e -
en senio i y quan um numbe s (νs,ν
d), lea ing in a ian hese
senio i y subspaces. Wi hin each subspace he Hamil onian
ma ices a e idiagonal and can be easily diagonalized o
e y la ge N alues. We will label s a es wi hin each subspace
010203040
N
2
2.1
2.2
E(4+
1,2)/E(2+
1,1)
0 10203040
N
2
3
4
E(0+
2,0)/E(2+
1,1)
010203040
N
0.6
0.8
1
E(0+
2,0)/E(0+
1,3)
0 10203040
N
3.2
3.4
3.6
E(0+
1,3)/E(2+
1,1)
10 20 30 40
N
1.4
1.6
1.8
2
R1(E2)
10 20 30 40
N
0
0.5
1
1.5
R2(E2)
E(5)
E(5) E(5)
E(5)
E(5)
E(5)
β4
β4β4
β4
β4
β4
FIG. 2. Va ia ion wi h he
numbe o bosons (up o N=40)
o selec ed ene gy and B(E2) a ios
o IBM calcula ions pe o med a
he c i ical poin s o Hamil onians
(1) (b oken line) and (2) ( ull line).
The co esponding E(5) and β4
alues a e ma ked wi h ho izon al
do ed lines.
037301-2
BRIEF REPORTS PHYSICAL REVIEW C 72, 037301 (2005)
101102103104
N
2
2.1
2.2
E(4+
1,2)/E(2+
1,1)
101102103104
N
2
3
4
E(0+
2,0)/E(2+
1,1)
101102103104
N
0.6
0.8
1
E(0+
2,0)/E(0+
1,3)
101102103104
N
3.2
3.4
3.6
E(0+
1,3)/E(2+
1,1)
101102103104
N
1.4
1.6
1.8
2
R1(E2)
101102103104
N
0
0.5
1
1.5
R2(E2)
E(5)
E(5) E(5)
E(5)
E(5) E(5)
β4
β4β4
β4
β4
β4
FIG. 3. Same as Fig. 2 bu he e he numbe o bosons uns up o 10,000 in bo h he ene gy and he B(E2) a ios. No e he loga i hmic scale
o he Naxis.
by he quan um numbe ξ. I is wo hwhile o no e he e
ha d-boson senio i y, νd, is equi alen o he O(5) quan um
numbe τ[2]. The cons uc ion o he spec um o a sys em
wi h e en numbe o bosons is as ollows: One s a s wi h
he subspace τ=0(νs=0,ν
d=0), whe e all he bosons a e
coupled in pai s o ze o angula momen um. Consequen ly,
s a es wi hin his subspace will ha e o al angula momen um
L=0. The lowes eigen alue (ξ=1) is he g ound s a e 0+
1,0
( he no a ion is Lπ
ξ,τ [3]), he second lowes eigen alue is
he i s exci ed s a e τ=0, Lπ=0+, which is labeled 0+
2,0,
e c. The nex block wi h τ=1(νs=1,ν
d=1) has one pai
b oken in o an sboson and a dboson. Co espondingly, all
s a es in his block ha e L=2. The lowes eigen alue (ξ=1)
is he lowes 2+, which is labeled as 2+
1,1, he nex one (ξ=2)
is 2+
2,1, e c. The nex block is o τ=2(νs=0,ν
d=2) and
co esponds o one boson pai b oken in o wo dbosons. I
p o ides s a es wi h angula momen a L=4,2. (No ice ha
L=0 is excluded om his subspace since i is included in he
νs=0,ν
d=0 subspace.) One can con inue in his way wi h
he nex block, τ=3(νs=1,ν
d=3), which co esponds
o wo boson pai s b oken in o one sboson and h ee
dbosons and gi es ise o L=6,4,3,0 s a es and so on. The
g ound-s a e band is o med by all lowes (ξ=1) eigens a es
o τ=0,1,2,3,.... The i s exci ed band (ξ=2) is
o med by he nex lowes τ=0,1,2,3,... eigens a es, e c.
Following his sequence one inds he well-known iangula
s uc u e associa ed o O(5). All his is shown schema ically in
Fig. 1.
The diagonaliza ion o he Hamil onian in each subspace
p o ides he necessa y in o ma ion o calcula e elec omag-
ne ic ansi ion a es. We will be in e es ed he e in he elec ic
quad upole ansi ions, which, apa om an unimpo an scale
ac o , a e desc ibed by he quad upole ope a o (5). The ac ion
o his ope a o on he basis s a es wi hou b oken pai s is
ˆ
Qµ|˜
ns˜
nd,0,0= 1
C˜
ns
s,0C˜
nd
d,0˜
ns(K+
s)˜
ns−1(K+
d)˜
nds†d†
µ|0
+˜
nd(K+
s)˜
ns(K+
d)˜
nd−1s†d†
µ|0,(15)
whe e |0s ands o he boson acuum.
The ma ix elemen s o in e es i one wan s o e alua e
ansi ion a es om he g ound s a e o he i s exci ed s a e
a e
(˜
ns−1)˜
nd,1,1|ˆ
Qµ|˜
ns˜
nd,0,0=2˜
ns(2˜
nd+5)
5,(16)
˜
ns(˜
nd−1),1,1|ˆ
Qµ|˜
ns˜
nd,0,0=2˜
nd(2˜
ns+1)
5.(17)
I we w i e he eigens a es as
|, νsνd=
˜
ns,˜
nd
ςνsνd
˜
ns,˜
nd|˜
ns˜
nd,ν
sνd,(18)
he ma ix elemen o he ˆ
Qope a o be ween he g ound s a e,
|, 00, and he i s exci ed s a e, |, 11,is
, 11|ˆ
Qµ|, 00=
˜
ns,˜
nd2˜
ns(2˜
nd+5)
5ς00
˜
ns,˜
ndς11
˜
ns−1,˜
nd
+2˜
nd(2˜
ns+1)
5ς00
˜
ns,˜
ndς11
˜
ns,˜
nd−1.(19)
The ma ix elemen s o he elec ic quad upole ope a o
be ween he i s exci ed s a e (νs=1,ν
d=1) and he s a es
wi h νs=0,ν
d=2 can be calcula ed in a simila way.
037301-3
BRIEF REPORTS PHYSICAL REVIEW C 72, 037301 (2005)
In Fig. 2 we p esen some selec ed low-ene gy eigen alues
and B(E2) a ios o boson numbe s up o N=40 a he
c i ical poin s o bo h IBM Hamil onians (1) and (2). We
emphasize he e ha he c i ical poin s o he wo Hamil onians
a e di e en . The ou ene gy a ios p esen ed a e w i en
explici ly in he igu e and he wo displayed B(E2) a ios
a e R1=B(E2; 4+
1,2→2+
1,1)/B(E2; 2+
1,1→0+
1,0) and R2=
B(E2; 0+
2,0→2+
1,1)/B(E2; 2+
1,1→0+
1,0), whe e we a e using
he no a ion Lπ
ξ,τ o indica e he s a es. The pu pose o his
igu e is o co ec a mis ake we had in Fig. 3 o Re . [1],
whe e he esul s o he Hamil onian (2) we e calcula ed wi h
a w ong alue o xc. As can be seen in he igu e he e, he e
a e sizable di e ences in he spec um and ansi ion a es
be ween bo h Hamil onians a he c i ical poin s. Though om
Fig. 2 a gene al endency o con e gence o he solu ion o
he Boh equa ion wi h a β4po en ial a he han o he E(5)
symme y is obse ed, he esul s, especially om he B(E2)’s,
a e no ye conclusi e. In Fig. 3 we show he new esul s o his
epo . The same quan i ies as in Fig. 2 a e plo ed o N alues
up o 10,000, including ansi ion a es. In Re . [1] ene gy
eigen alues we e calcula ed up o N=1000 and ansi ion
a es up o N=40. We can now clea ly app ecia e, bo h om
he ene gies and B(E2) ansi ions, ha he IBM Hamil onians
a he U(5) o O(6) c i ical poin in he la ge-Nlimi con e ge
o he Boh Hamil onian wi h a β4po en ial.
In his B ie Repo we make use o he p ope y ha he
IBM Hamil onian along he ansi ional line om U(5) o
O(6) is block diagonal wi h espec o he boson senio i y
quan um numbe and idiagonal wi hin each subspace. This
educ ion allows us o ob ain exac solu ions up o e y la ge
numbe o bosons o ene gies and wa e unc ions. We ha e
applied his o malism o con i m p e ious s udies abou he
co espondence be ween he IBM Hamil onians a he c i ical
poin in he U(5)–O(6) ansi ion and he Boh Hamil onian
wi h a β4po en ial o he low-ene gy p ope ies. This issue
has also been s udied ecen ly om a di e en poin o iew
by Rowe e al. [10]. The o malism p esen ed he e o he IBM
can be easily gene alized o o he wo-le el boson models [11].
This wo k was suppo ed in pa by he Spanish DGICYT
unde P ojec s Nos. BFM2002-03315, BFM2003-05316-C02-
02, BFM2003-05316, and FPA2003-05958.
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037301-4