Quan um Phase T ansi ions in he In e ac ing Boson Model:
In eg abili y, Le el Repulsion, and Le el C ossing
J. M. A ias,
1
J. Dukelsky,
2
and J. E. Ga cı
´a-Ramos
3
1
Depa amen o de Fı
´sica A o
´mica, Molecula y Nuclea , Facul ad de Fı
´sica, Uni e sidad de Se illa,
Apa ado 1065, 41080 Se illa, Spain
2
Ins i u o de Es uc u a de la Ma e ia, CSIC, Se ano 123, 28006 Mad id, Spain
3
Depa amen o de Fı
´sica Aplicada, Uni e sidad de Huel a, 21071 Huel a, Spain
(Recei ed 20 June 2003; published 14 Oc obe 2003)
We s udy he quan um phase ansi ion mechanisms ha a ise in he in e ac ing boson model. We
show ha he second-o de na u e o he phase ansi ion om U(5) o O(6) may be a ibu ed o
quan um in eg abili y, whe eas all he i s -o de phase ansi ions o he model a e due o le el
epulsion wi h one singula poin o le el c ossing. We p opose a model Hamil onian wi h a ue i s -
o de phase ansi ion o ini e sys ems due o le el c ossings.
DOI: 10.1103/PhysRe Le .91.162502 PACS numbe s: 21.60.Fw, 21.10.Re, 64.60.F
Quan um phase ansi ions (QPTs) ha e a ac ed g ea
a en ion om he heo e ical and expe imen al com-
muni ies in ecen yea s. Expe imen s on high-Tcsupe -
conduc o s, on quan um-Hall sys ems, on Bose o Fe mi
dilu e gases in di e en ap geome ies, and on nuclei
close o a c i ical poin challenge heo e icians o de elop
eliable app oaches wi h which o desc ibe he c i ical
p ope ies o hese sys ems.
A QPT desc ibes a s uc u al change in he p ope ies
o he g ound s a e o he sys em associa ed wi h he
a ia ion o a con ol pa ame e ( he empe a u e being
he usual one in classical phase ansi ions). This pa ame-
e migh be he le el o doping in high-Tcsupe conduc-
o s, he magne ic ield in quan um-Hall sys ems, he
sca e ing leng h in dilu e gases, he nucleon numbe in
nuclea phase ansi ions, e c.
The phase diag am o he in e ac ing boson model
(IBM) [1] o nuclei has been s udied a he mean- ield
le el [2,3], using bo h ca as ophe heo y [4] and he
Landau heo y o phase ansi ions [5]. Bo h ea men s
show ha he model displays i s -o de phase ansi ions
om sphe ical o de o med shapes and om obla e o
p ola e de o med shapes, wi h an excep ional poin o
an isola ed second-o de phase ansi ion. In his wo k,
we s udy in de ail he phase ansi ions associa ed wi h
he IBM.
The IBM is a phenomenological model o nuclea
s uc u e wi h a deep connec ion o he unde lying mic o-
scopic shell model. In his model, which has been e y
success ul in desc ibing nuclea spec oscopic p ope ies
o collec i e nuclei, pai s o co ela ed nucleons o an-
gula momen um L0and L2a e ep esen ed by s
and dbosons, espec i ely. In i s simples e sion he
model does no dis inguish be ween p o ons and neu ons
and has an unde lying g oup s uc u e based on he dy-
namical g oup U(6).
The mos gene al IBM Hamil onian can be w i en in
e ms o six ee pa ame e s, as a linea combina ion o
he linea and quad a ic Casimi ope a o s o U5and he
quad a ic Casimi ope a o s o O6,SU3,O5,and
O3. A con enien and equen ly used o m o he
IBM Hamil onian ha keeps all he main ing edien s
ela ed o he s uc u e o he g ound s a e is he
consis en -QHamil onian [6]
Hxndx1
NQQ;(1)
whe e ndPdy
d,Q
dy~
ss sy~
dd2
dy~
dd2
,
and Nis he o al numbe o bosons, which is equal o
he numbe o nucleon pai s in he alence space.
One o he mos impo an ea u es o he IBM is he
exis ence o ou dis inc dynamical symme ies (DS),
each ep esen ing a well-de ined phase o nuclea collec-
i e mo ion. A quan um sys em has a DS i he Hamil-
onian can be exp essed as a linea combina ion o he
Casimi ope a o s o a subg oup chain o he dynamical
g oup. The ou dynamical symme ies a e he ollowing:
he U5symme y o sphe ical ib a ional nuclei (x
1), he SU3symme y o p ola e de o med nuclei (x
0, 
7
p=2), he O6symme y o uns able de-
o med nuclei (x0,0), and he SU3symme y
o obla e de o med nuclei (x0,
7
p=2).
The symme ies o he IBM and he ansi ions be-
ween hem a e illus a ed in he ex ended Cas en iangle
[7] o Fig. 1, along wi h he co esponding alues o he
pa ame e s xand . Dis inc phases associa ed wi h h ee
‘‘shapes,’’deno ed S o sphe ical, P o p ola e de o med,
and O o obla e de o med, can be seen in he igu e. The
h ee di e en shape phases a e sepa a ed by wo lines o
i s -o de phase ansi ions, x0:8and 0, espec-
i ely, wi h he excep ion o a iple poin [5] a x0:8
and 0, which is ep esen ed by a solid squa e and o
which he phase ansi ion is second o de . Recen ly, wo
e ec i e models based on he Boh Hamil onian, called
X5and E5;ha e been p oposed o desc ibe he physics
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a he c i ical poin s along he U5SU3and U5
O6lines, espec i ely [8].
The aim o his Le e is o go beyond he mean- ield
desc ip ion o he phase diag am in Fig. 1. We will show
ha he i s -o de QPTs a ise due o le el epulsion
be ween wo s a es wi h di e en p ope ies, as was ea -
lie ecognized in [9]. Howe e , he e a e wo excep ional
singula c i ical poin s, ep esen ed by he bold squa e
and he bold diamond in Fig. 1, o which he mechanism
igge ing he phase ansi ion is comple ely di e en .
We begin wi h a discussion o he ansi ion om O6
o U5. I is desc ibed wi hin he Hamil onian (1) by
se ing 0and a ying x om 0 o 1. Ea lie in es-
iga ions o he p ope ies o he le el spacings along his
leg o he Cas en iangle showed a Poisson dis ibu ion.
This indica es ha he model is in eg able along his leg
[10], e en hough i is no cha ac e ized by a global
dynamical symme y.
Quan um in eg abili y equi es he exis ence o a com-
ple e se o mu ually commu ing He mi ian ope a o s.
These ope a o s a e he in eg als o mo ion o quan um
in a ian s and hei eigen alues a e he conse ed quan-
i ies ha comple ely label a unique basis o common
eigens a es. A DS is a pa icula class o in eg able model
in which each one o he Casimi ope a o s in he sub-
g oup decomposi ion chain is a quan um in a ian . Bu
he e a e also o he in eg able and exac ly sol able mod-
els ha ha e played impo an oles in he unde s anding
o quan um many-body sys ems. Examples include he
Heisenbe g and Hubba d models, bo h o which a e sol -
able by he Be he ansa z. The Hamil onian (1) o 0
is a pa icula case o a gene al class o pai ing models
ha ha e been shown o be quan um in eg able [11]. The
quan um in a ian s o boson pai ing Hamil onians and
he co esponding exac solu ions we e discussed in
Re . [12]. He e we will speci y he o m o he quan um
in a ian s in he O6U5 ansi ion. Since we a e
in e es ed only in p ope ies o he g ound s a e, we ocus
ou ea men on he case in which all bosons a e pai ed
o ze o angula momen um.
We i s no e ha his ansi ion is embedded in
he decomposi ion U6O5O3O2,
whe e a missing pa ame e dependen ‘‘symme y’’ o
quan um in a ian commu ing wi h he Casimi ope a-
o s o all he g oups in he chain is esponsible o he
in eg abili y o he model.
As no ed in Re s. [10,13], he Hamil onian i sel , in-
e pola ing be ween he Casimi ope a o s o U5and
O6, cons i u es he addi ional independen quan um
in a ian ensu ing he in eg abili y o he sys em.
The eigen alues o he Hamil onian (1) o 0a e
[12]
E20x1
NX
N=2
1
1
2e
;(2)
whe e he pai ene gies ea e he solu ions o he coupled
se o nonlinea Richa dson equa ions
xN
21x1
e5
2e4X

1
ee0:(3)
The Hamil onian eigen alues (2) a e analy ic unc ions
o he con ol pa ame e x. The e o e, in he absence o
le el c ossings a he g ound-s a e le el, a i s -o de
phase ansi ion is p ecluded. Le el c ossings can occu
only o in eg able models wi h mo e han one pa ame e
dependen in eg al o mo ion, an example being he sdg
boson model [14].
Because o he exac sol abili y o he model, we a e
able o ob ain exac esul s nume ically o a la ge num-
be o bosons. In Fig. 2, we show esul s o 1000 bosons
and compa e hem wi h he mean- ield solu ion. In he le
0.6 0.8 1
x
-300
-200
-100
0
0.6 0.8 1
x
0
100
200
300
400
500
0.6 0.8 1
x
0
500
1000
1500
Eg.s. <nd>dEg.s./dx
FIG. 2. T ansi ion om O6 o U5. The g ound-s a e en-
e gy (le panel), expec a ion alue o he nd(cen al panel),
and de i a i e o he g ound-s a e ene gy ( igh panel) a e
p esen ed as a unc ion o he con ol pa ame e x o a sys em
wi h 1000 bosons. The exac esul s a e ep esen ed by ci cles
and he mean ield esul s by a ull line.
U(5)
SU(3)
O(6)
SU(3)
x
x=0 , χ=√7/2
x=0 , χ=0
x=0 , χ=−√7/2
x=1 , χ=0 x
x
χ
χ
x=0.8 ♦
S
O
P
FIG. 1. Phase diag am o he in e ac ing boson model in he
phase space o he con ol pa ame e s xand . S, O, and P s and
o he sphe ical, obla e, and p ola e phases, espec i ely.
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panel, we plo he g ound-s a e ene gy e sus he con ol
pa ame e x; in he cen al panel, he expec a ion alue o
he numbe o dbosons hndiis p esen ed as a unc ion o
x; in he igh panel, we plo he de i a i e o he ene gy
e sus x. Because o he exac sol abili y o he model, we
can make use o he Hellmann-Feynman heo em o
exp ess he i s de i a i e o he ene gy as he expec a ion
alue o he de i a i e o he Hamil onian (1) wi h
espec o x, i.e., @H=@xnd1=NQ0Q0. Bo h
ope a o s, ndand @H=@x, a e ze o in one phase (diso -
de ed) and di e en om ze o in he o he phase
(o de ed), ul illing he equi emen s o being o de pa-
ame e s. Bo h expec a ion alues a e con inuous a he
ansi ion poin sa is ying bo h he Landau and Eh en es
de ini ions o a second-o de phase ansi ion.
We will now discuss he phenomenon o le el epulsion
o nonin eg able sys ems. Fo an a bi a y ini e alue o
, we selec a alue o xx0in he icini y o which we
can use nondegene a e pe u ba ion heo y. De ining x0
xx0, we can ew i e he Hamil onian as Hx0nd
x01=NQQx0nd1
NQQH0x0V.
The i s and second de i a i es o he Hamil onian ei-
gen alues in pe u ba ion heo y a e [15]:
@En
@x0@En
@x Vnnx0;(4)
@2En
@x02@2En
@x22X
mn
V2
nmx0
Enx0Emx0:(5)
Assuming ha he e is a pai o ene gy le els ha ge
e y close in ene gy, he ene gy gap be ween hem is
xE2xE1x. Neglec ing he con ibu ion o all
o he s a es, pe u ba ion heo y o he gap de i a i e
gi es
@
@x V22 V11;@2
@x24V2
12
:(6)
We can in e p e E1and E2as he posi ions o wo
ic i ious classical pa icles in a one-dimensional space.
Then (6) a e he equa ions o hei ela i e eloci y and
he o ce be ween hem. Clea ly, i he in e ac ion ma ix
elemen V12 emains ini e as !0, he e is an in ini e
epulsion and his p e en s he pa icles om colliding.
This is a pic o ial iew o he phenomenon o le el
epulsion o nonin eg able Hamil onians. On he o he
hand, i he e is an addi ional conse ed quan i y and
bo h le els ha e di e en symme ies, hen V12 is s ic ly
ze o and he wo le els can c oss. In wha ollows, we
discuss he applica ion o hese ideas i s o he U5 o
SU3 ansi ion and hen o he SU3 o SU3 ansi ion.
In Fig. 3, we show in he le panel he en lowes 0
s a es o a sys em o 50 bosons as a unc ion o xalong
he U5 o SU3 ansi ion. The e a e se e al cases in
which wo 0le els ge e y close in ene gy bu , as jus
discussed, he close hey ge he la ge he le el epulsion
be ween hem is. In he igh panels, we magni y wo o
he cases o closes app oach o show ha indeed he
le els epel and do no c oss. In pa icula , he e is a
smoo h c osso e in he g ound-s a e ene gy a he poin
o closes app oach wi h he i s exci ed s a e. The dy-
namics o hese wo le els is well desc ibed by Eq. (6)
wi h ini e ma ix elemen s Vij. The c osso e will change
g adually as he numbe o bosons inc eases owa ds a
nonanaly ic poin in he he modynamic limi , de ining a
i s -o de phase ansi ion. F om he abo e, we conclude
ha le el epulsion is indeed esponsible o he i s -
o de phase ansi ion in his case [9].
Now we u n o he hi d leg o he iangle (x0), he
ansi ion om de o med obla e [SU3shapes, 

7
p=2] o de o med p ola e [SU3shapes,  
7
p=2]
shapes, passing h ough he -uns able O6case wi h
0.In hiscase, he i s -o de phase ansi ion akes
place p ecisely a he O6dynamical symme y [7]. A
his c i ical poin , he 0s a es can be classi ied wi h he
quan um numbe o an O5symme y. As discussed
abo e, he ealiza ion o his new symme y allows he
c ossing o le els wi h di e en quan um numbe s. The
Hamil onian (1) a his poin is
H1
NQ0Q01
2NC2O6C2O5;(7)
wi h eigen alues E 43=2N.
The lowes O6band co esponds o Nand 
0;3;.... In he la ge Nlimi , he O5s a es wi hin he
g ound-s a e O6band will be degene a e, since he
ene gy spacings go as N1. Acco ding o he abo e analy-
sis, in he he modynamic limi an in ini e se o 0s a es
wi h di e en senio i y quan um numbe s c oss. The
c ossing is allowed because o he O5symme y a he
c i ical poin , implying ha he in e ac ion ma ix ele-
men s in (6) a e ze o.
As jus no ed, he exac degene acy in he le els o he
g ound-s a e band a x0and 0occu s only in he
la ge-No he modynamic limi . To be e isualize his
QPT in he con ex o ini e sys ems, we he e o e con-
side a Hamil onian simila o ha in (1), bu wi h an
0.7 0.8 0.9
x
-8
-4
0
4
Ene gy (a bi a y uni s)
0.770 0.771 0.772
-1.52
-1.48
0.726 0.728 0.73
-4.00
-3.90
FIG. 3. Ten lowes 0s a es in he ansi ion om U5 o
SU3 o N50 bosons as a unc ion o x. On he igh panels
we show in an enla ged scale wo examples o le el epulsion.
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addi ional e m ha cancels he O5 e m in i ha
p oduces ene gy spacings wi hin he g ound-s a e band
o ini e N[see Eq. (7)]. Wi h his in mind, we conside
he ollowing Hamil onian associa ed wi h he x0
ansi ion leg:
H1
NQQ1
2jj

7
pC2O5:(8)
Fo  
7
p=2, he Hamil onian (8) is in he SU3
limi ; o  
7
p=2,i isin heSU3limi ; and o
0i is p opo ional o he quad a ic Casimi ope a o
o he O6g oup. The second e m in he modi ied
Hamil onian (8) hus cancels ou he O5b eaking o
he degene acies in he g ound-s a e band ha occu s o
ini e N.
In Fig. 4, we show all o he 0eigens a es o he
Hamil onian (8) o a sys em wi h N9bosons as a
unc ion o .A 0, he ou le els ha co espond o
he g ound-s a e band wi h 9c oss. This c ossing
gi es ise o a i s -o de QPT o a ‘‘ ini e’’ sys em.
This is he unique poin along he x0leg a which
le el c ossings a e allowed due o he ealiza ion o he
O5symme y. Away om his poin he mechanism o
le el epulsion se s in. In he uppe igh panel o Fig. 4,
we show in an enla ged scale an appa en le el c ossing a
0. The mechanism o le el epulsion p edic ed in
Eq. (6) is nume ically e i ied. On he con a y a 0,
he ealiza ion o he O5symme y allows he c ossing
o le els wi h di e en quan um numbe s, as shown in
he lowe igh panel.
In summa y, he e a e wo singula poin s o QPTs in
he IBM phase diag am. The i s is in he ansi ion om
U5 o O6and co esponds o a second-o de QPT due
o he in eg able na u e o he Hamil onian along his
line, which ensu es ha he g ound-s a e ene gy is an
analy ic unc ion o he con ol pa ame e . The second
singula poin is loca ed a he QPT om SU3 o SU3,
p ecisely a he O6c i ical symme y. In he he mody-
namic limi , his QPT is associa ed wi h he c ossing o
an in ini e numbe o 0s a es. By adding o he Hamil-
onian (7) a e m p opo ional o he O5Casimi ope a-
o , we we e able o ealize his QPT o a ini e sys em.
This is an unusual, bu especially in e es ing, case o
s udy u he , since mos phase ansi ions ake place in
he he modynamic limi . The e o e, he model desc ibed
by he Hamil onian (8) may be use ul o explo e he
uni e sal p ope ies o i s -o de phase ansi ions.
These wo isola ed c i ical poin s ep esen excep ional
poin s in he phase diag am o he IBM. Along he lines
sepa a ing he sphe ical, p ola e, and obla e phases, he
mechanism unde lying he i s -o de phase ansi ions is
le el epulsion be ween he g ound s a e and he i s
exci ed s a e. Fo ini e N, his mechanism is e lec ed
by a so c osso e , bu in he la ge Nlimi he model
de elops a singula i y in he de i a i e o he gap con-
sis en wi h a i s -o de phase ansi ion. This conclusion
is also alid o mo e gene al IBM Hamil onians, since
le el epulsion is a uni e sal phenomenon associa ed wi h
quan um nonin eg abili y.
This wo k was suppo ed in pa by he Spanish DGI
unde P ojec s No. BFM2002-03315 and No. BFM2000-
1320-C02-02. We acknowledge use ul discussions wi h
S. Pi el, P. Van Isacke , J. Jolie, and F. Iachello.
[1] F. Iachello and A. A ima, The In e ac ing Boson Model
(Camb idge Uni e si y P ess, Camb idge, 1987).
[2] A. E. L. Diepe ink, O. Schol en, and F. Iachello, Phys.
Re . Le . 44, 1747 (1980).
[3] D. H. Feng, R. Gilmo e, and S. R. Deans, Phys. Re . C 23,
1254 (1981).
[4] E. Lo
´pez-Mo eno and O. Cas an
˜os, Phys. Re . C 54, 2374
(1996).
[5] J. Jolie e al., Phys. Re . Le . 89, 182502 (2002).
[6] R. F. Cas en and D. D. Wa ne , Re . Mod. Phys. 60, 389
(1988).
[7] J. Jolie, R. F. Cas en, P. on B en ano, and V. We ne ,
Phys. Re . Le . 87, 162501 (2001).
[8] F. Iachello, Phys. Re . Le . 85, 3580 (2000); 87, 052502
(2001).
[9] N.V. Zam i , P. on B en ano, R. F. Cas en, and J. Jolie,
Phys. Re . C 66, 021304 (2002).
[10] Y. Alhassid and N. Whelan, Phys. Re . Le . 67, 816
(1991).
[11] J. Dukelsky. C. Esebbag, and P. Schuck, Phys. Re . Le .
87, 066403 (2001).
[12] J. Dukelsky and S. Pi el, Phys. Re . Le . 86, 4791 (2001)
[13] P. Cejna and J. Jolie, Phys. Re . E 58, 387 (1998).
[14] J. Dukelsky, J. M. A ias, J. E. Ga cı
´a-Ramos, and
S. Pi el, nucl- h/0310031.
[15] E. A. Yuzbashyan, B. L. Al shule , and B. S. Shas y,
J. Phys. A 35, 7525 (2002).
-1 -0.5 00.5 1
χ
-20
-15
-10
-5
0
Ene gy (a bi a y uni s)
0.38 0.4
-9.8
-9.7
-9.6
-9.5
-0.001 0 0.001
-13.00
-12.99
FIG. 4. T ansi ion om SU3 o SU3 h ough he O6
limi . All 0eigens a es o he Hamil onian (8) o a sys em
wi h N9bosons a e ep esen ed as a unc ion o he con ol
pa ame e . The igh panels show in a la ge scale he le el
epulsion o wo close le els a 0( igh - op panel) and he
le el c ossing a 0( igh -bo om panel).
PHYSICAL REVIEW LETTERS week ending
17 OCTOBER 2003
VOLUME 91, NUMBER 16
162502-4 162502-4