No el esul s om an algeb aic app oach o
molecula bending dynamics
F ancisco P´e ez-Be nal1, Osi is ´
Al a ez-Bajo1, Jos´e M A ias2, Miguel
Ca ajal1, Jos´e E Ga c´ıa-Ramos1, Danielle La ese3, and Ped o
P´e ez-Fe n´andez2
1Depa amen o de F´ısica Aplicada, Fac. de CC. Expe imen ales, Uni e sidad de Huel a,
Huel a 21071 (SPAIN)
2Depa amen o de F´ısica A ´omica, Molecula y Nuclea , Facul ad de F´ısica, Uni e sidad de
Se illa, Apdo. 1065, Se illa E-41080 (SPAIN)
3Depa men o Chemis y, Yale Uni e si y, P.O. Box 208107, New Ha en, CT 06520-8107
(USA)
E-mail: [email p o ec ed]
Abs ac . We p esen a b ie e iew o esea ch opics o cu en in e es ha depend on an
algeb aic app oach o molecula bending dynamics. This app oach is based on a u(3) spec um
gene a ing algeb a. In pa icula , we b ie ly p esen esul s on h ee opics: he calcula ion o
ini e-size analy ical co ec ions o mean ield esul s, he applica ion o he model o he la ge-
ampli ude ib a ional bending mode o he NCNCS molecule, and he analysis o he in luence
o quad a ic Casimi ope a o s on exci ed s a e quan um phase ansi ions.
1. In oduc ion
The modeling o n-dimensional sys ems using a u(n+ 1) Lie algeb a as he sys em’s spec um
gene a ing algeb a (SGA) has p o ed success ul in an ample se o physical si ua ions [1]. This
app oach has caused a majo impac in nuclea s uc u e s udies, whe e he in e ac ing boson
model (IBM), based on a u(6) SGA and o mula ed by Iachello and A ima in he se en ies, has
become a s anda d heo e ical ool [2].
Following he ail o he IBM, he Vib on Model was bo n in he eigh ies, ea ing molecules
as bosonic sys ems whe e bosons ep esen ib a ional and o a ional exci a ions ( ib ons) [3].
The SGA in his case is a u(4) Lie algeb a, due o he h ee-dimensional na u e o he p oblem.
The Vib on Model, hough use ul, p o ed qui e cumbe some when applied o molecula
species composed by mo e han ou a oms. In he cases ha do no equi e a simul aneous
ea men o molecula o a ions and ib a ions, a possible al e na i e app oach is he one-
dimensional (1D) limi o he ib on model, based on a u(2) SGA [4]. In his limi o a ional
deg ees o eedom a e neglec ed and each local ib a ional deg ee o eedom has an associa ed
u(2) Lie algeb a. The e o e his model conside s anha monici y om he ou se , due o an
isomo phism be ween he u(2) Lie algeb a and he 1D Mo se po en ial [5]. In addi ion, his
app oach na u ally embeds disc e e symme y conside a ions [6].
The bending ib a ional dynamics o linea and quasi-linea molecules imply he in e play
o ib a ional and o a ional deg ees o eedom. In his case an app op ia e model is he wo-
dimensional (2D) limi o he ib on model, wi h a u(3) Lie algeb a as he sys em’s SGA [7].
GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing
Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049
Published unde licence by IOP Publishing L d
1
The 2D limi o he ib on model has success ully modeled spec al e m ene gies and
in ensi ies o a ious molecula species, om igidly-linea o igidly-ben molecules, including
quasi-linea , and non- igid in e media e si ua ions [8, 9, 10]. A ho ough desc ip ion o bo h
quan al and classical aspec s o his model can be ound in Re . [11].
In ecen yea s he s udy and cha ac e iza ion o quan um phase ansi ions (QPTs) in
algeb aic models has a ac ed much a en ion, a line o esea ch ha can be aced back o
he seminal wo k o Gilmo e [12]. These ansi ions, also called shape phase ansi ions and
g ound s a e ansi ions, a e ze o- empe a u e phase ansi ions be ween di e en geome ic
limi s o he sys em’s g ound s a e upon a ia ion o one o se e al con ol pa ame e s in he
Hamil onian. This esea ch line has been ecen ly bols e ed up wi h he disco e y o p ecu so s
o QPTs in nuclea sys ems. Re e ence [13] is a excellen and ecen e iew on his opic.
Mo e ecen ly, he s udy o exci ed s a e quan um phase ansi ions (ESQPTs), ansi ions
ha ake place upon he a ia ion o he sys em’s exci a ion ene gy ins ead o he Hamil onian
pa ame e s, has been p oposed [14]. In his case he ansi ion happens o some pa icula
exci ed s a es o he sys em. This is especially ele an in he ield o molecula spec oscopy,
whe e no el spec oscopic echniques allow he access o highly-exci ed s a es. In ac , quan um
monod omy e ec s, whose appea ance in se e al molecula sys ems has been expe imen ally
con i med [15, 16], can be ega ded as an ESQPT om he algeb aic model pe spec i e [11].
We inish his sec ion wi h a b ie ou line o he p esen wo k. Sec ion 2 is de o ed o
a sho p esen a ion o he bosonic algeb aic app oach o 2D sys ems and i s classical limi .
Analy ical co ec ions o he classical limi o he model, going beyond i s mean ield limi a e
p esen ed in sec ion 3. An applica ion o a ealis ic case can be ound in sec ion 4, whe e we i
ib a ional bending ene gies o a la ge ampli ude bending mode o NCNCS. Sec ion 5 con ains
some in e es ing esul s conce ning he in luence o quad a ic Casimi ope a o s on he model
ESQPT. In sec ion 6 we inish by gi ing some concluding ema ks.
2. Algeb aic app oach o 2D sys ems
In his sec ion we p esen , in an ab idged o m, he 2D limi o he ib on model and i s classical
limi .
2.1. Basic esul s
The building b icks o he u(3) Lie algeb a used o modeling 2D sys ems a e h ee bosons
o wo di e en ypes. One o hem is a scala boson, σ†, and he o he wo a e Ca esian
bosons, {τ†
x, τ†
y}. The ope a o s ollow he usual bosonic commu a ion ela ions: [σ, σ†] = 1 and
hτi, τ†
ji=δi,j wi h i, j =x, y. All o he commu a o s a e ze o.
Fo he sake o con enience we in oduce ci cula bosons [11]
τ†
±=∓τ†
x±iτ†
y
√2, τ±=∓τx∓iτy
√2.(1)
The nine u(3) gene a o s a e buil wi h he possible bilinea p oduc s o σ†and τ†
x,τ†
yc ea ion
and annihila ion ope a o s. They a e con en ionally exp essed as [7]:
ˆn=τ†
+τ++τ†
−τ−,ˆns=σ†σ
ˆ
l=τ†
+τ+−τ†
−τ−
ˆ
D+=√2(τ†
+σ−σ†τ−),ˆ
D−=√2(−τ†
−σ+σ†τ+)
ˆ
Q+=√2τ†
+τ−,ˆ
Q−=√2τ†
−τ+
ˆ
R+=√2(τ†
+σ+σ†τ−),ˆ
R−=√2(τ†
−σ+σ†τ+).
(2)
GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing
Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049
2
The ope a o ˆ
lis he 2D angula momen um, as one can easily see once i is exp essed in e ms
o Ca esian boson ope a o s.
The e a e wo possible dynamical symme ies ha conse e 2D angula momen um, s a ing
in he u(3) SGA and ending in he so(2) symme y algeb a.
u(3) ⊃u(2) ⊃so(2) Chain I ,(3)
u(3) ⊃so(3) ⊃so(2) Chain II .(4)
Nume ical calcula ions in he p esen a icle ha e been ca ied ou using he basis associa ed
wi h dynamical symme y I. A de ailed discussion o bo h dynamical symme ies can be ound
in [11].
Ano he ing edien o he algeb aic app oach is he Casimi (o in a ian ) ope a o s associa ed
o each subalgeb a chain [5]. The i s and second o de Casimi ope a o s o he subalgeb as
in chains (3) and (4) a e
ˆ
C1[u(2)] = ˆn , ˆ
C2[u(2)] = ˆn(ˆn+ 1) ,
ˆ
C2[so(3)] = ˆ
W2= ( ˆ
D+ˆ
D−+ˆ
D−ˆ
D+)/2 + ˆ
l2,
ˆ
C1[so(2)] = ˆ
l , ˆ
C2[so(2)] = ˆ
l2.
(5)
The so(3) Casimi ope a o ˆ
W2can be eplaced by he pai ing ope a o ˆ
P=N(N+1)−ˆ
W2,
whe e he o al numbe ope a o , ˆ
N= ˆns+ ˆn, has been eplaced by i s alue, N, due o he
ac ha we conside sys ems wi h a ixed numbe o bosons.
The mos gene al o a ion and pa i y-in a ian , one and wo-body Hamil onian can be w i en
as a linea combina ion o he ou possible i s and second o de Casimi ope a o s (5)
ˆ
H=E0+ǫˆ
C1[u(2)] + αˆ
C2[u(2)] + βˆ
C2[so(2)] + Aˆ
C2[so(3)] .(6)
Ma ix elemen s in he wo possible bases o he Casimi ope a o s (5) can be ound in Re .
[11].
2.2. Mean ield limi o he model
The mean ield limi o he model, also called he he modynamical limi , he la ge Nlimi , o
he classical limi o he model, can be ound ollowing an algo i hm es ablished by Gilmo e [17]
known as he cohe en (o in insic) s a e app oach.
In making use o he cohe en s a e app oach, we use p ojec i e cohe en s a es ha de ine
an in insic g ound s a e
|[N]; , θi=1
√N!b†
cN|0i,(7)
whe e and θa e pola coo dina es associa ed o Ca esian coo dina es xand y, and b†
cis he
boson condensa e
b†
c=1
√1 + 2σ†+xτ†
x+yτ†
y.(8)
The cohe en s a e (7) is he numbe p ojec ed gene alized cohe en s a e o u(3) [2]. The
cohe en s a e app oach p o ides an ene gy unc ional ha is an app oxima ion o he g ound
s a e ene gy o a gi en Hamil onian. This app oxima ion becomes exac o N→ ∞.
The exis ence o a second o de QPT be ween he wo possible dynamical symme ies can
be easily p o en using he a o emen ioned cohe en s a e app oach [11]. In o de o do so, i
is con enien o de ine a simpli ied model Hamil onian, wi h one ope a o o each chain and
p ope ly no malized
ˆ
H=ε(1 −ξ)ˆn+ξ
N−1ˆ
P.(9)
GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing
Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049
3
The pa ame e εis an ene gy scale and ξis a con ol pa ame e ha akes alues be ween ze o
(u(2) dynamical symme y) and one (so(3) dynamical symme y).
The ene gy pe boson o he essen ial Hamil onian (9) is [9, 11]
Eξ( )≡1
Nh[N]; , θ|ˆ
H|[N]; , θi
h[N]; , θ|[N]; , θi=ǫ
(1 −ξ) 2
1 + 2+ξ 1− 2
1 + 2!2
.(10)
The ex emiza ion o he ene gy unc ional (10) e eals ha he e a e wo di e en geome ic
limi s: symme ic (o linea ) and de o med (o ben ). In he symme ic case he ene gy minimum
is a e= 0, while in he second case e=p(5ξ−1)/(3ξ+ 1) 6= 0. The symme ic phase akes
place o con ol pa ame e alues ξ≤ξc= 0.2, and he de o med phase o ξ > ξc= 0.2. When
e alua ed a = e, he ene gy unc ional is
Eξ( e) = (ξ0≤ξ≤ξc
−9ξ2+10ξ−1
16ξξc< ξ ≤1.(11)
By e alua ing he de i a i es o Eξ( e) wi h espec o ξ, one inds ha , acco ding o Eh en es ’s
classi ica ion scheme, he u(2) −so(3) phase ansi ion is o second o de [11].
Ano he obse able o in e es is he expec a ion alue o he u(2) numbe ope a o in he
sys em g ound s a e
h[N]; , θ|ˆn|[N]; , θi=N 2
e
1 + 2
e
=(0 0 ≤ξ≤ξc
5ξ−1
8ξξc< ξ ≤1.(12)
This ope a o can ac as a classical o de pa ame e o he ansi ion be ween he symme ic
and de o med phases [11].
3. Beyond mean ield analy ical co ec ions
We e alua e analy ical co ec ions o he mean ield limi o he algeb aic app oach o go beyond
he esul s p esen ed in he p e ious sec ion. The mean ield o classical limi is only exac in
he la ge Nlimi , and he calcula ed co ec ions allow us o calcula e he N0co ec ions o he
g ound s a e ene gy (11) and he numbe o τbosons (12).
In o de o compu e hese co ec ions we pe o m a Hols ein-P imako expansion and a shi
ans o ma ion, ollowed by a Bogoliubo ans o ma ion, ollowing Re . [18]. This e e ence is
a gene al wo k, aimed a bosonic wo-le el Hamil onians wi h SGA u(2L+ 2) wi h L= 1,2,....
In he symme ic egion Dusuel and collabo a o s go a s ep u he han us, making use o he
con inuous uni a y ans o ma ions app oach (CUTS) [19].
The Hols ein-P imako expansion implies he de ini ion o a new pai o Ca esian bosons,
b†
i(bi), i=x, y, wi h he usual bosonic commu a ion ela ions, [bi, b†
j] = δij. The inclusion o
his boson emo es he dependence on he scala σ†boson o he Hamil onian
τ†
iτj=b†
ibj,
τ†
iσ=√N b†
iq1−ˆnb/N =σ†τi†,(13)
ˆnσ=σ†σ=N−ˆnb,
whe e i, j =x, y, and ˆnb=b†
xbx+b†
yby. A hi d se o bosons, c†
i(ci), wi h i=x, y is de ined ia
a shi ans o ma ion
b†
i=√Nλδix +c†
i;i=x, y . (14)
GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing
Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049
4
The pa ame e λ, known as he shi pa ame e , is ze o in he sphe ical phase and nonze o in
he de o med phase.
Once he model Hamil onian (9) is ans o med, we ob ain an expansion in powe s o N, wi h
e ms ˆ
H=ˆ
H1+ˆ
H1/2+ˆ
H0+O(1/√N),(15)
whe e ˆ
Hiinco po a es he e ms wi h a Nidependence.
The i s e m is ˆ
H1=Nhξ+ (1 −5ξ)λ2−4ξλ4i.(16)
Se ing λ= /√1 + 2 his e m is no hing mo e han he mean ield esul (10). The nex o de
in he expansion o he model Hamil onian, ˆ
H1/2, is ze o once he equilib ium alues o (o
λ) a e subs i u ed. This is due o he ela ion ˆ
H1/2=2
√N
dˆ
H1
dλ . The e o e, he i s ini e-size
co ec ion o he mean ield limi is he ˆ
H0 e m, ha is quad a ic in he cboson ope a o s
H0=λ2(4λ2−1)ξ+ (1 −3ξ+ 14λ2ξ)c†
xcx+ (1 −3ξ+ 4λ2ξ)c†
ycy
+ (5λ2−1)ξc†
xc†
x+cxcx+ (2λ2−1)ξc†
yc†
y+cycy.(17)
The Hamil onian (17) can be diagonalized wi h a Bogoliubo ans o ma ion
c†
i=uia†
i+ iai,
ci=uiai+ ia†
i,(18)
ha should be independen ly pe o med in he symme ic and de o med phases [18, 20]. The
inal esul p o ides he co ec ion o he g ound s a e ene gy pe pa icle in he symme ic and
de o med egions
Esym
0=ξ+N−1h3ξ−1 + Ξsym(ξ)1/2i,(19)
Ede
0=−9ξ2+ 10ξ−1
16ξ+N−1"1−6ξ−27ξ2+ 8ξΞde (ξ)1/2
16ξ#,(20)
whe e Ξsym = 5(ξc−ξ)(1 −ξ) and Ξde (ξ) = 5(ξ−ξc)(1 + 3ξ). In he de o med phase he y
coo dina e con ibu ion is a spu ious Golds one boson, associa ed wi h a g ound s a e o a ion.
Compa ed o he mean ield limi , he deduced ini e-size co ec ions g ea ly imp o e he
ag eemen wi h nume ical esul s. This can be clea ly seen in Fig. 1 whe e he mean ield and
beyond mean ield (BMF) co ec ions o he g ound s a e ene gy a e compa ed o a nume ical
calcula ion o N= 20.
The ini e-size co ec ion can be calcula ed o o he obse ables. We include esul s o he
expec a ion alue in he g ound s a e o ˆn(5). In his case we can make use o he Hellman-
Feynman heo em, de ining a new con ol pa ame e xsuch ha ξ= 1/(1 + x), we ob ain o
he symme ic and de o med phases
hˆni
N=d
dx [(1 + x)Esym
0(x)] = 1
N
1−3ξ−Ξsym(ξ)1/2
Ξsym(ξ)1/2+O(N−2),(21)
hˆni
N=d
dx h(1 + x)Ede
0(x)i=5ξ−1
8ξ+1
N
4ξ(ξ−1) + (1 −3ξ)Ξde (ξ)1/2
8ξΞde (ξ)1/2+O(N−2).(22)
We compa e his esul , he mean ield limi and he nume ical calcula ion o N= 20 in Fig.
2. As in he p e ious case, he ob ained co ec ion no ably imp o es he mean ield esul , in
pa icula in he icini y o he c i ical con ol pa ame e .
GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing
Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049
5

0.0 0.2 0.4 0.6 0.8 1.0
Con ol Pa ame e (ξ)
0.00
0.05
0.10
0.15
0.20
0.25
G ound S a e Ene gy pe Pa icle (ε0=E0/N)
Mean Field
BMF Symme ic N = 20
BMF De o med N = 20
Nume ical, N = 20
Figu e 1. (Colo online) G ound
s a e ene gy pe pa icle (E0) in
a bi a y uni s as a unc ion o he
con ol pa ame e ξ.
0.0 0.2 0.4 0.6 0.8 1.0
Con ol Pa ame e (ξ)
0.0
0.1
0.2
0.3
0.4
0.5
No malized O de Pa ame e (〈n(ξ)〉/N)
Mean Field
BMF Symme ic N = 20
BMF De o med N = 20
Nume ical N = 20
Figu e 2. (Colo online) Expec a ion
alue o he numbe ope a o ˆnin he
g ound s a e o Hamil onian (9) as a
unc ion o he con ol pa ame e ξ.
The esul s p esen ed a e encompassed in an open line o esea ch [21]. We a e cu en ly
wo king on compu ing he BMF co ec ion o o he obse ables, on analy ically de i ing ini e-
size scaling exponen s o he u(2) −so(3) second o de phase ansi ion, and on es ablishing a
connec ion be ween Re . [18] esul s and ou s.
4. Applica ion o he ν7bending mode o cyanogen iso hiocyana e
The bending dynamics o se e al molecula species [7, 9, 10, 22, 23, 24, 25] ha e been modeled
using he wo-dimensional limi o he ib on model. We a e cu en ly in e es ed in he analysis
o he spec a o non- igid molecules, as hese p obably p o ide examples o ESQPT. As he
exci a ion ene gy inc eases, he non- igid molecule’s exci ed s a es ha e o o e come a po en ial
hump in he o igin, and change om a ben -like o a linea -like cha ac e . This ac was known
long ago [26], bu nowadays such sys ems ha e ocused an inc easing deg ee o a en ion due o
quan um monod omy e ec s and i s implica ions [16].
We ha e pe o med a i ing o he bending ene gy spec um o cyanogen iso hiocyana e
(NCNCS), a non- igid molecule whose la ge ampli ude bending spec um displays monod omy
e ec s and is expe imen ally accessible [27]. We i Hamil onian (6) o he NCNCS e m alues
ob ained wi h he Gene al Semi igid Bende Hamil onian model in Re . [27]. The inal ms is
2.2 cm−1, in a i o 70 e m ene gies. The op imized pa ame e s a e gi en in Table 1. The
calcula ions we e pe o med using he FORTRAN code ia u3 [28].
Table 1. Op imized pa ame e s o he one- and wo-body Hamil onian (6) and hei associa ed
unce ain ies ob ained in he i o ν7 e m alues o NCNCS [27]. All pa ame e s, excep ing N,
a e in uni s o cm−1. The i oo mean squa e de ia ion is ms = 2.2 cm−1.
N ǫ α β A
70 203.6(19) -2.58(3) 1.496(10) 0.813(6)
The ob ained esul s a e p omising, especially i he simplici y o he model is conside ed.
GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing
Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049
6
The esul s in Re . [27] and ou i a e compa ed in a quan um monod omy plo in Fig. 3 whe e
he good ag eemen be ween bo h app oaches is ema kable. We ep oduce he change in slope
in he o igin ha cha ac e izes he c i ical monod omy poin .
-8 -6 -4 -2 0 2 4 6 8
Vib a ional Angula Momen um (K)
0
200
400
600
800
1000
Ene gy (cm-1)
This wo k
PRL 95 243002
Figu e 3. (Colo online) Quan-
um monod omy plo o calcula ed
ν7mode bending ib a ional le els o
NCNCS. The ib a ional angula mo-
men um lis labelled K ollowing he
usual spec oscopic no a ion.
We a e cu en ly wo king in he i o he ib a ional bending spec um o wa e , whe e
quan um monod omy e ec s ha e been expe imen ally epo ed [15].
5. In luence o he quad a ic u(2) Casimi ope a o in ESQPT
As shown in Table 1, he second o de Casimi ope a o o he u(2) subalgeb a plays an impo an
ole when i ing expe imen al da a. This also happens in o he cases [9, 10] and has sugges ed
us o explo e he ole o his anha monic e m in he model, in pa icula i s e ec s on g ound
s a e QPT and ESQPT [29].
This can be accomplished wi h a new model Hamil onian, con enien ly scaled, ha
inco po a es a second con ol pa ame e
ˆ
H=ε(1 −ξ)ˆn+α
N−1ˆn(ˆn+ 1) + ξ
N−1ˆ
P.(23)
The cohe en s a e app oach, when applied o his case, gi es as a esul ha he g ound s a e
ansi ion is mos ly unpe u bed by he addi ion o he new e m. The ene gy unc ional E0(ξ, α)
in he mean ield limi becomes
E0(ξ, α; ) = εξ+ (1 −3ξ) 2+ (1 + α) 4
(1 + 2)2.(24)
The minimiza ion o he ene gy unc ional (24) p o ides he equilib ium alues e= 0 ,q5ξ−1
3ξ+2α+1
ha implies a lowe bound α > −(1+3ξ)/2 i ξ > 0.2. The c i ical alue o he con ol pa ame e
is, once mo e, ξc= 0.2. Fo ξ≤ξc he ene gy unc ional minimum lies a he o igin, while o
alues o ξ > ξca new minimum appea s and he minimum a he o igin becomes a maximum.
When e alua ed o = eand ε= 1, he ene gy unc ional is
E0(ξ, α; e) = (ξ0≤ξ≤ξc
−9ξ2+10ξ+4αξ−1
16ξ+4αξc< ξ ≤1,(25)
which has a discon inuous second o de de i a i e in ξ=ξc.
Al hough he second o de g ound s a e ansi ion is basically una ec ed by he new con ol
pa ame e , he e a e conspicuous e ec s in he sys em’s ESQPT o nega i e α alues. This
GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing
Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049
7
is specially no iceable once he exci a ion ene gy diag am o he Hamil onian (23) is plo ed
o di e en α alues. Ins ead o one sepa a ix be ween he con igu a ion wi h linea and ben
cha ac e , like in he α= 0 case [11], he e a e wo sepa a ices. In Figs. 4 and 5 we plo he
exci a ion ene gy diag am o ze o angula momen um (l= 0) s a es o and α= 0 and −0.4,
espec i ely. The numbe o bosons in bo h cases is N= 100 and he ene gy is plo ed as a
unc ion o he con ol pa ame e ξ. The colo code in he cu es is assigned in acco dance o
he magni ude o he maximum squa ed-componen o he eigen ec o in he wo bases o choice.
I he eigen ec o is close o dynamical symme y (3), he maximal componen will be close o
one in he u(2) basis and he poin will be plo ed in ed colo . I he maximal componen is in
he so(3) basis he mapped colo is blue. This way o ep esen ing he da a clea ly ma ks he
s a es ha co espond o he sepa a ix in he monod omy plo , in he high le el densi y egion
ha sepa a es he linea - and ben -like s a es. The appea ance o wo sepa a ices in Fig. 5 is
ma kedly clea .
Figu e 4. (Colo online) Exci a ion
ene gy diag am in a bi a y uni s
(ε= 1) o ze o angula momen um
eigens a es o Hamil onian (23) as a
unc ion o he con ol pa ame e ξ
o α= 0. The numbe o bosons is
N= 100 and ene gies a e no malized
by N. The colo code indica es he
s a e p oximi y o he u(2) o so(3)
dynamical symme ies (see ex ).
Figu e 5. (Colo online) Exci a ion
ene gy diag am in a bi a y uni s
(ε= 1) o ze o angula momen um
eigens a es o Hamil onian (23) as a
unc ion o he con ol pa ame e ξ o
α=−0.4. The numbe o bosons is
N= 100 and ene gies a e no malized
by N. The colo code indica es he
s a e p oximi y o he u(2) o so(3)
dynamical symme ies (see ex ).
The posi ion o he wo sepa a ices in Fig. 5 depends on he alues o he ene gy unc ional
(25) maximum a he o igin and i s asymp o ic alue [29]. The alue a he o igin is only a
unc ion o ξ, while he asymp o ic alue depends only on α. In pa icula , he e is a c i ical
alue o α,αc=ξ−1, whe e he wo sepa a ices c oss and i is associa ed wi h equal alues o
he ene gy unc ional a ze o and o la ge alues.
In he case o ESQPT he e is no a clea de ini ion o an o de pa ame e [14]. The
expec a ion alue o he numbe ope a o ˆnin he di e en eigens a es p o ides a possible
app oxima ion. This obse able changes ab up ly in he c i ical exci a ion ene gy bu i does
no go o ze o in any o he phases [14]. We plo in Fig. 6 he expec ed alue o ˆn o he
eigens a es o Hamil onian (9) as a unc ion o he no malized s a e exci a ion ene gy. We ha e
GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing
Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049
8
ixed he numbe o bosons (N= 800) and he con ol pa ame e ξ(ξ= 0.6). Fo α= 0 ( ed
cu e in Fig. 6) we eco e he esul s published in [14], whe e he c i ical ene gy is ma ked by
he wa ped minimum in he plo ed unc ion. Fo 0 > α > αc, in addi ion o he displacemen
o la ge ene gies o he ini ial minimum, a maximum appea s o la ge exci a ion ene gies
(o ange cu e in Fig. 6). As α ends o αcbo h ex emes app oach. We ound he su p ising
esul ha o α=αc he expec a ion alue o ˆnbecomes cons an o he ull ene gy ange
excep a sudden a ia ion a ound he c i ical ene gy ha co esponds o he c ossing poin o
he wo sepa a ices (g een cu e in Fig. 6).
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Exci a ion Ene gy (E/N)
0.2
0.4
0.6
0.8
1.0
O de Pa am. [〈n(E)〉/Ν]
N = 800, ξ = 0.6
α = 0.0
α = -0.25
α = -0.4
Figu e 6. (Colo online) Expec a-
ion alue o he numbe ope a o ˆn
in he eigens a es o model Hamil-
onian (23) o N= 800 and ξ= 0.6
as a unc ion o he exci a ion en-
e gy o he s a e no malized by N
o h ee di e en alues o α.
6. Concluding ema ks
F om ou poin o iew, he wo dimensional limi o he ib on model is a simple algeb aic model
ha s ill encompasses enough complexi y o be applied o in e es ing eal physical sys ems.
The calcula ion o ini e-size co ec ions o he mean ield limi g ea ly deepens he knowledge
o he model. These co ec ions a e a undamen al help o cha ac e ize he QPTs be ween he
di e en geome ic limi s o a model and o analy ically ex ac he scaling exponen s o he
model, as i will be shown o he model unde s udy in a o hcoming publica ion [21].
We ha e applied he model o he bending dynamics o a non- igid molecule. Despi e he
simplici y o he app oach, i is capable o coping wi h complica ed si ua ions, as i has been
shown in he case o NCNCS, whe e quan um monod omy e ec s appea . We a e especially
in e es ed in he desc ip ion wi h his model o wa e ’s ib a ional bending le els.
Being he simples bosonic wo-le el model wi h a non- i ial angula momen um, his limi
o he ib on model is a e y con enien aid in he s udy o ESQPTs and hei implica ions.
The expe imen al access o bending exci a ion ene gies abo e he monod omy c i ical ene gy
makes his ea u e especially a ac i e. The ob ainmen o expe imen al da a o such s a es is
no possible in o he ields, as nuclea spec oscopy, which ha e been he adi ional es g ound
o g ound s a e ansi ions.
Acknowledgmen s
This wo k was suppo ed in pa by he Spanish Jun a de Andaluc´ıa unde p ojec s P07-
FQM-02962, P07-FQM-03014, and P07-FQM-02894 and by he Spanish MICINN and he
Eu opean egional de elopmen und (FEDER) unde p ojec s FIS2008-04189 and CPAN-
Ingenio (CSD2007-00042). PPF acknowledge he Spanish MEC o a FPU g an . The au ho s
hank Jo ge Dukelsky, F ancesco Iachello, Pie e an Isacke , and Ami am Le ia an o aluable
commen s and sugges ions.
GROUP 28: Physical and Ma hema ical Aspec s o Symme y IOP Publishing
Jou nal o Physics: Con e ence Se ies 284 (2011) 012049 doi:10.1088/1742-6596/284/1/012049
9