quantum phase transitions in nuclear collective models€¦ · · 2010-07-23quantum phase...

Quantum Phase Transitions in Nuclear Collective Models

(Links to Other Concepts)

Pavel CejnarInstitute of Particle & Nuclear Physics, Charles University, Prague

[email protected]

NIL DESPERANDUM !(every suffering has end (when new suffering starts (but through suffering we grow)))

“Now, Nina, do you think you could throw something into the sea?”“I think I could,” replied the child, “but I am sure that Pablo would throw it a great deal further than I can.”“Never mind, you shall try first.”Putting a fragment of ice into Nina’s hand, he addressed himself to Pablo: “Look out, Pablo; you shall see what a nice little fairy Nina is! Throw, Nina, throw, as hard as you can.”

Jules Verne: Off on a CometHector Sarvedac (1877)

Nina balanced the piece of ice two or three times in her hand, and threw it forward with all her strength.A sudden thrill seemed to vibrate across the motionless waters to the distant horizon, and the Gallian Sea had become a solid sheet of ice!

Quantum phase transitions (QPTs)phase transitions at zero temperature driven by interaction strengths

Lattice systems:Studied since 1920’s in connection with magnetism. Usually 2D spin grids that

exhibit a continuous order-disorder transition. Ising, Potts and many other models. Recent overviews: S Sachdev (1999), M Vojta (2003).

Many-body systems: mostly atomic nuclei • DJ Thouless (1961): “collapse of RPA”• R Gilmore (late 1970’s): “ground state (energy) phase transition”

Lipkin model (toy many-body model formulated in terms of quasispin operators) Dicke model (laser phase transition in an interacting atom-field system)

• AEL Dieperink, O Scholten, F Iachello (1980): Interacting boson model (nuclear shape transitions)

• WM Zhang, DH Feng, JN Ginocchio (1987): Fermion dynamical symmetry model(fermionic counterpart of IBM)

• DJ Rowe et al (1998), A Volya, V Zelevinsky (2003)....: various shell-like models describing the ground-state pairing transition

1) Phenomenology (properties of the mean field)Landau theory (1930’s): analysis of the free energy in terms of order parametersCatastrophe theory (1960’s): classification of structurally unstable potentials

2) Mathematical physics (mechanism responsible for QPTs)Pechukas-Yukawa approach: Coulomb-gas analogy for level dynamicsBranch points: level crossings in complex-extended parameter spaceMonodromy (integrable systems): singular class of orbits that disallows analytic description

3) Statistical physics (relation to thermodynamics)Excited-state (finite-temperature) phase transitionsThermodynamic phase transitions

Links to:

Part 1/4:

Nuclear “shape phases”- phenomenology

RF Casten et al (1993)

V Zamfir et al (2002)

Geometric model

Potential energy:

neglect higher-order terms neglect …

depends on 2 internal shape variables

A

Boblate

prolate

spherical432

2222322

3cos)()3()(

bgbb CBAyxCxyxByxAV

++=

++-++=

x

y

…corresponding tensor of momenta

βγ

gba

gba

sin|Re2

cos|Re

PAS2

PAS0

==

==

±y

x

=> nuclear shapes appear as “phases”

quadrupole tensor of collective coordinates (2 shape param’s, 3 Euler angles )

[ ] ( ) .....][5][2

35][5.....][2

5 2)0()0()2()0()0( + ́+´ ́- ́++ ́= aaaaaaapp CBAK

H

Interacting boson model

åå +++ +=lkji

lkjiijklji

jiij bbbbvbbuH,,,,

2,...,2{

+-=+

++ =

mmds

bis-bosons (l=0)

d-bosons (l=2)

• “nucleon pairs with l = 0, 2”• “quanta of collective excitations”

Dynamical algebra: U(6) jiij bbG +=

Subalgebras: U(5), O(6), O(5), O(3), SU(3), [O(6), SU(3)]

]SU(3)[]O(3)[]O(5)[]O(6)[]U(5)[]U(5)[ 2625242322110 CkCkCkCkCkCkkH ++++++=

…generators: …conserves: å +=i

ii bbN

F Iachello, A Arima (1975)

ÉÉÉÉÉÉÉÉ

SU(3)O(5)O(6)

O(3)O(5)U(5)U(6)Dynamical symmetries (extension of standard, invariant symmetries):

063 == kk

0621 === kkk

04321 ==== kkkk

U(5)

O(6)SU(3)

[O(6), SU(3)]

D Warner, Nature 420, 614 (2002).

SU(3)

U(5)

SU(3) O(6)

O(6)

triangle(s)

The simplest, one-component version of the model, IBM-1

21

0

00

||÷÷ø

öççè

æ -µ

lllb

1st order

2nd order

Phase diagram for axially symmetric quadrupole deformation

4

1

32 3cos bgbb ++=±321BAE

order parameter:β=0 spherical, β>0 prolate, β<oblate.

I II III

Triple point

ground-state = minimum of the potential

critical exponent

1st order

Landau (1930’s)

Catastrophe theory

Robert Gilmore: Catastrophe Theory for Scientists and Engineers (1981).“ground-state energy phase transitions”classical & quantal models

Further development:EC Zeeman, V Arnol’d, I Stewart...

The notion of structural stability:“Perturb the Hamiltonian and the topology does not change.”

Theory showing how “continuous causes” can lead to “discontinuous effects”

René Thom (1960’s)

Example:(from EC Zeeman’s lecture, 1995)

many other examples:physics, biology, medicine,economy, sociology, psychology...

bxaxxV ++= 24

Cusp catastrophe

Germ = x4

Codim (no.of free param’s) = 2KbLa µµ ,

1

2

3

4

3

3

4

Germ codim Universal unfolding

Arnol’d classification:

Name

fold

cusp

swallowtail

hyperbolic umbilic

butterfly

elliptic umbilic

parabolic umbilic

For parameter-dependent families of functions with codimension ≤ 5 there are just a few generic families to which the others can be reduced.

+ 4 others with codim=5For codim>5 the classification becomes infinite.

Cusp catastrophe:

432 bbb +-» BAVBy chosing (i) shift of V and (ii) shift of x (both depending on a,b) we can convert V to the standard cusp form. However,.....

In the shape-phase case we have:

-0.4 -0.2 0.2 0.4

-0.5

0.5

1

...the relevant sign of β is positive, so its negative branches must be disregarded:

spinodal

antispinodal

critical

−a

b

mapping of the shape evolution with),(,1 +¥-¥Î= AB onto the cusp parameter space

deformed

spherical

Part 2/4:

Branch points

Ground-state quantum phase transition ( T=0 QPT )

( ){ {0],[ 0

00

)1()0()1(0]1,0[ ¹ +Î

+-=+=

VH VHH

HHVHH lllll

321

02)(

)(

0 0

00

0000

)()(

2)()(

)()(

||2

2£-=

º=

å> -

YY

YY

i i

i

EEdd

dd

VE

VVE

ll

lll

lll

l

l

For two typical QPT forms:

The ground-state energy E0 may be a nonanalytic function of λ (for ).

0V

0V

l l

2nd order QPT 1st order QPT

0 0

¥®dim

There are more possibilities and the Ehrenfest classification is not always applicable...

00

³V

úûù

êëé ×

--= )()(1),( d cchhwch QQ

NnH h

ddn ~d ×= +

)2(]~[~)( dddssdQ ´++= +++ cc

Pairing+Quadrupole Hamiltonian

d-boson number operator

quadrupole operator

scaling constant ħω=1 MeV

control parameters η, χsymmetry triangle

)()()()(

1d

10

0

cccc

h

QQnVQQH

VHH

N

N

×+º

×-º

+=

ensures that the thermodynamiclimit exists: N→∞

c

c-

l

No-crossing rule: In generic situations, eigenvalues of H(λ) with the samesymmetry quantum numbers do not cross:

Instead, one can observe avoided crossings of levels.Rapid structural changes of eigenstates take place atthese sites:

Finite-size precursors of the QPTs are connected with avoided crossings of levels involving the ground-state. These crossings become infinitely closeas the size of the system asymptotically increases...

l[ ]å

¹ --»+

)(2

2

22

)()(

)()()(1)()(

ij ji

jiii EE

V

ll

lylydldllyly

)()( 1 ll +¹ ii EE

)(ly i

)(liE

eigenfunctions

eigenvaluesHamiltonian VHH ll += 0)(

Hill, Wheeler (1953)

QPT mechanism?

QPT mechanism? Multiple avoided crossing of levels

structure I

structure IIg.s.

lcl

Example

Branch pointsTrue degeneracies in the complex plane of λ.They are determined by the conditions

that after elimination yield:

discriminant

2)1( -nn

This has complex conjugate pairs of solutions, where n is dimension of the Hilbert space. For large n numerical calculations become prohibitively difficult.

partial discriminant

The simplest case(dim=2)

0=c

27-=c

20=N

2nd order QPT

1st order QPT

IBM Hamiltonian

withBranch points for J=0

SU(3)-U(5)

O(6)-U(5)

P Cejnar, S Heinze, J Dobeš, AIP Conf. Proc. (2005)

Solution of det[E−H(λ)]=0 → single function living on n interconnected Riemann sheetsCrossings = complex square root type singularitiesBranch points can be sorted to n Riemann sheets

associated with individual levels for λ real (each sheet contains n−1 complex conjugate pairs)

For the ground-state QPT onlythe ground-state Riemann sheetis relevant.

Excursion to thermodynamic phase transitions (TPTs):Zeros of partition function in the complex-temperature plane

CN Yang, TD Lee (1952)S Grossmann, W. Rosenhauer, V Lehmann (1967,69) .....P Borrmann et al (2000) ......

Specific heat = an indirect measure of the density of complex zeros close to real T

Let us assume that there exists an λ↔T analogy between the branch points on the g.s. Riemann sheet and complex zeros of Z, namely, let us consider

1) the ground-state partial discriminant D0 being an analog of Ωth power of thepartition function (where ):

2) the 2D “Coulomb-gas energy” being proportional to free energy:

3) the following expression being an analog of specific heat:

1-µW n

P Cejnar, S Heinze, J Dobeš, PRC 71, 011304(R), (2005).

SU(3)-U(5) 27-=c

O(6)-U(5) 0=c1st order QPT

2nd order QPT

N=10,20,40,80, ...

growth of the maximal value for 1-µW n

1=W

1=W

P Cejnar, S Heinze, J Dobeš, PRC 71, 011304(R), (2005).

1) Branch points accumulateclose to real-λ axis at QPTs

2) The rate of accumulationis quantitatively the same as the accum.of Z=0 pointsclose to real-T axis at TPTs !!!

Part 3/4:

Finite temperatures

Thermodynamic phase transitions (temperature-driven)

Helmholtz free energy 43421321rr

rrrr

ˆˆ

)ˆlnˆ(rt)ˆˆ(rt)(ˆ

SE

THTF --=

)(ln)(0 TZTTF -=

Thermal population of states [ ])ˆexp(tr)()ˆexp()(

1)(ˆ 110 HTTZHT

TZT -- -=-=r

canonical partition function

equilibrium free energy

0)]ˆˆ(tr)ˆˆ(tr[)(

)ˆlnˆ(tr)(

222

022

00

000

322 1

0

³-=

º=

-ºD

-

444 3444 21

TTTEEE

TdTd

dTd

HHTF

TF Srr

rrEntropy is a nondecreasing function of T

In the thermodynamic limit ( ), the entropy (thus F0) may be nonanalytic

2nd order TPT 1st order TPT0S 0S

TT

VNN

/|¥®

Thermodynamic phase transition Quantum phase transition (T=0)

What happens at T>0 ?

( ){ {0],[ 0

00

)1()0()1(0]1,0[ ¹ +Î

+-=+=

VH VHH

HHVHH lllll

321

λ

T F(λ,T)

λ

E

unavoided crossingof the lowest states

isolated structural phase transition at T=0

Possibility 1: isolated QPT at T=0… nongeneric situation

Example:

Finite-T extension of a quantum phase transition

λ

T F(λ,T)

continuous phase separatrix in the (λ,T) plane

TETF

STF

TT

T2

0

00

322 1),(

),(

D=

=

¶¶

¶¶ -

l

l

Possibility 2: QPT survives inclusion of finite T

[ with possible endpoint (λt,Tt) ]

?),(

?),(

0

0

22

=

=

¶¶¶¶

TF

TF

l

l

l

l

… expected situation


Thermodynamic phase transition Quantum phase transition (T=0)


“Excited-state quantum phase transition” (T>0)

λ

T F(λ,T)

Nonzero density of Z=0 pointsas ImT→0 (thermodynamic limit)

Nonzero density of branch pointsas Imλ→0 (thermodynamic limit)Surmise: )(

),()( br

)()(

br

1

lrl

lrl

(i)

i

ETT

TZe i

å--

=density on the i th Riemann sheet,or:

T>0 relevant density

?


)(lim),(lim )(brIm0Im

)(br0Im

lrlr lll

ii¶

¶

®®

Part 4/4:

Monodromy andexcited-state QPTs

in integrable systems

N=80all levels with J=0

ground-state phase transition (2nd order)

What about “phase transitions” for excited states (if any) ???

E

η

in the sense of nonanalytic evolutions of and

2nd order

1st order

O(6)-U(5)

)()( lyl iiE

General analytic approach: To employ the basis(condensate of N−k bosons) plus

(k bosons in single-particle states)with k=0...N and evaluate the dependence ofeigenvalues on λ. (Work in progress: F Iachello, M Caprio...)

Approach discussed here: based on some analogies in classical mechanics (monodromy)and on some specific analytic approximations(shifted oscillator approximation). Not systematicand applicable only in integrable systems [i.e., along O(6)−U(5) in the IBM], nevertheless still yielding some new insights...

N=80all levels with J=0

E

ηground-state phase transition (2nd order)

Integrable systems

{ } { } 0,,0,...1),(

),(

===

lkk

iik

ii

CCCHfkpxC

pxH Hamiltonian for f degrees of freedom:

),...,,(

),...,,(

21

21

fi

fi

ppppxxxx

º

º

f integrals of motions “in involution”(compatible)

Action-angle variables:

const)()0()(

0 =Q+=Q

==Q

þýü

îíìQ

®þýü

îíì Þ

tItt

IIpx

i

iii

idtd

iidtd

i

i

i

i ww

The motions in phase space stick onto surfaces that are topologically equivalent to tori

Monodromy in classical and quantum mechanics

Invented: JJ Duistermaat, Commun. Pure Appl. Math. 33, 687 (1980).

Promoted: RH Cushman, L Bates: Global Aspects of Classical Integrable Systems(Birkhäuser, Basel, 1997).

Simplest example: spherical pendulum

x

z

y

Etymology: Μονοδρoμια = “once around”

ρ

( ) zpppH zyx +++= 22221

01222

=++=++

zyx zpypxpzyx

Hamiltonian

constraints

Conserved angular momentum: xyz ypxpL -=

{ } Þ= 0, PoissonzLH 2 compatible integrals of motions, 2 degrees of freedom(integrable system)

-1 -0.5 0.5 1

-2

-1

1

2

r

rp

trajectories with E=1, Lz=0

Singular bundle of orbits: point of unstable equilibrium(trajectory needs infinite time to reach it)

“pinched torus”

…corresponding lattice of quantum states:

It is impossible to introduce a global system of 2 quantum numbersdefining a smooth grid of states:

q.num.#1: z-component of ang.momentum mq.num.#2: ??? candidates: “principal.q.num.” n, “ang.momentum” l, combination n+m

K Efstathiou et al., Phys. Rev. A 69, 032504 (2004).“crystal defect”of the quantum lattice

m m m

low-E high-E

Another example: Mexican hat (champagne bottle) potential

x

y

V

E=0

principal q.num. 2nrad+mradial q.num. nrad

Pinched torus of orbits: E=0, Lz=0

Crystal defect of the quantum lattice

-1 -0.5 0.5 1

-0.6

-0.4

-0.2

0.2

0.4

0.6

rrp

( ) 422221 rr bappH yx +-+=

MS Child, J. Phys. A 31, 657 (1998).

( ))

~~)(()3(

~)0(

~

dd2

21

d

)]5([ ddddnnC

ssddPdssdQ

ddn

O ××-+=

-×=

+=

×=

++

++++

++

+

O(6)-U(5) transition

hhhh

34'34

-»-»a

{ })4(1'1'

)0()0(1)0,(

)]5(O[222d

2d

+--

+÷øö

çèæ -

+=

×-

-==

+ NNCN

PPN

nN

a

QQN

nNN

H

hhh

hhch

)3( +vv “seniority”

42222cl )1(

245)1(

2bhbhpbhph

-+-

+-+=® HNH

kinetic energy Tcl potential energy Vcl

22222

÷÷ø

öççè

æ+=+=

bp

pppp gbyx

222 yx +=b

Classical limit for J=0 :

J=0 projected O(5) “angular momentum”

xyNv yx pppg -=«

22

10

)]5([O2 gp¾¾ ®¾=J

CN

The O(6)-U(5) transitional system is integrable: the O(5) Casimir invariant remains an integral of motion all the way and seniority v is a good quantum number.

]1,0[],2,0[],2,0[ ÎÎÎ gb ppb

422422

cl

11 )~

()~

(2

bbpb +-=

+×+-= ++++

H

dssddssdHNN

2212

21

cl

11

bp +=

=

H

nH dNN

422

452222cl

11

)1()1(

)0()0(2

bhbpbhp hh

hh

-++-+=

×-=-

-

H

QQnHNdNN

O(6) U(5)

0 14/5h

O(6)-U(5) transition

deformed g.s. spherical g.s.

η=0.6

pinched torus

Poincaré surfaces of sections:

22-

Μονοδρoμια

M Macek, P Cejnar, J Jolie, S Heinze, Phys. Rev. C 73, 014307 (2006).

( ) )()(ˆTr )( atoryllioscsmooth

Ω

EEHEE(E)

rrdr +=-=

µ43421

Available phase-space volume at given energyconnected to the smooth component of quantum level density

ò=W)(

)(

max

min

);(2 max

);(2)(E

E

dEhE

Ep

b

b

bbbp

bb

321

)(EW

minb maxb

β

E

0

E0

singular tangent

)(EdEd

W

( )ò -=W ff dqdpqpHEE ),()( d

Volume of the “enveloping” torus:

P Cejnar et al., subm.to J.Phys.A

g

b

b

g

w

w==

TT

RClassification of trajectories by the ratio of periods associated with

g

b

mm

=Roscillations in β and γ directions. For rational the trajectory is periodic:

M Macek, P Cejnar, J Jolie, S Heinze,Phys. Rev. C 73, 014307 (2006).

R

E

E=0

Spectrum of orbits(obtained in a numerical simulation involving ≈ 50000 randomly selectedtrajectories)

η=0.6

R≈2“bouncing-ball orbits”(like in spherical oscillator)

R>3“flower-like orbits”(Mexican-hat potential)

At E=0 the motions change theircharacter from O(6)- to U(5)-liketype of trajectories

M Macek, P Cejnar, J Jolie, S Heinze,Phys. Rev. C 73, 014307 (2006).

O(6) transitional U(5)

→seniority

ener

gy

Lattice of J=0 states(N=40)


E=0

J=0 level dynamics across the O(6)-U(5) transition (all v’s)

N=40

S Heinze, P Cejnar, J Jolie, M Macek,Phys. Rev. C 73, 014306 (2006).

Wave functions in an oscillator approximation:DJ Rowe, Phys. Rev. Lett. 93, 122502 (2004), Nucl. Phys. A 745, 47 (2004).

)()()()(

)()()()(

2

2

22111

11 22

xxxx

xExxx

EH

idxd

Nidxd

NiNi

iiNiddNiddidd

iii

nHnnHnnHn

yyyy

yyyy

+±=±

=-+++

Y=Y

-+

)(12 xnNnx iid

d yºY-=

H oscillator with x-dependent mass:

N=60, v=0i=1 i=2

O(6) limit

O(6)-U(5)

x may be treated as a continuous variable (N→∞)

Method applicable along O(6)-U(5) transition for N→∞ and states with rel.seniority v/N=0 :

O(6) quasi-dynamical symmetry breaks down once the edge of semiclassical wave function reaches the nd=0 or nd=N limits.

nd

¾¾¾ ®¾¥®N ( ) 0

20osc )( ExxL

dxdxK

dxdH +-+-=

)0()0(1)(2d QQ

Nn

NNH

×-

-=hhh

}

( )44444 844444 76

443442143421

876h

)(

)()(

)(

2

2

2osc11

00

1

2)1(4

452)1(4

14)( ][][xV

Ex

xm

xxd

dxxd

dN

H

h

hh

h hh

hhh

µ

µ

µ

- --

--

-+--=

-

For v=0 eigenstates of we obtain:

h

-0.2

0

0.2 0

0.25

0.5

0.75

1

-1-0.75-0.5

-0.250

-0.2

0

0.2

h

0)( £xVh

x

0.2

0.4

0.6

0.8

-0.2

-0.10.1

0.2 x

h

[ ] [ ]maxmin2 ,1,11 xxxx N

nd º+-ÎÞ-=

0.2 0.4 0.6 0.8

-1

-0.8

-0.6

-0.4

-0.2

ground-state phasetransition

=> approximation holds for energies below 0)()( minup == xVE hh

η=0.8

)(0 hE1-


At E=0 all v=0 states undergo a nonanalytic change.

-0.25 -0.2 -0.15 -0.1 -0.05 0.05

5

10

15

20

25

30

35

x1-

0=E 0<E

x-dependence of velocity–1( classical limit of |ψ(x)|2 )

)(1

minxx -µ 2/1

end )(1xx -

µ

At E=0 all v=0 states undergo a nonanalytic change.

Effect of m(x)→∞ for x → –1

Similar effect appears in the β-dependence of velocity–1 in the Mexican hat at E=0

1/β-divergence 0001 =«=«-= bdnx

ßIn the N→∞ limit the average <nd >i →0 (and <β >i →0) as E→0.


i=1 i=10

i=20 i=30|Ψ(nd )|2

å-=dn

didii nPnPS )(ln)(U(5)

U(5) wave-function entropy

quasi-O(6)quasi-U(5)

v=0↓

Eup=0

S Heinze, P Cejnar, J Jolie, M Macek, PRC 73, 014306 (2006).

N=80

4444444 34444444 21

444 8444 76

)(

)1(2

452

)1()1(2

eff

)(

422

2222

cl

b

bhbhb

hddhpbhhb

b

V

H

V

-+-

++-+úûù

êëé -+=

( ) 2222

10

)]5([O2 gpd ¾¾ ®¾º«=JN

vN

C

constant & centrifugal terms

è!!!!20.1

0.2

0.3

0.4

0.5

0.6

è!!!!2-0.1

0.1

0.2

0.3

è!!!!2-0.1

0.1

0.2

0.3

0.4

0.5

0.6

4Nv = 2

Nv =0=veffV

b

422

22cl

1 )1(2

45)1(20

bhbhbp

pbhh gb -+

-+

úúû

ù

êêë

é÷÷ø

öççè

æ+úû

ùêëé -+=¾¾ ®¾

=HH

JN

Any phase transitions for non-zero seniorities?

0 0.2 0.4 0.6 0.8 1

0.5

0.6

0.7

0.8

0.9

1

d

10 =b

h 10

For δ≠0 fully analytic evolution of the minimum β0 and min.energy Veff(β0)=> no phase transition !!!

effV


N=80

v=0

v=18

2nd ordercontinuousground stateexcited states

0up =E

J=0 level dynamics for separate seniorities

no phase transition

(maybe with noEhrenfest classif.)


Surprising analogy:

M Macek et al, in preparation

O(6)-U(5)

“semi-regular arc”

O(6)SU(3)

U(5)

Does this imply similar QPT behavior in the arc? How to define monodromy in nonintegrable regimes?Phase transitions for other paths in the triangle?

(Alhassid et al, 1991)

J=0N=40

η

Collaborators: Michal Macek, Pavel Stránský (Prague)

Jan Dobeš (Řež)

Stefan Heinze, Jan Jolie (Cologne)

Thanks to: Vladimir Zelevinsky (Michigan)

David Rowe (Toronto)Franco Iachello, Rick Casten, Yoram Alhassid (Yale)

Zdeněk Pluhař (Prague)

.................

APPENDICES

n1=nrad+v/3

n2=nrad+2v/3

radial quantum number nrad

principal quantum number N=2nrad+m

angular-momentum quantum number m0 2 3 41

0

01

1

2

2

3

3

4

4… yes, but only for nd=3k

differences between the O(2) and J=0 projected O(5) angular momenta

*

*m

E

[V(V+3)]½

E

2212

21

cl

11

bp +=

=

H

nH dNN

Quantum lattice of states: U(5) limit

Analogy with standard isotropic 2D harmonic oscillator:_________________________________________________________

N=40

O(5) quantum number: seniority v

O(6) quantum number: σ

O(6) limit

422422

cl

11 )~

()~

(2

bbpb +-=

+×+-= ++++

H

dssddssdHNN

[V(V+3)]½

N=40

Transitional case

O(6)-liketype of cells

U(5)-liketype of cells

Redistribution of levels between O(6) and U(5) multiplets

E=0

422

45

2222cl

11

)1()1(

)0()0(2

bhbpbhp

h

h

hh

-++-+=

×-=

-

-

H

QQnHNdNN

[V(V+3)]½

ΜονοδρoμιαPinched torus of E=v=0orbits connected with the unstable equilibrium at β=0


N=40

≈2x

v=0

v=18

EnergySpacing of neighboring levels

N=80

J=0 level dynamics across the O(6)-U(5) transition (separate v’s)

“shock wave”

S Heinze, P Cejnar, J Jolie, M Macek,Phys. Rev. C 73, 014306 (2006).

( )( )

( )[ ]421

1,,3

2

1fluct cos p

mp

mm

mp n

mr --= åå

¥

=

rrh

r

r

hh

rESrgr

TE

r E

Berry-Tabor trace formula(an analog of the Gutzwiller formula, but for 2D integrable systems)

( )Efluctr

mrT

( ) ( )mpmrr

r ×= IES 2

( )21, mmm =r

2

1

2

1

mm

ww

=

… fluctuating part of level density

…pair of integers characterizing periodic orbit with ratio of frequencies

…period of the primitive orbit

r …number of repetitions

…action per period

( )1IgE ( )2

1,21, w

w-=Þ== EE gEgIIH…function defined by

mn r …Maslov index of the primitive orbit

1st10th

20th30th

average numbers of d-bosons

v=0

v=18

Heinze, Cejnar, Jolie, Macek,PRC (2006).

IBM classical limitGeneral method: coherent states (Schrödinger, Glauber, Gilmore, Perelomov,....) Specific realization: RL Hatch, S Levit, Phys. Rev. C 25, 614 (1982)

Y Alhassid, N Whelan, Phys. Rev. C 43, 2637 (1991)____________________________________________________________________________________

● use of Glauber coherent states

aa HH =cl

å +=ºm

maaaa 22s ||||NN

● boson number conservation (only in average)

complex variables contain coordinates & momentaa

● classical limit:

[ ]mmmaa -- +-=® ipq

N)(

21 ( ) 2£+å 22

mmm qp

¥®N

fixed

10 real variables:(2 quadrupole deformation parameters, 3 Euler angles,5 associated momenta)

0)exp( s|| 2

21

å ++- +=m

ma aaa dse

● angular momentum J=0 Euler angles irrelevant only 4D phase space

(12 real variables)

2 coordinates (x,y) or (β,γ)

restricted phase-space domainÞ Þ

● classical Hamiltonian

● result: ( )4

232

22

3cos1

),,,(21

2

bgbb

ppgbp

b

gbbp

bg

CBAV

fK

T

+-+=

+úûù

êëé += Similar to GCM but with position-dependent

kinetic terms and higher-order potential terms

[ ]2,0Îb

quantum phase transitions in nuclear collective models€¦ · · 2010-07-23quantum phase...

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