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Pavel Cejnar Institute of Particle & Nuclear Physics, Charles University, Prague

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### Monodromy & Excited-State Quantum Phase Transitions in Integrable Systems Collective Vibrations of Nuclei

Pavel Cejnar

Institute of Particle & Nuclear Physics, Charles University, Prague

NIL DESPERANDUM !

Monodromy(in classical & quantum mechanics): singularity in the phase space of a classical integrable system that prevents introduction of global analytic action-angle variables. Important consequences on the quantum level...

Quantum phase transitions: abrupt changes of system’s ground-state properties with varying external parameters. The concept will be extended to excited states...

Monodromy

Hamiltonian for f degrees of freedom:

f integrals of motions “in involution”

(compatible)

Action-angle variables:

The motions in phase space stick onto surfaces

that are topologically equivalent to tori

Monodromy in classical and quantum mechanics

Etymology:Μονοδρoμια= “once around”

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

Promoted: RH Cushman, L Bates: Global Aspects of Classical Integrable Systems

(Birkhäuser, Basel, 1997).

Simplest example:spherical pendulum

z

Hamiltonian

constraints

y

ρ

x

Conserved angular momentum:

2 compatible integrals of motions, 2 degrees of freedom

(integrable system)

point of unstable equilibrium

(trajectory needs infinite time to reach it)

trajectories with E=1, Lz=0

“pinched torus”

…corresponding lattice of quantum states:

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

q.num.#1: z-component of ang.momentum m

q.num.#2: ??? candidates: “principal.q.num.” n, “ang.momentum”l, combination n+m

low-E

high-E

m

m

m

“crystal defect”

of the quantum lattice

K Efstathiou et al., Phys. Rev. A 69, 032504 (2004).

Another example:Mexican hat (champagne bottle) potential

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

V

E=0

y

Pinched torusof orbits: E=0, Lz=0

x

radial q.num. n

principal q.num. 2n+m

Crystal defect of the quantum lattice

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

The ground-state energyE0 may be a nonanalytic function of η(for ).

For two typical QPT forms:

2nd order QPT

1st order QPT

But the Ehrenfest classification is not always applicable...

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

…corresponding tensor of momenta

neglect higher-order terms

neglect …

B

oblate

For zero angular momentum:

spherical

prolate

motion in principal coordinate frame

A

y

β

γ

2D system

x

Interacting boson model(from now on)

F Iachello, A Arima (1975)

s-bosons (l=0)

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

d-bosons (l=2)

Dynamical algebra:U(6)

…generators:

…conserves:

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

Dynamical symmetries (extension of standard, invariant symmetries):

U(5)

O(6)

SU(3)

[O(6), SU(3)]

triangle(s)

D Warner, Nature 420, 614 (2002).

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

Method by: RL Hatch, S Levit, Phys. Rev. C 25, 614 (1982)

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

____________________________________________________________________________________

● use of Glaubercoherent states

● classical Hamiltonian

complex variables contain coordinates & momenta

(12 real variables)

● boson number conservation (only in average)

10 real variables:

(2 quadrupole deformation

parameters, 3 Euler angles,

5 associated momenta)

fixed

● classical limit:

restricted phase-space domain

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

2 coordinates(x,y) or (β,γ)

● result:

Similar to GCM but with position-dependent

kinetic terms and higher-order potential terms

Phase diagramfor axially symmetric

quadrupole deformation

ground-state = minimum of the potential

IBM

critical exponent

1st order

Triple point

2nd order

Order parameter for axisym. quadrup. deformation:

β=0 spherical, β>0 prolate,β<oblate.

I II III

1st order

O(6)-U(5) transition (…from now on)

“seniority”

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.

Classical limit for J=0 :

kinetic energy Tcl

potential energy Vcl

J=0 projected O(5)

“angular momentum”

Available phase-space volume at given energy

connected to the smooth component of

quantum level density

Volume of the “enveloping” torus:

singular tangent

E

0

E0

β

Classification of trajectories by the ratio

of periods associated with

oscillations 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).

“bouncing-ball orbits”

(like in spherical oscillator)

Spectrum of orbits

(obtained in a numerical simulation

involving ≈ 50000 randomly selected

trajectories)

E

η=0.6

At E=0 the motions change their

character from O(6)- to U(5)-like

type of trajectories

E=0

R>3

“flower-like orbits”

(Mexican-hat potential)

M Macek, P Cejnar, J Jolie, S Heinze,

Phys. Rev. C 73, 014307 (2006).

R

O(6)

transitional

U(5)

(N=40)

energy

Μονοδρoμια

M Macek, P Cejnar, J Jolie, S Heinze,

Phys. Rev. C 73, 014307 (2006).

→seniority

all levels with J=0

1st order

2nd order

E

O(6)-U(5)

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

This problem (independently of the model) solved at most for the lowest states. Difficulty: in the classical limit excited states loose their individuality...

η

ground-state phase transition(2nd order)

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

N=40

E=0

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).

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

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

H oscillator with x-dependent mass:

O(6) quasi-dynamical symmetrybreaks down

once the edge of semiclassical wave function

reaches thend=0ornd=Nlimits.

O(6) limit

O(6)-U(5)

nd

i=1

i=2

N=60, v=0

For v=0eigenstates of

ground-state phase

transition

η=0.8

=> approximation holds for energies

below

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

( classical limit of |ψ(x)|2 )

Effect of m(x)→∞for x → –¼

Similar effect appears in the

β-dependence of velocity–1 in

the Mexican hat at E=0

1/β-divergence

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

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

i=1

i=10

N=80

|Ψ(nd )|2

i=20

i=30

v=0

↓

Eup=0

S Heinze, P Cejnar, J Jolie,

M Macek, PRC 73, 014306 (2006).

1 maximum

10 maxima

i=10

|Ψ(nd )|2

i=20

20 maxima

v=0

↓

Eup=0

i=30

30 maxima

S Heinze, P Cejnar, J Jolie,

M Macek, PRC 73, 014306 (2006).

i=1

i=10

quasi-O(6)

quasi-U(5)

N=80

|Ψ(nd )|2

i=20

i=30

v=0

↓

Eup=0

S Heinze, P Cejnar, J Jolie,

M Macek, PRC 73, 014306 (2006).

Any phase transitions for nonzero seniorities?

constant & centrifugal terms

For δ≠0 fully analytic evolution of the minimum β0 and min.energy Veff(β0)

=>no phase transition !!!

J=0 level dynamics for separate seniorities

N=80

excited states

ground state

v=0

continuous

2nd order

(probably without

Ehrenfest classif.)

no phase transition

v=18

- Quantum phase transitionsin integrablesystems: connection with monodromy
- Testing example: γ-soft nuclear vibrations [O(6)-U(5) IBM] - relation to other systems with Mexican-hat potential (Ginzburg-Landau model)
- Concrete results on quantum phase transitions for individual excited states:
- Open questions:
- Connection with thermodynamic description of quantum phase transitions?
- Extension to nonintegrable systems: is there an analog of monodromy?

- E=0 phase separatrix for zero-seniority states
- analytic evolutions for nonzero-seniority states

Collaborators:Michal Macek(Prague), Jan Dobeš(Řež),

Stefan Heinze, Jan Jolie(Cologne).

Thanks to:David Rowe(Toronto), Pavel Stránský (Prague)…

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