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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
[email protected]
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...
Part 1/4:
Monodromy
Integrable systems
H ( xi , pi )
Hamiltonian for f degrees of freedom:
xi  ( x1 , x2 ,..., x f )
pi  ( p1 , p2 ,..., p f )
Ck ( xi , pi )
k  1... f
f integrals of motions “in involution”
(compatible)
H , Ck   0, Ck , Cl   0
Action-angle variables:
 xi  i 
  
 pi   I i 
i  i
d
dt I i  0
d
dt

i(t )  i t  i(0)
I i (t )  const
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:
Promoted:
JJ Duistermaat, Commun. Pure Appl. Math. 33, 687 (1980).
RH Cushman, L Bates: Global Aspects of Classical Integrable Systems
(Birkhäuser, Basel, 1997).
spherical pendulum
Simplest example:
Hamiltonian
constraints

z

H  12 px2  p y2  pz2  z
x2  y 2  z 2  1
xpx  yp y  zp z  0
Conserved angular momentum:
H , Lz Poisson  0

ρ
Lz  xpy  ypx
y
x
2 compatible integrals of motions, 2 degrees of freedom
(integrable system)
p
2
Singular bundle of orbits:
point of unstable equilibrium
1
(trajectory needs infinite time to reach it)
trajectories with E=1, Lz=0
“pinched torus”
-1
-0.5
0.5
-1
-2
…corresponding lattice of quantum states:
1

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
m
“crystal defect”
of the quantum lattice
high-E
m
m
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


H  12 px2  p y2  a 2  b 4
E=0
y
Pinched torus of orbits: E=0, Lz=0
0.6
0.4
x
p

0.2
-1
-0.5
0.5
-0.2
-0.4
-0.6
Crystal defect of the quantum lattice
1
radial q.num. n
principal q.num. 2n+m
Part 2/4:
Quantum phase transitions
& nuclear collective motions
Ground-state quantum phase transition ( T=0 QPT )
H 
   H0   V  (1   ) 
H (0)  H
(1)



 [ 0,1]
H 0V
H0
[ H0 , V ]  0
The ground-state energy E0 may be a nonanalytic function of η (for dim  ).
d E ( )   ( ) V  ( )  V
0
0
d 0
d2
d 2
For V
V
0
E0 ( )  2 
|  ( ) V
i 0
i
0
0 ( )
Ei ( )  E0 ( )
|2 0
 0 two typical QPT forms:
2nd order QPT
0
0

V
1st order QPT
0
0
But the Ehrenfest classification is not always applicable...

Geometric collective model
quadrupole tensor of collective coordinates (2 shape param’s, 3 Euler angles )
…corresponding tensor of momenta
H

5
35
[   ]( 0 )  .....  5 A[   ]( 0 ) 
B [   ]( 2 )  
2K
2

(0)

 5C [   ]( 2 )
For zero angular momentum: J  i 10[   ]  0
* (1)
B
V  A 2  B 3 cos 3  C 4
motion in principal coordinate frame
x  Re     cos 
y
y  2 Re  2   sin 
2D system
x
 .....
oblate
1
( x2   y2 )  A( x 2  y 2 )  B( x 3  3 y 2 x)  C ( x 2  y 2 ) 2
2K
β
γ
2
neglect …
neglect higher-order terms
H

spherical
prolate
A
Interacting boson model
(from now on)
H   uijbi b j   vijklbi b j bk bl
i, j

i
i , j , k ,l
s
{d
b 
F Iachello, A Arima (1975)


s-bosons (l=0)
• “nucleon pairs with l = 0, 2”
• “quanta of collective excitations”
  2,...,2 d-bosons (l=2)
Dynamical algebra: U(6)
…generators: Gij  bib j
…conserves: N 
b b

i i
i
Subalgebras: U(5), O(6), O(5), O(3), SU(3), [O(6), SU(3)]
H  k0  k1C1[ U(5)]  k2C2 [ U(5)]  k3C2 [O(6)]  k4C2 [O(5)]  k5 C2 [O(3)]  k6 C2 [SU(3)]
Dynamical symmetries (extension of standard, invariant symmetries):
U(6)  U(5)  O(5)  O(3)
 O(6)  O(5) 
 SU(3) 
k3  k 6  0
U(5)
k1  k2  k6  0
k1  k2  k3  k4  0
O(6)
SU(3)
[O(6), SU(3)]
inherent structure:
triangle(s)
U(5)
D Warner, Nature 420, 614 (2002).
The simplest, one-component
version of the model, IBM-1
O(6)
SU(3)
SU(3)
O(6)
IBM classical limit
Method by: 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
 e
 12 | |2
exp( s s      d  ) 0

● classical Hamiltonian
Hcl   H 
complex variables
 contain coordinates & momenta
(12 real variables)
● boson number conservation (only in average)
N   N  |  s |   |   |
2
● classical limit:
N

10 real variables:
(2 quadrupole deformation
parameters, 3 Euler angles,
5 associated momenta)
fixed

N 
  
2
1
2
()

q  ip 

  p  q   2


restricted phase-space domain
● angular momentum J=0  Euler angles irrelevant  only 4D phase space
2 coordinates (x,y) or (β,γ)
● result:
T
1
2K
 2 
 
    f ( ,  , 
 2


,  )
V  A 2  B 3 1  2 cos 3  C 4
2
Similar to GCM but with position-dependent
kinetic terms and higher-order potential terms

  0, 2

Phase diagram for axially symmetric
quadrupole deformation
GCM
ground-state = minimum of the potential
 |   0 | 


0


 0  
1
2
IBM
critical exponent
1st order
Triple point
2nd order
E  A 2  B 3 cos 3  C 4  ...

1
Order parameter for axisym. quadrup. deformation:
β=0 spherical,
I
β>0 prolate,
II
β<oblate.
III
1st order
Part 3/4:
Monodromy for integrable
collective vibrations
O(6)-U(5) transition
~
nd  d   d
~
Q ( 0)  d  s  s  d
P 
1
2
d

 d   ss

(…from now on)
H ( ,   0) 
1 
 nd  2 Q(0)  Q(0)
N
N
N
1  '
 '
 1 
 a  nd  2 P  P   2 C2 [ O(5)]  N ( N  4)
N
N
 N
~ ~ a  4  3
C2 [O (5)]  nd (nd  3)  (d   d  )( d  d )
' 

v(v  3) “seniority”
4  3
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 :
H
 H cl    2  (1   )  2 2  5  4  2  (1  )  4
N
2
2
kinetic energy Tcl
potential energy Vcl
 2  x2  y 2

 2   x2   y2   2   




2
 [0, 2 ],   [0, 2 ],   [0,1]
1
N2
v
N
C2 [O(5)]  2
J 0
 
 x y  y x
J=0 projected O(5)
“angular momentum”
O(6)-U(5) transition
1
N
~
~
H   N12 (d  s  s  d )  (d  s  s  d )
1
N
O(6)

0
1
N

H 
H cl 
N

2
1
N
nd
H cl  12  2  12  2
H cl   2 2  42  2   4
deformed g.s.
H
spherical g.s.
4/5
nd  1N2 Q(0)  Q(0)
 2  (1   )  2 2  524  2  (1   )  4
U(5)
1
Μονοδρoμια
Poincaré surfaces of sections:
η=0.6
pinched torus
 2
2
Available phase-space volume at given energy
( E )    E  H ( p, q) dp f dq f
connected to the smooth component of
quantum level density


 ( E )  Tr  E  Hˆ  smooth ( E )   osci llatory( E )



 Ω(E)
Volume of the “enveloping” torus:
 max( E)
( E ) 

2  h(  ; E ) d

 min (E )
2 pmax( ; E )
 max
 min
d
(E )
dE
singular tangent
E
0
E0
β
(E )

T
Classification of trajectories by the ratio R 

T

oscillations in β and γ directions. For rational
R


of periods associated with
the trajectory is periodic:
M Macek, P Cejnar, J Jolie, S Heinze,
Phys. Rev. C 73, 014307 (2006).
R≈2
“bouncing-ball orbits”
(like in spherical oscillator)
Spectrum of orbits
(obtained in a numerical simulation
involving ≈ 50000 randomly selected
trajectories)
η=0.6
E
At E=0 the motions change their
character from O(6)- to U(5)-like
type of trajectories
E=0
R>3
“flower-like
M Macek, P Cejnar,
J Jolie, Sorbits”
Heinze,
R
Phys. Rev. C 73,(Mexican-hat
014307 (2006).
potential)
O(6)
transitional
U(5)
energy
Lattice of J=0 states
(N=40)
Μονοδρoμια
M Macek, P Cejnar, J Jolie, S Heinze,
Phys. Rev. C 73, 014307 (2006).
→seniority
Part 4/4:
Excited-state quantum phase
transitions for integrable vibrations
N=80
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=0 level 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:
1 n 1
x  d  
2 N 2
nd i   i ( x) x may be treated as a continuous variable (N→∞)
H i  Ei i
nd H nd
 i ( x) 
nd H nd  2
 i ( x  N1 )   i ( x)  N1
H

N 
d
dx
 i ( x  N1 ) 
 i ( x)  2 N1
2
d2
dx 2
nd H nd  2
 i ( x  N1 )  Ei  i ( x)
 i ( x)
oscillator with x-dependent mass:
d2
2
H osc   K ( x) 2  Lx  x0   E0
dx
O(6) limit
O(6) quasi-dynamical symmetry breaks down
once the edge of semiclassical wave function
O(6)-U(5)
i=1
nd
N=60, v=0
i=2
reaches the nd=0 or nd=N limits.
For v=0 eigenstates of
H ( ) 
1 

nd  2 Q(0)  Q(0)
N
N
N
2
1
1 2 m ( x )
1
1
we obtain:
1
1
V ( x)





2
2
2 5  4 2
1  16 x d

H osc ( )  

16
[
x

]
[
]
4N 2 d x2
16(  1)
4 (1   )


 


1
x0 ( )
1 E0 ( )
V ( x )  0
ground-state phase
transition

0.8

0.2
0.6
0
η=0.8

-0.8
0.2
0.75
0.5
-1
E0 ( )
-0.2
x  12

nd
N

-0.2
x
-0.1
0
0.1
0.2
0.2
x
0.25
0
0.8
-0.6
0.4
1
0.6
-0.2
-0.4
-0.25
-0.5
-0.75
-1
0.4
 21  x   14 , 14   xmin , xmax 
=> approximation holds for energies
below Eup ( )  V ( xmin )  0
At E=0 all v=0 states undergo a nonanalytic change.
x-dependence of velocity–1
( classical limit of |ψ(x)|2 )
E 0
E0
35
30
1

( x  xmin )
∞
Effect of m(x)→

25
1
( x  xend )1/ 2
20
for x → –¼
15
Similar effect appears in the
β-dependence of velocity–1 in
the Mexican hat at E=0
x   14  nd  0    0
1/β-divergence
10
5

-0.25
1
4

-0.2
-0.15
-0.1
-0.05
x
0.05
0
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.
SiU(5)   Pi (nd ) ln Pi (nd )
nd
i=1
i=10
U(5) wave-function entropy
N=80
i=20
i=30
|Ψ(nd )|2
v=0
↓
Eup=0
S Heinze, P Cejnar, J Jolie,
M Macek, PRC 73, 014306 (2006).
i=1
i=10
1 maximum
10 maxima
|Ψ(nd )|2
i=20
20 maxima
v=0
↓
i=30
30 maxima
Eup=0
S Heinze, P Cejnar, J Jolie,
M Macek, PRC 73, 014306 (2006).
SiU(5)   Pi (nd ) ln Pi (nd )
nd
i=1
i=10
U(5) wave-function entropy
quasi-O(6)
quasi-U(5)
N=80
i=20
i=30
|Ψ(nd )|2
v=0
↓
Eup=0
S Heinze, P Cejnar, J Jolie,
M Macek, PRC 73, 014306 (2006).
Any phase transitions for nonzero seniorities?
1
N
 2  

2
H  H cl    (1  )      
J 0
2
 

C2 [O(5)] 
1
N2
 Nv 2   2
  2



2
 5  4 2
  (1  )  4

2

J 0
( )
V

2

5  4 2


H cl    (1   )  2  2  (1   ) 2 

  (1   )  4
2
 2
2

2
Veff (  )
Veff
constant & centrifugal terms
0  1
1
Veff
0.9
v0
0.6
0.5
v
0.6
N
4
0.5
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
v
N
2
0.8
2
-0.1
2
-0.1

0.7
For δ≠0 fully analytic evolution of the
minimum β0 and min.energy Veff(β0)
0.6
0.5
0
0
0.2
0.4

0.6
0.8
1
1
=> no phase transition !!!

2
J=0 level dynamics for separate seniorities
N=80
Eup  0
excited states
continuous
v=0
ground state
2nd order
(probably without
Ehrenfest classif.)
no phase transition
v=18
Conclusions:
•
•
•
Quantum phase transitions in integrable systems: 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:
•
•
E=0 phase separatrix for zero-seniority states
analytic evolutions for nonzero-seniority states
Open questions:
•
•
Connection with thermodynamic description of quantum phase transitions?
Extension to nonintegrable systems: is there an analog of monodromy?
Collaborators: Michal Macek (Prague), Jan Dobeš (Řež),
Stefan Heinze, Jan Jolie (Cologne).
Thanks to: David Rowe (Toronto), Pavel Stránský (Prague)…