interacting boson model s-bosons (l=0) d-bosons (l=2) interpretation: “nucleon pairs with l = 0,...
DESCRIPTION
Simplified Hamiltonian d-boson number operator quadrupole operator scaling constant ħω=1 MeV control parameters η, χ symmetry triangle ensures that the thermodynamic limit exists: N→ ∞TRANSCRIPT
Interacting boson modelInteracting boson model
lkji
lkjiijklji
jiij bbbbvbbuH,,,,
2,...,2{
ds
bis-bosons (l=0)
d-bosons (l=2)
Interpretation:• “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: i
ii bbN
F Iachello, A Arima (1975)
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)]See eg.: F.Iachello, A. Arima : The Interacting Boson Model, Cambridge University Press, 1987
D Warner, Nature 420, 614 (2002).
SU(3)
U(5)
SU(3) O(6)
O(6)
triangle(s)
Parameter space of the model simplest version (IBM-1) can be imaged as the surface of a “symmetry pyramide”.
Corresponding points in various triangles are connected by similarity transformations (parameter symmetries), so it is sufficient to investigate dynamics in one of the triangles.
)()(1),( d QQ
NnH
ddn ~d
)2(]~
[~
)( dddssdQ
Simplified Hamiltonian
d-boson number operator
quadrupole operator
scaling constant ħω=1 MeV
control parameters η, χsymmetry triangle
)()()()(
1d
10
0
QQnVQQH
VHH
N
N
ensures that the thermodynamic
limit exists: N→∞
21
0
00
||
1st order
2nd order
Phase diagram
4
1
32 3cos BAE
critical exponent
1st order
0cos 0N
ds
obtained from variational procedure based on condensate-type ground-state wave functions
order parameter:
β=0 spherical, β>0 prolate, β<0 oblate. I II III
vacdss
exp
2847
)1(
3sin23cos2)(127
)1(
)1(2)1(221),(
2222
3222
222cl
pT
ppppT
TH
HH clˆ
)()(1),(ˆ
2d QQN
nN
aH
d-boson number
quadrupole operator
ddn ~d
)2(]~
[~
)( dddssdQ
The Hamiltonian with s- and d- bosons:
Classical Limit obtained by Glauber coherent states:
If restricted to L=0 states, Hamiltonian is solely a function of quadrupole deformation parameters β – γ (in the intrinsic frame) -> 2D system
Classical limit of IBM1
NN ˆ
N
ssnN ~ˆˆd
]1,0[]2,0[]2,0[]2,0[
pp
],0[]1,0[ 27 scaling constant a = N/10 MeV
1,5.0
2,lim VE1,5.0
Classical potential in case of prolate deformation
section through the plane y = 0
Particularly important values of energy:
0E
,sadE
,minE
2141
72
21 11145 CBA
432
2cl 3cos1),(
2
CBAV
xy
x
yγ
sincos
yx
Chaos within the TriangleStandard classical measures of chaos:• Lyapounov exponents λ
D(t) ... separation of two neighbouring trajectories at time t
• Fraction of chaotic phase space volume σ
)()(ln1lim 0tDtDtt
New highly regular arc discovered [Alhassid,Whelan PRL 67 (1991) 816]
using both measures : chaotic volume σ
and average maximal Lyapounov exponent λ.
Fraction of chaotic phase space σ at two values of angular momentum. The arc is clearly visible in both pictures.
adapted from Alhassid,Whelan PRL 67 (1991) 816
Chaos within the TriangleStandard quantum measures of chaos:
• Brody parameter ω (distribution of nearest neighbor level separation S):
• Distribution of B(E2) strengths (Porter-Thomas distribution) (Alhassid,Whelan)
• Δ3 statistics (long range spectral correlations) (Alhassid,Whelan)
• Wave function entropy (localisation in dynamical-symmetry bases) (Cejnar,Jolie)
1exp)( SNSSPinterpolates between Poisson (ω=0, regular
dynamics) and Wigner distribution (ω=1, chaos)