Chaotic properties of the interacting boson model

Nuclear Physics A556 ( 1993) 42-66 North-Holland

NUCLEAR PHYSICS A

Chaotic properties

of the interacting

N. Whelan

boson model

and Y. Alhassid

Center for Theoretical Physics, Sloane Physics Laboratory, and A. W. Wright Nuclear Structure Laboratory, Yale University, New Haven, CT06511, USA Received

25 February

1992

Abstract: We study classically and quantum mechanically the chaotic properties of low-lying collective states of nuclei by using the interacting boson model. Classical and quanta1 diagrams are constructed for several measures of chaos in a general parameter space of the hamiltonian known as Casten’s triangle. Strong correlations are found between the classical and quanta1 results. A new nearly regular region is discovered.

1. Introduction Since the connection between statistical fluctuations of energy levels and classical chaos was first noticed ‘32), there has been a growing interest in the quantum signatures of chaos 3m5).Most numerical studies, however, were restricted to models with two degrees of freedom *-‘). With few exceptions such as Rydberg atoms in strong magnetic fields ‘), these models were unrealistic. Of particular interest is the study of the onset of chaos in the atomic nucleus. The neutron resonances were known *-‘O) to obey the statistics predicted by the gaussian orthogonal ensemble (GOE) of random matrices ‘I). These resonances lie in a region of high nuclear level density and are expected to obey such statistics due to the complexity of the nuclear hamiltonian. In view of the new interpretation 172) which relat es GOE fluctuations to an underlying classical chaos, these studies were recently extended to the low-lying collective states of nuclei ‘2m’5).To see whether chaos can prevail in low-lying collective states, Weidenmuller et al. 12) analyzed experimentally known levels. To obtain reasonable statistics it was necessary to analyze together levels of different spin-parity so that only partial conclusions were reached. Raman ef al. 14) investigated a complete set of levels in “%n and Garrett et al. 15) analyzed many near yrast states of heavy nuclei with mass 155 C A c 185. To understand whether the statistics of low-lying levels of nuclei are associated with underlying chaotic dynamics it is necessary to study a realistic theoretical model of the nucleus. Since the quadrupole deformation plays a major role in Correspondence to: Prof. Y. Alhassid, Yale University, Laboratory, Box 6666, New Haven, CT 06511, USA. 0375-9474/93/$06.00

0

1993 - Elsevier

Science

Publishers

Center for Theoretical

Physics, Sloane

B.V. All rights reserved

Physics

N. Whelan,

collective

nuclear

dynamics,

Y. Alhassid

43

/ Chaotic properties

any such model requires

at least five degrees of freedom.

The solution of the Schrtidinger equation and the analysis of the classical dynamics are technically more difficult than the usual problems studied in two degrees of freedom. We have recently studied lh,“) such a model, the interacting boson model of levels and electromag(IBM) ‘8,‘9) which is known to provide a good description netic transitions in many heavy nuclei. Our study was restricted to a particular set of nuclei which belong to a transition class between rotational and y-unstable nuclei. The purpose of this paper is to study the full parameter space of the IBM. This is important if the results of our study are to be applied to realistic nuclei which do not necessarily belong to any of the transition classes. The outline of the papers is as follows. In sect. 2, we review the model and discuss its classical limit which is obtained through the use of coherent states. The classical dynamics is found to depend on two parameters. In sect. 3, we review both classical and quanta1 measures of chaos. Results are discussed in sect. 4, where diagrams that show the degree of chaoticity are constructed in the parameter space of the hamiltonian. We find strong correlations between quanta1 fluctuations of energy levels and E2 intensities and the underlying classical dynamics. Of particular interest is the discovery “) of a nearly regular region which was not known previously.

2. Model We model the collective dynamics of the nucleus by the interacting boson model (IBM) which has been successful in describing phenomenologically the low-lying energy levels and electromagnetic transition intensities of heavy nuclei ‘*,19).

2.1. QUANTAL

MODEL

The model’s degrees of freedom are one monopole s-boson (spin-parity 0’) and five d,(p = -2,. . . ,2) bosons (spin-parity 2+). The 36 bilinear combinations (sls, s.‘d@, d:s, d:d,) form a U(6) dynamical algebra. The most general IBM hamiltonian is built from all one- and two-body scalars which conserve the total number of bosons N = sis +C, d:d,. The most economical parametrization of the IBM is achieved in the consistent Q-formalism 2’,22), H=Eo+co&,+c,QX~QX+clL’.

In eq. (2.1) nd = d; . 2 is the number Q” is a quadrupole operator

of d-bosons,

(2.1) L is the angular

momentum

and

(2.2) which depends on a parameter x. We have used the notation & transform under rotation like d:.

& = (-)@d_,

so that

44

N. Whelan, Y. Alhassid / Chaotic properties

The E2 transition

operator

is given by T( E2) = (yZQX,

and the B(E2)

values

are calculated

from the reduced

B(E2,i+f)=

2.2. CLASSICAL

(2.3) matrix

elements

&/(fll~(E2)lli)~2.

through (2.4)

LIMIT

The inverse boson number, l/N, plays the role “) of h so that the classical limit is obtained for N+ 00. The mean-field dynamics 23-25) is derived from the timedependent variational principle Sf dt($Iia/at - HI+) = 0 by using a trial family of coherent states 26) /+!I)= lol( t)) where

Ia)= exp Herecu={ru,,cu,;p=-2,... equations

have the hamiltonian

exp ( czy,st+C cu,dL) IO) . \ / P

(-+I’)

,2} are six complex form “)

i&, =*,

i&T=

numbers.

The resulted

(2.5) mean-field

aSY dLrj

--,

aYe

j = s, -2,.

. . ,2,

(2.6)

%

where the hamiltonian

SY is SY=

crj and ia?

play the role of canonically

If H is normal

ordered,

then

dp + ap, N=(N)=cr,*a,+C, To take properly

(aJH[a) . conjugate

(2.7) variables.

SY in eq. (2.7) is found

d;+a;,

p = -2,.

LY~CY,is conserved by (2.6). the limit N + KJ we scale the variables Gj =

a;-/V% .

by the simple

. . ) 2.

substitution

(2.8)

ai_ by v%, (2.9)

At the same time it is necessary to scale the coupling parameters accordingly. Denoting by h the hamiltonian per boson h = X/N, we obtain (after renaming (Y by a) h(cr,~*)=a,,+~,,n~+C~q~~q~+~,1~+0(1/N),

(2.10)

N. Whelan, Y. Alhassid / Chaotic properties

45

where Zi=Ncj The classical L, respectively,

(i=l,2),

e,,= E,,/N.

(2.11)

quantities nd, qx and I are obtained from the operators &, Q” and through the replacement (2.8). Notice that they are also given by nd = (~~n^&)lN, $’ = (&%)IN, l=(alL[(~)/N.

(2.12)

The terms in eq. (2.10) which are of order l/N are obtained since H in eq. (2.1) is not normal ordered. These terms however are negligible in the limit N + CO. The variables (Y, and LX: obey the canonical Poisson brackets {cx,,cr,}={(Y~,(YI*}=0.

to,, a;> = &,,

(2.13)

It follows that the 36 quantities {(-uta,, (Y~(w,, Cuba,, c~;a,} form a U(6) algebra with respect to the Poisson brackets. To any quantum-mechanical operator which is a function of the U(6) generators, there will be a classical counterpart constructed similarly from the classical generators. Any algebraic commutation relation satisfied by the quantum operators will have a similar Poisson bracket relation satisfied by the corresponding classical quantities. In particular a quantum-mechanical constant of the motion will have a corresponding classical constant. This will be important for the discussion of integrability in sect. 2.4. We will reparametrize the classical hamiltonian (2.10) as h=&g+~[77~4-(1-77)qX.qX]+F,12,

(2.14)

where -= 77

-- CO

1-n

NC,

(O
E=c,/v.

Since 1* is a constant of the motion, E, does not affect the character of the dynamics so we set E, = 0. By proper scaling of the hamiltonian we may set Z = 1 so that the only important 2.3. DYNAMICAL

parameters

are x and 7, where

-@‘s

x s 0 and 0 s n i 1.

SYMMETRIES

A dynamical symmetry in an algebraic model occurs when the hamiltonian can be written as a function of the Casimir invariants C of a chain of subalgebras of the original algebra G, H = LX(G) + ,‘C(G’“)

+ ,“C(G’*‘)

G-J G(‘) 3 G(*) -J . . . ,

+. . . , (2.16)

46

N. Whelan, Y. Alhassid / Chaotic properties

The eigenstate can then be labeled by algebraic quantum numbers and the spectrum is known analytically through the eigenvalues of the Casimir operators. The IBM has three dynamical symmetries (with O(3) being the last algebra in the chain) 19),

U(6) 1

U(5) = O(5) SU(3)

(I) = O(3)

1 O(6) = O(5) I

(II)

.

(2.17)

(III)

Chain (I) (U( 5)) corresponds to vibrational nuclei and is obtained in eq. (2.1) when c2 = 0. Chain (II) (SU(3)) describes rotational nuclei and is obtained for cO= 0, x = -&,J? since Q”. Q” is then related to the Casimir invariant of SU(3). Finally chain (III) (O(6)) which describes y-unstable nuclei is obtained when co = 0 and x = 0 (since Q" . Qx]X=O= CJO(6)) - C,(O(5))).

2.4. INTEGRABILITY

A system is completely integrable when it has a complete set of constants of the motion in involution with each other. A dynamical symmetry necessarily implies complete integrability 16,*‘). To show that, we note that all the Casimir invariants in eq. (2.16) are commuting constants of the motion, [H 9C(G”‘)] [C(G”‘)

2C(G”‘)]

= 0, = 0.

(2.18)

If the set {C(G”‘)} is not complete, there must be a missing label in one (or more) of the reductions C(G”‘) 3 C(G(“‘) ). It is usually possible to find an invariant of G(‘+‘) which is built from the generator of G(‘) and therefore commute with C(G”‘). With these invariants we obtain a complete set of constants. Through the substitution (2.8), we obtain the classical counterpart of these constants. They will satisfy the same relations as in eq. (2.18) but with the Poisson brackets replacing the commutators. Therefore the classical limit will also be completely integrable. In the case of U(6), we have six degrees of freedom (six a’s). N, L2 and L, are always three commuting constants. To get a complete set we need three more. In the various dynamical symmetries these constants are: (i) U(5): Two of the constants are the number of d-bosons i& and the quadratic Casimir invariant of O(5). The reduction from O(5) to O(3) has a missing label. Accordingly there will be an O(3) invariant which is built from the O(5) generators. This invariant is the third constant. (ii) SU(3): The quadratic and cubic Casimir invariants of SU(3) and the invariant (LxQ) (I1 . L . The latter is a scalar (O(3) invariant) which is not a Casimir invariant of SU(3) but commutes with the SU(3) Casimir operators. It is related to the missing

N. Whelan, Y. Alhassid / Chaotic properties

label

(K)

in the reduction

47

from SU(3)

to O(3). Alternatively the three constants can be chosen as Q. Q, (Q x Q)‘2’ . Q and (L x Q) (‘I * L In the classical limit the corresponding constants are 9. q, (q x q)‘*’ . q, and (Ix q)* (1) . l . (iii) O(6): Two constants are the quadratic O(6) Casimir invariant and the quadratic Casimir invariant of O(5). The third is as discussed in (i). The dynamical symmetry hamiltonians have a particular property beyond integrability. They do not depend on all the independent mutually commuting constants of the motion. This is the case when there is a missing label. The hamiltonian (2.16) does not depend on the invariant associated with this missing label. A missing label occurs in all three chains in eq. (2.17). A similar situation occurs when a representation label is associated with a Casimir invariant whose order is higher than second. This is the case of SU(3) where the cubic Casimir invariant of SU(3) does not appear in eq. (2.1). Such situations may lead to additional degeneracies and can be termed “overintegrable”.

2.5. CASTEN’S

TRIANGLE

The family of hamiltonians (2.1) or (2.14), can be described by a triangle whose vertices are the three dynamical symmetries. The base of the triangle is the x-axis, and its height is the q-axis. Each point inside the triangle describe a hamiltonian (2.14) with parameters (x, 7). To read the value of x for a point inside the triangle we read the intersection on the x-axis of the line which originates at the upper vertex and passes through the given point. This is a rotated version of Casten’s triangle. A colored map of the triangle was used to characterize the approximate symmetries of various nuclei in the periodic table. A main purpose of the current work is to construct a “chaotic” map of the triangle “) which will characterize the degree of regularity (or chaoticity) of various regions inside the triangle.

2.6. LEVEL

DENSITY

As is explained in sect. 3.2, the study of the level statistics of the level density into an average and fluctuating part,

requires

the separation

(2.19)

P(E)=P,“(E)+Pfl”,,(E). In the limit N-f ~0, the quanta1 level density density which is given by

should

coincide

with the classical

level

(2.20) The (Y’Sin eq. (2.20) are the ones before the scaling (2.9). Fig. 1 shows (solid line) the classical level density [calculated by Monte Carlo I’)] for n = 0 and x = -0.7. In the limit of large N, pav will approach (2.20). For finite N, we expect corrections

48

N. Whelan, Y. Alhassid / Chaotic properties

in fig. 1 show the quanta1 level density for N = 10 bosons PEW. The histograms (dashed) and N = 25 bosons (solid). In the various dynamical symmetry limits it is possible to evaluate p,,(E) to leading order in N analytically. 2.6.1. Vibrational limit (U(5)). In this limit H = c$, and we have to find the to

number

of states with a given number of nd bosons. n,) such that c nr = nd is given by (n-,,n-I,...,

The number

of sequences

(2.21) where the approximation we have

is to leading

P(E)

In terms of the energy

per boson P(E)

order in N. Since the increment =hE4.

in nd is 1,

(2.22)

F = El N we find -&N5~4=./V&5~4),

(2.23)

where Jtot -AN5 is the total number of states in the model. To calculate the classical level density (2.20) which is a 12-dimensional integral we introduce polar coordinates in the 2-dimensional LY,plane and the lo-dimensional {a,1 space, 2rrr, dr, 6( N - rz - ri)S( E - r;)

-

riS(&?-

rd) = &r6E4.

(2.24)

Fig. 1. Level density distribution for an hamiltonian with 7 = 0 and x = -0.7. The solid line is the classical level density (N+ CO). The histograms are quanta1 level densities for N = 10 (dashed) and N = 25 (solid).

N. Whelan, Y. Alhassid / Chaotic properties

Thus,

the classical

result

agrees with the leading

order

49

of the quanta1

calculation

(2.23). 2.6.2. Rotational limit (SU(3)). We choose H to be proportional to the quadratic Casimir invariants of SU(3) [obtained in (2.1) for cO= 0, c, =& c2= -11 so that E=-~(A*t/1~+h~+3(h+~)), where (A, p) are the representation To count A =

ZK,

so

the number that

labels of SU(3).

of states

with energy

E = 4(3K*+ The ranges

of the allowed

(2.25)

quantum

numbers

E we change

variables

p = r] -K,

72).

(2.26)

labeling

the SU(3) states are

7=N,N-3,...

SU(3)

K=Tf,7,'-2,...

K=min(n-~,2~),min(n-K,~K)-2

Counting

O(3)

L=K,K+l,...,

O(2)

M=-L,...,L.

N

K-t-rnax(~-K,2K)

(2.27)

1)

dKK(vj2-K2)6(E+;(3K'+$))

dv

I =

I

;;-2E)l'

-2‘N26E<0

-2N’c

{ &(N-GF)*(N+~ZI?)

2.6.3.

0

states in the limit of large N we find

p(E)=;

tonian

,...,

y-unstable

limit (O(6)).

Choosing

E s -;N’.

(2.28)

c0 = 0, c, = 0, c2 = -1, A = 0 in the hamil-

(2.1) we have E =--(T(u++)+~(T+~)=-(T*+T-~,

where the various

quantum

numbers

in the chain

O(5)

o=N,N-2

O(5)

7.=o,u-l,...,

(2.29)

(III)

are

3. * . ,

A=r,r-3,...,

Counting

O(3)

L=A,A+l,...,

O(2)

M=-L,...,L.

2A, (2.30)

states we find drT3S(E

-~*+a~) (2.31)

where -N2c

E ~0.

50

N. Whelan, Y. Alhassid / Chaotic properties

3. Chaotic properties Our purpose is to study the character of the classical dynamics of a nucleus described by (2.14) and its signatures in the quanta1 model (2.1). We shall briefly review the classical and quanta1 measures of chaos and present the results of our study to the particular model.

3.1. CLASSICAL

MEASURES

OF CHAOS

The onset of chaos in systems with two degrees of freedom is easily demonstrated in terms of the corresponding Poincare sections. In our studies the phase space is 12 dimensional so that the PoincarC sections are impractical. Instead we detect chaos through the linear instability of the classical trajectories. A Lyapunov exponent A is defined in terms of the separation D(t) between two neighboring trajectories &j(t) and ~~~(t)+d~;(t), h=limfln[D(r)/D(O)]. ,+@Z

(3.1)

Depending on the choice of da;-(O), there are several such exponents whose number is equal to the dimension of phase space. Due to the symplectic structure of the Hamilton equations they come in pairs of opposite signs. In our case there are six pairs of Lyapunov exponents but since the hamiltonian, boson number and the angular momentum are always constants of the motion, at most two pairs can be non-zero. Due to numerical errors, any choice for da(O) will lead to a Au(t) whose length is dominated by the largest exponent. To find other exponents the following method is used 29). Suppose that we want to calculate n Lyapunov exponents. We choose an initial set of n orthonormal vectors Aa( We evolve them to the next time step so that the new set of vectors ACYis not orthonormal. We calculate D, as the length of one such vectors. We subtract from the second vector its projection on the first one and denote its length by D2. By continuing this Gram-Schmidt procedure we generate a new set of orthonormal vectors and a set of n distances Di. Repeating the above procedure at each time step we find Q( 1) which upon substitution in eq. (3.1) will give the exponent Ai. An example is shown in fig. 2 where In [Di( t)/Di(O)] is plotted versus f. The slope of the curves are the Lyapunov exponents. A trajectory is chaotic if its maximal Lyapunov exponent is positive. Otherwise it is regular. We use two measures of classical chaos, the fraction of chaotic volume u and the average largest Lyapunov exponent h. Defining for each point (Yiin phase space, p(czj) to be 1 or 0 for a chaotic or regular trajectory, respectively, we have

(3.2)

51

N. Whelan, Y. Alhassid / Chaotic properties

50

Di(t) 0

-50 0

200

400

600

t Fig. 2. In( D,( 1)/D, (O)), versus time, where D(t) is the euclidean distance between neighboring trajectories. The various curves correspond to the same intitial point in phase space but different choices for D,(O). The slopes are the Lyapunov exponents A. There are six pairs of A’s and only two are non-zero.

at a given energy largest Lyapunov

E() and angular momentum LOper boson. Similarly exponent associated with the point LYJ,then

if A((Y,) is the

(3.3) Sometimes we also calculate the Kolmogorov-Sinai be the average sum over all positive exponents A,, K=

* nd2+(h-&5(I-I,) I X:a,u,=l

entropy

K which is known

1 Aj(a). h,>”

to

(3.4)

The integrals are calculated by Monte Carlo methods through selecting the initial points randomly on the surface defined by the constraints on the boson number, I’ and lz. For each initial point, we calculate its Lyapunov exponent using the BulirschStoer method for solving the classical differential equations. See ref. “) for details.

3.2. QUANTAL

SIGNATURES

OF CHAOS

Quanta1 signatures of classical chaos are searched for in the statistical fluctuations of both the spectrum and the E2 transition intensities. To analyze fluctuations of the spectrum {Ei}, we separate its smooth part (whose behavior is non-universal). This is done by constructing the staircase function N(E) and separating it into average and fluctuating parts N(E) = N,,(E) + NRuct(E). N,, is found by fitting a sixth-order polynomial to N(E). The unfolded levels E are defined by the mapping l$ = NJE,). Since spin-parity (J”) are good quantum numbers, we do our analysis separately in each spin-parity class of levels. Two statistical measures are used for the unfolded spectrum: the nearest-neighbor level

N. Whelan, Y. Alhassid / Chaotic properties

52

spacing distribution P(S) and the A3 statistics of Dyson and Metha. The distribution P(S) is fitted to a Brody distribution ‘) characterized by a parameter w : Z’,(s) = AS”’ exp ( --cxS’+~) , with cx and A chosen such that (3.5) interpolates between the regular system and the Wigner system. The A, statistics measures the a straight line,

(3.5)

P is normalized to 1 and (S) = 1. The distribution Poisson distribution (w =0) which characterizes a distribution (w = 1) which characterizes a chaotic deviation

J

of the (unfolded)

staircase

function

from

a+L

A,(a, To obtain a smooth overlap successively

L) =nj;$ .

[iV(&-(AE+B)12dk

(3.6)

ct

&CL), we average by ;L:

A,(a,

L) over n, intervals

(ty, LY+ L), which

(3.7) For the Poisson

statistics d;(L)=-&L,

(3.8)

3;(L)L;,

$1,

(3.9)

is also available

lo).

and for the GOE statistics

An exact expression

L-0.007.

To describe an intermediate A3 statistics, we assume a superposition of a Poisson spectrum and a GOE spectrum with weights 1 - q and q, respectively 30). We obtain a A!(L) which interpolates between the Poisson (q = 0) and GOE (q = 1) limits, A~(L)=A,P”i”“““((l-q)L)+A~oE(qL). Another useful signature of chaos is the statistical intensities “s3* ) which probe the system’s wave functions. T we define Y=

KflTIN*,

(3.10) fluctuations of transition For a transition operator

(3.11)

where 1i> and / f) are initial and final eigenstates of the hamiltonian. We then construct a distribution P(y) such that P(y) dy is the probability to have an intensity in the interval dy around y.

N. Whelan, Y. Alhassid / Chaotic properties

The fluctuations

of y are to be considered

with respect

that describes the secular variation of the intensity we define an average intensity for an initial energy

53

to the smooth

envelope

with energy. For that purpose E and final energy E’,

y(E E,j=Ci,tI(dTb)12 w HE -EJ2/2y21exp [-(E'-EJ2/2y21 > ’ (3’12) Ci,feXp [-(E- Ei)*/2y2] exp [-(E’Ef)2/2y2] The gaussian

width

are then defined

y should

be chosen

properly

yfi=IflTli)12/j(E

intensities

= Ei, E’= EC),

and are used in the statistical analysis. In the GOE (chaotic) limit we obtain P(Y)

More generally

32). The renormalized

by

a Porter-Thomas

= (274~))~“‘y-“’

we use a x2 distribution

ew

(-Y/~(Y))

in v degrees

P,(y) = Ay”“-’

(3.13)

distribution

(3.14)

.

of freedom

33)

3’), (3.15)

exp (-vy/2(y)),

where .

A = (v/~(Y))“‘~/~(;v)

(3.16)

(3.15) reduces to the Porter-Thomas distribution for v = 1. It is usually found that as the system dynamics becomes more regular, v decreases towards small positive values 32). In an actual calculation, P(S) and P(y) are found by histogramming the calculated spacings and intensities. To find the best valuex of w and Y for these distributions we have followed two different methods: (i) We do a least-square fit by minimizing with respect to v the quantity (3.17) The error expected

(T; in bin i is estimated

number

of intensities

to be proportional

to the square

in the bin, (3.18)

RaJFJZ9

where v0 is our recent estimate v. This procedure is equivalent

root of the

for Y. By iterating to minimizing

we converge

(e -PAY;))'

c

to a final value for

(3.19)

P"(Y,)

I

The error SV in u is estimated 34) such that xZ is increased by 1 when v is changed to v+av. (ii) We minimize the information content of P with respect to Pv, 1=x

I

Pi In

P, [ P”(Yi) I .

(3.20)

N. Whelan, Y. Alhassid / Chaoiic properties

54

The advantage of (3.20) is that it is a convex function and has a unique minimum. It can be shown that the conditions for minimizing (3.20) are that (y) and (lny), calculated from Py(y) are equal to those calculated with P(y). To leading order in APIP,, I coincides with x2 in eq. (3.19). Therefore if one can obtain a good fit to P with a distribution of the form Pu, the two methods are essentially

equivalent.

To find the error in V, notice that I/ is a function propagate the errors in (y) and (In y).

of (y) and (lny),

so we can

4. Results The quanta1 fluctuations which correlate well with the classical results are independent of the number of bosons N. In order to obtain good statistics (by analyzing a given nucleus) we chose a relatively large N, N = 25. The total number of states for each spin (J s 20) is given in table 1.

4.1. TRANSITION

CLASSES

We first discuss the transitions between the various dynamical symmetry limits of the model. (A) The transition between deformed rotational nuclei (SU(3)) and spherical vibrational nuclei (U(5)) is obtained by taking x = -ifi and 0~ 7~G 1. Fig. 3 compares two classical measures of chaos h and o, with two quanta1 measures w and V. The various quantities are plotted versus 7~ at fixed spin J = 2. The classical value of I= 0.1 is chosen to correspond to the above spin value. For 17= 0 and 77= 1 (SU(3)) and (U(5)) limits) h and c are zero as expected for a regular system. Maximal chaos is achieved for n = 0.5-0.7. The quanta1 results show strong correlation with the character of the classical dynamics, where w and v are largest around n = 0.5-0.7. The errors shown are statistical. It is seen that the error in v is smallest due to the large number of transitions. This number is proportional to the square of the number of states at the given spin. TABLE

1

The number of states for each spin J G 20 and for N = 25 bosom J

No. of states

J

No. of states

J

No. of states

J

No. of states

0

65 0 117 52 156 92

6

I 8 9 10

184 121 202 140 211

11 12 13 14 15

151 212 154 207 151

16 17 18 19 20

196 143 181 131 163

1 2 3 4 5

N. Whelan, Y. Alhassid

Q=O.l X

/ Chaotic properties

55

J=2

0.1

0.0 CT 0.5

Fig. 3. Classical and quanta1 measures Average maximal Lyapunov exponent momentum per boson of I = 0.1. Right: the parameter v characterizing the B(E2; of bosons is N = 25. Notice that 1 and J

of chaos for the SU(3) (7 =0) to U(5) (7 = 1) transition. Left: h and fraction of chaotic trajectories (J for classical angular Brody parameter w of the level-spacing distribution (3.5) and J + J) distribution (3.15) for the levels with J = 2+. The number are selected to approximately correspond to each other through l=JfN.

(B) The transition between rotational nuclei (SU(3)) and -y-unstable nuclei XG 0. It was already discussed in detail in (O(6)) is obtained for n = 0, -$?‘s ref. “) where it was found that chaos sets in at intermediate values of x. (U( 5)) nuclei (C) The transition between y-unstable (O(6)) and vibrational (x = 0; 0~ n G 1) is always completely regular. This is expected since O(5) is a common subalgebra in the two symmetry limits. The Casimir invariant of O(5) and an invariant that corresponds to a missing label in the O(5) I> O(3) reduction, together with N, L2, L, and the hamiltonian H form a complete set of constants in involution.

4.2. CASTEN’S

TRIANGLE

To explore the collective dynamics of realistic nuclei, it is important to extend the investigation to the full Casten triangle ‘“). We thus allow both x and n to change in the range -$fi < x < 0, 0 < n G 1. Figs. 4 and 5 summarize our classical calculations at angular momentum Z= 0.4 for the quantities u and h, respectively. Regions of various ranges of these parameters are shown by different shades in the triangle. In the a-diagram, for example, the unshaded region (0 < u < 0.2) is highly regular while the dotted region (0.8 <(T< 1) is highly chaotic. In the quanta1 calculations we have analyzed the 211 J = 10 states (in a nucleus with N = 25). The two quanta1 measures, w and V, are shown in figs. 6 and 7, respectively. In the w-diagram both the densely dotted and unshaded regions are highly regular. In fact, in the densely dotted region w < 0 due to degeneracies which occur at the dynamical symmetry limits (this is due to “overintegrability” as is explained in sect. 4.6).

N.

56

Whelan,

Y. AIhassid

/ Chnofic properties

I.oor lz!

0 4-O

6

06-08

q

0.8-10

0.25

Fig. 4. a-diagram of the Casten triangle. Each point (,y, 7) in the triangle corresponds to a classical to different hamiltonian (2.14) (with e0 = 0, C = 1 and C, = 0). Regions of different shades correspond ranges of cr. The dotted regions are the most chaotic. v is calculated for angular momentum I= 0.4.

x 0

0.00

0 03

I .oo

003-006

0.75

0.09-0.12

0.06-

rl

0.09

>

0.12

0.50

0.00

Fig. 5. h-diagram

- I.2

-0.8

X

in the (x, 7) plane for

-0.4

0.0

1= 0.4. Regular regions have h = 0.

N. Whelan,

Y. Alhassid

/ Chaotic properties

57

0.0-0.2 0.2-0.4 0.4 -0.6 0.6 -0.8 0.0-1.0

0.50

0.25

0.00

Fig. 6. w-diagram in the (,y, 7) plane. The number of bosom is N = 25 and the parameters of the quanta1 hamiltonian are related to (,y, 7) through eq. (2.15). The level-spacing statistics from which w is determined is for the J = 10 levels.

All four diagrams show strong correlations. In the regular regions CT,A, w and v are all small and close to zero, and in the chaotic regions they are largest with CT, w and v close to 1. Our results suggest that the correspondence between quanta1 fluctuations and the underlying classical dynamics, which has been conjectured in the study of model systems in two degrees of freedom, seems to hold in a larger number of degrees of freedom.

Fig. 7. u-diagram

of Casten’s

triangle.

See fig. 6 for details.

N. Whelan, Y. Alhassid / Chaotic properties

58

Several horizontal

cuts across the triangle

are shown in figs. 8 and 9 for spin J = 2

(I = 0.1 classically) and J = 10 (I = 0.4), respectively. The figures show the classical measures i, u and the quanta1 measures of chaos w (level spacing), v (B(E2) distribution) and q (A3 statistics) versus x for several strong correlations between all of these quantities.

values

of 7. Again

we see

Instead of x one may use the Kolmogrov entropy K as a classical measure of chaos. The calculation of K is more time consuming since two exponents need to be calculated. A comparison between i and K is made in fig. 10. We see that they have a similar behavior. Therefore the variations of i give us a good idea of how K is changing. Typical results of the analysis of quanta1 fluctuation are shown in figs. 11 and 12. Each figure has a fixed spin and n, and the distributions P(S), A,(L) and P(y) are shown for several values of x in the range -$fi s x s 0. In addition to the actual distributions we show the chaotic (GOE) and regular (Poisson) limits. 4.3. A NEW

NEARLY

REGULAR

REGION

One of the surprises of the “dynamical” diagrams is the discovery of a previously unknown nearly regular region which is a narrow band connecting the SU(3) and U(5) vertices inside the triangle. It is clearly observed in the classical diagrams

!I=0.1 .30

I

fh

I .15

1.0

0.5

0.5

0.0

0.0

w

’

.oo

‘I ‘I

I

‘I

0.5

0.5

”

,I’,,

0.0

0.5

I

0.5

“X1 _“... .oo _ ,,.... ‘. 0.0 0.5

.15 ‘. .....

1

0.0

X

0

.....

0.5 -..’

’

-1

0.0

X

“‘. ..._

0.0

0.0

0.5 ~ll’ll 1

0.5 - ......_..,:.-.._, ..._

0.0 0

.. .-

I 0.5 -....’ o.5 E I [I’ III1 1,

0.5

.15

iI

0.0

s 0.0

.oo I..

..-

I III 1

O.O

.. ..

0.5 -:

E 0.0 I’

‘.-

0.0

’ IF4

‘.

0.5 -:

:

.oo

.oo

“1

I

’

.15

I v ,...

1.0.

I

. r-7

.15

J=2

1.0

-1

X

.._

0

0.0

’ -1

X

0

Fig. 8. Classical measures of chaos A, (r and quanta1 measures w, Y versus x for several cuts across the triangle: 7) = 0, 0.1, 0.3, 0.5 and 0.7 (from top to bottom). The quanta1 analysis is done for J = 2 levels (N = 25) and the classical calculation is for I = 0.1.

N. Whelan, Y. Alhassid

/ Chaotic properties

59

J=lO .20

1.0

.lO

0.5

.oo

0.0

.lO

0.5

.oo

0.0

.lO

0.5

.oo

0.0

.lO

0.5

.oo

0.0

.10

0.5

.oo

0.0

X

lJ

q

X

Fig. 9. As in fig. 8 but for the J = 10 levels and 1= 0.4. Also shown is the parameter the A, statistics (see eq. (3.10)).

q which characterizes

x I

I

+

0.2

0.2

I

Ii

.

rl=o.o

I I I

0.0

0.0

0.2

0.2

?---i--

q=o.5 I I

0.0

-1-0.5 X

0.0 0

x

I

II

f* I

-1-0.5

0 X

Fig. 10. Comparison between the average maximal Lyapunov exponent h and the Kofmogorov entropy K. They are plotted versus x for the hamiltonian (2.14) for r) = 0 (top) and 7 = 0.5 (bottom).

60

N. Whelan,

(figs. 4 and 5). The fraction

Y, Alhnssid

/ Chaotic

of chaotic trajectories

properties

in this region is CTs 0.3 so it is not

completely regular. The signatures of this nearly regular region are clearly observed in the quanta1 diagrams (figs. 6 and 7) as well. In figs. 8 and 9 it is identified by the rather sharp minimum in all five measures of chaos. See also the distribution in figs. 11 and 12 at x = -1 (for n = 0.5). This nearly regular region may be connected with an unknown new approximate sytnmetry of the model. In fig. 13 (left-hand side) we indicate by a line the bottom

of this nearly

regular

valley. The value xmin of x for which w is minimal at a given 7, is plotted versus n on the right-hand side of fig. 13. The dependence is essentially linear. The variations of 1, a; w and v along the regular line of fig, 13 are shown in fig. 14. 4.4. SPIN

DEPENDENCE

An interesting issue is the spin dependence of the degree of chaoticity. The results are shown in figs. 15 and 16 where both classical (i, CT)and quanta1 (w, V, q) measures

x=-1.3

x=-1.0 0.0

0.0

0.5

0.5

0.0

0.0

0.5

0.5

0.0

0.0

0.5

0.5

0.0

0.0 -5

0

Ln(Y)

/ *....*.++

L----Lll~ /“G--

o,5

o,. ;

x=-o.7

x=-o.4

x=

0

5

L

10

0

2

0.0

4

S

Fig. 11. The level-spacing distribution P(S), Dyson-Metha statistics A,(L) and the B(E2, J+J) intensities distribution P(y) for the quanta1 hamiltonian (2.1) with N = 25 and several values of x. The solid line for P(S) is the fit to the Brody distribution P_,(S) [eq. (3.5)], the dashed line is the Wigner distribution and the dot-dashed line is the Poisson dist~bution. The GOE A, statistics is the dashed line while A3 for the Poisson statistics is the dot-dashed line. The solid line for P(y) is the fit to a x’distribution in v degrees of freedom (3.15) and the dashed line is the Porter-Thomas (GOE) distribution. n = 0.5 and the level analyzed are J = 2.

N. Whelan, Y. Alhassid / Chaotic properties

q=O.5

J=lO P(Y) 0.5

0.0

0.0

0.5

0.5

0.0

0.0

0.5

0.5

0.0

0.0

0.5

0.5

0.0

0.0

0.5

0.5

0.0

x=-1.3

x=-

0

1.0

x=-O.?

x=-o.4

x=

MO.0 -5

p(s)

_I 53w

0.5

61

0

5

10

0

L

My)

2

0.0

4

S

Fig. 12. As in fig. 11 but for 17= 0.5 and J = 10 states.

of chaos and pfotted versus spin. We chose two points in the Casten triangle: a chaotic one (x = -0.66, n = 0, see fig. 15) and an intermediate one (x = -1, n = 0, see fig. 16). In all measures (except h) we see a weak dependence on the spin at low and medium spins. However, at high spins (J~2Ofi, I3 1) there is a rapid descrease of chaoticity and the motion becomes more regular.

4.5. DYNAMICAL

SYMMETRIES

As explained The hamiltonian

in sect. 2.4., the dynamical symmetry does not depend on all the mutually

limits are “overintegrable”. commuting constants. This

1.0

77 0.5

0.0

X Fig. 13. A newly regular (maximal

rl

region inside the Casten triangfe. Right: The value of x for which F is minimized regularity) versus q. Left: The regular valley inside the triangle.

62

N. Whelan, Y. Alhassid Iz

/ Chaotic properties

J=lO

=0.4

X

Fig. 14. Chaotic

measures

A, a, w and v along the regular valley of fig. 13. In the quanta1 used J = 10 states. Classically I = 0.4.

analysis

we

leads to additional degeneracies. For example, the SU(3) hamiltonian is independent of K (or alternatively of (Lx Q)“’ . L), so states with the same angular momentum and which belong to the same SU(3) representation (A, p) but have different K’s are degenerate. Missing labels occur also in the U(5) and O(6) limits and are denoted by Za and v,, respectively 19). The U(5) limit, obtained in (2.1) for c2=0, has even more degeneracies since it is also independent of the O(5) quantum number r. Similarly the O(6) limit of (2.1) (c,=O,x =0) where E = --(~((~+4)+7(7+3) has additional degeneracies due to the particular rational combination of the O(6) and O(5) Casimir invariants. Such overintegrability leads to a non-generic behavior of the level statistics. We often find for the level-spacing distribution (3.5), a negative value of w that indicates

.., El 7-/=0.0

0.2

0.1 0.0 0.5

x=-.66

I

I

0.0

0

1

2

Q

Fig. 15. Chaos versus spin for 17= 0, x = -0.66 and N = 25 bosons. Left: Classical 1. Right: quanta1 measures w, v and q versus J.

measures

h; u versus

N. Whelan,

Y. Alhnssid

/ Chaotic properties

63

0.2

5; 0.1 0.0 cl

0.5

Fig.16.

Asinfig.l5butforq=OandX=-1

a larger number of small (or zero) spacings than in the Poisson distribution. See for example in fig. 12 the case T= 0.5 and x = 0 (which has an O(5) symmetry). However, these distributions are non-generic due to correlations between levels that exist in an overintegrable situation and are not always well desecribed by a Brody distribution (3.5). An interesting case is the SU(3) limit (c,, = 0, x = -ifi). Usually for a given J there is a K-degeneracy which will lead to a negative w. For the special values J = 0,3 there is only one K-value possible (K = 0 or K = 2, respectively). There is thus no K-degeneracy, and the missing label K should not effect the statistics. However, since the hamiltonian is also independent of the cubic Casimir invariant behavior. C,(SU(3)) (or alternatively (Q x Cl)“’ . Q) we still expect a non-generic We distinguish two cases: (i) The number of bosons N is divisible by 3. In this case for every representation (A, p) with A > p there is also a representation (p, A) which appears in the symmetric representation [N] of U(6). Since the energy (2.25) is symmetric in A and p, the states which belong to the above representations are degenerate. About f of the states are degenerate in this manner. We then expect a level-spacing distribution with a large number of zero spacings. This is the case shown in the first row of fig. 17 (N = 24) for 6 = 0. (ii) The number of bosons N is not divisible by 3 so that the degeneracy in (i) does not occur. It is possible to show that the difference between any two levels is a multiple of 3 (note that the relevant SU(3) representations are of the form A = 2N - 4i - 6j, p = 2i with i,j Z=0 integers). The energy range of the model is -N’s E s 0. For E 2 -$N* the level density is approximately constant (p = ~-/18&= 0.1008). Therefore the level-spacing distribution (for E 1 -$N*) will be a series of &functions at S = (l/p) 3k (where k b 0 is an integer). These &functions will be superimposed on the background which comes for the levels with - N2 G E s -fN’. This non-generic P(S) is clearly seen in the third row of fig. 23 for the case

N. Whelan, Y. Alhassid

64

N = 301 bosons.

It appears

/ Chaotic properties

that for large N the delta function

at S = 0 will dominate

and its amplitude is -In N. However, when N = 25 (second row) the &functions at S # 0 dominate and if the bin’s width is not too small we obtain a GOE-like distribution (though the system is regular!). One should be careful not to associate such distributions with a chaotic behavior 35). To reproduce the generic behavior in the level statistics,

one can add 36) to the

hamiltonian terms which depend on the missing constants. For example, we show in fig. 17 the distributions obtained when an additional term which depends on the cubic Casimir invariant C3 is added to the SU(3) hamiltonian H’=-QX.QX-X,(SU(3))/N.

(4.1)

with x = -&‘?. H’ is still integrable with the same constants of the motion as the SU(3) hamiltonian. Now we see that the generic Poisson statistics is reproduced in all three cases of fig. 17. Fig. 18 gives examples from the U(5) and O(6) limits. The top row corresponds to a hamiltonian: H’= nd-SC,(0(5))/N. For 6 = 0, eq. (4.2) However, for large that the integrability The last two rows

(4.2)

is a U(5) hamiltonian and we see a very large peak at S = 0. enough 6 # 0 we obtain the generic Poisson statistics (notice is preserved for any 6). of fig. 18 correspond to an hamiltonian Q”-SC,(O(5))/N,

-9”.

H’=

p(s)

6=.00

(4.5)

A3(L) 6=.00

6=.05

1.

6=.05

0.5

0.

0.0 0.5

1. 0.

0.0 0.5

1.

0. 0

2

0

S

2

4

0.0

0

5

0

5

10

L

Fig. 17. The SU(3) limit. The hamiltonian is eq. (4.1) and N = 24 (top row), N = 25 (middle row) and and we obtain a non-generic N = 301 (bottom row). When 6 = 0 the hamiltonian is “overintegrable” distribution (notice in particular the N = 301 case shown in a high resolution). For S = 0.05 the generic Poisson statistics is recovered.

N. Whelan,

Y. Alhassid

/ Chaotic

p(s)

d=O

6=0

6#0

1.

65

properties

*304

6#0

0.5

0.

0.0 0.5

1.

0.

0.0 0.5

1.

0. 0

2

0

2

4

0.0

0

5

0

5

10

L

S

Fig. 18. The U(5) and O(6) limits. Top row: The hamiltonian is eq. (4.2) with N = 25, S = 0 and S = 10. Middle and bottom row: the hamiltonian is eq. (4.3) with N = 25 (middle row) and N = 301 (bottom row), the parameters S are 6 = 0 and 0.1. Notice that the 6 = 0 non-generic statistics becomes the generic Poisson statistics for 6 # 0.

which for 6 = 0 is an O(6) hamiltonian. For N = 25 (middle row) the distribution happened to be Poisson but for N = 301 (bottom row) it is not. Adding a 8 f 0 term to produce a generic integrable hamiltonian at that limit, we again recover the Poisson statistics. An important conclusion from the above examples is that a GOE-like statistics for P(S) and A3 is not sufficient to infer that the dynamics is chaotic.

5. Conclusions We presented

the results for classical

and quantum

chaos in the general parameter

space of the interacting boson model of nuclei. We have constructed detailed diagrams of classical and quanta1 measures of chaos of Casten’s triangle. Strong correlations are observed between the onset of classical chaos and of quantum chaos. An interesting result is the discovery of a newly regular region inside the triangle which is not related to any of the known dynamical symmetries of the model. The study of the general IBM parameter space will be useful in classifying the degree of chaoticity of low-lying collective states of nuclei within the nuclear periodic table. We remark that at higher spin and/or energy, the bosons may break into quasi-particles and it is important to include these additional fermionic degrees of freedom to obtain a realistic description “).

We thank F. Iachello for interesting discussions. by DOE Grant No. DE-FG-0291ER-40608.

This work was supported

in part

66

N. Whelan, Y. Alhassid / Chaotic properties

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