Physics Letters B 527 (2002) 55–61

www.elsevier.com/locate/npe

Singular character of critical points in nuclei

V. Wernera, P. von Brentanoa, R.F. Castenb, J. Joliea

a Institut für Kernphysik, Universität zu Köln, Germanyb WNSL, Yale University, New Haven, CT 06520-8124, USA

Received 22 June 2001; received in revised form 12 December 2001; accepted 3 January 2002

Editor: W. Haxton

Abstract

The concept of critical points in nuclear phase transitional regions is discussed from the standpoints ofQ-invariants, simpleobservables and wave function entropy. It is shown that these critical points very closely coincide with the turning points of thediscussed quantities, establishing the singular character of these points in nuclear phase transition regions between vibrationaland rotational nuclei, with a finite number of particles. 2002 Elsevier Science B.V. All rights reserved.

PACS: 21.60.-n; 21.60.Ev; 21.60.Fw

Keywords: Critical point symmetry; Phase transition; Shape transition; Quadrupole shape invariants; Wave function entropy; IBA

Nuclear structural evolution in transitional regionsis often thought of as a continuous variation of proper-ties, as a function of nucleon number, from one ideal-ized limit (e.g., vibrator, rotor) to another. The rapid-ity of structural change may vary across a transitionalsequence of nuclei, and different mass regions exhibitdifferent rates of change but, until recently, no individ-ual point along these evolutionary trajectories could besingled out with special observational properties.

In the last years, however, the concept of criticalpoints in shape/phase transition regions has been muchdiscussed [1–5]. While the concept itself is wellknown in nuclei (in the context of the coherent stateformalism [6,7] of the IBA model [8]), it is onlyvery recently that analytic descriptions of critical pointnuclei have been given [9,10]. This is a significant

E-mail address: vw@ikp.uni-koeln.de (V. Werner).

point since, historically, such nuclei have been themost difficult to treat: they exhibit competing degreesof freedom, and one has had to resort to numericalcalculations.

Two critical point symmetries, called E(5) andX(5), have been proposed [9,10], giving analyticexpressions for observables which are exactly at thecritical points of a vibrator to axially asymmetric(γ -soft) rotor transition region, and of a vibrator tosymmetric rotor transition region, respectively, for aninfinite number of nucleons. An important aspect ofthis is that, for the first time, one is able to associatespecial observational characteristics to a specific pointalong a trajectory from one structural limit to another.Recently [11], using the methods presented here, thewell-known O(6) limit of the IBA has also beenidentified as another, heretofore unrecognized, criticalpoint symmetry, for the transition between prolateand oblate nuclei. This is an important result since

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(02)01160-7

56 V. Werner et al. / Physics Letters B 527 (2002) 55–61

the O(6) symmetry can be calculated in the IBA forfinite nucleon numbers, in contrast to the non-IBAsymmetries E(5) and X(5). So far only two examplesfor nuclei [12,13] which lie close to the X(5) and E(5)symmetries are known while, interestingly, there aremany examples for O(6) like nuclei. In the presentwork we will restrict our discussion to prolate nuclei.

To understand the evolution of structure in realnuclei, with a finite number of nucleons, it is importantto gather information about systematic changes ofobservables at or near such critical points. This aimcan be achieved by the use of a model that isable to describe limiting cases of nuclear structure—vibrators, rotors andγ -soft nuclei—and a large varietyof nuclei between these limits. Such a model is givenby the IBA, which—in the expansion of the coherentstate formalism—exhibits critical points as has beendiscussed in Refs. [6,7,14,15]. We stress that thecritical point descriptions X(5) and E(5) are definedin terms of a geometrical approach, not the IBA.Nevertheless the IBA provides a convenient tool tospan a range of structure, including phase transitions,and also to assess effects of finite particle numbers.

It is the purpose of this Letter to show, from sev-eral complementary theoretical approaches, that thereis independent evidence for the singular character ofthese critical points, and independent ways of identify-ing them in observables calculated in collective mod-els. To do so we bring together three major themes:the already mentioned study of phase transitional re-gions and critical point nuclei, the behavior of quadru-pole shape (Q)-invariants, and the study of chaos andentropy in nuclear systems. We show that the criticalpoints occur very near to the turning points (points ofsteepest descent or ascent) of theseQ-invariants—thatis, at the extrema of their first derivatives. The samebehavior will also be shown to hold for some moreeasily accessible observables.

To span the transition regions, it is convenient touse the IBA Hamiltonian in the following form

(1)H = a

[(1− ζ )nd − ζ

4NQ · Q

],

whereQ = s†d̃ + d†s + χ[d†d̃](2) and we considerthe well known parameter space of the extendedconsistentQ formalism (ECQF) varyingζ between 0and 1, andχ from 0 to −√

7/2 = −1.32, whilea isa scaling factor. This parametrization is equivalent to

Fig. 1. Symmetry triangle of the IBA model. The U(5) ↔ O(6) legis characterized byχ = 0 and varyingζ , while the U(5) ↔ SU(3)

transition region hasχ = −√7/2 and ζ is varied. The dashed

line indicates the phase transitional region where critical points arefound.

the more commonly encountered (equivalent) ECQF[16,17] form of H , which includes the parametersεandκ .

Fig. 1 illustrates the three dynamical symmetries ofthe IBA in terms of a triangle. With the Hamiltonianof Eq. (1) it is easy to calculate the structure for anypoint in the triangle. Forζ = 0 one obtains a U(5)structure (for anyχ ), and ζ = 1, χ = −√

7/2 givesSU(3). Thus, a U(5) ↔ SU(3) transition region isdefined byχ = −√

7/2 andζ varying from 0 to 1,while a U(5) ↔ O(6) region hasχ = 0 andζ varyingfrom 0 to 1.

One can use the coherent state formalism [6,7] ofthe IBA model to identify the critical points in theECQF space. In this approach, the energy functionalfor the ECQF Hamiltonian is given by

E(ζ,χ,β, γ )

= Nβ2(1− ζ(χ2−3)

4N−4Nζ+ζ

)1+ β2

− N(N − 1)ζ

4N − 4Nζ + ζ

×(

4β2 − 4√

2/7χβ3 cos 3γ + 2

7χ2β4

)

(2)× (1+ β2)−2

.

The variation of ζ changes the structure betweenthe vibrator limit and rotational nuclei—both axially

V. Werner et al. / Physics Letters B 527 (2002) 55–61 57

symmetric and axially asymmetric—which are thetransitions we will focus on. Critical points inζ arefound whereE becomes flat atβ = 0. These points,which we refer to asζc, can be derived by evaluatingthe condition

(3)

∣∣∣∣∂2E(ζc)

∂β2

∣∣∣∣β=0

= 0.

On the transition path from U(5) to O(6) (forχ =0) exactly one critical point is found, namely, wherea second, deformed, minimum inβ of the energyfunctional emerges.

The situation becomes more complicated for tran-sitions withχ �= 0. In these cases, the spherical min-imum is joined by a deformed minimum and bothminima coexist in a very close parameter range inζ ,converging to one point when approachingχ = 0.Thus, in general there exist three critical points, whichis illustrated in Fig. 2 forN = 10 bosons for the lim-iting case ofχ = −√

7/2. The thick lines in Fig. 2give points in the(ζ,β) plane, which are local min-ima of the energy functional (2). The shaded area isthe parameter range ofζ , where two local minima ofthe energy functional coexist. The lower dashed linegives the criticalζ value where a deformed minimumappears, while the upper dashed line gives the criticalpoint in ζ where the spherical minimum disappearsand only the deformed minimum is left. The dottedline gives the criticalζ value where two coexisting

Fig. 2. The thick lines represent the locus in the (ζ ,β) parameterspace where the energy functional of the coherent state formalismhas a local minimum. The thick line atβ = 0 extends downwardsto ζ = 0. The results are shown for the case ofN = 10 bosons.Dashed lines mark criticalζ values where one minimum disappears(the spherical one at and above the larger value, the deformed oneat and below the lower value). Only in the shaded area two minimacoexist. The dotted line marks the criticalζ value where these twominima are equally deep.

minima are equally deep. The parameter region in be-tween is small for any boson number.

Thus, as it is the aim of this work to identify thecritical points in observables, and we do not expectto be able to distinguish between these three points(close lying inζ ) in real nuclei, we restrict ourselvesto the critical point given by condition (3) where thespherical minimum disappears, and which is given by

(4)ζc = 4N

8N − 8+ χ2N→∞−→ 0.5.

The χ dependence is just a finiteN effect, and thusit is convenient to vary only the parameterζ for theinvestigation of phase transitions between vibrationaland rotational nuclei. Additionally we note that thechoice of our parametrization has the convenientfeature that in the largeN limit we getζc = 0.5.

While, due to their physical meaning, the endpointsof the line of critical points betweenχ = 0 andχ =−√

7/2 in Fig. 1 can be approximately related to thenon-IBA symmetries E(5) and X(5), we see that amuch richer structure shows up in the IBA, wherecritical points occur over the whole transitional regionbetween these legs of the symmetry triangle.

Since we are interested in obtaining signatures forcritical points in observables including matrix ele-ments, we now survey the behavior ofQ-invariants[18,19] in the transition regions. Recently, the conceptof Q-invariants has been re-investigated in the frame-work of the IBA model and theQ-phonon approach[20,21], and the behavior of these moments across thegamut of nuclear collective structures has been elu-cidated [22–24]. These invariants represent quadraticand higher order moments of the quadrupole operator.The invariants are denotedqn andKn ≡ qn/q

n/22 , and

are defined by expressions of the generic type

(5)qn ∼ 〈Ψ0|Q1 · Q2 · · ·Qn|Ψ0〉,where Ψ0 is the ground state wave function, andwhere intermediate angular momentum couplings inthe operator are omitted for simplicity.

For the IBA [8], theQ-invariants have been eval-uated over the entire symmetry triangle of Fig. 1. Toshow the extreme cases, we first focus on the twotransition paths U(5) ↔ SU(3) (χ = −√

7/2) andU(5) ↔ O(6) (χ = 0). We note that the invariantsq2,K3, K4, and σγ ≡ K6 − K2

3 represent, respectively,

58 V. Werner et al. / Physics Letters B 527 (2002) 55–61

Fig. 3. Behavior ofq2, K4 andσγ , and their first derivatives with respect toζ , for the U(5) ↔ SU(3) transition region, calculated forN = 10bosons.

the quadrupole deformation, the triaxiality, the soft-ness of the nuclear shape inβ , and inγ .

We first study the U(5) ↔ SU(3) transition andobtain the results shown forN = 10 in the top rowof Fig. 3 for q2,K4 andσγ . Each of these exhibits arapidly changing behavior which has a turning pointζt near ζ = 0.5. To investigate this in more detail,the second row of Fig. 3 shows the first derivativeswith respect toζ . Again there is a striking consistencyof behavior: the first derivative has an extremum atessentially the same point for each invariant.

Specifically, the turning points (the zeros of the sec-ond derivatives) are:ζt = 0.54 for q2; ζt = 0.53 forK4; and ζt = 0.52 for σγ . In the coherent state for-malism, for N = 10, one obtainsζc = 0.54 for theU(5) ↔ SU(3) case. This is very close to the turningpoints in q2,K4 and σγ : that is ζt ∼ ζc. This corre-spondence between the turning points and the criticalpoints is the main result of this work. The small differ-ences probably represent a finite boson number effect.

This identification of a special point along thestructural evolution from vibrator to rotor is apparenteven in the simplest observables as well. In Fig. 4we show the behavior of the structural observablesR4/2 ≡ E(4+

1 )/E(2+1 ) andB(E2 : 2+

1 → 0+1 ) for the

U(5) ↔ SU(3) transition, again forN = 10. Clearly,as seen in the first derivative plots in the second row,both quantities exhibit their steepest rates of changenear the critical points. Here, the first derivative has anextremum atζt = 0.54 for bothR4/2 and theB(E2)

Fig. 4. Similar to Fig. 3 (forN = 10) for the observablesR4/2 and

B(E2 : 2+1 → 0+

1 ) for the U(5) ↔ SU(3) transition region.

value. In this latter case, this result is not surprisingsince thisB(E2) value andq2 are directly related.

The existence of three critical points on the U(5) ↔SU(3) transition path seems not to be reflected in theQ-invariants, which may be explained by the verycompact parameter region inζ where these criticalpoints occur, while the peaks in the derivatives havea certain width. Also note that fluctuations, resultingfrom the limited numerical accuracy of the PHINT

code used for these calculations, have been smoothedby the use of splines. Thus, perhaps the three criticalpoints just cannot be resolved in the observables dueto numerical truncations.

V. Werner et al. / Physics Letters B 527 (2002) 55–61 59

Fig. 5. Similar to Fig. 3 (forN = 10), forq2 andK4, for the U(5) ↔ O(6) transition region (left panels), and forq2 in the O(6) ↔ SU(3) (notehere with respect toχ ) transition region (right).

Returning to theQ-invariants, similar results applyin the U(5)→O(6) region. Fig. 5 (left panels) showsthis for q2 and K4. In this case the turning points(determined from the rates of change), are:ζt = 0.60for q2 andζt = 0.56 forK4. From Eq. (2), the coherentstate formalism givesζc = 0.56 forN = 10. Again theζt and ζc values obtained from the behavior of theQ-invariants and from the coherent state formalismare quite close. Lastly, we note that the rate of changeof q2 andK4 in the U(5) ↔ O(6) case is much lessthan in the first order U(5)→SU(3) transition region.For example,(dq2/dζ )max ∼ 800 for U(5) ↔ SU(3)

while it is only ∼200 for U(5) ↔ O(6). Also, thewidths of the first derivative curves are much wider(corresponding to a more gradual structural evolution)in the U(5) ↔ O(6) case.

Using the IBA, it is also possible to investigateinternal paths in the symmetry triangle. In particular,internal straight line trajectories, starting from U(5),will correspond toχ values between 0 and−√

7/2,allowing a full mapping of transitional trajectories. Weillustrate such results by showing the change of thefirst derivative of the shape invariantK4 for variousvalues ofχ in Fig. 6. The minima of the derivativesfollow the line of critical points that is also givenin the coherent state formalism, with only a smallχ

dependence.Finally, in regard toQ-invariants, we look at the

O(6) ↔ SU(3) transitional region. The right panels of

Fig. 6. First derivative ofK4 (for N = 10), but for various valuesof the parameterχ . A peak indicating a phase transition occurs forevery value ofχ . The dependence of its position onχ is a finiteN

effect.

Fig. 5 show the behavior ofq2 and its derivative. Notethat the shape is qualitatively different than in the othertransition regions, showing a gradually asymptoticcurve and a first derivative againstχ (the appropriatevariable for this region) which is monotonic. Nocritical point is definable in this region ofχ values,except when O(6) itself is reached (see Ref. [11]).

Another theme in nuclear structure recently hasbeen the study of order and chaos for different struc-tures. It was shown in Ref. [25] that nuclear systemsdisplay ordered spectra at and near the three symme-try limits of the IBA, but that there is a rapid onset ofchaotic behavior away from these benchmark regions.(See Fig. 1 of Ref. [25] but note that the symmetry tri-

60 V. Werner et al. / Physics Letters B 527 (2002) 55–61

angle is differently defined therein.) Recently, Cejnarand Jolie [26,27] have developed the concept of wavefunction entropy as an alternate (and physically intu-itive) way of studying the relative complexity of nu-clear wave functions. Basically, the entropy of a stateis a measure of its spreading within a given basis. Notethat this is not the same as the chaoticity (which is ba-sis invariant) since a wave function may have high en-tropy in one basis [e.g., U(5)] and low entropy in an-other [e.g., SU(3)].

Now that we showed a visible effect of criticalpoints in various observables, it is interesting to seewhether effects of a phase transition can also be seen inthe wave functions and thus the wave function entropy.A rise of the wave function entropy can be expected inmoving from one limit to another, but the question iswhether it also appears in a close region with turningpoints which coincide with the turning points of thepreviously mentioned observables. Thus, we define[26] a quantity, calledWB

Ψ , for a stateΨ , that can bewritten in the basisB asΨ = ∑n

iB aiB |ΨB〉, as

(6)WBΨ ≡ −

n∑iB=1

|aiB |2 ln∣∣aBiB

∣∣2,wheren is the number of basis vectors. IfΨ coincideswith a basis vector, thenWB

Ψ = 0. If Ψ is uniformlyspread out over the basisB, thenWB

Ψ ≈ lnn.A physically intuitive expression of the entropy is

the quantity [27]

(7)nBeffΨ ≡ expWBΨ

which expresses a kind of “effective number” of wavefunction components. For a “pure” stateΨ , nBeffΨ

= 1

and for a fully de-localized statenBeffΨ≈ n.

To properly normalize the entropies we define theentropy ratio

(8)rB ≡ expWBΨ − 1

exp〈WGOE〉 − 1

relative to that for the Gaussian Orthogonal Ensemble[27]. The ratiorBΨ varies from 0 for a pure (localizedin the basisB) state to∼1 for a highly mixed state(see Ref. [27] for a more detailed discussion of thisnormalization).

We show the results in Fig. 7 forrB0+

1and its

derivative as a function of the order parameterζ

Fig. 7. The entropy ratio (forN = 10) for the 0+1 state (top row) inthe three transition regions, plotted againstζ and given, for eachregion, in two bases as indicated (e.g., U(5) and SU(3) for theU(5) ↔ SU(3) transition). The lower panels give the derivative ofthe entropy ratio againstζ in the appropriate basis.

for the U(5) ↔ SU(3) and U(5) ↔ O(6) transitionregions (all forN = 10). The entropy ratio for theground state undergoes a very rapid change nearζc forboth transition regions. We note that for larger bosonnumbersN the transition becomes much sharper (seeFig. 6 in [11]). For the U(5) ↔ SU(3) and U(5) ↔O(6) phase transitions, it is easy to read the turningpoints,ζt , values from the derivative plots, obtainingζt = 0.52 andζt = 0.59 (in a U(5) basis), respectively,compared to values ofζc = 0.54 and ζc = 0.56from the coherent state formalism. We note that thesteepness of the entropy functions againstζ increaseswith boson numberN , as pointed out in Ref. [28]. Thisalso holds true for the observables studied above.

To conclude, from the behavior of several ratherdifferent quantities, theQ-invariants, the simple ob-servablesR4/2 andB(E2 : 2+

1 → 0+1 ), and the wave

function entropy, we have shown that critical pointsof the phase transitional regions U(5) ↔ SU(3) andU(5) ↔ O(6) are reflected in the behavior of theseobservables along these evolutionary trajectories. Thisresult was obtained for finite boson numbers, making itpossible to investigate effects of valence particle num-ber on the singularities.

Acknowledgements

We are grateful to N.V. Zamfir, F. Iachello, J. Eberthand K. Heyde for useful discussions, and to P. Cejnar

V. Werner et al. / Physics Letters B 527 (2002) 55–61 61

for the entropy calculations. Work supported by theUS DOE under Grant number DE-FG02-91ER40609and by the DFG under Project number Br 799/10-1and by NATO Research Grant No. 950668. One of us[R.F.C.] is grateful to the Institut für Kernphysik inKöln for support.

References

[1] A. Wolf et al., Phys. Rev. C 49 (1994) 802.[2] R.F. Casten, N.V. Zamfir, D.S. Brenner, Phys. Rev. Lett. 71

(1993) 227.[3] F. Iachello, N.V. Zamfir, R.F. Casten, Phys. Rev. Lett. 81

(1998) 1191.[4] R.F. Casten, D. Kusnezov, N.V. Zamfir, Phys. Rev. Lett. 82

(1999) 5000.[5] J. Jolie, P. Cejnar, J. Dobes, Phys. Rev. C 60 (1999) 061303.[6] A.E.L. Dieperink, O. Scholten, F. Iachello, Phys. Rev. Lett. 44

(1980) 1747.[7] J.N. Ginocchio, M.W. Kirson, Phys. Rev. Lett. 44 (1980) 1744.[8] F. Iachello, A. Arima, The Interacting Boson Model, Cam-

bridge Univ. Press, Cambridge, 1987.

[9] F. Iachello, Phys. Rev. Lett. 87 (2001) 052502.[10] F. Iachello, Phys. Rev. Lett. 85 (2000) 3580.[11] J. Jolie, R.F. Casten, P. von Brentano, V. Werner, Phys. Rev.

Lett. 87 (2001) 162501.[12] R.F. Casten, N.V. Zamfir, Phys. Rev. Lett. 87 (2001) 052503.[13] R.F. Casten, N.V. Zamfir, Phys. Rev. Lett. 85 (2000) 3584.[14] E. Lopez-Moreno, O. Castanos, Phys. Rev. C 54 (1996) 2374.[15] E. Lopez-Moreno, O. Castanos, Rev. Mex. Fis. 44 (1998) 48.[16] D.D. Warner, R.F. Casten, Phys. Rev. Lett. 48 (1982) 1385.[17] P.O. Lipas, B.P. Toivonon, D.D. Warner, Phys. Lett. B 155

(1985) 295.[18] D. Cline, Ann. Rev. Nucl. Part. Sci. 36 (1986) 683.[19] K. Kumar, Phys. Rev. Lett. 28 (1972) 249.[20] G. Siems et al., Phys. Lett. B 320 (1994) 1.[21] T. Otsuka, K.-H. Kim, Phys. Rev. C 50 (1994) 1768.[22] R.V. Jolos et al., Nucl. Phys. A 618 (1997) 126.[23] Yu.V. Palchikov, P. von Brentano, R.V. Jolos, Phys. Rev. C 57

(1998) 3026.[24] V. Werner et al., Phys. Rev. C 61 (2000) 021301.[25] Y. Al-hassid, N. Whelan, Phys. Rev. Lett. 67 (1991) 816.[26] P. Cejnar, J. Jolie, Phys. Lett. B 420 (1998) 241.[27] P. Cejnar, J. Jolie, Phys. Rev. E 58 (1998) 387.[28] P. Cejnar, J. Jolie, Phys. Rev. E 61 (2000) 6237.