THE BERRY PHASE AND DIABOLIC PAIR TRANSFER IN ROTATING NUCLEI

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THE BERRY PHASE AND DIABOLIC PAIR TRANSFER IN ROTATING NUCLEI

P. Ring, R. Nikam

To cite this version:

P. Ring, R. Nikam. THE BERRY PHASE AND DIABOLIC PAIR TRANSFER IN ROTATING NU- CLEI. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-339-C2-342. �10.1051/jphyscol:1987251�.

�jpa-00226520�

JOURNAL D E PHYSIQUE

Colloque C 2 , supplement au n o 6, Tome 48, j u i n 1987

P. R I N G and R . S . NIKAM

Physik-Department der Technischen Universitat Munchen, 0-8046 Garching, F.R.G.

Rdsumd

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Abstract - A new manifestation of the Berry-Phase is presented in the theory of fast rotating superfluid nuclei: Diabolic Pair Transfer, i.e. oscillations of the pair transfer matrix elements a s a function of the angular velocity, is shown to be the direct consequence the Berry-Phase, giving a non-trivial contribution a t the diabolical points of cranked Hartree-Fock-Bogoliubov (HFB) spectra.

Considering a general Hamiltonian H(X,w, ...), which depends on more than one para- meters X,o,... Berry discovered recently1), that a quantum system described as an eigenstate of this Hamiltonian, acquires a topological phase factor exp{i.d(C)>, when transported adiabatically around a path C in parameter space. This phase is of special importance in cases where the path includes a so called diabolical point, i.e. a point' in parameter space where two eigen energy surfaces E1(X,w) and E,(h,u) with the same symmetries of the Hamiltonian touch each other, a s indicated schematically in Fig.1. These points are exceptional points in the sense, that they violate the no crossing rule of von Neumann and Wigner2)

A

E,lh,wl

Fig.1 Schematic representation of the energy surfaces formed by two eigenvalues E(h,w) of an Hamil- tonian, which depends on two pa- rameters h and w. A path C is chosen on the lower surface in such a way that it includes a Dia- bolical Point, i.e. a point where the two surfaces touch each other

' l ' ~ o r k supported in part by t h e Alexander von Humboldt Poundation and by t h e Bundesrninisterium fllr Porschung und Technologic

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987251

C2-340 JOURNAL DE PHYSIQUE

This phase has the form of a generalized magnetic flux going through a surface sur- rounded by the path C. The corresponding generalized magnetic field 3 is the curva- t u r e of t h e line bundle defined by the eigenfunctions Q(h,o) on this path. I t s sources a r e magnetic monopols sitting in the diabolical points. The phase one half of

the solid angle, under which the path-is- seen from the diabolical point. In the case of only two parameters, where the parameter space i s a plane, it i s therefore -1 in cases, where the path includes a diabolical point and +1 otherwise.

Recently the manifestation of this phase has appeared in many diverse areas of phy- sics (c.f. ref.3 and references given therein). A s Berry himself has pointed out, this phase factor i s quite universal, and appears for any system described by a Herrnitian operator. I n this talk we present an example in nuclear physics. We show that the recently predicted Diabolic Pair Transfer4) in rotating superfluid nuclei is a direct manifestation of a Berry phase.

I t has been already realized in refs. 5 and 6 t h a t t h e Cranked Shell Model Ha- miltionian7 )

The diabolical points (indicated by circles) in the (X,o)-plane for the j=13/2 model. Full lines correspond to lines of constant average partic- le number, indicated by the numbers 1,2.. above the abscissa

has in i t s spectrum several diabolical points. I t s eigenfunctions a r e many-body wave- functions of the HFB type. They describe rotating superfluid nuclei and depend on two parameters: The chemical potential X and the angular velocity o. These parameters a r e determined by the particle number A and the angular momentum I and a r e closely connected to deformations and pairing correlations, t h e two most important degrees of freedoms of nuclear collective motion.

The diabolical points of the Iiamiltonian (1) correspond to alignment processes of particles in the intruder orbits. They indicate s h a r p level crossings between certain configurations, a s for instance between the ground state band and an aligning two- quasiparticle band or between other multi-quasiparticle configurations. I n the Rare Earth region the intruder orbit i s the neutron l i l , / , orbit. We therefore use a model of particles moving in a deformed and superfluid single j-shell with j = 13/2, with

8~ = ~ . ( 3 K = / j ( j + l ) - l ) . I n particular we use A/K = 0.45.

I n Fig.2 we show the diabolical points in the (X,w)-plane. The pattern of these points i s v e r y regular, a s is shown i n a schematic way in Fig.3. I t can be understood8) in terms of vanishing spatial overlap integrals between rotating single particle orbits with different signature. In Fig.:! we also give the lines of constant average particle number and we find an interesting bunching of these lines a t the diabolical points.

ENTRANCE EXIT

Fig.3 Schematic representation of t h e pattern of diabolical points in t h e (X,w) plane.

Full horizontal arrows indicate pair transfer matrix elements with positive sign and dashed arrows indicate those with negative sign. Below t h e abscissa we indicate t h e K-quantum numbers in the deformed j=13/2 shell. On t h e right hand side experimental trajectories a r e shown, which correspond to two dif- ferent paths around a diabolical point which yield destructive interference.

W e now s t u d y t h e influence of the Berry-Phase a t a specific diabolical point. We have to choose a closed path in t h e (h,w)-plane around this point. Having in mind, that changes in h a t constant w mean changes in t h e particle number a t constant angular momentum, i.e. transitions from the nucleus A with spin I to t h e nucleus At2 with the same spin I by transfering a pair of nucleons coupled t o angular momentum zero, and t h a t change& in w for constant X indicate changes of the angular momentum within the same nucleus, we see t h a t we can go on two different pathes from t h e point (A,I) to t h e point (A+2,1+2): either (A,I) + (At2,I) + (A+2,1+2) or (A,I) + (A,It2) + (A+2,1+2). Both pathes enclose t h e diabolical point, i.e. we have to end u p with wavefunctions Q(A+2,1+2), which differ by t h e phase -1. Since, by convention we choose a continuous connection for the phases of t h e wavefunctions along t h e lines with constant A, we see that t h e pair transfer matrixelement

has to change i t s sign when crossing t h e diabolical point (st is a pair of particles coupled to angular momentum zero ( S t = ( C * C ~ ) ~ = ~ )). We t h u s have a v e r y natural explanation for t h e recently predicted effect of Diabolic Pair Transfer'), namely t h e oscillating bahavior of t h i s matrix element shown in Fig.5.

A s pointed out in ref.4 this effect i s a nuclear analogue of the DC-Josephson effect in solid s t a t e physics. Obviously t h e nucleus i s a finite system and therefore t h e number of oscillations is finite. A s we see from Fig.3 it depends on t h e number of diabolical points between t h e t o pathes (X(A),o) and (A(A+2),o). I n Fig.2 ws find, t h a t we have no such diabolical point and no oscillation, if the pair of nucleons i s trans- fered to t h e neighborhood of the K=1/2 o r of t h e K=13/2 orbit in the i,,/, shell. We have one diabolical point and one sign change of the pair t r a n s f e r matrix element, if t h e pair i s transfered to t h e K=3/2 o r to the K=11/2 orbit. For the K=5/2 and t h e K=9/2 we have two and for t h e K=7/2 orbit we have t h r e e oscillations.

The effect is not only restricted t o t h e i=13/2 model, b u t i t also shows u p in fully realistic configuration spaces a s shown in Fig.4. I n fact all t h e arguments given in t h e present paper apply t o t h e general case too. I t i s therefore certainly t h e most interesting question to ask if this new effect can be discovered experimentally. The most direct evidence f o r the phase change would certainly be a destructive interfe- rence between t h e two trajectories, viz. (A,I) + (At2,I) + (At2,1+2) and (A,I) +

(A,1+2) + (A+2,It2) which enclose a diabolical point, a s showm in Fig.4.

JOURNAL DE PHYSIQUE

Fig.4 Diabolic Pair Transfer in the Nuc- leus 168Hf. The calculations a r e carried out in t h e realistic con- figurations space and with the re- sidual interaction of Baaner and Kumar. The pairing-deformation of the potential in eq.14 i s deter- mined in such a way, that the pairing-deformation of t h e wave- functions A s t a y s contant. Cal- culations for different values of A a r e shown.

On the right hand side of Fig.3 we show, a t least schematically, the experiment, which would correspond to these two paths. It represents Coulomb excitation using heavy ions in connection with the transfer of a pair of particles. In a classical appro- ximation the transfer takes place a t the distance of closest approach, a t which half the final angular momentum i s transfered. We therefore propose the following: choose an appropriate initial energy and look only for the largest final angular momenta, such that t h e two most important reaction contributions come from the paths on two different sides of a diabolical point. In such a case one should be able to study the destructive interference between the wave functions corresponding to these two trajectories.

Since this is probably a very difficult experiment, we would propose only looking a t the square of the pair transfer matrix element: In the diabolic region i t goes through zero, which means one should observe a reduction of this quantity in t h e this region.

There a r e essentially two ways to search for such a reduction: i) One could investigate t h e behavior of P(1) for a fixed value of A a s a function of the angular momentum I. In this case one crosses the diabilical region in Fig.4 in a vertical di- rection. ii) One also could investigate the behavior of P(1,A) for fixed "diabolical" I in a chain of isotopes, i.e. a s a function of A. Now we cross t h e diabolical regions in a horizontal direction. One should observe a reduction for those nuclei, which show a particularly strong backbendingl1).

Since the diabolical points lie a t relatively low angular momenta in many nuclei, which can be reached by Coulomb excitation using heavy ions in connection with pair transfer1'), we hope that this new manifestation of the Berry-Phase will be seen experimentally in near future.

References:

1) M.V.Berry, Proc.Roy.Soc.(London) A392 (1984) 45 2) J.von Neumann and E.P.Wigner, Phys.2. 30 (1929) 467 3) R.Y.Chiao and Yong-Shi Wu, Phys.Rev.Lett. 57 (1986) 933 4) R.S.Nikam, P.Ring and L.F.Canto, Z.Phys. (1986) 241

5) R.Bengtson, 1.Hamamoto and B.R.Mottelson, Phys.Lett. 73B (1978) 259 6) H.Frisk and Z.Szymanski, Phys.Lett. (1986) 272

7) P.Ring and H.J.Mang, Phys.Rev.Lett. 3 (1974) 1174

8) R.S.Nikam, P.Ring and L.F.Canto, Phys.Lett. B (1987) in print 9) B.R.Mottelson and J.G.Valatin, Phys.Rev.Lett. 5 (1960) 511 10) B.Simon, Phys.Rev.Lett. 51 (1983) 2167

11) We thank Dr.Ch.Lauterbach for bringing this point to our attention.

12) M.W.Guidry et al, Nucl.Phys. &36J (1981) 275