Schwinger boson mapping of SO(8) fermion model

Schwinger boson mapping of SO(8) fermion model

R S NIKAM

Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India

Abstract. The Schwinger representation of the SO(8) fermion pair algebra in terms of d and quasispin vector (u, s, v) bosons is used in deriving a microscopic boson coherent state having both particle-hole and pair excitations. The coherent state is the exact boson image of the HFB variational solution. We can study the shape phase transition and pairing behaviour of the nuclear ground states using the coherent states.

Keywords. Schwinger boson mapping; SO(8) boson coherent state; phase transition properlies.

PACS No. 21.60

I. Introduction

The interacting boson model (IBM) of the atomic nucleus has got some phenomeno- logical success, but its connection with the underlying fermionic degrees of freedom is not well understood and therefore it is important to study a fermionic system which has an exact bosonic counterpart. Ginocchio's SO(8) model provides such a possibility and therefore we study its bosonized version from a geometric variable point of view and its properties under phase transition. IBM has also been interpreted in terms of geometrical shape variables (lachello 1981) by considering a condensate of deformed bosons having a fixed boson number (Ginocchio and Kirson 1980; Dieprink et al 1980). More recently it was shown that the IBM Hamiltonian can be divided exactly into an intrinsic and a collective part, the intrinsic part having the condensate as an eigenstate (Kirson and Leviatan 1985; Leviatan 1985) and defining collective bands as zero-energy excitation of Goldstone bosons associated with spontaneously broken symmetries.

Recently a fermionic SO(8) symmetry model with pairing and quadrupole interac- tions has been proposed (Ginocchio 1979; 1980) which has quite an essential piece of physics in common with the IBM, namely it is a model of scalar and quadrupole, or S- and D-pairs. Evidence for SO(8) symmetry in actual nuclei has been formally investigated within various boson mapping methods (Arima etal 1981; Geyer etal 1987; Kim and Vincent 1987; Kaup 1987). Standard methods yield finite hermitean forms of boson operators only for a restricted range of interaction parameters.

However, it has been discovered that the SO(8) fermion algebra can be mapped exactly on a Schwinger representation (Kaup 1987) of collective bosons. This new method thus can be regarded as a fully microscopic pairing-plus-quadrupole boson model.

The SO(8) model can be interpreted in terms of geometrical shape variables using the Hartree-Fock-Bogoliubov (HFB) mean-field approximation. Using Schwinger re- presentation (Kaup 1987) we show that the fermion HFB variational state corresponds 331

exactly to a boson coherent state of the IBM-type. The only difference is that in the SO(8) coherent state the additional collective mode of the pairing rotation is built in. This is reflected by the spontaneous breakdown of the particle number SO(2) symmetry. Using this coherent state shape phase transition properties of this model will be investigated.

2. Schwinger representation of SO(S)

Similar to Schwinger's realization of SO(3) in terms of SU(2) generators the bosonization of SO(8) is carried out by embedding SO(8) into U(8). Therefore we need eight bosons to realize SO(8). These are three angular momentum zero bosons (u, s, v), and five angular momentum two bosons (d~). We write down all 28 generators of the SO(8) in Schwinger representation (Kaup 1987; Kaup et al 1988).

S + = v/2(s+u + v+s), S = ,¢/2(u+s + s+v), (la) O+ = x / ~ ( d + u - v+ Z[u), FJu= x/~(u+ ~[u-d+ v), (lb)

So = v+v - u+u, Qu = 2(d2s + s+~lu), (lc)

L~, = x / ~ ( d + ~ , P3 = ~ ( d + ~ . (ld)

Standard notations are used for angular momentum coupling. Equations (1 a)-(l d) are an isomorphic boson representation of the shell model fermion operators defined by Ginocchio (1979). S, D~ are pair operators having angular momentum 0, 2. L~ is the angular momentum, Qu the qUadrupole, and _~P°) the octupole operator respectively.

So = (Nv - f~)/2, where Nr is the fermion number operator and ~ is half the number of available s.p.-levels. The boson image "lvac)" of the fermion vacuum is not the true boson vacuuml) but

Ivac) = ( ~ . t ) - '(u + )NI). (2)

All the fully paired fermion states can be constructed by applying the pair operators on vacuum. Since the Schwinger representation (1) conserves the boson number N and since S o = -[1/2 for the fermion vacuum, the evaluation of Solvac) in boson space yields N = fl/2 in (2). The SD pair states span the irrep (f~/2, 0,0,0) of SO(8) the boson realization of which is the "physical subspace" of the boson space. There are three dynamical subalgebra-chains (Ginocchio 1979, 1980) having the angular momentum SO(3) and its particle number SO(2) subalgebras namely

SU(2) ® SO(5)

SO(8) SO(7) SO(2) ® SO(5)

~ S O ( 2 ) ® SO(6) /

where the generators of the subalgebras are, SU(2):{S, So, S+ },

SO(7):{So, D~, + D~, L~, P~}, 3

so(2):{s0},

S0(2) ® S0(3), (3)

(4a) (4b) (4e)

S0(6): {L., P], Q.}, (4d)

S0(5): {L~,, P~ }, (4e)

S0(3): {L.}. (40

From the expressions in (4) it is seen that SO(6) and its subalgebras SO(5) and SO(3) are realized only in terms ofs and d bosons which makes this chain identical to IBM SO(6).

We therefore identify our s and d bosons with IBM ones.

The general one-plus two-body hamiltonian having SO(8) dynamical symmetry is (Ginocchio 1979, 1980).

,,,

Since this hamiltonian commutes with the subalgebras SO(5), SO(3) and SO(2), only the first two terms of the hamiltonian decide the subgroup chain. Therefore assigning.

bl = b3 = bo = e = 0, (6a)

Go = - f c o s 0, (6b)

b 2 = - f s i n 0, (6c)

we get the one-parameter hamiltonian

1 sin 0

h(O) = f H = - cos 0 S + S - - - 4 - Q " Q. (7)

The parameter 0 determines the three subgroup chains as

SU(2): 0 = 0 °, (8a)

SO(7): 0 = 45 °, (8b)

S0(6): 0 = 90 °, (8c)

another useful parameter is the normalized fermion number defined as:

r/= (1"1 - N~)tFI, (9a)

Therefore

- ! ~<r/~< I. (9b)

3. S O ( 8 ) B o s o n c o h e r e n t s t a t e

Let us now turn to the HFB approximation and consider the most general H F B variational state carrying the SO(8)-irrep (f~/2, 0, 0, 0) which is a function of 6 variational parameters.

IP, ~.) = exp {p(S + - S) + ~ ~.(D + - D~)}lvac).

/J

(10) The state can be equivalently expressed in terms of the collective bosons. However, we

are actually interested in a less general H F B state, namely a semiclassical "intrinsic state" which is "at rest" with respect to collective rotations (strictly, the expectation values of collective momenta vanish)• There are two kinds of collective motions in the SO(8) model: spatial and pairing rotations• Time reversal invariance assures that the intrinsic state does not rotate in space. Particle-hole conjugation invariance is the analogue eliminating the pairing rotation. Imposing these requirements we introduce a body-fixed frame of reference both in coordinate and particle number ("gauge") space.

The Schwinger representation takes the intrinsic state at rest in coordinate and gauge space into a boson state of the form

]p, ct, 7 ) = exp

[ipSy +

with the SO(8) rotation R(0t, ~,) being generated by hermitean operators

= 2(S - S +) = i(b+s - s+b.),

Sy

i 1

= + ----~sin ~,(D 2 + D_ 2) - h.c.] =

i[b:b~ - b~-b~].

The Schwinger representation is here expressed in terms of deformed bosons defined in a similar way as in the IBM (Leviatan 1985) by

b~ = ~2(u + v), b. = ~22(u- v),

1 - - i = - ~ ( d 2 - d _ 2),

b x = ~ ( d l + d - x ) , b y = - - ~ ( d l - d - l ) , bz x~ ~

sin ~1.~ cos 1,1. ~

b ~ = c o s ) " d o + x / ~ , 2 + d - 2 ) ,

b y = - - ~ - 2 + d _ 2 ) - s i n ) , ' d o .

Carrying out explicitly the transformation we obtain the coherent state

1 +

Ip,~,~> = ~t l-R(ct,)')u R-l(~t,)')]NI) = ~N.r(b~+)NI),

where

b~ + = ~22(b, + cos p + s + sin p + cos ~t b, + + sin ~tb~+).

04)

being the deformed boson built by four rather than three constituents since the b~-boson is itself a linear combination depending on the variational parameter. The remaining four bosons are forbidden to appear because they are odd, i.e. change sign, either under time reversal (bx,y.z), or under particle-hole conjugation (bn). They are the Goldstone bosons associated with spurious angular and gauge rotations:

Lk lp, ct, ), > = x/~ sin ~t'sin( ~, - V ) b~ bclp, ot, ~, >,

(15)

l +

- , - c o s ~ b . bclp,~,~).

S°IP'='~') = x / 2 (16b)

The cartesian components x, y, z have been enumerated here with k = 1, 2, 3 to save space. The condensate (8) is automatically invariant under a 180°-rotation a b o u t the x-axis and thus has a conserved positive signature. Allowing for b~ or b, in the condensate would make sense in gauge and angular cranking models, respectively, breaking particle-hole or time reversal symmetry but still conserving signature.

4. Phase transition properties

The coherent state of (15) is treated as a variational ansatz with p and ct as the variational parameters. The equilibrium values of these parameters are found by minimizing the energy expectation

E(~) = <a, ~, ~lnlp, ~,, ~>. ⁽¹⁷⁾

It can be shown that since the hamiltonian (7) has SO(5) symmetry the energy (17) is independent of ?. This procedure is completely anologous to the H F B m e t h o d in fermionic space where quite generally in the H F B theory one minimizes, for separable interactions, only the direct term viz.

(A+A) = (A+)(A) + [ ( A + A ) - (A+)(A)] (18) i.e. neglect the second term in the square bracket. This term is the fluctuating part containing contributions from exchange and other lower order terms. Since th SO(8) hamiltonian (7) is in the required separable form in order to evaluate its expectation value we need to know the expectation values of the following operators

( Q o ) = 2N sin p sin atcos),, (19a)

(Q± 2) = x / ~ N sin p sin ,t sin ),, (19b)

( S + ) = ( S ) = N sin p cos =, (19c)

( S o ) = - N cos p cos ~,, (19d)

( D ~ ) = ( D o ) = N cos p sin ct cos ),, (19e) ( D ~ + 2 ) = ( D ± 2 )

=iN

(Lz> = ( p a> = 0. (19g)

Using these the direct term of the energy expectation value is calculated to be E(p, ~) = N 2 {b 2 + (Go - be) cos e ~ - b2 cos 2 p + (b2 - G 0) cos 2 p cos 2 ~}

- N~ cos p cos • (20)

with the constraint that the particle number expectation value be fixed at a given fermion number,

cos p cos a = r/. (21) Because of the constraint (21) the energy depends upon only one parameter ct.

Rewriting (20)

E(~t) = ~ E(p, ct) = sin 0(r/2 sec 2 a -- 1) + (sin 0 - cos 0)(cos 2 ct - ~/2) 1 (22)

where the range of • is constrained by (21) as

I=1 ~< c o s - ~ (1~1). (23)

In the expression (22) the energy depends upon two kinds of parameters. The parameters ~t and p are the variational parameters with respect to which the energy is minimized, whereas the parameters 0 and r/ are to be fixed at the outset and are determined by fitting energy spectra. The variation of energy as a function of latter parameters (Go and b2) is studied to find changes in the ground state shapes.

At this juncture one would like to establish the connection between the variational parameter ~t and the geometrical shape parameter (deformation) fl by using the expectation value of the intrinsic quadrupole moment. From (19a)

( Q o ) = 2N tan a(cos 2 ~t - ~/2)1/2, (24a)

= 2N fl'((1 + f12)-1 _ r/2)1/2, (24b)

where,

fl = tan a. (25)

In figures 1-5 the energy as a function of = is drawn for various values of ~/and 0.

Using (22) the variational equation

OE(a)lOa = 0 (26)

can be solved analytically which gives the following two regions of 0 values having different properties under ground state shapes.

(S) G O ~ b2(O ~ 45°): spherical solution for all r/.

~to = 0 °, cos2 Po = r/2, (27a)

( Q o ) = 0 , ( S ) = N ( 1 _ ~/2)1/2, (27b)

E(t/, 0) = cos 0(~/2 - I). (27c)

The SU(2) (b 2 = 0) and SO(7) (Go = b2) limits are both contained as special cases. SO(7) plays the role of a borderline between the spherical and the deformed regime. It deserves some additional discussion. As a matter of fact, the energy does not depend explicitly on 0t in this limit. 0t is only determined by the particle number constraint as long as cos p = 0. At r/= 0 it is even completely undermined; at this point there is neither a stable quadrupole nor a stable pair deformation. Therefore the shape fluctuations will become very large towards the middle of the shell. As a matter of fact, the S-pair transfer strength I ( A - - 2 [ S I A ) I 2 deviates strongly from an SU(2) behaviour towards the middle of the shell, while the pair deformation A/G o = (S+ + S ) calculated in H F B

0.0

- 0 . 2

A v - 0 . 4 laJ

- 0 . 6

- - 0 . 8 O.C

I

- I 0 0 -50 I 0 0

2

i//

,

0 50

0.0 - 0 . 2 - 0 . 4 - 0 . 6 - O . 8

- I 0 0

Figure lb.

Figure la. Energy E(~,) as a function of ,~ for SU(2)

limit 0 = 0 °. SO(7) limit.

0:45 °

C .•/1012

J

I I I

- 5 0 0 5 0 I 0 0

II

Same as figure la, except for 0 = 45 °,

:60° [ fl:90°

o o

4 - 0 . 6 14

- I 0 0 - 5 0 0 SO I0 -IOO - 5 0 0 50 IO0

Q I1

Figure le. Same as figure la, except for 0 = 60 °, Figure ld. Same as figure la, except for 0 = 90 °,

mixed SO(7) + SO(6) limit. SO{6) limit.

is just still at its SU(2) value, except at tl = 0 where it is not defined by the H F B solution. However, an infinitesimal change of 0 would be sufficient to produce a strong minimum in (S+ + S_), characteristic of the deformed regime, at this point.

The intrinsic quadrupole moment behaves in an analogous manner. At 0 = 0 °, ~1 = 0, it is zero, while an infinitesimal displacement in 0 would already produce a large deformation. The exact values of B(E2; 2 ~ --} 0 +) vary fast but smoothly in the vicinity of 0 = 45 °. Thus the HFB intrinsic state is unstable against shape fluctuations in the SO(7) limit.

(D) b2 < Go(O < 45°): phase transition at ~h = + x/1 - cot 0. (28) (i) I~1 < I~,1 deformed solution

~o = c o s - 1 ( ~ ) (29)

(ii) Ir#l t> Irhl spherical solution.

• / 0

- I u(z) 0

Figure 2.

SPHERICAL PHASE

I sot;') so(s)

45 90

8

The critical curve ~/(0) = :1:(1 - cot 0) 1/2.

At the transition point the energy is smooth, i.e.,

Ea(r/,) = E~(rh) = N2G2o/b2. (30)

There is a discontinuity only in the second derivative ofenergy with respect to t/, while it is in the first derivatives of ( Q ) and ( S ) . In the limiting case Go = 0, i.e. the SO(6) limit, the deformation persists for all values of r/.

Therefore the curve

r/= _ (I - c o t O ) ~/2 (31)

in the (r/,0) plane is the critical curve (figure 2). Along this curve deformation changes its property. The interesting thing about SO(7) limit is that it falls on this curve for the specific value of r/= 0. Therefore it can be said that SO(7) has r/= 0 as a critical point.

On this point the energy becomes independent of the variational parameter which means that the intrinsic state is undefined. However, the energy is continuous across this curve.

5. Conclusions

The bosonization of fermionic SO(8) model using Schwinger representation is used to study the IBM content of this model.

The intrinsic state of bosonized SO(8) model can be given a geometric interpretation using deformation parameter. SO(8) can have spherical and axially symmetric deformed shapes. The change in the shape is sharp and indicative of phase change. The SO(7) is a critical limit in that one cannot define a definite nuclear shape. In this respect this model shows that the concept of dynamical symmetry can describe critical behaviour.

Acknowledgements

This paper was presented at the Nuclear/Particle Physics Symposium in honour of Prof S P Pandya. This work is done in collaboration with U K a u p and Peter Ring of T U Munich.

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