Low-lying collective excited states in non-integrable pairing models based on stationary-phase approximation to path integral
aa r X i v : . [ nu c l - t h ] N ov Low-lying collective excited states in non-integrable pairing models based onstationary-phase approximation to path integral
Fang Ni, Nobuo Hinohara,
2, 1 and Takashi Nakatsukasa
1, 2, 3 Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Japan Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan iTHES Research Group, RIKEN, Wako 351-0198, Japan For a description of large-amplitude collective motion associated with nuclear pairing, requantiza-tion of time-dependent mean-field dynamics is performed using the stationary-phase approximation(SPA) to the path integral. We overcome the difficulty of the SPA, which is known to be applicableto integrable systems only, by developing a requantization approach combining the SPA with theadiabatic self-consistent collective coordinate method (ASCC+SPA). We apply the ASCC+SPA tomulti-level pairing models, which are non-integrable systems, to study the nuclear pairing dynam-ics. The ASCC+SPA gives a reasonable description of low-lying excited 0 + states in non-integrablepairing systems. I. INTRODUCTION
Pairing correlation plays an important role in open-shell nuclei. Effect of the pairing is prominent in manyobservables for the ground states, such as the odd-evenmass difference, moment of inertia of rotational bands,and common quantum number J π = 0 + for even-evennuclei [1]. Collective excitations associated with the pair-ing, such as pair vibrations and pair rotations, have beenobserved in a number of nuclei [2]. Most of these statesthat are “excited” from neighboring even-even systemsare associated with the ground J π = 0 + states in even-even nuclei. In contrast, properties of excited J π = 0 + states are not clearly understood yet [3, 4], for which thepairing dynamics plays an important role in a low-energyregion (a few MeV excitation) of nuclei. In this paper, weaim to understand the dynamics associated with pairingcorrelation in nuclei from a microscopic view point.The time-dependent mean-field (TDMF) theory is astandard theory to describe the dynamics of nuclei fromthe microscopic degrees of freedom. Inclusion of the pair-ing dynamics leads to the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory, which has been utilized fora number of studies of nuclear reaction and structure[5]. The small-amplitude approximation of the TDHFBwith modern energy density functionals, the quasiparti-cle random-phase approximation (QRPA), has success-fully reproduced the properties of giant resonances innuclei. In contrast, the QRPA description of low-lyingquadrupole vibrations is not as good as that of giant res-onances [5]. A large-amplitude nature of the quantumshape fluctuation is supposed to be important for theselow-lying collective states. Five-dimensional collectiveHamiltonian (5DCH) approaches have been developedfor studies of low-lying quadrupole states, in which thecollective Hamiltonian is constructed from microscopicdegrees of freedom using the mean-field calculation andthe cranking formula for the inertial masses [6–8]. The5DCH model is able to take into account fluctuations ofthe quadrupole shape degrees of freedom which are im-portant in many nuclear low-energy phenomena, such asshape coexistence and anharmonic quadrupole vibration. However, the calculated inertia is often too small to re-produce experimental data, due to the lack of time-oddcomponents in the cranking formula [1]. This deficiencycan be remedied in the adiabatic self-consistent collec-tive coordinate (ASCC) method [9], in which the time-odd effect is properly treated. In addition, the ASCCmethod enables us to identify a collective subspace ofinterest. The ASCC was developed from the basic ideaof the self-consistent collective coordinate (SCC) methodby Marumori and coworkers [10]. It has been applied tothe nuclear quadrupole dynamics including the shape co-existence [11–13].The TDMF (TDHFB) theory corresponds to an SPAsolution in the path integral formulation [14]. It lacksa part of quantum fluctuation that is important in thelarge-amplitude dynamics. To introduce the quantumfluctuation based on the TDHFB theory, the requantiza-tion is necessary [14–19]. A simple and straightforwardway of requantization is the canonical quantization. Thisis extensively utilized for collective models in nuclearphysics. For instance, the canonical quantization of the5DCH was employed for the study of low-lying excitedstates in nuclei [5, 20, 21]. The similar quantization wasalso utilized for the pairing collective Hamiltonian [22–25]. In our previous work [26], we studied various requan-tization methods for the two-level pairing model, to inves-tigate low-lying excited 0 + states. Since the collectivityis rather low in the pairing motion in nuclei, the canonicalquantization often fails to produce an approximate valueto the exact solution. In contrast, the stationary-phaseapproximation (SPA) to the path integral [27] can givequantitative results not only for the excitation energies,but also for the wave functions and two-particle-transferstrengths. The quantized states obtained in the SPA havetwo advantages; first, the wave functions are given di-rectly in terms of the microscopic degrees of freedom,and second, the restoration of the broken symmetries areautomatic. In the pairing model, the quantized statesare eigenstates of the particle-number operator. On theother hand, applications of the SPA have been limited tointegrable systems. This is because we need to find sepa-rable periodic trajectories on a classical torus. Since thenuclear systems, of course, correspond to non-integrablesystems, a straightforward application of the SPA is notpossible.In this paper, we propose a new approach of the SPAapplicable to the non-integrable systems, which is basedon the extraction of the one-dimensional (1D) collectivecoordinate using the ASCC method. Since the 1D sys-tem is integrable, the collective subspace can be quan-tized with the SPA. The optimal degree of freedom as-sociated with a slow collective motion is determined self-consistently inside the TDHFB space, without any as-sumption. Thus, our approach of the ASCC+SPA to thepairing model basically consists of two steps: (1) find adecoupled 1D collective coordinate of the pair vibration,in addition to the pair rotational degrees of freedom. (2)apply the SPA independently to each collective mode.The paper is organized as follows. Section II introducesthe theoretical framework of the ASCC, SPA, and theircombination, ASCC+SPA. In Sec. III, we provide somedetails in the application of the ASCC+SPA to the multi-level pairing model. We give the numerical results in Sec.IV, including neutron pair vibrations in Pb isotopes. Theconclusion and future perspectives are given in Sec. V. II. THEORETICAL FRAMEWORKA. Adiabatic self-consistent collective coordinatemethod and 1D collective subspace
In this section, we first recapitulate the ASCC methodto find a 1D collective coordinate, following the notationof Ref. [28].As is seen in Sec. III, the TDHFB equations can beinterpreted as the classical Hamilton’s equations of mo-tion with canonical variables { ξ α , π α } . Each point in thephase space ( ξ α , π α ) corresponds to a generalized Slaterdeterminant (coherent state). Assuming slow collectivemotion, we expand the Hamiltonian H ( ξ, π ) with respectto the momenta π up to second order. The TDHFBHamiltonian is written as H = V ( ξ ) + 12 B αβ ( ξ ) π α π β (2.1)with the potential V ( ξ ) and the reciprocal mass param-eter B αβ ( ξ ) defined by V ( ξ ) = H ( ξ, π = 0) , (2.2) B αβ ( ξ ) = ∂ H ( ξ, π ) ∂π α ∂π β (cid:12)(cid:12)(cid:12)(cid:12) π =0 . (2.3)For multi-level pairing models in Sec. III, there is aconstant of motion in the TDHFB dynamics, namely theaverage particle number q n ≡ h ˆ N i /
2. Since the parti-cle number ˆ N is time-even Hermitian operator, we treatthis as a coordinate, and its conjugate gauge angle, p n ,as a momentum. Since q n is a constant of motion, theHamiltonian does not depend on p n . On the other hand,the gauge angle p n changes in time, which corresponds to the pair rotation, a Nambu-Goldstone (NG) mode asso-ciated with the breaking of the gauge (particle-number)symmetry. We assume the existence of 2D collective sub-space Σ (4D phase space), described by a set of canonicalvariable ( q , q = q n ; p , p = p n ), which is well decou-pled from the rest of degrees of freedom, { q a , p a } with a = 3 , · · · . The collective Hamiltonian is given by impos-ing q a = p a = 0, namely by restricting the space into thecollective subspace H coll ( q, p ; q n ) = ¯ V ( q , q n ) + 12 ¯ B ( q , q n ) p . (2.4)Since there exist two conserved quantities, q n and H coll ,this 2D system is integrable. We can treat the collec-tive motion of ( q , p ) separately from the pair rotation( q n , p n ).In the collective Hamiltonian (2.4), the variable q n istrivially given as the particle number, which is expandedup to second order in the momenta π , q n = h ˆ N i f n ( ξ ) + 12 f (1) nαβ π α π β . (2.5)To obtain the non-trivial collective variables ( q , p ), weassume the point transformation ∗ , q = f ( ξ ) , (2.6)and ξ α on the subspace Σ is given as ξ α = g α ( q , q n , q a = 0). The momenta on Σ are transformedas p = g α, π α , p n = g α,n π α , (2.7) π α = f ,α p + f n,α p n , (2.8)where the comma indicates the partial derivative ( f ,α = ∂f /∂ξ α ). The Einstein’s convention for summation withrespect to the repeated upper and lower indices is as-sumed hereafter. The canonical variable condition leadsto f i,α g α,j = δ ij , (2.9)where i, j = 1 and n . The collective potential ¯ V ( q , q n )and the collective mass parameter ¯ B ( q , q n ) =[ ¯ B ( q , q n )] − can be given by¯ V ( q , q n ) = V ( ξ = g ( q , q n , q a = 0)) , (2.10)¯ B ( q , q n ) = f ,α ˜ B αβ ( ξ ) f ,β , (2.11)where ˜ B αβ are defined as˜ B αβ ( ξ ) = B αβ ( ξ ) − ¯ V ,n f (1) nαβ ( ξ ) . (2.12) ∗ We may lift the restriction to the point transformation, as Eq.(2.5) [29]. In this paper, we neglect these higher-order terms,such as f (1)1 αβ π α π β / Decoupling conditions for the collective subspace Σ lead to the basic equations of the ASCC method [9, 28],which determine tangential vectors, f ,α ( ξ ) and g α, ( q ). δH M ( ξ, π ) = 0 , (2.13) M βα f ,β = ω f ,α , M βα g α, = ω g β, . (2.14)The first equation (2.13) is called moving-frame Hartree-Fock-Bogoliubov (HFB) equation. The moving-frameHamiltonian H M is H M ( ξ, π ) = H ( ξ, π ) − λ q ( ξ ) − λ n q n ( ξ, π ) . (2.15)The second equation (2.14) is called moving-frame QRPAequation. The matrix M βα in the moving-frame QRPAequation (2.14) can be rewritten as M βα = ˜ B βγ (cid:0) V ,γα − λ n f n,γα (cid:1) + 12 ˜ B βγ,α V ,γ . (2.16)The NG mode, f n,α and g α,n , corresponds to the zero modewith ω = 0. Therefore, the collective mode of our inter-est corresponds to the mode with the lowest frequencysquared except for the zero mode.In practice, we obtain the collective path according tothe following procedure:1. Find the HFB minimum point ξ αi ( i = 0) by solvingEq. (2.13) with λ = 0. Let us assume that thiscorresponds to q i = 0.2. Diagonalize the matrix M βα to solve Eq. (2.14) us-ing Eq. (2.16).3. Move to the next neighboring point ξ αi +1 = ξ αi + dξ α with dξ α = g α, dq . This corresponds to thecollective coordinate, q i +1 = q i + dq .4. At ξ αi +1 ( q i +1 ), obtain a self-consistent solution ofEqs. (2.13) and (2.14) to determine ξ αi +1 , f ,α , and g α, .5. Go back to Step 3 to determine the next point onthe collective path.We repeat this procedure with dq > dq <
0, andconstruct the collective path. In Steps 2 and 4, we choosea mode with the lowest frequency squared ω . Note that ω can be negative. In Step 4, when we solve Eq. (2.13),we use a constraint on the magnitude of dq = f ,α dξ α .Since the normalization of f ,α and g α, is arbitrary as faras they satisfy Eq. (2.9), we fix this scale by an additionalcondition of ¯ B ( q ) = 1. B. Stationary-phase approximation to path integral
For quantization of integrable systems, we can applythe stationary-phase approximation (SPA) to the pathintegral. In our former study [26], we have proposed and tested the SPA for an integrable pairing model. Sincethe collective Hamiltonian (2.4) that is extracted fromthe TDHFB degrees of freedom is integrable, the SPAis applicable to it. In this manner, we may apply theASCC+SPA to non-integrable systems in general.
1. Basic idea of ASCC+SPA
Because the Hamiltonian H coll of Eq. (2.4) is separable,it is easy to find periodic trajectories on invariant tori.Since the pair rotation corresponds to the motion of p n with a constant q n , all we need to do is to find classi-cal periodic trajectories C k in the ( q , p ) space (witha fixed q n ) which satisfy the Einstein-Brillouin-Keller(EBK) quantization rule with a unit of ~ = 1, I C k p dq = 2 πk, (2.17)where k is an integer number.At each point in the space ( q , q n ; p , p n ) correspondsto a generalized Slater determinant | q , q n ; p , p n i = | ξ, π i where ( ξ, π ) are given as ξ α = g α ( q , q n , q a = 0)and π α = f ,α p + f n,α p n . According to the SPA, the k -thexcited state | ψ k i is constructed from the k -th periodictrajectory C k , given by ( q ( t ) , p ( t )), of the Hamiltonian H coll . | ψ k i ∝ I dp n I C k ρ ( q, p ) dt | q, p i e i T [ q,p ] , (2.18)where ( q, p ) means ( q , q n ; p , p n ), and the weight func-tion ρ ( q, p ) is given through an invariant measure dµ ( q, p )as dµ ( q, p ) = ρ ( q, p ) dEdtdq n dp n . (2.19)The invariant measure dµ ( q, p ) is defined by the unitycondition R dµ ( q, p ) | q, p i h q, p | = 1. An explicit form of dµ ( q, p ) for the present pairing model is shown in Eq.(3.21). The action integral T is defined by T [ q, p ] ≡ Z t h q ( t ′ ) , p ( t ′ ) | i ∂∂t ′ | q ( t ′ ) , p ( t ′ ) i dt ′ . (2.20)The SPA quantization is able to provide a wave func-tion | ψ k i in microscopic degrees of freedom, which isgiven as a superposition of generalized Slater determi-nants | q, p i . In addition, the integration with respectto p n over a circuit on a torus automatically recoversthe broken symmetry, namely the good particle number.However, it relies on the existence of invariant tori. Inthe present approach of the ASCC+SPA, we first derivea decoupled collective subspace Σ and identify canonicalvariables ( q, p ). Because of the cyclic nature of ( q n , p n ),it is basically a 1D system and becomes integrable. Inother words, we perform the torus quantization on ap-proximate tori in the TDHFB phase space ( ξ, π ), whichis mapped from tori in the 2D collective subspace ( q, p ).
2. Notation and practical procedure for quantization
For the application of the ASCC+SPA method to thepairing model in Sec. III, we summarize some notationsand procedures to obtain quantized states.In Sec. III, the time-dependent generalized Slater de-terminants (coherent states) are written as | Z i with com-plex variables Z α ( t ). The variables Z α are transformedinto real variables ( j α , − χ α ) that correspond to ( ξ α , π α )in Sec. II. χ α and j α correspond to the “angle” and the“number” variables, respectively. Although it is custom-ary to take the time-odd angle χ α as a coordinate, wetake the number j α as a coordinate and the angle − χ α as a momentum with an additional minus sign. Simi-larly, the gauge angle Φ and the total particle number J correspond to variables of the pair rotation, − p n and q n ,respectively.According to the EBK quantization rule (2.17), theground state with k = 0 corresponds to nothing but theHFB state with a fixed particle number J (= q n ). For the k -th excited states, we perform the following calculations:1. Obtain the 1D collective subspace with canoni-cal variables ( q , p ) according to the ASCC inSec. II A.2. Find a trajectory ( q ( t ) , p ( t )) which satisfies the k -th EBK quantization condition (2.17).3. Calculate the action integral (2.20) for the k -th tra-jectory.4. Construct the k -th excited state using Eq. (2.18).The ASCC provides the 2D collective subspace ( q , J )and the generalized coherent states | Φ = 0 , J ; q, p = 0 i .For finite values of momenta, we use Eq. (2.8) to obtainthe state | Φ , J ; q, p i . III. PAIRING MODEL
We study the low-lying excited 0 + states in the multi-level pairing model by applying the ASCC+SPA. TheHamiltonian of the pairing model is given in terms of thesingle-particle energies ǫ l and the pairing strength g asˆ H = X α ǫ α ˆ n α − g X α,β ˆ S + α ˆ S − β = X α ǫ α (2 ˆ S α + Ω α ) − g ˆ S + ˆ S − , (3.1)where we use the SU(2) quasi-spin operators, ˆ S = P α ˆ S α , withˆ S α = 12 X m ˆ a † j α m ˆ a j α m − Ω α ! , (3.2)ˆ S + α = X m> ˆ a † j α m ˆ a † j α m , ˆ S − α = ˆ S + † α . (3.3) Each single-particle energy ǫ α possesses 2Ω α -fold degen-eracy (Ω α = j α + 1 /
2) and P m> indicates the sum-mation over m = 1 / , / , · · · , and Ω α − /
2. Theoccupation number of each level α is given by ˆ n α = P m ˆ a † j α m ˆ a j α m = 2 ˆ S α + Ω α . The quasi-spin operatorssatisfy the commutation relations[ ˆ S α , ˆ S ± β ] = ± δ αβ ˆ S ± α , [ ˆ S + α , ˆ S − β ] = 2 δ αβ ˆ S α . (3.4)The magnitude of quasi-spin for each level is given by S α = (Ω α − ν α ), where ν α is the seniority quantum num-ber, namely the number of unpaired particles at the level α . In the present study, we consider only seniority-zerostates with ν = P α ν α = 0. The residual two-body inter-action consists of the monopole pairing interaction whichcouples two particles to zero angular momentum. We ob-tain the exact solutions either by solving the Richardsonequation [30–32] or by diagonalizing the Hamiltonian us-ing the quasi-spin symmetry. A. Classical form of TDHFB Hamiltonian
The time-dependent coherent state for the seniority ν = 0 states ( S α = Ω α /
2) is constructed with time-dependent complex variables Z α ( t ) as | Z ( t ) i = Y α (cid:0) | Z α ( t ) | (cid:1) − Ω α / exp[ Z α ( t ) ˆ S + α ] | i , (3.5)where | i is the vacuum (zero-particle) state. TheTDHFB motion is given by the time development of Z α ( t ). In the SU(2) quasi-spin representation, | i = Q α | S α , − S α i . The coherent state | Z ( t ) i is a superposi-tion of the states with different particle numbers withoutunpaired particles. In the present pairing model, the co-herent state is the same as the time-dependent BCS wavefunction with Z α ( t ) = v α ( t ) /u α ( t ), where ( u α ( t ) , v α ( t ))are the time-dependent BCS u, v factors.The TDHFB equation can be derived from the time-dependent variational principle, δ S = 0, where S ≡ Z L ( t ) dt = Z h Z ( t ) | i ∂∂t − ˆ H | Z ( t ) i dt (3.6)( ~ = 1 applies hereafter). After the transformationof the complex variables into the real ones with Z α =tan θ α e − iχ α (0 ≤ θ α ≤ π ), the Lagrangian L and theexpectation value of the Hamiltonian are written as L ( t ) = X α Ω α − cos θ α ) ˙ χ α − H ( Z, Z ∗ ) , (3.7)with H ( Z, Z ∗ ) ≡ h Z | ˆ H | Z i = X α ǫ α Ω α (1 − cos θ α ) − g X α Ω α [Ω α (1 − cos θ α ) + (1 − cos θ α ) ] − g X α = β Ω α Ω β q (1 − cos θ α )(1 − cos θ β ) e − i ( χ α − χ β ) . (3.8)We choose χ α as canonical coordinates, and their conju-gate momenta are given by j α ≡ ∂ L ∂ ˙ χ α = Ω α − cos θ α ) . (3.9) χ α represents a kind of gauge angle of each level, and j α corresponds to the occupation number of each level,2 j α = h Z | ˆ n α | Z i . As we mention in Sec. II B 2, we switchthe coordinates and momenta, ( χ α , j α ) → ( j α , − χ α ), tomake the coordinates time even. The TDHFB equationis equivalent to classical Hamilton’s equations − ˙ χ α = − ∂ H ∂j α , ˙ j α = ∂ H ∂ ( − χ α ) . (3.10) B. Application of ASCC
We construct a 2D collective subspace Σ from theASCC method. We expand the classical Hamiltonian upto second order with respect to the momenta, − χ α H ( j, χ ) ≈ V ( j ) + 12 B αβ ( j ) χ α χ β , (3.11)where the potential V ( j ) and the reciprocal mass param-eter B αβ ( j ) are given as V ( j ) = H ( j, χ = 0)= X α ǫ α j α − g X α (cid:20) Ω α j α − ( j α ) + ( j α ) Ω α (cid:21) − g X α = β q j α j β (Ω α − j α )(Ω β − j β ) , (3.12) B αβ ( j ) = ∂ H ∂χ α ∂χ β (cid:12)(cid:12)(cid:12)(cid:12) χ =0 (3.13)= ( g P γ = α p j γ j α (Ω γ − j γ )(Ω α − j α ) for α = β − g p j α j β (Ω α − j α )(Ω β − j β ) for α = β . We may apply the ASCC method in Sec. II A by regard-ing ξ → j and π → − χ .The TDHFB conserves the average total particle num-ber N . We adopt J ≡ N/ X α j α , (3.14) as a coordinate q n . Since this is explicitly given as theexpectation value of the particle-number operator, cur-vature quantities, such as f n,αβ and f (1) nαβ , are explicitlycalculable. On the other hand, the gauge angle Φ = − p n is not given a priori. Since the ASCC solution provides g α,n as an eigenvector of Eq. (2.14), we may construct itas Eq. (2.7) in the first order in π = − χ . We confirmthat the pair rotation corresponds to an eigenvector ofEq. (2.14) with the zero frequency ω = 0.In the present pairing model, from Eq. (3.14), we find J does not depend on χ . This means f (1) nαβ = 0 inEq. (2.5), thus, ˜ B αβ = B αβ . The second derivative of J with respect to j also vanishes, which indicates f n,γα inEq. (2.16) is zero. The gauge angle Φ is locally deter-mined by the solution of Eq. (2.14).Φ = g α,n χ α , (3.15)It should be noted [28] that the definition of the col-lective variables ( q , p ) is not unique, because it can bearbitrarily mixed with the pair rotation ( q n , p n ) as q → q + cq n , p n → p n − cp (3.16)with an arbitrary constant c . Numerically, this arbitrari-ness sometimes leads to a problematic behavior in itera-tive procedure of the ASCC. In order to fix the parameter c , we adopt a condition called “ETOP” in Ref. [33]. Werequire the following condition to determine c : X α f ,α = 0 , (3.17)where f ,α is replaced as f ,α → f ,α + cf n,α (3.18)with Eq. (3.16). C. Application of SPA
After deriving the collective subspace Σ , we performthe quantization according to the SPA in Sec. II B. Calcu-lating a trajectory in the ( q , p ) space, we can identifya series of states {| Φ , J ; q ( t ) , p ( t ) i} on the trajectory,in the form of Eq. (3.5) with parameters Z α given at(Φ , J, q , p ) and q a = p a = 0 for a ≥
3. Since thevariables (Φ , J ) and ( q , p ) are separable, we may takeclosed trajectories independently in (Φ , J ) and ( q , p )sectors, which we denote here as C Φ and C , respectively.The action integral is given by T (Φ , J ; q , p ) = Z C Φ h Φ( t ) , J ; q , p | i ∂∂t | Φ( t ) , J ; q , p i dt + Z C h Φ , J ; q ( t ) , p ( t ) | i ∂∂t | Φ , J ; q ( t ) , p ( t ) i dt = J Φ + Z C X α j α dχ α ≡ T Φ ( J, Φ) + T ( q , p ; J ) . (3.19)In fact, the gauge-angle dependence is formally given as | Φ , J ; q , p i = e − i Φ ˆ N/ | J ; q , p i , (3.20)where ˆ N = P α ˆ n α . Then, the action for the tra-jectory C can be also expressed as T ( q , p ; J ) = R C h J ; q , p | i ∂∂t | J ; q , p i dt .In the SU(2) representation, the invariant measure is dµ ( Z ) = Y α Ω α + 1 π (1 + | Z α | ) − d (Re Z ) d (Im Z )= Y α − (Ω α + 1)4 π d (cos θ α ) dχ α = Y α − α π dχ α dj α = "Y α − α π d Φ dJdq dp Y a dq a dp a , (3.21)where ( q a , p a ) are the intrinsic canonical variables decou-pled from the collective subspace Σ . In the last line inEq. (3.21), we used the invariance of the phase-space vol-ume element in the canonical transformation. Accordingto Eq. (3.21), the weight function ρ ( q, p ) in Eq. (2.18) isjust a constant, thus, treated as the normalization of thewave function.The coherent state | Φ , J ; q , p i = | Z i is expanded inthe SU(2) quasispin basis as | Z i = X { m α } A m ( Z ) |· · · ; S α , − S α + m α , · · ·i , (3.22)where the summation is taken over all possible combina-tions of integer values of { m α } with A m ( Z ) = Y α Z m α α (1 + | Z α | ) Ω α / m α ! s Ω α ! m α !(Ω α − m α )!= Y α (cid:18) − cos θ α (cid:19) m α / (cid:18) θ α (cid:19) (Ω α − m α ) / × s Ω α ! m α !(Ω α − m α )! e − im α χ α , (3.23)where the lower index m indicates a combination of { m α } . The integer number m α corresponds to the num-ber of pairs in the level α .Using Eq. (3.23), the k -th excited state is calculatedas | ψ k i ∝ I C Φ d Φ I C dt | Φ , J ; q , p i e i T (Φ ,J ; q ,p ) = X { m α } Z π d Φ e i ( J − P α m α )Φ × I dte i T ( t ) B m ( Z ) |· · · ; S α , − S α + m α , · · ·i≡ X { m α } J C m |· · · ; S α , − S α + m α , · · ·i , (3.24) where B m ( Z ) are identical to A m in Eq. (3.23) except forreplacing χ α with the relative angles φ α ≡ χ α − Φ. Thecoefficients C m are given by C m = I C dte i T ( t ) B m ( Z ( t )) . (3.25)In the last line of Eq. (3.24), the summation is restrictedto { m α } that satisfy P α m α = J . It is easy to find that J must be integer, according to the quantization rule (2.17)for the ( J, Φ) sector.The SPA for the ground state ( k = 0) is given by thestationary point in the ( q , p ) sector, namely, the HFBstate | Φ , J ; q, p i = e − i Φ ˆ N/ | HFB i . Nevertheless, the ro-tational motion in Φ( t ) is present, which leads to thenumber quantization (projection). Therefore, Eq. (3.24)becomes | ψ g . s . i ∝ X { m α } Z π d Φ e i ( J − P α m α )Φ | HFB i , (3.26)which is identical to the wave function of the particle-number projected HFB state. IV. RESULTS
In the pairing model in Sec. III, the number of TDHFBdegrees of freedom equals that of single-particle levels. Asthere are two constants of motion, that is, the particlenumber and the energy, the system is integrable for one-and two-level models. We first apply the ASCC+SPAmethod to an integrable two-level model, then, to non-integrable multi-level models.
A. Integrable case: Two-level pairing model
The two-level pairing model corresponds to the 2D TD-HFB system. Explicitly separating the gauge angle Φ andfixing the particle number J , the 2D TDHFB is reducedto the 1D system with the relative angle φ ≡ χ − χ andthe relative occupation j ≡ ( j − j ) / = Ω = 8, the pairing strength g/ǫ = 0 .
2, andthe particle number N = 16. In this two-level case, weuse the level spacing, ǫ ≡ ǫ − ǫ , as the unit of en-ergy. The moving-frame QRPA produces the zero modeand another eigenvector with a finite frequency squared ω = 0. We follow the latter mode to construct thecollective path. In the ASCC calculation, we set the in-crement of the collective coordinate, dq = 0 .
01, in unitsof 1 / √ ǫ . We confirm that the pair rotation always has -4-2 0 2 4 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 j φ ASCCTDHFB2TDHFB
FIG. 1: Classical trajectories satisfying the EBKquantization condition (2.17) with k = 1 and 2 in the( φ, j ) phase space. The crosses, solid and dashed linescorrespond to the results of the ASCC+SPA,TDHFB2+SPA, and TDHFB+SPA, respectively. Thecrosses for the ASCC+SPA trajectories are plottedevery ten calculations ( δq = 10 dq = 0 . / √ ǫ ). π π Τ ( t ) t [ ε ]ASCCTDHFB2TDHFB FIG. 2: Calculated action integrals for the | +2 i and | +3 i states as functions of time t . The crosses, solid anddashed lines correspond to the ASCC+SPA, theTDHFB2+SPA, and the TDHFB+SPA, respectively.The action integrals are calculated on each trajectory inFig. 1 from ( φ, j ) = (0 , j max ) in the clockwise direction.The crosses for the ASCC+SPA trajectories are plottedevery ten calculations ( δq = 10 dq = 0 . / √ ǫ ).a zero frequency on the collective path. On the obtainedcollective path, we calculate a classical trajectory for theHamiltonian H coll ( q , p ; J ) = 12 p + V ( q , J ) (4.1)with J = N/ | C m | m -m -8 -6 -4 -2 0 2 4 6 8 -m ASCCTDHFB2TDHFB
FIG. 3: Occupation probabilities for the 0 +2 and 0 +3 states. The horizontal line indicates the j = ( m − m ) / m − m from left to rightrepresent | C m | of Eq. (3.25) in the ASCC+SPA,TDHFB2+SPA and TDHFB+SPA calculations,respectively.the EBK quantization condition (2.17) for the first andsecond excited states (0 +2 and 0 +3 ) are mapped onto the( φ, j ) plane and shown in Fig. 1. We also calculate thetrajectories using the explicit transformation of the vari-ables to (Φ , J ; φ, j ), which are shown by dashed lines inFig. 1. We call this “TDHFB trajectories”. Small devi-ation in large φ is due to the absence of the higher-orderterms in χ α in the ASCC. In fact, if we calculate the tra-jectories in the variables (Φ , J ; φ, j ) using the Hamilto-nian truncated up to the second order in χ α (“TDHFB2trajectories”), we obtain the solid lines in Fig. 1, whichperfectly agree with the ASCC trajectories.The action integrals T ( t ) corresponding to these closedtrajectories are shown in Fig. 2. For the 0 +2 state, allthree calculations agree well with each other, while wesee small deviation between the full TDHFB and theASCC/TDHFB2 calculations for the 0 +3 state.The calculated wave functions for the excited 0 + statesare shown in Fig. 3. We show the occupation prob-abilities which are decomposed into the 2 n -particle-2 n -hole components. The left end of the horizontal axis at m − m = − j = −
4) corresponds to a state with( m , m ) = ( N/ ,
0) where all the particles are in thelower level α = 1. The next one at m − m = − j = −
3) corresponds to the one with ( m , m ) =(( N − / , -20-10 0 10 20 30 40 -1 0 1N=14 ω ( q ) [ ε ] q [ ε ] -1 0 1 2N=16q [ ε ] -1 0 1N=18q [ ε ] -1 0 1N=20q [ ε ] -1 0 1N=22q [ ε ] -1 0 1N=24q [ ε ] FIG. 4: Eigenvalues of moving-frame QRPA equation as a function of the collective coordinate q , from N = 14 to N = 24. The thick (green) lines are the modes we choose as the collective coordinate q , while the dashed linescorrespond to the zero modes ( q n ). In each panel, both ends of the horizontal axis corresponds to the ending pointsof the collective path q . o cc upa t i on nu m be r q [ ε ] -1 0 1 2N=161 23 q [ ε ] -1 0 1N=181 23 q [ ε ] -1 0 1N=201 23 q [ ε ] -1 0 1N=221 23 q [ ε ] -1 0 1N=241 23 q [ ε ] FIG. 5: The occupation numbers 2 j α in each single-particle level α as a function of the collective coordinate q ,from N = 14 to N = 24. The purple, green, and blue lines correspond to α = 1, 2, and 3, respectively. At the leftend point of the collective coordinate in each panel, the configuration corresponds to the “HF-like” states. See textfor details. B. Non-integrable case (1): Three-level pairingmodel
In contrast to the two-level model, the TDHFB for thethree-level model is non integrable. We set the parame-ters of the system as follows: Ω = Ω = Ω = Ω = 8, ǫ = − ǫ , ǫ = 0, ǫ = 1 . ǫ , and g = 0 . ǫ . We use theparameter ǫ as the unit energy. For the sub-shell closedconfiguration of N = 2Ω = 16, the HFB ground statechanges from the normal phase to the superfluid phaseat g c = 0 . ǫ . We calculate a chain of systems witheven particle numbers from N = 14 to N = 24.We obtain three eigen frequencies for the moving-frameQRPA equation, on the collective path (Fig. 4). Firstof all, we clearly identify the zero mode with ω = 0everywhere along the collective path. This means thatthe pair rotation is separated from the other degrees offreedom in the ASCC. The frequency could become imag-inary ( ω < N = 2Ω = 16), the frequency rapidly increases near theend points. The end points are given by points where thesearch for the next point on the collective path in Sec. II Afails.We choose the lowest frequency squared mode, exceptfor the zero mode, as a generator of the collective path ( q ). Figure 5 shows variation of the occupation probabil-ity of each single-particle state, as functions of the collec-tive coordinate q on the collective path. The most strik-ing feature is that the collective path terminates with spe-cial configurations which are given by the integer numberof occupation. This is the reason why the search for thecollective path fails at both the ends. At the end points,the occupation of the level 3 ( ǫ ) vanishes, while those ofthe levels 1 and 2 become either maximum or minimum.The left end of each panel in Fig. 5 corresponds to a kindof “Hartree-Fock” (HF) state which minimizes the single-particle-energy sum, P α j α ǫ α . The pairing correlationis weakened in both ends of the collective path.The collective mass with respect to the coordinate q is normalized to unity. The collective potential is shownin Fig. 6. The range of q is the largest for the systemwith N = 16. This is because the variation of j and j is the largest in this case.Based on the collective path determined by the ASCCcalculation, we perform the requantization according tothe SPA. Table I shows the excitation energies of the firstand second excited states, determined by the EBK quan-tization condition (2.17). Comparing the result of theASCC+SPA with that of the exact calculation, we findthat the excitation energies are reasonably well repro- V ( q ) [ ε ] q [ ε ]N=14N=16N=18N=20N=22N=24 FIG. 6: Collective potential V ( q ) obtained from theASCC. We adjust the energy minimum point as q = 0and V = 0.TABLE I: Calculated excitation energies of the first andthe second excited states in units of ǫ . In the exactcalculation, the second excited state in the ASCC+SPAcorresponds to the 0 +4 state. See text for details. N
14 16 18 20 22 24ASCC+SPA (1st exc.) 3 .
87 3 .
90 3 .
97 4 .
09 4 .
23 4 . .
09 4 .
13 4 .
20 4 .
30 4 .
44 4 . .
42 7 .
42 7 .
60 7 .
92 8 .
26 8 . .
65 7 .
71 7 .
88 8 .
15 8 .
49 8 . duced. The ASCC+SPA underestimates the excitationenergies only by about 5 %.It should be noted that the second excited state in thecollective path corresponds to the 0 +4 state, not to the 0 +3 state, in the exact calculation. We examine the interband( k = k ′ ) pair-addition transition, B ( P ad ; k → k ′ ) ≡ (cid:12)(cid:12)(cid:12) h + k ′ ; N + 2 | ˆ S + | + k ; N i (cid:12)(cid:12)(cid:12) , (4.2)in the exact solution. B ( P ad ; 0 +2 → +3 ) is 10 ∼ B ( P ad ; 0 +1 → +3 ), while B ( P ad ; 0 +1 → +2 ) and B ( P ad ; 0 +1 → +3 ) are in the same order.The ASCC+SPA produces states in the same family,namely, those belonging to the same collective subspace(path). In the phonon-like picture, we expect similarmagnitude of the strengths for B ( P ad ; g . s . → ω phon )and B ( P ad ; ω phon → ω phon ), but smaller values of B ( P ad ; g . s . → ω phon ). Thus, the 0 +4 state in the exactcalculation corresponds to the two-phonon state in theASCC+SPA. The 0 +3 state in the exact calculation maycorrespond to a collective path associated with anothersolution of the moving-frame QRPA (thin black line inFig. 4).Next, we calculate the wave functions, according toEqs. (3.24) and (3.26). The ground state corresponds to ] ASCC+SPA ] Exact B ( P ad ) Initial particle numbergs-->gs0 -->0 gs-->gs0 -->0 ] ASCC+SPA ] Exact B ( P ad ) Initial particle numbergs-->0 -->gsgs-->0 -->gs FIG. 7: Calculated strength of pair-addition transition(4.2) from N = 14 to N = 22. The solid (dashed) linescorrespond to the ASCC+SPA (exact) calculation. Thehorizontal line indicates the particle number of theinitial states. The upper panel shows the intrabandtransitions, 0 +1 → +1 and 0 +2 → +2 , while the lowerpanel shows the interband transitions, 0 +1 → +2 and0 +2 → +1 .the number-projected HFB state (variation before pro-jection). In contrast, the excited states are given as su-perposition of generalized Slater determinants in the col-lective subspace, The pair-addition transition strengthscomputed using these wave functions of the excited statesare shown in Fig. 7. For the intraband transition ( k = k ′ in Eq. (4.2)), the ASCC+SPA method well reproducesthe strengths of the exact calculation. The ground-to-ground transitions, B ( P ad ; 0 +1 → +1 ), are perfectly re-produced, while B ( P ad ; 0 +2 → +2 ) are underestimated byabout 10% ∼ k = k ′ ), which are far smallerthan the intraband transitions. Although the increasing(decreasing) trend for B ( P ad ; 0 +2 → +1 ) ( B ( P ad ; 0 +1 → +2 )) as a function of the particle number is properly re-produced, the absolute magnitude is significantly under-0 Pb ω ( q ) [ M e V ] q [MeV -1/2 ] -1 0 1 Pbq [MeV -1/2 ] -1 0 1 Pbq [MeV -1/2 ] -1 0 1 Pbq [MeV -1/2 ] FIG. 8: The same as Fig. 4 but for Pb isotopes. Pb o cc upa t i on nu m be r q [MeV -1/2 ] h f i p f p Pbq [MeV -1/2 ] -1 0 1 Pbq [MeV -1/2 ] -1 0 1 Pbq [MeV -1/2 ] FIG. 9: The same as Fig. 5 but for Pb isotopes.estimated in the ASCC+SPA. This is due to extremelysmall collectivity in the interband transitions. Almost allthe strengths are absorbed in the intraband transitions.Even in the exact calculation, the pair addition strengthis about two orders of magnitude smaller than the intra-band strength. Remember that the non-collective limit( g →
0) of this value is B ( P ad ; 0 +1 → +2 ) = Ω. Therefore,the pairing correlation hinders the interband transitionsby about one order of magnitude. For such tiny quanti-ties, perhaps, the reduction to the 1D collective path isnot well justified.We should remark here that there is a difficulty inthe present ASCC+SPA requantization for weak pair-ing cases. In such cases, the potential minimum is closeto the left end ( q = q L ) of the collective path, andthe potential height at q = q L , V ( q L ) − V (0), becomessmall. Then, a classical trajectory with E > V ( q L ) hitsthis boundary ( q = q L ). In construction of wave func-tions, the boundary condition at q = q L significantlyinfluences the result. In the present study, we choosea strong pairing case to avoid such a situation. As inFig. 6, the potential height at q = q L has about 10 ǫ which is larger than the excitation energies of the secondexcitation. Therefore, all the trajectories are “closed” in the usual sense. C. Non-integrable case (2): Pb isotopes
Finally, we apply our method to neutrons’ pairing dy-namics in neutron-deficient Pb isotopes. The spheri-cal single-particle levels of neutrons between the magicnumbers 82 and 126 are adopted and their energies arepresented in Table II. The coupling constant g = 0 . N ) = ( − N +1 [ B ( N + 1) − B ( N ) + B ( N − Pb. Theeven-even nuclei from
Pb to
Pb are studied.The TDHFB dynamics is described by six degrees offreedom. Figure 8 shows eigenvalues of the moving-frameQRPA equation. Again, we find that there is a zero modecorresponding to the neutron pair rotation. Among thefive vibrational modes, we choose the lowest one to con-struct the collective path in the ASCC. This lowest modenever crosses with other modes, though the spacing be-tween the lowest to the next lowest mode can be verysmall, especially for
Pb. The evolution of the occupa-1TABLE II: Single-particle energies of Pb isotopes usedin the calculation in units of MeV. These are obtainedfrom the spherical Woods-Saxon potential with theparameters of Ref. [34]. orbit h / f / i / p / f / p / energy(MeV) − . − . − . − . − . − . V ( q ) [ M e V ] q [MeV -1/2 ] FIG. 10: The same as Fig. 6 but for Pb isotopes.tion numbers along the collective path is shown in Fig. 9.Similarly to the three-level model, the end points of thecollective path indicate exactly the integer numbers, andthe left end of each panel corresponds to the “Hartree-Fock”-like state. On the collective path, the occupationnumbers of i / , p / , and f / mainly change.The collective potentials for these isotopes are shownin Fig. 10. The heights of the potentials at the left end, V ( q L ) − V (0), are 2 ∼ . Pb, the heightof the potential is not enough to satisfy the condition,
E < V ( q L ), to have a closed trajectory for the first ex-cited state (See the last paragraph in Sec. IV B). We en-counter another kind of problem for Pb, which will bediscussed in Sec. V. Therefore, in this paper, we calculatethe first excited states in , , , Pb.We show the calculated excitation energy of the firstexcited state in Table III. Experimentally, this pair vi-brational excited 0 + state is fragmented into several 0 + states due to other correlations, such as quadrupole cor-relation, not taken into account in the present model. Wemake a comparison with the exact solution of the multi-level pairing model. The ASCC+SPA method quantita-tively reproduces the excitation energy of the exact solu-tion.The pair-addition transition strengths are shown inFig. 11. The feature that is similar to the three-levelcase is observed: dominant intraband transition andvery weak interband transitions. The accuracy from theASCC+SPA method well reproduces B ( P ad ; 0 +1 → +1 )and qualitatively reproduces B ( P ad ; 0 +2 → +2 ) as well.The deviation for the latter is about 25%. The interband TABLE III: The same as Table I but for Pb isotopes.The energies are given in units of MeV. Pb Pb Pb Pb Pb PbASCC+SPA − .
31 2 .
21 2 .
12 2 . − Exact 2 .
58 2 .
44 2 .
34 2 .
25 2 .
20 2 . ] ASCC+SPA ] Exact B ( P ad ) Initial mass number of Pbgs-->gs0 -->0 gs-->gs0 -->0 ] ASCC+SPA ] Exact B ( P ad ) Initial mass number of Pbgs-->0 -->gsgs-->0 -->gs FIG. 11: The same as Fig. 7 but for Pb isotopes.transitions are smaller than the intraband transitions bymore than two orders of magnitude. This is also similarto the three-level model discussed in Sec. IV B. For suchweak transitions, the ASCC+SPA significantly underes-timates the strengths. We may say that the ASCC+SPAgives reasonable results for the intraband transitions inthe realistic values of the pairing coupling constant g andsingle-particle levels.Finally, we discuss the validity of the collective modelapproach assuming the pairing gap as a collective co-ordinate. The 5D collective Hamiltonian assuming thequadrupole deformation parameters α µ ( µ = ± , ± , and 0) as the collective coordinates is widely utilized toanalyze experimental data of quadrupole states. Simi-larly, we may construct the pairing collective Hamilto-nian in terms of the pairing gap ∆ and the gauge angle2 ∆ [ M e V ] q [MeV -1/2 ] FIG. 12: Pairing gap as a function of collectivecoordinate q in Pb.Φ. As far as there is a one-to-one correspondence be-tween ∆ and the collective variable q we obtained in thepresent study, we can transform the collective Hamilto-nian in ( q , Φ) into the one in (∆ , Φ). The pairing gap ∆is defined as∆( q ) ≡ g h Φ , J ; q, p | ˆ S − | Φ , J ; q, p i (cid:12)(cid:12)(cid:12) Φ= p =0 = g X α p j α (Ω α − j α ) . (4.3)Figure 12 shows the pairing gap ∆ in Pb as a functionof the collective coordinate q . The peak in ∆ is near q = 0 and it is not a monotonic function of q , thusno one-to-one correspondence exists. The same behavioris observed for other Pb isotopes too. Therefore, thepairing gap ∆ is not a suitable collective coordinate todescribe the pairing dynamics in the multi-level model. V. CONCLUSION AND DISCUSSION
Extending our former work [26], which demonstratedthe accuracy of the SPA for the requantization of theTDHFB dynamics in the two-level pairing model, wepropose the ASCC+SPA method for non-integrable sys-tems. In this approach, we use the ASCC method toextract the 2D collective subspace including the pair ro-tation. In other words, we extract an approximate inte-grable system in the non-integrable system described by( q , p ; J, Φ). We apply the ASCC+SPA method to the multi-levelpairing model. We investigate the three-level model andthe multi-level model simulating Pb isotopes with a real-istic pairing coupling constant g and single-particle levels.In both cases, the low-lying excited 0 + states obtainedwith the ASCC+SPA well reproduce the exact solutionsnot only of the excitation energies but also of the wavefunctions. In the ASCC+SPA, the pair-transition calcu-lation is straightforward, because we have a microscopicwave function for every quantized state. This overcomesa disadvantage in the conventional canonical requanti-zation in which we need to construct a pair-transitionoperator in terms of the collective variables only.Although the overall agreement between theASCC+SPA and the exact calculations is good ingeneral, we have encountered several problems remain-ing to be solved. First, we can calculate a classicaltrajectory bound by the pocket of a potential. However,it is not trivial how to treat “unbound” trajectories thathit the end point of the collective path. See the poten-tials in Figs. 6 and 10. This happens in the calculationof Pb (Sec. IV C). Probably, it is necessary to find aproper boundary condition in the collective subspace.For instance, the 5D quadrupole collective model hassuch boundary conditions imposed by the symmetryproperty of the quadrupole degrees of freedom [35].The second problem occurred in the calculation of
Pb, in which we have encountered complex eigenvaluesand eigenvectors of the moving-frame QRPA equation.This happens at a point where the two eigen frequenciesbecome identical, ω = ω , namely at a crossing point.We do not have a problem for the crossing between thepair rotational mode and the other modes. Currently, wedo not know exactly when the complex solutions emerge.Another problem we need to solve is a description ofthe quantum tunneling. The tunneling plays an essentialrole in spontaneous fission, sub-barrier fusion reaction,and shape coexistence phenomena [5, 12, 33, 36, 37]. Inthe present ASCC+SPA, the classical trajectory cannotpenetrate the potential barrier. Since the ASCC is ableto provide the 1D collective coordinate, the imaginary-time TDHF is feasible and may be a solution to thisproblem [14]. These remaining issues in the ASCC+SPAapproach should be addressed in future. ACKNOWLEDGMENTS
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