Linear response calculation using the canonical-basis TDHFB with a schematic pairing functional
aa r X i v : . [ nu c l - t h ] S e p Linear response calculation using the canonical-basisTDHFB with a schematic pairing functional
Shuichiro Ebata , , Takashi Nakatsukasa , , and Kazuhiro Yabana , , RIKEN Nishina Center, Wako-shi 351-0198, Japan Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571,Japan Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8571, Japan
Abstract.
A canonical-basis formulation of the time-dependent Hartree-Fock-Bogoliubov(TDHFB) theory is obtained with an approximation that the pair potential is assumed to bediagonal in the time-dependent canonical basis. The canonical-basis formulation significantlyreduces the computational cost. We apply the method to linear-response calculations for even-even nuclei. E
1. Introduction
The time-dependent Hartree-Fock (TDHF) theory has been extensively utilized to study nuclearmany-body dynamics [1]. Recently, it has been revisited with modern energy density functionalsand more accurate description of nuclear properties has been achieved [2, 3, 4, 5, 6, 7]. TheTDHF theory uses only occupied orbitals, number of which is equal to the number of particles( N ). However, it neglects the residual interactions in particle-particle and hole-hole channels,which are important for properties of open-shell heavy nuclei. It is well-known that the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory [8] properly takes into account the pairingcorrelations. The TDHFB equation is formulated in a similar manner to the TDHF, howeverit requires us to calculate the time evolution of quasi-particle orbitals, number of which is, inprinciple, infinite. Therefore, the practical calculations with the TDHFB are very limited [9, 10].Very recently, we have proposed a possible approximation for the TDHFB theory [11]. This isa time-dependent version of the BCS approximation [12] for the Hartree-Fock-Bogoliubov theory.Namely, we neglect off-diagonal elements of the pair potential in the time-dependent canonicalbasis. We show that this approximation results in significant reduction of the computationaltask. We call the equations obtained with this approximation, “Canonical-basis TDHFB” (Cb-TDHFB) equations. We apply the method to the linear-response calculations using the fullSkyrme functionals of the parameter set of SkM*, and discuss properties of the E . Formalism of the Cb-TDHFB theory In this section, we show the basic equations of the Cb-TDHFB and recapitulate their derivation.See Ref. [11] for more details.
Let us first show the Cb-TDHFB equations we derive in the followings. i ∂∂t | φ k ( t ) i = ( h ( t ) − η k ( t )) | φ k ( t ) i , i ∂∂t | φ ¯ k ( t ) i = ( h ( t ) − η ¯ k ( t )) | φ ¯ k ( t ) i , (1a) i ddt ρ k ( t ) = κ k ( t )∆ ∗ k ( t ) − κ ∗ k ( t )∆ k ( t ) , (1b) i ddt κ k ( t ) = ( η k ( t ) + η ¯ k ( t )) κ k ( t ) + ∆ k ( t ) (2 ρ k ( t ) − . (1c)These basic equations determine the time evolution of the canonical states, | φ k ( t ) i and | φ ¯ k ( t ) i ,their occupation, ρ k ( t ), and pair probabilities, κ k ( t ). The real functions of time, η k ( t ) and η ¯ k ( t ),are arbitrary and associated with the gauge degrees of freedom. The time-dependent pairinggaps, ∆ k ( t ), which are given in Eq. (12), are similar to the BCS pairing gap [12] except forthe fact that the canonical pair of states are no longer related to each other by time reversal.Although we use the same symbols, ( ρ, κ, ∆), for matrixes in Eqs. (3a) and (3b), the quantities inthe Cb-TDHFB equations are only their diagonal elements with a single index for the canonicalstates k . It should be noted that similar equations can be found in Ref. [13] for a simple pairingenergy functional. We now derive the Cb-TDHFB equations starting from the generalized density-matrix formalism.The TDHFB equation can be written in terms of the generalized density matrix R ( t ) as [8] i ∂∂t R ( t ) = [ H ( t ) , R ( t )] . (2)This is equivalent to the following equations for one-body density matrix, ρ ( t ), and the pairing-tensor matrix, κ ( t ). i ∂∂t ρ ( t ) = [ h ( t ) , ρ ( t )] + κ ( t )∆ ∗ ( t ) − ∆( t ) κ ∗ ( t ) , (3a) i ∂∂t κ ( t ) = h ( t ) κ ( t ) + κ ( t ) h ∗ ( t ) + ∆( t )(1 − ρ ∗ ( t )) − ρ ( t )∆( t ) . (3b)Here, h ( t ) and ∆( t ) are single-particle Hamiltonian and pair potential, respectively.At each instant of time, we may diagonalize the density operator ˆ ρ in the orthonormalcanonical basis, { φ k ( t ) , φ ¯ k ( t ) } with the occupation probabilities ρ k . Then, the TDHFB state isexpressed in the canonical (BCS) form as | Ψ( t ) i = Y k> n u k ( t ) + v k ( t ) c † k ( t ) c † ¯ k ( t ) o | i . (4)For the canonical states, we use the alphabetic indexes such as k for half of the total spaceindicated by k >
0. For each state with k >
0, there exists a “paired” state ¯ k < k >
0. The set of states { φ k , φ ¯ k } generate the wholeingle-particle space. We use the Greek letters µ, ν, · · · for indexes of an adopted representation(complete set) for the single-particle states. Using the following notations, hh µν | φ k ( t ) φ ¯ k ( t ) ii ≡ h µ | φ k ( t ) ih ν | φ ¯ k ( t ) i − h µ | φ ¯ k ( t ) ih ν | φ k ( t ) i , (5)ˆ π k ( t ) ≡ | φ k ( t ) ih φ k ( t ) | + | φ ¯ k ( t ) ih φ ¯ k ( t ) | , (6)the density and the pairing-tensor matrixes are expressed as ρ µν ( t ) = X k> ρ k ( t ) h µ | ˆ π k ( t ) | ν i , (7) κ µν ( t ) = X k> κ k ( t ) hh µν | φ k ( t ) φ ¯ k ( t ) ii , (8)where ρ k ( t ) = | v k ( t ) | and κ k ( t ) = u ∗ k ( t ) v k ( t ). It should be noted that the canonical pair ofstates, | φ k ( t ) i and | φ ¯ k ( t ) i , are assumed to be orthonormal but not necessarily related with eachother by the time reversal, | φ ¯ k i 6 = T | φ k i .We can invert Eqs. (7) and (8) for ρ k and κ k , ρ k ( t ) = X µν h φ k ( t ) | µ i ρ µν ( t ) h ν | φ k ( t ) i = X µν h φ ¯ k ( t ) | µ i ρ µν ( t ) h ν | φ ¯ k ( t ) i , (9) κ k ( t ) = 12 X µν hh φ k ( t ) φ ¯ k ( t ) | µν ii κ µν ( t ) . (10)The derivative of ρ k ( t ) with respect to time t leads to i ddt ρ k ( t ) = X µν h φ k ( t ) | µ i i dρ µν dt h ν | φ k ( t ) i + iρ k ( t ) ddt h φ k ( t ) | φ k ( t ) i = 12 X µν (cid:8) κ k ( t )∆ ∗ µν ( t ) hh νµ | φ k ( t ) φ ¯ k ( t ) ii + κ ∗ k ( t )∆ µν ( t ) hh φ k ( t ) φ ¯ k ( t ) | µν ii (cid:9) . (11)We used the assumption of norm conservation and the TDHFB equation (3a). This can berewritten in the simple form of Eq. (1b) with the definition of the pairing gap,∆ k ( t ) ≡ − X µν ∆ µν ( t ) hh φ k ( t ) φ ¯ k ( t ) | µν ii . (12)In the same way, we evaluate the time derivative of κ k ( t ) as i ddt κ k ( t ) = 12 X µν hh φ k ( t ) φ ¯ k ( t ) | µν ii i dκ µν dt + iκ k ( t ) (cid:18) h dφ k dt | φ k ( t ) i + h dφ ¯ k dt | φ ¯ k ( t ) i (cid:19) . (13)Then, using the TDHFB equation (3b), we obtain Eq. (1c) with the real gauge functions η k ( t ) ≡ h φ k ( t ) | h ( t ) | φ k ( t ) i + i h ∂φ k ∂t | φ k ( t ) i . (14)These functions control time dependence of phase for the canonical states, which are basicallyarbitrary.o far, the derivation is based on the TDHFB equations, and no approximation beyond theTDHFB is introduced. However, to obtain simple equations for time evolution of the canonicalbasis, we need to introduce an assumption (approximation) that the pair potential is written as∆ µν ( t ) = − X k> ∆ k ( t ) hh µν | φ k ( t ) φ ¯ k ( t ) ii . (15)This satisfies Eq. (12), but in general, Eq. (12) can not be inverted because the two-particlestates | φ k φ ¯ k i do not span the whole space. In other words, we only take into account thepair potential of the “diagonal” parts in the canonical basis, ∆ k ¯ l = − ∆ k δ kl . In the stationarylimit ( | φ ¯ k i = T | φ k i ), this is equivalent to the ordinary BCS approximation [12]. With theapproximation of Eq. (15), it is easy to see that the TDHFB equations, (3a) and (3b), areconsistent with Eqs. (1). The Cb-TDHFB equations, (1), are invariant with respect to the gauge transformation witharbitrary real functions, θ k ( t ) and θ ¯ k ( t ). | φ k i → e iθ k ( t ) | φ k i and | φ ¯ k i → e iθ ¯ k ( t ) | φ ¯ k i (16) κ k → e − i ( θ k ( t )+ θ ¯ k ( t )) κ k and ∆ k → e − i ( θ k ( t )+ θ ¯ k ( t )) ∆ k (17)simultaneously with η k ( t ) → η k ( t ) + dθ k dt and η ¯ k ( t ) → η ¯ k ( t ) + dθ ¯ k dt . The phase relations of Eq. (17) are obtained from Eqs. (10) and (12). It is now clear that thearbitrary real functions, η k ( t ) and η ¯ k ( t ), control time evolution of the phases of | φ k ( t ) i , | φ ¯ k ( t ) i , κ k ( t ), and ∆ k ( t ).In addition to the gauge invariance, the Cb-TDHFB equations possess the followingproperties.(i) Conservation law(a) Conservation of orthonormal property of the canonical states(b) Conservation of average particle number(c) Conservation of average total energy(ii) The stationary solution corresponds to the HF+BCS solution.(iii) Small-amplitude limit(a) The Nambu-Goldstone modes are zero-energy normal-mode solutions.(b) If the ground state is in the normal phase, the equations are identical to the particle-hole, particle-particle, and hole-hole RPA with the BCS approximation. We adopt a Skyrme functional with the SkM* parameter set for the particle-hole channels. Forthe pairing energy functional, we adopt a simple functional of a form E g ( t ) = − X k,l> G kl κ ∗ k ( t ) κ l ( t ) , = − X k> κ ∗ k ( t )∆ k ( t ) , ∆ k ( t ) = X l> G kl κ l ( t ) , (18)where G kl = Gf ( ǫ k ) f ( ǫ l ) with G = 0 . f ( ǫ k ), whose explicitform can be found in Ref. [11], depends on the single-particle energy of the canonical state k at able 1. Ground-state properties of Mg isotopes calculated with the SkM* functional;quadrupole deformation parameters ( β, γ ), pairing gaps for neutrons and protons (∆ n , ∆ p ),chemical potentials for neutrons and protons ( λ n , λ p ). In the case of normal phase (∆ = 0), wedefine the chemical potential as the single-particle energy of the highest occupied orbital. Thepairing gaps and chemical potentials are given in units of MeV. β γ ∆ n ∆ p − λ n − λ p Mg 0.31 0 ◦ Mg 0.0 − Mg 0.38 0 ◦ Mg 0.39 0 ◦ Mg 0.20 54 ◦ Mg 0.0 − η k ( t ) = ǫ k ( t ) = h φ k ( t ) | h ( t ) | φ k ( t ) i , η ¯ k ( t ) = ǫ ¯ k ( t ) = h φ ¯ k ( t ) | h ( t ) | φ ¯ k ( t ) i . (19)For numerical calculations, we extended the computer program of the TDHF in the three-dimensional coordinate-space representation [2] to include the pairing correlations. The groundstate is first constructed by the HF+BCS calculation. Then, we solve the Cb-TDHFB equationsin real time, under a weak impulse isovector dipole field, yielding a time-dependent E1 moment, D E ( t ). To obtain the E D E ( t ) → D E ( t ) e − Γ t/ . The readers should refer toRef. [11] for more details.
3. Electric dipole strength distribution in proton-rich Mg isotopes
In Table 1, the ground-state deformations, pairing gaps, and chemical potentials are listed forstable to proton-rich Mg isotopes. In the present calculation with SkM*, Mg turns out to bebound with a small binding energy of 200 keV. Thus, we include this nucleus in our calculationas a “fictitious” proton halo nucleus. The neutron pairing gap is absent for all these nuclei. Theproton gap also vanishes for nuclei with prolate shapes, β = 0 . ∼ . E , Mg. The K = 0 peak is located around 15 MeV and the K = 1 is near 22 MeV. In contrast, for Mg,although the ground state is deformed in a prolate shape with β ≈ .
3, the double-peak structureis not clearly seen. In this nucleus, the E K = 0 and K = 1 components arefragmented into a wide range of energy. Previously, we calculated the E E E <
10 MeV is negligible for stable nuclei ( , Mg). For theneutron-rich side, we see a small low-energy peak in Mg. Since the neutron separation energyis still sizable (about 9 MeV), we assume that this is due to the occupation of the neutron s / orbital which is spatially extended. For the proton-rich (neutron-deficient) side, the low-energy (a) Mg Total
K=0K=1 (b) Mg (c) Mg S ( E ; E ) [ e f m / M e V ] (d) Mg E [ MeV ] (e) Mg (f) Mg Figure 1.
Calculated E N = 8 ∼ K = 0 (green dashed line) and K = 1(blue dotted line) components. The z -axis is chosen as the symmetry axis for axially deformedcases. The smoothing parameter of Γ = 1 MeV is used.strength is seen in , Mg. This should be due to the weak binding of the last-occupied proton d / orbitals, since the calculated proton separation energies are less than 3 MeV for these nuclei.
4. Summary
We presented an approximate and feasible approach to the TDHFB; canonical-basis TDHFBmethod. Since the number of the canonical states we need to calculate is the same order as theparticle number, this method significantly reduces the computational task of the TDHFB. Wecalculated the E E Acknowledgments
This work is supported by Grant-in-Aid for Scientific Research(B) (No. 21340073) and onInnovative Areas (No. 20105003). The numerical calculation was performed on RICC at RIKEN,the PACS-CS at University of Tsukuba, and Hitachi SR11000 at KEK.
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