Spurious states and stability condition in extended RPA theories

Transcription

1 Spurious states and stability condition in extended RPA theories V. I. Tselyaev Citation: AIP Conference Proceedings 1606, 201 (2014); View online: View Table of Contents: Published by the American Institute of Physics

2 Spurious states and stability condition in extended RPA theories V. I. Tselyaev St. Petersburg State University, RU , St. Petersburg, Russia Abstract. The problems of the spurious (ghost) states and of convergence and stability of the model s solutions are considered as applied to the theories which can be classified as the versions of the extended random phase approximation. It is shown that stability is ensured by making use of the subtraction method proposed previously. The spurious states are eliminated with the help of the projection technique. The particular case of the dipole spurious states is considered. These results are illustrated by the calculations of the low-lying states in the nucleus 208 Pb. Keywords: extended RPA, stability condition, spurious states PACS: Jz 1. INTRODUCTION Theoretical description of the properties of the ground and excited states of atomic nuclei within the framework of the modern microscopic approaches leads to the problems of the spurious (ghost) states and of convergence and stability of the model s solutions. These problems are not new ones but they gain new character in these approaches. The spurious states arise because of the breaking of some symmetry. The problems of convergence and stability become relevant in the calculations using a large model configuration space. All these problems are resolved within the framework of the so-called mean-field self-consistent theories: in the Hartree-Fock (HF) approximation, density functional theory (DFT, see, e.g., Refs. [1 3]), and in the HF or DFT based self-consistent random phase approximation (RPA, see [4]). In particular, the problem of stability is resolved in the self-consistent RPA as was shown by Thouless in Refs. [5, 6]. However, if we go beyond the mean-field theories by taking into account additional correlations effects (that means in practice enlarging the configuration space), the aforementioned problems become actual and remain so far open. The main reason is that it is difficult to achieve the full self-consistency in the extended models. An example of such an extended model is the second RPA (SRPA, see [7 10]). In this model the configuration space includes two-particles two-holes (2p2h) states in addition to the oneparticle one-hole (1p1h) states incorporated in the RPA. In the recent calculations of nuclear vibrational excitations within the SRPA it was obtained (see [11]) that some low-lying states become unstable if the space of the 2p2h states is sufficiently large. Spurious dipole states also appear at low energies in the SRPA, while in the self-consistent RPA they have zero energy. This raises the question of the applicability of the SRPA in its conventional form in the low-energy region, that is in the region which is important from the point of view of astrophysical applications (see, e.g., Refs. [12 14]). In the present paper we consider the methods which allow us to resolve the problems of the spurious states and stability in the extended RPA (ERPA) theories. This common term will be used to refer to the following models: the SRPA and two versions of the quasiparticle-phonon coupling model formulated within the Green function method on the basis of the time-blocking approximation (TBA). These versions are the model including 1p1h phonon configurations [15 17] and the two-phonon model [18]. Both in the SRPA and in the TBA, the model equations have common structure (see [19]) that justifies their unification in the framework of the ERPA. 2. RPA AND EXTENDED RPA In many cases it is convenient to formulate nuclear-structure models in terms of the response function formalism. The response function R(ω) determines the distribution of the strength of transitions in the nucleus caused by some II Russian-Spanish Congress on Particle and Nuclear Physics at all Scales, Astroparticle Physics and Cosmology AIP Conf. Proc. 1606, (2014); doi: / AIP Publishing LLC /$

3 external field represented by the single-particle operator Q according to the formulas S(E)= 1 ImΠ(E + iδ), Π(ω)= Q R(ω) Q, (1) π where S(E) is the strength function, E is an excitation energy, Δ is a smearing parameter, and Π(ω) is the (dynamic) polarizability. The response function can be defined as a solution of the Bethe-Salpeter equation (BSE, see Ref. [20]). In the RPA it reads R RPA (ω)=r (0) (ω) R (0) (ω)vr RPA (ω), (2) where R (0) (ω) is uncorrelated particle-hole (ph) propagator and V is the amplitude of the residual interaction. All the matrices in Eq. (2) are defined in the 1p1h configuration space. The ph propagator R (0) (ω) is defined as R (0) (ω)= ( ω Ω (0)) 1 M RPA, (3) where M12,34 RPA = δ 13 ρ 42 ρ 13 δ 42, Ω (0) 12,34 = h 13 δ 42 δ 13 h 42. (4) M RPA is the metric matrix, ρ is the single-particle density matrix satisfying the condition ρ 2 = ρ, h is the singleparticle Hamiltonian, and it is implied that [h,ρ ]=0. The numerical indices here and in the following denote the set of the quantum numbers of some single-particle basis. In the self-consistent RPA based on the energy density functional E[ρ] one has h 12 = δe[ρ], V δρ 12,34 = δ 2 E[ρ], (5) 21 δρ 21 δρ 34 so the quantities h and V appear to be linked by these equations. The BSE for the response function in the ERPA is R ERPA (ω)=r (c) (ω) R (c) (ω)vr ERPA (ω). (6) It differs from Eq. (2) by replacement of the uncorrelated ph propagator R (0) (ω) by the correlated propagator R (c) (ω) satisfying the equation R (c) (ω)=r (0) (ω) R (0) (ω)w(ω)r (c) (ω). (7) The matrix W(ω) in Eq. (7) is the energy-dependent interaction amplitude that includes contributions of complex configurations (2p2h, 1p1h phonon or two-phonon configurations). As a matrix, it is defined in the 1p1h subspace and can be represented in the form W 12,34 (ω)= c, σ σ F c(σ) 12 F c(σ) 34, (8) ω σ Ω c where σ = ±1 and c is an index of the subspace of complex configurations. Explicit formulas for the amplitudes F c(σ) 12 and the energies Ω c in the case of the SRPA and TBA are given in Ref. [19]. Note that in the models considered in [19] all the energies Ω c are positive. 3. SUBTRACTION METHOD AND STABILITY CONDITION IN THE ERPA In the DFT-based approaches the energy density functional E[ρ] in Eqs. (5) is, as a rule, a functional with phenomenologically adjusted parameters. In this case E[ρ] already effectively contains a part of the contributions of those complex configurations which are explicitly included in the ERPA. Therefore, in the theory going beyond the RPA, the problem of double counting arises. This problem can be avoided by making use of the subtraction method. It consists in the replacement of the amplitude W(ω) in Eq. (7) by the quantity W(ω)=W(ω) W(0). Thus, in the general case we have instead of (7) R (c) (ω)=r (0) (ω) R (0) (ω)w (κ) (ω)r (c) (ω), (9) 202

4 where W (κ) (ω)=w(ω) κw(0), (10) κ = 1 when subtraction is used, κ = 0 for the ERPA without subtraction. A detailed justification for the subtraction method is given in Ref. [19] where it is also shown that, in addition to the elimination of double counting, this method ensures stability of solutions of the ERPA eigenvalue equations and leads to the acceleration of the convergence. To be more precise, stability, as applied to the description of the excited states, means that all the calculated excitation energies should be real and positive. The scheme of the proof is the following. Let us introduce the RPA matrix Ω RPA = Ω (0) + M RPA V. (11) Using Eqs. (3) and (11), we find the solution of Eq. (2) in the explicit form: R RPA (ω)= ( ω Ω RPA) 1 M RPA. From this we obtain that the poles and residua of the response function R RPA (ω) are determined by the following RPA eigenvalue equation: Ω RPA z n = ω n z n, (12) where the eigenvalues ω n correspond to the excitation energies of the many-body system in the RPA. The matrix Ω RPA can be represented in the form Ω RPA = M RPA S RPA, (13) where S RPA is the RPA stability matrix, and the matrices M RPA and S RPA are Hermitian. As was proved by Thouless in Refs. [5, 6] (the so-called Thouless theorem), all eigenvalues ω n in Eq. (12) are real if the matrix S RPA is positive semidefinite (i.e., z S RPA z 0 for any complex vector z ). The positive semidefiniteness of the matrix S RPA follows from the conditions of minimization of the energy density functional E[ρ] in the self-consistent theory (see [4, 5]). The poles and residua of the response function R ERPA (ω) are determined by the ERPA eigenvalue equation which can be written in the extended space including 1p1h and complex configurations in the form similar to Eq. (12): The matrix Ω ERPA in this equation can be also represented in the form Ω ERPA Z ν = ω ν Z ν. (14) Ω ERPA = M ERPA S ERPA, (15) where S ERPA is the ERPA stability matrix, and the matrices M ERPA and S ERPA are Hermitian. In Ref. [19] it was proved that the matrix S ERPA is positive semidefinite if: (i) S RPA is positive semidefinite; (ii) Ω c > 0 in Eq. (8); and (iii) the subtraction method is used [that is κ = 1 in Eqs. (9) and (10)]. In analogy to the Thouless theorem mentioned above, it means that, under these conditions (which constitute stability condition in the ERPA), all the eigenvalues ω ν in Eq. (14) are real. Without subtraction (κ = 0) this property of the solutions of the ERPA eigenvalue equation is not guaranteed. 4. PROBLEM OF THE SPURIOUS STATES IN THE ERPA The subtraction method described above ensures by construction zero energies of the spurious states in the ERPA if they appear at ω = 0 in the RPA. This property was one of the motivations for introducing this method in Ref. [18]. However, the subtraction method in itself does not provide full decoupling of the spurious states and the physical modes. As a result, though the main component of the spurious state appears at zero energy, its fragments are spread out over the energy range near zero. This fragmentation of the spurious states can be eliminated in the following way. As it was shown in Ref. [19], the solution of Eq. (2) in the self-consistent RPA can be represented in the form R RPA (ω)=r RPA(0) (ω) n sgn(ω n ) z n z n ω ω n, (16) where ω n and z n are solutions of Eq. (12), the excitation energies ω n are real and non-zero, the vectors z n are normalized by the condition z n M RPA z n = sgn(ω n )δ n,n. (17) 203

5 The function R RPA(0) (ω) represents the ghost part of the RPA response function. It consists of two terms: R RPA(0) (ω)= a(0,1) ω The matrices a (0,1) and a (0,2) are Hermitian and satisfy the equations a(0,2) ω 2. (18) Ω RPA a (0,1) = a (0,2), Ω RPA a (0,2) = 0, (19) a (0,1) M RPA a (0,k) = a (0,k), k = 1,2, (20) a (0,1) = Pa (0,1) P, a (0,2) = Pa (0,2) P, (21) where P is the permutation operator acting in the space of the pairs of the single-particle indices. Let us introduce a projection operator P = 1 a (0,1) M RPA. (22) From Eq. (20) it follows that P 2 = P, Pa (0,k) = 0, k = 1,2. (23) Let us denote W (ω) =P W(ω)P. From Eq. (22) and from orthogonality of the spurious states and the physical RPA-modes z n and z n it follows that z n W (ω) z n = z n W(ω) z n. (24) It enables one to replace in all the ERPA equations the amplitudes W(ω) and W(0) by W (ω) and W (0), respectively. Such a replacement is equivalent to the replacement of F c(σ) by F c(σ) = P F c(σ) in Eq. (8). Note that this replacement does not violate stability condition in the ERPA formulated in Sec. 3. In this way from Eqs. (2), (6), and (7) we obtain R ERPA (ω)=r RPA (ω) R RPA (ω) W (ω)r ERPA (ω), (25) where (with making use of the subtraction method) W (ω)=w (ω) W (0). From Eqs. (16), (18), (23), and (25) it follows that in the ERPA with projection the coupling of the spurious states to the physical modes is fully eliminated, since W (ω)r RPA(0) (ω)=r RPA(0) (ω) W (ω)=0. (26) The generalization of this method including pairing correlations was applied in Refs. [21, 22] to eliminate spurious 0 + states in the quasiparticle TBA (QTBA). 5. THE CASE OF THE DIPOLE SPURIOUS STATES The projection operator P is defined by Eqs. (19) (22) implicitly. It can be determined explicitly in the important particular case of the 1 spurious states. It is well known that these states arise because of the breaking of the translation symmetry in the mean-field approach. This symmetry is restored in the self-consistent RPA at the cost of the appearance of the zero-energy mode. Since the eigenvector of this mode is non-normalizable in the sense of Eq. (17) (see Ref. [4]) it is convenient to proceed in the following way. Let us represent the energy density functional E[ρ] in Eqs. (5) as a sum of two terms E[ρ]=Tr ( ρ h 0) + E int [ρ], (27) where h 0 is a single-particle operator and the term E int [ρ] contains all contributions to the total energy related to the interaction. Actually, h 0 is a simple kinetic-energy operator but in order to deal in the following with the normalizable solutions of the RPA equations we include an oscillator potential into h 0 by setting h 0 = p2 2m + ω2 0 mr2 2 h 2, (28) 204

6 where m is the nucleon mass (generally different for the neutrons and protons) and it is supposed that ω 0 > 0. The functional E int [ρ] is supposed to be invariant under the symmetry transformations of the type E int [e iαq ρ e iαq ]=E int [ρ], (29) where q is a Hermitian single-particle operator and α is an arbitrary real parameter. Differentiating Eq. (29) with respect to α and ρ and setting α = 0 we obtain [h,q] 12 + V 12,34 [q,ρ ] 34 =[h 0,q] 12, (30) 34 where Eqs. (5) and (27) were taken into account. Multiplying Eq. (30) from the left with the matrix M RPA and using definitions (4), (11), and equality [h,ρ ]=0, we get Ω RPA 12,34 [q,ρ ] 34 =[[h 0,q],ρ ] 12. (31) 34 Now we note that the functional E int [ρ] should be invariant under the translations and the Galilean transformations. This means that Eqs. (29) (31) should be fulfilled for those operators q which are the space components of the momentum operator (p = i h ) or of the coordinate operator multiplied by the nucleon mass (mr). In the case of the operator h 0 defined by Eq. (28) we have [h 0, ]= ω2 0 h 2 mr, [h0,mr]= h 2. (32) From Eqs. (31) and (32), after some algebra we arrive at the following equation Ω RPA 12,34 z(±) 34 = ±ω 0 z (±) 12, (33) 34 where z (±) 12 = h ([,ρ ] 12 ω ) 0 2ω0 M 0 h 2 [mr,ρ ] 12, (34) M 0 = Tr(ρm) is the total mass of the nucleus. The transition amplitudes z (±) 12 are normalized according to Eq. (17). These amplitudes represent the explicit solutions of the RPA eigenvalue equation (12) obtained from the symmetry properties of the energy density functional. They correspond to the spurious 1 excitations. If ω 0 0, we can substitute these solutions into the second term of the right hand side (r.h.s.) of Eq. (16). In the limit ω 0 0 the contribution of these solutions into the RPA response function takes the form of the r.h.s. of Eq. (18) with a (0,1) 12,34 = 1 ( [,ρ ] M 12 [mr,ρ ] 43 [mr,ρ ] 12 [,ρ ] 43 ), (35) 0 a (0,2) 12,34 = h2 [,ρ ] M 12 [,ρ ] 43. (36) 0 The matrices a (0,1) and a (0,2) defined by Eqs. (35) and (36) are Hermitian. It is not difficult to verify that they satisfy Eqs. (19) (21). Therefore, the projection operator defined by Eqs. (22) and (35) also satisfies all the conditions described in Sec. 4. Its use in Eq. (25) ensures full elimination of the 1 spurious states in the ERPA. 6. RESULTS FOR AND 3 1 STATES IN 208 Pb IN THE TBA Consider the question of convergence and stability of the ERPA solutions using, as an example, the calculations of the energies of the first 2 + and 3 levels in the nucleus 208 Pb. As the ERPA, we consider the quasiparticle-phonon coupling model within the TBA including 1p1h phonon configurations [15 17] (without ground state correlations beyond the RPA included in [15 17]). For brevity, this model will be referred to as the TBA. The starting point of the TBA is the self-consistent RPA which also determines the phonon basis of the model. The energy density functional E[ρ] in 205

7 Eqs. (5) was taken in the form of the Skyrme energy functional. In the calculations the T6 Skyrme force (see Ref. [23]) was used. In this parametrization the nucleon effective mass m is equal to the bare nucleon mass. The calculation scheme for this case is described in Ref. [21] (but here, in contrast to [21], the spin-orbit and Coulomb terms were not included in the residual interaction). In the TBA calculations the single-particle continuum was discretized (using the box boundary condition at 18 fm) and was included completely by making use of the coordinate representation technique (see, e.g., Refs. [16, 17]). In the RPA limiting case, when W(ω) =0 in Eq. (7), this calculation scheme provides the following energies of the first 2 + and 3 levels in 208 Pb: ω(2 + 1 )=4.044 MeV and ω(3 1 )=2.818 MeV. In the RPA calculations of the phonons the (discretized) single-particle basis was restricted by the states with energies below 20 MeV. To compensate the lack of the continuum in these calculations, the overall factor was introduced into the residual interaction V in Eqs. (11) and (12). With making use of this factor, the energies of the and 3 1 FIGURE 1. Dependence of the energy of the first 2 + level in 208 Pb on the maximal phonon s energy of the phonon basis in the TBA. Solid (red) line represents the results obtained with the use of subtraction method (κ = 1). The dashed (blue) line represents the results without subtraction (κ = 0). The HF mean field and the residual interaction are calculated with the T6 Skyrme force. FIGURE 2. The same as Fig. 1 for the squared energy of the first 3 level in 208 Pb. 206

8 levels obtained in the continuum RPA calculation are nicely reproduced. The phonon basis constructed in such a way included all the phonons with natural parity and with reduced transition probability B(EL) which is more than 0.1% of the maximal B(EL) for the given spin L. Under these conditions, the phonon basis restricted by the phonons with the energies ω n < 30 MeV contains 1816 phonons with 0 L 18. Dependence of the energies of the and 3 1 states on the maximal phonon s energy of the phonon basis ω max in the TBA is shown in Figs. 1 and 2. In the case, when the subtraction method is used [κ = 1 in Eqs. (9) and (10)], both states remain stable and the convergence of the results is fairly well (the difference between the TBA energies for ω max = 25 MeV and ω max = 30 MeV is 33 kev for state and 19 kev for 3 1 state). Without subtraction (κ = 0) the convergence is not observed and the energy of the 3 1 state becomes imaginary at ω max 20 MeV. 7. CONCLUSIONS In the paper the problems of the spurious states and of convergence and stability in the extended random phase approximation (ERPA) theories are considered. It is shown that stability of the ERPA solutions is ensured by making use of the subtraction method. This method was suggested previously to avoid double counting in the models going beyond the conventional RPA and based on the density functional theory with phenomenologically fitted energy density functionals. Acceleration of convergence and stability of the solutions in the ERPA with subtraction are illustrated by the calculations of the first 2 + and 3 states in 208 Pb within the framework of the ERPA version based on the quasiparticle-phonon coupling model. It is shown that the spurious states in the ERPA can be fully separated from the physical modes with the help of the projection technique. The explicit formulas for this method are obtained in the case of the dipole spurious states. ACKNOWLEDGMENTS The author acknowledges financial support from the St. Petersburg State University under Grant No REFERENCES 1. H. Eschrig, The Fundamentals of Density Functional Theory (Edition am Gutenbergplatz, Leipzig, 2003). 2. M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod. Phys. 75, 121 (2003). 3. J. E. Drut, R. J. Furnstahl, and L. Platter, Progr. Part. Nucl. Phys. 64, 120 (2010). 4. P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer-Verlag, New York, 1980). 5. D. J. Thouless, Nucl. Phys. 21, 225 (1960). 6. D. J. Thouless, Nucl. Phys. 22, 78 (1961). 7. H. Suhl and N. R. Werthamer, Phys. Rev. 122, 359 (1961). 8. J. Sawicki, Phys. Rev. 126, 2231 (1962). 9. C. Yannouleas, M. Dworzecka, and J. J. Griffin, Nucl. Phys. A397, 239 (1983). 10. S. Drożdż, S. Nishizaki, J. Speth, and J. Wambach, Phys. Rep. 197, 1 (1990). 11. P. Papakonstantinou and R. Roth, Phys. Rev. C 81, (2010). 12. S. Goriely, E. Khan, and M. Samyn, Nucl. Phys. A739, 331 (2004). 13. E. Litvinova, H. P. Loens, K. Langanke, G. Martínez-Pinedo, T. Rauscher, P. Ring, F.-K. Thielemann, and V. Tselyaev, Nucl. Phys. A823, 26 (2009). 14. I. Daoutidis and S. Goriely, Phys. Rev. C 86, (2012). 15. V. I. Tselyaev, Yad. Fiz. 50, 1252 (1989) [ Sov. J. Nucl. Phys. 50, 780 (1989) ]. 16. S. P. Kamerdzhiev, G. Ya. Tertychny, and V. I. Tselyaev, Fiz. Elem. Chastits At. Yadra 28, 333 (1997) [ Phys. Part. Nucl. 28, 134 (1997) ]. 17. S. Kamerdzhiev, J. Speth, and G. Tertychny, Phys. Rep. 393, 1 (2004). 18. V. I. Tselyaev, Phys. Rev. C 75, (2007). 19. V. I. Tselyaev, Phys. Rev. C 88, (2013). 20. J. Speth, E. Werner, and W. Wild, Phys. Rep. 33, 127 (1977). 21. V. Tselyaev, J. Speth, S. Krewald, E. Litvinova, S. Kamerdzhiev, N. Lyutorovich, A. Avdeenkov, and F. Grümmer, Phys. Rev. C 79, (2009). 22. V. I. Tselyaev, Izv. Ross. Akad. Nauk, Ser. Fiz. 74, 905 (2010) [Bull. Russ. Acad. Sci., Phys. (USA) 74, 865 (2010)]. 23. F. Tondeur, M. Brack, M. Farine, and J. M. Pearson, Nucl. Phys. A420, 297 (1984). 207