- Email: [email protected]
Monte Carlo shell model
Nuclear Physics A 693 (2001) 383–393 www.elsevier.com/locate/npe
Monte Carlo shell model Takaharu Otsuka a,b a Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan b RIKEN, Hirosawa, Wako-shi, Saitama 351-0198, Japan
Received 18 September 2000; accepted 27 November 2000
Abstract The development and limitations of the conventional shell-model calculations are mentioned. In order to overcome the limitations, the Quantum Monte Carlo Diagonalization (QMCD) method has been proposed. The basic formulation and features of the QMCD method are presented as well as its application to the nuclear shell model, as referred to as Monte Carlo Shell Model (MCSM). The MCSM provides us with a breakthrough in the shell-model calculation: the structure of low-lying states can be studied with realistic interactions for a wide, nearly unlimited basically, variety of nuclei. Thus, the MCSM will contribute significantly to the physics to be developed by means of Radioactive Ion Beams. Applications to 56 Ni, Ba isotopes and N ∼ 20 exotic nuclei far from the β stability line are mentioned. 2001 Elsevier Science B.V. All rights reserved. PACS: 21.60.Ka; 21.60.Cs; 24.10.Cn; 21.60.Fw; 27.40.+z; 27.30.+t Keywords: Shell model; Monte Carlo shell model; Quantum Monte Carlo Diagonalization (or QMCD) method; Exotic nuclei; Shape phase transition; Magic number
1. Introduction The nuclear shell model was started by Mayer and Jensen in 1949 [1] as a singleparticle model. Afterwards, many valence particles are treated in the shell model, which then became a many-body theory and a calculational method. A good example can be found in the sd shell [2]. The nuclear shell model has been successful in the description of various aspects of nuclear structure, partly because it is based upon a minimum number of natural assumptions, and partly because all dynamical correlations in the model space, beyond the mean-field calculations, can be incorporated appropriately. Although the direct diagonalization of the Hamiltonian matrix in the full valence-nucleon Hilbert space is desired, the dimension of such a space is too large in many cases, preventing us from performing the full calculations. E-mail address: [email protected] (T. Otsuka). 0375-9474/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 0 ) 0 0 6 0 5 - 9
384
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
In order to overcome this dimension problem, quantum Monte Carlo approaches have been introduced. As one of them, the Shell Model Monte Carlo (SMMC) method has been proposed successfully [3]. However, the SMMC is basically restricted to groundstate and thermal properties. Moreover, the SMMC suffers from the so-called minus-sign problem for realistic interactions. As a completely different approach, the Quantum Monte Carlo Diagonalization (QMCD) method has been proposed for solving quantum manybody systems with a two-body interaction [4–7]. The QMCD can describe not only the ground state but also excited states, including their energies, wave functions and hence transition matrix elements. The sign problem is irrelevant to the QMCD. Thus, based upon the QMCD method, we introduce the Monte Carlo Shell Model as its application to the nuclear shell-model calculation. The MCSM has become a new tool for clarifying the structure of the ground and low-lying states of nuclei, providing us with a quite powerful tool for theoretical studies of nuclei produced by Radioactive Ion Beams. 2. Development and limitations of conventional Shell Model In the shell-model calculation, one introduces single-particle state first. This state can be given as an eigenstate of a spherical single-particle potential, for instance, Harmonic Oscillator or Woods–Saxon potential. Single-particle states relevant to a given nucleus are grouped into the core part and the valence part. The core part is completely occupied, and is frozen like a vacuum. The valence part is usually one major shell on top of the core part, and called valence shell. The valence shell is partly occupied. In the shell-model calculation, one generates all possible Slater determinants in the valence shell. The Slater determinants can be classified according to the z-component of the angular momentum, denoted by M. The number of Slater determinants for a given value of M is usually called (M-scheme) shell-model dimension. In order to obtain the eigenstate of the Hamiltonian for the given valence shell, one calculates matrix elements of the Hamiltonian for Slater determinants. This is a straightforward calculation. However, this calculation should be done for all pairs of Slater determinants as bra and ket vectors of the matrix elements. Once all matrix elements are calculated, one should diagonalize the matrix. A schematic description of this process is shown in the upper part of Fig. 1. So, if the shell-model dimension is large, the number of matrix elements is certainly much larger (up to about half of the square of the dimension), and the actual calculation becomes very difficult. By recent (conventional) shell-model codes like ANTOINE by Caurier, VECSSE by Sebe or MSHELL by Mizusaki, one can handle up to shell-model dimension ∼ 100 million at technical edge, while practical calculations up to a few tens million dimension can be done. An example can be found in [8], where the levels of 52 Fe are calculated by the code ANTOINE. Although the conventional shell-model calculation has thus been developed significantly, the dimension can be much larger in many real nuclei and is indeed much beyond the reach of the future development. For instance, certain unstable nuclei to be discussed in this paper require calculations with more than 10 billion dimension. This is already very far beyond the limit of the existing shell-model codes.
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
∗ ∗ ∗ ∗ ∗ ∗ H= ∗ ∗ · · ·
∗ ∗ ∗ · ·
∗ ∗ ∗ · · · · · ·
· · · · · ·
ε
385
1
ε2 ε3
·
0 Conventional Shell Model all Slater determinants ∗
∗ H≈ ∗ ·
∗ ∗ ∗ ·
∗ · ∗ · ·
ε
1
0 Monte Carlo Shell Model bases important for a specific eigenstate
ε2
·
·
0 ·
0 ·
Fig. 1. Hamiltonian matrices in the conventional shell-model calculation and in the Monte Carlo Shell Model.
3. Basic formulation of QMCD We have seen the difficulties the conventional shell-model calculations are facing. In order to overcome those difficulties, one has to introduce an alternative approach. That is stochastic methods to many-body problems. We now turn to this subject. The Shell Model Monte Carlo (SMMC) method has been proposed first [3]. The SMMC method belongs to so-called auxiliary field Monte Carlo methods. It is designed to calculate the ground-state properties or to study thermal properties. Therefore, the SMMC is not very suitable for investigating level structure or transitions between eigenstates. It is also known that the so-called minus-sign problem prevent us from using realistic interactions in the SMMC calculations directly, although extrapolation procedures were successfully implemented [3]. The Quantum Monte Carlo Diagonalization (QMCD) method has been proposed several years later by Honma, Mizusaki and myself [4]. We shall outline very briefly the process of this method first. In the QMCD method, there are two major steps. In the first step, we generate basis states for the given many-body system, which is comprised of valence protons and neutrons in the case of the nuclear shell model. We should select only basis states important to the eigenstate to be obtained. In the second step, having those selected basis states, we calculate matrix elements of the Hamiltonian and diagonalize the Hamiltonian matrix. If we have all important basis states, the result should be a good approximation to the exact diagonalization in the entire Hilbert space. Thus we solve quantum many-body problems. So, the major question is how to generate important basis states. We shall begin with this question. Since the formulation of the QMCD has been presented first in [4] and has been explained later in simple terms in [9,10], we shall come directly to the equation by which the basis states are generated. It is written as
386
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
Φ(σ ) ∝ e−βh(σ ) Ψ (0) ,
(1)
where σ means a set of random numbers which control the basis generation, |Φ(σ ) is a basis state, |Ψ (0) is the initial state. Here h(σ ) is so-called one-body Hamiltonian, and is a linear combination of various one-body operators with certain weights. We note that Eq. (1) transforms a Slater determinant into another Slater determinant, because h(σ ) is a one-body operator. What are varied in the operation in Eq. (1) are single-particle wave functions constituting Slater determinants. Those single-particle wave functions are generally deformed. At this moment, we introduce a toy Hamiltonian as H = 12 V O 2 ,
(2)
where V is a coupling constant and O is a arbitrary one-body operator. For simplicity, we assume V < 0. The above-mentioned one-body Hamiltonian is then given by h(σ ) = V σ O.
(3)
Note that there is just one random variable σ in this equation because the present toy Hamiltonian in Eq. (2) consists of one term. By having different values of the random number, σ , we can generate different state vectors, |Φ(σ ) ’s, by Eq. (1). We diagonalize the Hamiltonian in a subspace spanned by those basis states. We evaluate how much each basis contributes for lowering the energy eigenvalue being calculated. We keep only those with larger contributions (i.e., important bases), whereas the others are thrown away. To improve the result, we generate more states by other values of σ , and keep only important ones. Thus, one can add bases, until certain convergence of the energy eigenvalue is reached. By having such selected basis vectors which are important for a specific eigenstate (not necessarily the ground state), its energy eigenvalue and wave function are obtained as a result of the diagonalization of small matrix with respect to these important basis vectors (see the lower part of Fig. 1). In realistic cases, a general shell-model Hamiltonian can be written as H=
Nf
Eα Oα + 12 Vα Oα2 ,
(4)
α=1
where α stands for an index, Eα and Vα denote single-particle energies and interaction strengths, respectively, for one-body operators Oα . The one-body Hamiltonian is then given as a combination of terms like the right-hand side of Eq. (3): (Eα + sα Vα σα )Oα , (5) h(σ ) = α
where sα = ±1 (±i) if Vα < 0 (> 0). By using this h(σ ), the basis states are generated by using Eq. (1) in essentially the same way as that for the toy Hamiltonian. The adopted basis states are called QMCD bases. The number of QMCD bases is referred to as the QMCD basis dimension. We emphasize that energies and wave functions are determined by the diagonalization, and that we can obtain excited states as well as the
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
387
ground state, for instance, by monitoring energy eigenvalues of excited states in the basis generation process. The above procedure is the first version of QMCD. Although it works quite well for simple systems [4], it turned out that, in order to carry out realistic large-scale shell-model calculations, we have to improve the method as discussed in the next section.
4. Improvement of QMCD 4.1. Basis generation around the mean-field solution Since the number of manageable bases is finite in practice, we should select bases of higher importance. We thus choose |Ψ (0) from a Hartree–Fock local minima or some states of similar nature, although |Ψ (0) is not specified in Eq. (1). In fact, the Hartree– Fock ground state is the “best” single Slater determinant by definition, and fits well to the present scheme. Such |Ψ (0) is adopted as one of the QMCD bases often. At the same time, h(σ ) in Eq. (3) is rewritten as h(σ ) = hMF + α Vα sα σα Oα . Here hMF can be set a mean-field potential, for instance, Hartree–Fock potential of the given shell-model Hamiltonian. Since hMF is independent of the random number σα and the sampling is made around σα = 0, the QMCD bases |Φ(σ ) are generated around local minima of the relevant mean field. This process is useful for yrast states by combining with the angular momentum projection discussed later. On the other hand, this process is not very relevant to other states, unless the states are in a pronounced local minima. 4.2. Restoration of angular momentum Since a nucleus has rotational symmetry, the restoration of the total angular momentum is quite crucial. The basis state in Eq. (1) is not an eigenstate of the angular momentum in general, however. This is because the initial state |Ψ (0) breaks the rotational symmetry in general, and moreover the operator e−βh(σ ) has nothing to do with any symmetry. In the QMCD generation of the basis vectors, the restoration of the rotational symmetry should be fulfilled because the eigenstate has this symmetry. However, this is a very slow process. We therefore restore the rotational symmetry by means of angular momentum projection of matrix elements [7]. 4.3. Compression of QMCD basis space Stochastically generated bases contain, in general, unnecessary fluctuation components which are nothing but random noise. We therefore revise the basis-generation method so that each basis contains more relevant components lowering the energy and less irrelevant components to be canceled eventually by other bases in the diagonalization process. For this purpose, for instance, we can shift the random number σα a bit, and compare how much the resultant energy eigenvalue is changed. We take the better state obtained before or after the shift. We keep shifting σα until we come to saturation of the improvement.
388
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
Each basis state now contains more relevant components and less irrelevant ones. Although the practical process of this improvement can vary, the original important bases should be refined so as to become even more important. Since the convergence of results of diagonalization becomes much faster, the Hilbert subspace of the QMCD bases is now compressed for the same quality of the result. This compression process differentiates the QMCD method from other quantum Monte Carlo approaches in its variational character.
5. Monte Carlo shell model By combining all the above improvements, the present version of QMCD has been constructed [7]. The application of the QMCD method to the nuclear shell model is called the Monte Carlo Shell Model (MCSM). 5.1. Test of MCSM for 48 Cr The validity of MCSM has been confirmed by comparing to the result of the exact diagonalization of the same Hamiltonian. Here, the nucleus 48 Cr is taken, and the exact result is obtained from Ref. [11]. Fig. 2 shows energies of yrast states of 48 Cr. As shown in Fig. 2, the ground-state energy has been reproduced within 130 keV with the QMCD dimension 40. This result is already ∼ 100 keV below the lower edge of the error bar of the SMMC result with the temperature T = 0.5 MeV [12]. Since the angular momentum projection is made for each basis, the addition of a new basis (i.e., increase of the QMCD dimension) implies inclusion of more dynamical degrees of freedom.
Fig. 2. Energies of yrast states of 48 Cr obtained by MCSM calculation compared with the energies obtained by the exact diagonalization [11]. The MCSM energy eigenvalues are shown as functions of the (QMCD) basis dimension. The point with error bar at far right is the ground-state energy of the SMMC calculation [12].
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
389
5.2. Features of MCSM There are two major advantages in the MCSM calculations. The first one is the feasibility of including many single-particle states. Because of this, one can describe drastic excitations within a nucleus. For instance, one can describe spherical yrast states, deformed rotational band and nearly superdeformed band at the same time with the same Hamiltonian in the same model space. This example is shown in Fig. 3 where those three kinds of states are nicely presented by an MCSM calculation [13] with the full pf shell and the g9/2 orbit. One finds quite good agreement with experiment [14]. This kind of description over a wide variety of states may be characterized as the feasibility along the energy axis. The yrast states and (normally) deformed band can be described basically within the pf + shell [7,15], while the g9/2 orbit is needed for the description of 16+ 1 and 181 states and negative-parity states. In fact, these states show quite large deformation, in particular, 16+ 1 and 18+ 1. The second major advantage of the MCSM calculation is the feasibility of handling many valence particles. The maximum number of valence particles is rather limited in the conventional calculations. However, if one wants to describe a long chain of isotopes entering the region of exotic nuclei far from the β stability line, the number of particles should change significantly. So, this capability plays an indispensable role in studying the
Fig. 3. Spherical yrast states, deformed band and nearly superdeformed bands in 56 Ni.
390
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
Fig. 4. Low-lying levels of Ba isotopes. Symbols are experiments [17], while lines are MCSM calculations [16].
structure of such exotic nuclei. This feature can be characterized as the feasibility along the isospin axis, and will be discussed in the next section. This second advantage is essential also for describing the spherical-deformed phase transition in heavy nuclei [16]. This is partly because the collective motion involves many valence nucleons in general, and also because the phase transition occurs as a result of large change of the valence-nucleon number. Fig. 4 shows a recent MCSM calculation for the phase transition in Ba isotopes [16]. In this calculation, the Hamiltonian and the model space are fixed, and the nucleus is driven, by the increase of valence neutrons, from a sphere to an ellipsoid due to the proton–neutron correlation. The agreement with experiment [17] is remarkable including the transitional region, where dynamical correlations can be crucial.
6. Exotic nuclei in MCSM In exotic nuclei, two major shells are mixed rather often breaking the magic structure, and states of various characters appear at low energy. Even the ground state can be of quite exotic nature. In this situation, the above two feasibilities combined together play really crucial roles in clarifying the structure of exotic nuclei far from the β stability line. As an example, we shall discuss the structure of nuclei in the vicinity of 32 Mg. + Fig. 5 indicates energy levels of 2+ 1 and 41 states of even-A Ne and Mg isotopes obtained by MCSM calculation [18] compared with experiment [17,19–21]. The calculated results contain many predictions and seven of them have been seen by experiments carried out after the MCSM calculation, although the spin–parity assignments for the 4+ 1 levels for 32,34 Mg are only likely possibilities to be confirmed [20,21].
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
391
+ Fig. 5. Excitation energies of 2+ 1 and 41 states of even-A Ne and Mg isotopes obtained by MCSM calculation [18] compared with experiment [17,19–21]. Symbols are experimental data, while open symbols are those before the calculation of [18] and closed symbols are those after the calculation. Dashed lines are results of calculations within the sd shell.
In the N ∼ 20 neutron-rich unstable nuclei, the N = 20 magic number is broken in certain low-lying states. The extent of this breaking depends on nuclei and states, however. Therefore, calculations which can mix all relevant states are crucial in clarifying the exotic structure in this region. In the MCSM calculation in [18] and its extension, all particle– hole excitations from the N = 20 closed shell are fully included within the model space, and the same Hamiltonian is used for all nuclei investigated. In conventional calculations, such mixings cannot be included fully due to difficulties of large matrix dimension. In fact, the mixing between the 0p0h configurations and the 2p2h configurations occurs in some nuclei rather strongly due to varying shell gap [18], in a consistent manner to experiment. The lowering of the 2+ 1 level in Ne isotopes just below N = 20 is related to this mixing [18], while such lowering is not seen in Mg isotopes. Many of exotic properties in nuclei far from stability are due to interplay between T = 0 and T = 1 effective nucleon–nucleon interactions. The T = 0 interaction is strongly attractive, and is responsible for the strong mean attractive potential in nuclei. The T = 1 interaction is very weak or repulsive, but dominates the structure of nuclei with high isospin. An example can be seen in the structure of neutron-rich oxygen isotopes. From the same Hamiltonian used for the results presented above, one can obtain the magic structure for N = 14 and 16 [22], whereas this magic structure is rapidly washed away by having
392
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
only a few valence protons. Thus, a variety of new unknown structure can be expected in nuclei far from stability, and the MCSM should provide us with quite sound theoretical results because of its advantages.
7. Summary In summary, we have discussed a new method for solving quantum many-body problems where particles are interacting through a two-body interaction. In the QMCD calculation, basis vectors are selected according to their “importance” as expressed by importance truncation to the full calculation [9]. The number of such QMCD bases is usually 30–50, even if the original Hilbert space has 1 billion dimensions or more. The application of QMCD to the nuclear shell model is called Monte Carlo Shell Model (MCSM). Because one can handle many valence orbits and many valence particles in the MCSM calculations, one can describe a wide variety of states within a given nucleus, and also can move over a large region of the nuclear chart with the same Hamiltonian and the same model space. Thus, the dream of the shell-model physicists is to describe all nuclei but the lightest ones within the shell model and it seems to come true. The MCSM is of particular importance for the study of exotic nuclei far from the β stability line. For instance, one can explore effects of isospin by moving far away along the isotopic chain, or investigate what interaction can produce possible exotic deformation. It may be quite fortunate that RIB experiments and MCSM calculations came out in the same era.
Acknowledgements The MCSM calculations were performed in part by the Alphleet computer in RIKEN. This work was supported in part by Grant-in-Aid for Scientific Research (A)(2) (10304019) from the Ministry of Education, Science and Culture.
References [1] M.G. Mayer, Phys. Rev. 75 (1949) 1969; O. Haxel, J.H.D. Jensen, H.E. Suess, Phys. Rev. 75 (1949) 1766. [2] B.A. Brown, B.H. Wildenthal, Annu. Rev. Nucl. Part. Sci. 38 (1988) 29. [3] S.E. Koonin, D.J. Dean, K. Langanke, Phys. Rep. 278 (1997) 1, references therein. [4] M. Honma, T. Mizusaki, T. Otsuka, Phys. Rev. Lett. 75 (1995) 1284. [5] T. Mizusaki, M. Honma, T. Otsuka, Phys. Rev. C 53 (1996) 2786. [6] M. Honma, T. Mizusaki, T. Otsuka, Phys. Rev. Lett. 77 (1996) 3315. [7] T. Otsuka, M. Honma, T. Mizusaki, Phys. Rev. Lett. 81 (1998) 1588. [8] C.A. Ur et al., Phys. Rev. C 53 (1996) 2786. [9] T. Otsuka, T. Mizusaki, M. Honma, J. Phys. G 25 (1999) 699. [10] T. Otsuka, in: H. Rebel, D. Poenaru, J. Wentz (Eds.), Proceedings of the Advanced Study Inst. 2000, Predeal, Kluwer, Dordrecht, 2000. [11] E. Caurier, A.P. Zuker, A. Poves, G. Martinez-Pinedo, Phys. Rev. C 50 (1994) 225.
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
393
[12] K. Langanke, D.J. Dean, P.B. Radha, Y. Alhassid, S.E. Koonin, Phys. Rev. C 52 (1995) 718. [13] T. Mizusaki, T. Otsuka, M. Honma, B.A. Brown, Proc. of RIKEN Symp. Shell Model 2000, Nucl. Phys. A (2001). [14] D. Rudolph et al., Phys. Rev. Lett. 82 (1999) 3763. [15] T. Mizusaki, T. Otsuka, Y. Utsuno, M. Honma, T. Sebe, Phys. Rev. C 59 (1999) R1846. [16] N. Shimizu, T. Otsuka, T. Mizusaki, M. Honma, submitted. [17] R.B. Firestone et al. (Eds.), Table of Isotopes, Wiley, New York, 1996. [18] Y. Utsuno, T. Otsuka, T. Mizusaki, M. Honma, Phys. Rev. C 60 (1999) 054315. [19] D. Guillemaud-Mueller et al., Nucl. Phys. A 426 (1984) 37. [20] A. Azaiez et al., AIP Conference Proceedings, Vol. 481, p. 243. [21] K. Yoneda et al., submitted. [22] A. Ozawa et al., Phys. Rev. Lett. 84 (2000) 5493.
Monte Carlo shell model Takaharu Otsuka a,b a Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan b RIKEN, Hirosawa, Wako-shi, Saitama 351-0198, Japan
Received 18 September 2000; accepted 27 November 2000
Abstract The development and limitations of the conventional shell-model calculations are mentioned. In order to overcome the limitations, the Quantum Monte Carlo Diagonalization (QMCD) method has been proposed. The basic formulation and features of the QMCD method are presented as well as its application to the nuclear shell model, as referred to as Monte Carlo Shell Model (MCSM). The MCSM provides us with a breakthrough in the shell-model calculation: the structure of low-lying states can be studied with realistic interactions for a wide, nearly unlimited basically, variety of nuclei. Thus, the MCSM will contribute significantly to the physics to be developed by means of Radioactive Ion Beams. Applications to 56 Ni, Ba isotopes and N ∼ 20 exotic nuclei far from the β stability line are mentioned. 2001 Elsevier Science B.V. All rights reserved. PACS: 21.60.Ka; 21.60.Cs; 24.10.Cn; 21.60.Fw; 27.40.+z; 27.30.+t Keywords: Shell model; Monte Carlo shell model; Quantum Monte Carlo Diagonalization (or QMCD) method; Exotic nuclei; Shape phase transition; Magic number
1. Introduction The nuclear shell model was started by Mayer and Jensen in 1949 [1] as a singleparticle model. Afterwards, many valence particles are treated in the shell model, which then became a many-body theory and a calculational method. A good example can be found in the sd shell [2]. The nuclear shell model has been successful in the description of various aspects of nuclear structure, partly because it is based upon a minimum number of natural assumptions, and partly because all dynamical correlations in the model space, beyond the mean-field calculations, can be incorporated appropriately. Although the direct diagonalization of the Hamiltonian matrix in the full valence-nucleon Hilbert space is desired, the dimension of such a space is too large in many cases, preventing us from performing the full calculations. E-mail address: [email protected] (T. Otsuka). 0375-9474/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 0 ) 0 0 6 0 5 - 9
384
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
In order to overcome this dimension problem, quantum Monte Carlo approaches have been introduced. As one of them, the Shell Model Monte Carlo (SMMC) method has been proposed successfully [3]. However, the SMMC is basically restricted to groundstate and thermal properties. Moreover, the SMMC suffers from the so-called minus-sign problem for realistic interactions. As a completely different approach, the Quantum Monte Carlo Diagonalization (QMCD) method has been proposed for solving quantum manybody systems with a two-body interaction [4–7]. The QMCD can describe not only the ground state but also excited states, including their energies, wave functions and hence transition matrix elements. The sign problem is irrelevant to the QMCD. Thus, based upon the QMCD method, we introduce the Monte Carlo Shell Model as its application to the nuclear shell-model calculation. The MCSM has become a new tool for clarifying the structure of the ground and low-lying states of nuclei, providing us with a quite powerful tool for theoretical studies of nuclei produced by Radioactive Ion Beams. 2. Development and limitations of conventional Shell Model In the shell-model calculation, one introduces single-particle state first. This state can be given as an eigenstate of a spherical single-particle potential, for instance, Harmonic Oscillator or Woods–Saxon potential. Single-particle states relevant to a given nucleus are grouped into the core part and the valence part. The core part is completely occupied, and is frozen like a vacuum. The valence part is usually one major shell on top of the core part, and called valence shell. The valence shell is partly occupied. In the shell-model calculation, one generates all possible Slater determinants in the valence shell. The Slater determinants can be classified according to the z-component of the angular momentum, denoted by M. The number of Slater determinants for a given value of M is usually called (M-scheme) shell-model dimension. In order to obtain the eigenstate of the Hamiltonian for the given valence shell, one calculates matrix elements of the Hamiltonian for Slater determinants. This is a straightforward calculation. However, this calculation should be done for all pairs of Slater determinants as bra and ket vectors of the matrix elements. Once all matrix elements are calculated, one should diagonalize the matrix. A schematic description of this process is shown in the upper part of Fig. 1. So, if the shell-model dimension is large, the number of matrix elements is certainly much larger (up to about half of the square of the dimension), and the actual calculation becomes very difficult. By recent (conventional) shell-model codes like ANTOINE by Caurier, VECSSE by Sebe or MSHELL by Mizusaki, one can handle up to shell-model dimension ∼ 100 million at technical edge, while practical calculations up to a few tens million dimension can be done. An example can be found in [8], where the levels of 52 Fe are calculated by the code ANTOINE. Although the conventional shell-model calculation has thus been developed significantly, the dimension can be much larger in many real nuclei and is indeed much beyond the reach of the future development. For instance, certain unstable nuclei to be discussed in this paper require calculations with more than 10 billion dimension. This is already very far beyond the limit of the existing shell-model codes.
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
∗ ∗ ∗ ∗ ∗ ∗ H= ∗ ∗ · · ·
∗ ∗ ∗ · ·
∗ ∗ ∗ · · · · · ·
· · · · · ·
ε
385
1
ε2 ε3
·
0 Conventional Shell Model all Slater determinants ∗
∗ H≈ ∗ ·
∗ ∗ ∗ ·
∗ · ∗ · ·
ε
1
0 Monte Carlo Shell Model bases important for a specific eigenstate
ε2
·
·
0 ·
0 ·
Fig. 1. Hamiltonian matrices in the conventional shell-model calculation and in the Monte Carlo Shell Model.
3. Basic formulation of QMCD We have seen the difficulties the conventional shell-model calculations are facing. In order to overcome those difficulties, one has to introduce an alternative approach. That is stochastic methods to many-body problems. We now turn to this subject. The Shell Model Monte Carlo (SMMC) method has been proposed first [3]. The SMMC method belongs to so-called auxiliary field Monte Carlo methods. It is designed to calculate the ground-state properties or to study thermal properties. Therefore, the SMMC is not very suitable for investigating level structure or transitions between eigenstates. It is also known that the so-called minus-sign problem prevent us from using realistic interactions in the SMMC calculations directly, although extrapolation procedures were successfully implemented [3]. The Quantum Monte Carlo Diagonalization (QMCD) method has been proposed several years later by Honma, Mizusaki and myself [4]. We shall outline very briefly the process of this method first. In the QMCD method, there are two major steps. In the first step, we generate basis states for the given many-body system, which is comprised of valence protons and neutrons in the case of the nuclear shell model. We should select only basis states important to the eigenstate to be obtained. In the second step, having those selected basis states, we calculate matrix elements of the Hamiltonian and diagonalize the Hamiltonian matrix. If we have all important basis states, the result should be a good approximation to the exact diagonalization in the entire Hilbert space. Thus we solve quantum many-body problems. So, the major question is how to generate important basis states. We shall begin with this question. Since the formulation of the QMCD has been presented first in [4] and has been explained later in simple terms in [9,10], we shall come directly to the equation by which the basis states are generated. It is written as
386
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
Φ(σ ) ∝ e−βh(σ ) Ψ (0) ,
(1)
where σ means a set of random numbers which control the basis generation, |Φ(σ ) is a basis state, |Ψ (0) is the initial state. Here h(σ ) is so-called one-body Hamiltonian, and is a linear combination of various one-body operators with certain weights. We note that Eq. (1) transforms a Slater determinant into another Slater determinant, because h(σ ) is a one-body operator. What are varied in the operation in Eq. (1) are single-particle wave functions constituting Slater determinants. Those single-particle wave functions are generally deformed. At this moment, we introduce a toy Hamiltonian as H = 12 V O 2 ,
(2)
where V is a coupling constant and O is a arbitrary one-body operator. For simplicity, we assume V < 0. The above-mentioned one-body Hamiltonian is then given by h(σ ) = V σ O.
(3)
Note that there is just one random variable σ in this equation because the present toy Hamiltonian in Eq. (2) consists of one term. By having different values of the random number, σ , we can generate different state vectors, |Φ(σ ) ’s, by Eq. (1). We diagonalize the Hamiltonian in a subspace spanned by those basis states. We evaluate how much each basis contributes for lowering the energy eigenvalue being calculated. We keep only those with larger contributions (i.e., important bases), whereas the others are thrown away. To improve the result, we generate more states by other values of σ , and keep only important ones. Thus, one can add bases, until certain convergence of the energy eigenvalue is reached. By having such selected basis vectors which are important for a specific eigenstate (not necessarily the ground state), its energy eigenvalue and wave function are obtained as a result of the diagonalization of small matrix with respect to these important basis vectors (see the lower part of Fig. 1). In realistic cases, a general shell-model Hamiltonian can be written as H=
Nf
Eα Oα + 12 Vα Oα2 ,
(4)
α=1
where α stands for an index, Eα and Vα denote single-particle energies and interaction strengths, respectively, for one-body operators Oα . The one-body Hamiltonian is then given as a combination of terms like the right-hand side of Eq. (3): (Eα + sα Vα σα )Oα , (5) h(σ ) = α
where sα = ±1 (±i) if Vα < 0 (> 0). By using this h(σ ), the basis states are generated by using Eq. (1) in essentially the same way as that for the toy Hamiltonian. The adopted basis states are called QMCD bases. The number of QMCD bases is referred to as the QMCD basis dimension. We emphasize that energies and wave functions are determined by the diagonalization, and that we can obtain excited states as well as the
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
387
ground state, for instance, by monitoring energy eigenvalues of excited states in the basis generation process. The above procedure is the first version of QMCD. Although it works quite well for simple systems [4], it turned out that, in order to carry out realistic large-scale shell-model calculations, we have to improve the method as discussed in the next section.
4. Improvement of QMCD 4.1. Basis generation around the mean-field solution Since the number of manageable bases is finite in practice, we should select bases of higher importance. We thus choose |Ψ (0) from a Hartree–Fock local minima or some states of similar nature, although |Ψ (0) is not specified in Eq. (1). In fact, the Hartree– Fock ground state is the “best” single Slater determinant by definition, and fits well to the present scheme. Such |Ψ (0) is adopted as one of the QMCD bases often. At the same time, h(σ ) in Eq. (3) is rewritten as h(σ ) = hMF + α Vα sα σα Oα . Here hMF can be set a mean-field potential, for instance, Hartree–Fock potential of the given shell-model Hamiltonian. Since hMF is independent of the random number σα and the sampling is made around σα = 0, the QMCD bases |Φ(σ ) are generated around local minima of the relevant mean field. This process is useful for yrast states by combining with the angular momentum projection discussed later. On the other hand, this process is not very relevant to other states, unless the states are in a pronounced local minima. 4.2. Restoration of angular momentum Since a nucleus has rotational symmetry, the restoration of the total angular momentum is quite crucial. The basis state in Eq. (1) is not an eigenstate of the angular momentum in general, however. This is because the initial state |Ψ (0) breaks the rotational symmetry in general, and moreover the operator e−βh(σ ) has nothing to do with any symmetry. In the QMCD generation of the basis vectors, the restoration of the rotational symmetry should be fulfilled because the eigenstate has this symmetry. However, this is a very slow process. We therefore restore the rotational symmetry by means of angular momentum projection of matrix elements [7]. 4.3. Compression of QMCD basis space Stochastically generated bases contain, in general, unnecessary fluctuation components which are nothing but random noise. We therefore revise the basis-generation method so that each basis contains more relevant components lowering the energy and less irrelevant components to be canceled eventually by other bases in the diagonalization process. For this purpose, for instance, we can shift the random number σα a bit, and compare how much the resultant energy eigenvalue is changed. We take the better state obtained before or after the shift. We keep shifting σα until we come to saturation of the improvement.
388
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
Each basis state now contains more relevant components and less irrelevant ones. Although the practical process of this improvement can vary, the original important bases should be refined so as to become even more important. Since the convergence of results of diagonalization becomes much faster, the Hilbert subspace of the QMCD bases is now compressed for the same quality of the result. This compression process differentiates the QMCD method from other quantum Monte Carlo approaches in its variational character.
5. Monte Carlo shell model By combining all the above improvements, the present version of QMCD has been constructed [7]. The application of the QMCD method to the nuclear shell model is called the Monte Carlo Shell Model (MCSM). 5.1. Test of MCSM for 48 Cr The validity of MCSM has been confirmed by comparing to the result of the exact diagonalization of the same Hamiltonian. Here, the nucleus 48 Cr is taken, and the exact result is obtained from Ref. [11]. Fig. 2 shows energies of yrast states of 48 Cr. As shown in Fig. 2, the ground-state energy has been reproduced within 130 keV with the QMCD dimension 40. This result is already ∼ 100 keV below the lower edge of the error bar of the SMMC result with the temperature T = 0.5 MeV [12]. Since the angular momentum projection is made for each basis, the addition of a new basis (i.e., increase of the QMCD dimension) implies inclusion of more dynamical degrees of freedom.
Fig. 2. Energies of yrast states of 48 Cr obtained by MCSM calculation compared with the energies obtained by the exact diagonalization [11]. The MCSM energy eigenvalues are shown as functions of the (QMCD) basis dimension. The point with error bar at far right is the ground-state energy of the SMMC calculation [12].
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
389
5.2. Features of MCSM There are two major advantages in the MCSM calculations. The first one is the feasibility of including many single-particle states. Because of this, one can describe drastic excitations within a nucleus. For instance, one can describe spherical yrast states, deformed rotational band and nearly superdeformed band at the same time with the same Hamiltonian in the same model space. This example is shown in Fig. 3 where those three kinds of states are nicely presented by an MCSM calculation [13] with the full pf shell and the g9/2 orbit. One finds quite good agreement with experiment [14]. This kind of description over a wide variety of states may be characterized as the feasibility along the energy axis. The yrast states and (normally) deformed band can be described basically within the pf + shell [7,15], while the g9/2 orbit is needed for the description of 16+ 1 and 181 states and negative-parity states. In fact, these states show quite large deformation, in particular, 16+ 1 and 18+ 1. The second major advantage of the MCSM calculation is the feasibility of handling many valence particles. The maximum number of valence particles is rather limited in the conventional calculations. However, if one wants to describe a long chain of isotopes entering the region of exotic nuclei far from the β stability line, the number of particles should change significantly. So, this capability plays an indispensable role in studying the
Fig. 3. Spherical yrast states, deformed band and nearly superdeformed bands in 56 Ni.
390
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
Fig. 4. Low-lying levels of Ba isotopes. Symbols are experiments [17], while lines are MCSM calculations [16].
structure of such exotic nuclei. This feature can be characterized as the feasibility along the isospin axis, and will be discussed in the next section. This second advantage is essential also for describing the spherical-deformed phase transition in heavy nuclei [16]. This is partly because the collective motion involves many valence nucleons in general, and also because the phase transition occurs as a result of large change of the valence-nucleon number. Fig. 4 shows a recent MCSM calculation for the phase transition in Ba isotopes [16]. In this calculation, the Hamiltonian and the model space are fixed, and the nucleus is driven, by the increase of valence neutrons, from a sphere to an ellipsoid due to the proton–neutron correlation. The agreement with experiment [17] is remarkable including the transitional region, where dynamical correlations can be crucial.
6. Exotic nuclei in MCSM In exotic nuclei, two major shells are mixed rather often breaking the magic structure, and states of various characters appear at low energy. Even the ground state can be of quite exotic nature. In this situation, the above two feasibilities combined together play really crucial roles in clarifying the structure of exotic nuclei far from the β stability line. As an example, we shall discuss the structure of nuclei in the vicinity of 32 Mg. + Fig. 5 indicates energy levels of 2+ 1 and 41 states of even-A Ne and Mg isotopes obtained by MCSM calculation [18] compared with experiment [17,19–21]. The calculated results contain many predictions and seven of them have been seen by experiments carried out after the MCSM calculation, although the spin–parity assignments for the 4+ 1 levels for 32,34 Mg are only likely possibilities to be confirmed [20,21].
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
391
+ Fig. 5. Excitation energies of 2+ 1 and 41 states of even-A Ne and Mg isotopes obtained by MCSM calculation [18] compared with experiment [17,19–21]. Symbols are experimental data, while open symbols are those before the calculation of [18] and closed symbols are those after the calculation. Dashed lines are results of calculations within the sd shell.
In the N ∼ 20 neutron-rich unstable nuclei, the N = 20 magic number is broken in certain low-lying states. The extent of this breaking depends on nuclei and states, however. Therefore, calculations which can mix all relevant states are crucial in clarifying the exotic structure in this region. In the MCSM calculation in [18] and its extension, all particle– hole excitations from the N = 20 closed shell are fully included within the model space, and the same Hamiltonian is used for all nuclei investigated. In conventional calculations, such mixings cannot be included fully due to difficulties of large matrix dimension. In fact, the mixing between the 0p0h configurations and the 2p2h configurations occurs in some nuclei rather strongly due to varying shell gap [18], in a consistent manner to experiment. The lowering of the 2+ 1 level in Ne isotopes just below N = 20 is related to this mixing [18], while such lowering is not seen in Mg isotopes. Many of exotic properties in nuclei far from stability are due to interplay between T = 0 and T = 1 effective nucleon–nucleon interactions. The T = 0 interaction is strongly attractive, and is responsible for the strong mean attractive potential in nuclei. The T = 1 interaction is very weak or repulsive, but dominates the structure of nuclei with high isospin. An example can be seen in the structure of neutron-rich oxygen isotopes. From the same Hamiltonian used for the results presented above, one can obtain the magic structure for N = 14 and 16 [22], whereas this magic structure is rapidly washed away by having
392
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
only a few valence protons. Thus, a variety of new unknown structure can be expected in nuclei far from stability, and the MCSM should provide us with quite sound theoretical results because of its advantages.
7. Summary In summary, we have discussed a new method for solving quantum many-body problems where particles are interacting through a two-body interaction. In the QMCD calculation, basis vectors are selected according to their “importance” as expressed by importance truncation to the full calculation [9]. The number of such QMCD bases is usually 30–50, even if the original Hilbert space has 1 billion dimensions or more. The application of QMCD to the nuclear shell model is called Monte Carlo Shell Model (MCSM). Because one can handle many valence orbits and many valence particles in the MCSM calculations, one can describe a wide variety of states within a given nucleus, and also can move over a large region of the nuclear chart with the same Hamiltonian and the same model space. Thus, the dream of the shell-model physicists is to describe all nuclei but the lightest ones within the shell model and it seems to come true. The MCSM is of particular importance for the study of exotic nuclei far from the β stability line. For instance, one can explore effects of isospin by moving far away along the isotopic chain, or investigate what interaction can produce possible exotic deformation. It may be quite fortunate that RIB experiments and MCSM calculations came out in the same era.
Acknowledgements The MCSM calculations were performed in part by the Alphleet computer in RIKEN. This work was supported in part by Grant-in-Aid for Scientific Research (A)(2) (10304019) from the Ministry of Education, Science and Culture.
References [1] M.G. Mayer, Phys. Rev. 75 (1949) 1969; O. Haxel, J.H.D. Jensen, H.E. Suess, Phys. Rev. 75 (1949) 1766. [2] B.A. Brown, B.H. Wildenthal, Annu. Rev. Nucl. Part. Sci. 38 (1988) 29. [3] S.E. Koonin, D.J. Dean, K. Langanke, Phys. Rep. 278 (1997) 1, references therein. [4] M. Honma, T. Mizusaki, T. Otsuka, Phys. Rev. Lett. 75 (1995) 1284. [5] T. Mizusaki, M. Honma, T. Otsuka, Phys. Rev. C 53 (1996) 2786. [6] M. Honma, T. Mizusaki, T. Otsuka, Phys. Rev. Lett. 77 (1996) 3315. [7] T. Otsuka, M. Honma, T. Mizusaki, Phys. Rev. Lett. 81 (1998) 1588. [8] C.A. Ur et al., Phys. Rev. C 53 (1996) 2786. [9] T. Otsuka, T. Mizusaki, M. Honma, J. Phys. G 25 (1999) 699. [10] T. Otsuka, in: H. Rebel, D. Poenaru, J. Wentz (Eds.), Proceedings of the Advanced Study Inst. 2000, Predeal, Kluwer, Dordrecht, 2000. [11] E. Caurier, A.P. Zuker, A. Poves, G. Martinez-Pinedo, Phys. Rev. C 50 (1994) 225.
T. Otsuka / Nuclear Physics A 693 (2001) 383–393
393
[12] K. Langanke, D.J. Dean, P.B. Radha, Y. Alhassid, S.E. Koonin, Phys. Rev. C 52 (1995) 718. [13] T. Mizusaki, T. Otsuka, M. Honma, B.A. Brown, Proc. of RIKEN Symp. Shell Model 2000, Nucl. Phys. A (2001). [14] D. Rudolph et al., Phys. Rev. Lett. 82 (1999) 3763. [15] T. Mizusaki, T. Otsuka, Y. Utsuno, M. Honma, T. Sebe, Phys. Rev. C 59 (1999) R1846. [16] N. Shimizu, T. Otsuka, T. Mizusaki, M. Honma, submitted. [17] R.B. Firestone et al. (Eds.), Table of Isotopes, Wiley, New York, 1996. [18] Y. Utsuno, T. Otsuka, T. Mizusaki, M. Honma, Phys. Rev. C 60 (1999) 054315. [19] D. Guillemaud-Mueller et al., Nucl. Phys. A 426 (1984) 37. [20] A. Azaiez et al., AIP Conference Proceedings, Vol. 481, p. 243. [21] K. Yoneda et al., submitted. [22] A. Ozawa et al., Phys. Rev. Lett. 84 (2000) 5493.