Playing Dice with Nuclei: Pattern out of Randomness?

Playing Dice with Nuclei: Pattern out of Randomness?∗ arXiv:nucl-th/0111010v1 5 Nov 2001 Roelof Bijker 1 2 1† and Alejandro Frank 1,2 ICN-UNAM, AP 70-543, 04510 México, DF, México CCF-UNAM, AP 139-B, 62251 Cuernavaca, Morelos, México September 19, 2001 Many a shaft, at random sent, finds mark the archer little meant! Sir Walter Scott It’s chaos, but it’s organized chaos. Charles Mingus Introduction In a 1926 letter to Max Born, Albert Einstein wrote: I am convinced that God does not play dice, referring to his dissatisfaction with the intrinsic randomness of quantum mechanics. But recent work in nuclear structure physics with random interactions, suggests that Nature may weave some of its patterns by throwing dice. Low-lying spectra of many-body quantum systems often display a high degree of order and regularity. In the case of atomic nuclei, despite their complexity and the large number of degrees of freedom involved, they often exhibit simple features, such as pairing properties, surface vibrations and rotational motion in even-even nuclei. A recent analysis of experimental energy systematics of medium and heavy even-even nuclei suggests that these nuclei can be classified into just three families, namely the pairing, vibrator and rotor regimes [1]. These areas were identified in a plot of the excitation energies of the first-excited states + E(4+ 1 ) against E(21 ), which show characteristic slopes of 1.0, 2.0 and 3.3, respectively (see Figure 1). ∗ To † On be published as a feature article in Nuclear Physics News, Vol. 11, No. 4 (2001) sabattical leave at Università degli Studi di Genova, Dipartimento di Fisica, Via Dodecaneso 33, I-16146 Genova, Italy 1 + Figure 1: Plot of the excitation energies E(4+ 1 ) against E(21 ), showing the tripartite classification into seniority, anharmonic vibrator (AHV), and rotor (R) regions [1]. Conventional wisdom is that regularities arise from symmetries of the Hamiltonian, which lead to invariances that severely constrain the many-body motion. While some of these symmetries are exact (e.g. rotational and reflection invariance), others are approximate (e.g. isospin). These global symmetries, however, do not explain by themselves the regular patterns observed. Further assumptions about the nature of the nucleon-nucleon interaction are required. Thus, a strongly attractive pairing force between like nucleons has been shown to be responsible for the remarkable constancy of the energy of the first excited 2+ states in the tin isotopes [2], while deformation and rotational behavior is known to arise from an attractive quadrupole-quadrupole interaction between neutrons and protons [2]. These striking patterns as well as many other correlations have been shown to be robust features of low-energy nuclear behavior, which signal the emergence of order and collectivity. In every case the patterns arise as a consequence of particular forms of the nucleon-nucleon interaction. Thus the general belief is that the main features of low-energy nuclear spectroscopy and their underlying causes are understood. It came as a surprise, therefore, that recent studies of even-even nuclei in the nuclear shell model [3] and in the interacting boson model [4] with random interactions displayed a high degree of order. Both models showed a marked statistical preference (> 60 %) for ground states with angular momentum and parity J P = 0+ , despite the random nature of the interactions. 2 It is the aim of this article to review these new developments in random studies of nuclear structure. We first present a brief summary of some ensembles of random matrices, and then discuss the main results and their possible implications for our understanding of nuclear structure. Random matrix ensembles Physicists have long known that generic spectral properties, such as average distributions and fluctuations of peaks in neutron-capture experiments, or the statistical properties of small metallic particles and quantum dots, seem to be independent of the interactions involved, even if the data does not precisely match the properties of any one system. In order to describe statistical properties of nuclear spectra, Wigner developed Random Matrix Theory [5, 6], in which the Hamiltonian matrix elements are chosen at random, but keeping some global symmetries, e.g. the matrix should be hermitean, and be invariant under time-reversal, rotations and reflections. In his own words: ... the Hamiltonian which governs the behavior of a complicated system is a random symmetric matrix, with no particular properties except for its symmetric nature. Why is this? In nuclear physics, for example, high energy dynamics is assumed to have lost track of its correlations: there is no memory of a particular mechanism, or this mechanism turns out to be irrelevant in the determination of statistical or generic properties. Specifically, the Gaussian Orthogonal Ensemble (GOE) of real-symmetric random Hamiltonian matrices closely describes the level repulsion found in the distribution of nearest-neighbor spacings of states with the same quantum numbers, such as that found in neutron-capture resonances in 167 Er, proton resonances in 47 V, and in shell model calculations, among many other examples [7]. It should be noted, however, that the GOE gives rise to a semicircular energy level distribution, in contrast to the Gaussian distribution found for shell model calculations. An arbitrary choice of the Hamiltonian matrix elements, as in GOE, corresponds effectively to random many-body interactions in which the maximum order k is equal to the number of particles n. On the other hand, realistic shell model Hamiltonians should contain mostly one- and two-body interactions. A different ensemble of random interactions is the so called Two-Body Random Ensemble (TBRE) in which the two-body interactions are taken from a distribution of random numbers [8, 9]. For n = 2 particles, TBRE becomes identical to GOE, but for n > 2 particles, the many-body matrix elements of the Hamiltonian are correlated and can be expressed in terms of the two-body random matrix elements. The expansion coefficients are determined by the many-body dynamics of the model space (i.e. angular momentum coupling coefficients, coefficients of fractional parentage, etc.). The TBRE gives rise to a Gaussian energy level distribution. The transition between GOE and TBRE has been studied for the case of seven identical nucleons in the 3 f5/2 f7/2 shell and states with angular momentum and parity J P = 7+ 2 . As the order k of the random interactions decreases, the level distribution shows a transition from semicircular for k = 7 (GOE) to Gaussian for k = 2 (TBRE) [8]. For a detailed discussion of the these and other properties of GOE and TBRE we refer the reader to the review articles by Brody et al. [7] and by Guhr et al. [10]. An unexpected result Most studies with random matrix ensembles, either GOE or TBRE, involved sets of highly excited states with the same quantum numbers, such as angular momentum, parity and isospin. In effect, these studies probed the eigenvalue distributions near their middle part, and supported the general validity of random matrix methods regarding averages and fluctuations. Until recently, however, very little attention had been paid to correlations among different symmetries and to low-energy behavior, i.e. the tails of the distributions. Unlike in the GOE, in the TBRE the energy eigenvalues of states with different quantum numbers are strongly correlated, since they arise from the same Hamiltonian. In a recent development, Johnson, Bertsch and Dean carried out shell model calculations for eveneven nuclei in the sd shell and the pf shell with randomly distributed two-body interactions [3]. An analysis of the entire energy spectrum, i.e. states of all allowed values of the angular momentum and other quantum numbers, showed a remarkable statistical preference for ground states with J P = 0+ , plus the appearance of energy gaps, and other indicators of ordered behavior, despite the random nature of the two-body matrix elements (both in sign and relative magnitude). The unexpected strong dominance of 0+ ground states amazed nuclear physicists and motivated a large number of investigations. These studies explore low-lying features of nuclear structure by addressing the following problem. Consider that a certain nucleus is described by means of an ensemble of Hamiltonians acting on a given single-particle space of valence orbits. The interactions are restricted to two-body, satisfy hermiticity, and are invariant under time-reversal, rotations and reflections. The two-body matrix elements are taken from a (Gaussian) distribution of random numbers with zero mean, so that they are arbitrary and equally likely to be attractive or repulsive. Next, the many-body Hamiltonian matrices are calculated for each value of the angular momentum and diagonalized. The resulting spectrum is analyzed for its spectral properties, such as the angular momentum of the ground state, the relative position of other yrast states and quadrupole transitions among them. This procedure is repeated, let’s say 1,000 times. What is the result of such a numerical experiment? As an example, here we consider the nucleus 22 O which is described by six valence neutrons in the sd shell, which consists of single-particle orbitals with s1/2 , d3/2 and d5/2 [3, 11]. In Table 1 we show that 4 Table 1: Comparison of the percentage of J = 0 states in the basis and obtained as ground states in TBRE for (i) 6 neutrons in the sd shell (the nucleus 22 O) and (ii) for the IBM with N = 16 bosons. Basis TBRE O 9.9 % 67.7 % IBM 3.3 % 63.4 % 22 the percentage of the total number of runs for which the ground state has angular momentum J = 0 is 67.7 %. This is a seven-fold enhancement with respect to the percentage of J = 0 states in the model space of only 9.9 %. This result is rather insensitive to the model space and to the ensemble of two-body interactions [3, 11, 12, 13]. In subsequent studies, however, it was shown that the overlap of these states with realistic shell model wave functions is small [14]. For the cases with a J P = 0+ ground state, it is of interest to calculate the probability distribution of the ratio of excitation energies R = + E(4+ 1 )/E(21 ) , which constitutes a measure of the fundamental properties of random nuclei. This energy ratio has characteristic values of 1, 2 and 10/3 for the pure seniority, vibrational and rotational regions, respectively. Figure 2 shows that for six neutrons in the sd shell, the probability distribution P (R) has very little structure. There is a broad peak between 1 ≤ R ≤ 2, with a maximum around 1.3. This suggests that a system of identical nucleons on average tends to behave in accord with the seniority regime of [1]. Since the distribution extends to R = 2, there is some evidence, although hardly convincing, for the appearance of vibrational structure. On the other hand, for this system there is no occurrence of rotational bands [3, 15]. Recently it was shown, that neutron-proton systems do evince such collective behavior if the ensemble of two-body interactions is taken from a displaced Gaussian distribution (with negative mean) [16]. To gauge the robustness of these results, tests were carried out for other nuclear models as well. A similar preponderance of J P = 0+ ground states was found in an analysis of the Interacting Boson Model (IBM) with random interactions [4]. In the IBM, collective nuclei are described as a system of N 5 0.6 P(R) 0.4 0.2 0 0 1 2 R 3 4 + Figure 2: Probability distribution P (R) of the energy ratio R = E(4+ 1 )/E(21 ) for six neutrons in the sd shell with random two-body interactions. interacting monopole and quadrupole bosons [17]. These bosons reflect the dominant angular momentum components of the pairing interaction between identical nucleons. In spite of its simplicity, the IBM displays a wide range of collective behavior, which includes shape transitional regions. For the case of N = 16 bosons, a twenty-fold enhancement was found, since in 63.4 % of the cases the ground state has J P = 0+ , compared to only 3.3 % of the total number of basis states (see Table 1). Just as for the shell model, the probability distribution P (R) of the energy ratio R can be used to look for evidence for vibrational and/or rotational structure. Fig. 3 shows a surprising result: the probability distribution P (R) displays two very pronounced peaks, one at R ∼ 1.9 and a narrower one at R ∼ 3.3. These values correspond almost exactly to the harmonic vibrator and rotor values of 2 and 10/3. A plot of the energy ratio and the corresponding ratio of B(E2) values, shows a strong correlation between the first peak and the vibrator value for the ratio of B(E2) values, as well as between the second peak and the rotor value [4]. In this case, the evidence for regular patterns is not only based on energy systematics, but also on the structure of the wave functions. The unexpected results found for both the shell model and the IBM, have raised fundamental questions in nuclear structure physics and have led to a revival of random matrix studies. Naively one might expect most properties of these random nuclei to occur stochastically, as conventional wisdom suggests that they 6 3 P(R) 2 1 0 0 1 2 R 3 4 + Figure 3: Probability distribution P (R) of the energy ratio R = E(4+ 1 )/E(21 ) in the IBM with random one- and two-body interactions. The number of bosons is N = 16. depend strongly on the particular form of the interactions. But are there fundamental properties that remain essentially invariant under arbitrary interactions? One of the most striking features of these studies is that some of the most hallowed aspects of nuclear structure, such as the appearance of 0+ ground states, and the occurrence of vibrational and rotational motion, can emerge with a strong statistical preference from randomly distributed interactions. These results, however, were obtained from numerical studies. It is necessary to gain a better understanding as to why this happens. What is the origin of the regular features which arise from random interactions? I am very happy to learn that the computer understands the problem, but I would like to understand it too, as Wigner once quipped. Search for an explanation The observed dominance of J P = 0+ ground states was certainly not anticipated, considering that there is no obvious pairing character in the aleatory forces. The ingredients of these numerical simulations, both for the shell model and for the IBM, are the structure of the model space, the ensemble of random Hamiltonians, the order of the interactions (one- and two-body), and the global symmetries, i.e. time-reversal, 7 hermiticity and rotation and reflection symmetry. The latter three symmetries of the Hamiltonian cannot be modified, since we are studying many-body systems whose eigenstates have real energies and good angular momentum and parity. In this section, we discuss the role of some of the other ingredients, in search for a possible explanation of this phenomenon. (i) The dominance of 0+ ground states has been shown to be a robust property that arises from many different ensembles of random interactions, such as the Random Quasi-particle Ensemble which was used in the original article [3] or the TBRE in subsequent studies [11, 12, 13]. The inclusion of realistic singleparticle energies hardly changes the results [12, 13]. Even for ensembles in which the pairing interactions have been excluded (set to zero), still more than half of the cases have a 0+ ground state [12]. (ii) Another possibility is that this behavior is related to the time-reversal invariance of the random Hamiltonian. After all, nuclear pairing, as in electron Cooper pairs in BCS theory, involves the filling of time-reversed pairs in doubly degenerate single particle states. Since time-reversed states play an important role in these favored collective states, it is conceivable that time-reversal invariant Hamiltonians induce a built-in preference for 0+ ground states. To see whether this is indeed the case, it is possible to break time-reversal invariance in the random two-body interactions, while still maintaining hermiticity and rotational and reflection invariance [7]. The corresponding ensemble of random two-body matrix elements is the well-known Gaussian Unitary Ensemble (GUE). For the case of six identical nucleons in the sd shell, the breaking of time-reversal invariance of the two-body interactions increases the number of independent random matrix elements from 30 to 46. The dominance of 0+ ground states turns out to increase from 67.7 % (see Table 1) to 76.8 % [11]. On the basis of these results, one may conclude that time-reversal invariance is not the origin of the dominance of 0+ ground states. (iii) The dependence on the order of the random interactions has been investigated in the framework of the IBM [18]. It was found that whenever the number of bosons N is sufficiently large compared to the rank k of the interactions, the spectral properties are characterized by a dominance of 0+ ground states and the occurrence of both vibrational and rotational bands. These band structures appear gradually with increasing values of N/k. Essentially the same behavior is found for random two- and three-body interactions. The observed spectral order in systems with random interactions cannot be explained by the timereversal symmetry of the interactions, or by the choice of a specific ensemble, nor by the order of the many-body interactions. These results suggest that such features arise as robust properties of the manybody dynamics of the model space and/or of the general statistical properties of random interactions. In this respect, we mention two recent developments which may shed some light on the problem, and may help to lead toward a possible solution. 8 100 J=0 Percentage 80 60 40 J=2 20 0 6 8 10 12 14 16 N Figure 4: Percentages of ground states with angular momentum J = 0 and J = 2 in the IBM with random one- and two-body interactions calculated exactly (solid lines) and in mean-field approximation (dashed lines). For the case of identical nucleons occupying a single-j shell, an explanation has been suggested based on the idea of geometric chaoticity, which appears in finite many-fermion systems with complex interactions due to exact rotational invariance [19]. It was shown that statistical correlations of fermions drive the ground state spin to its minimum or its maximum value. The statistical approach predicts a smooth behavior of the 0+ ground state probability which, for a j 4 configuration, decreases from ∼ 50 % to ∼ 40 % with increasing values of the angular momentum j of the single-particle orbit. However, such a simple statistical approach cannot explain the large oscillations observed in the exact numerical results. A recent analysis of the structure of the random wave functions lends further support to some aspects of this interpretation [14]. In a different development, a Hartree-Bose mean-field analysis of the IBM Hamiltonian has been used to associate regions of the parameter space with particular intrinsic vibrational states, which in turn correspond to definite geometric shapes [20]. The results of this analysis indicate that there are three basic shapes: a spherical one carried by a single state with J = 0, a deformed shape which corresponds to a rotational band with J = 0, 2, . . . , 2N , and a condensate of quadrupole bosons which has a more complicated angular momentum content. The ordering of rotational energy levels depends on the sign of the corresponding moments of inertia. In Fig. 4 we show the percentages of ground states with J = 0 9 and J = 2 as a function of the total number of bosons N . A comparison of the results of the mean-field analysis (dashed lines) and the exact ones (solid lines) shows good agreement. There is a dominance of J = 0 ground states for ∼ 63-77 % of the cases. The large oscillations with N are due to the contribution of the condensate of quadrupole bosons. The sum of the percentages of J = 0 and J = 2 ground states hardly depends on the number of bosons. The mean-field analysis explains both the distribution of ground state angular momenta and the occurrence of vibrational and rotational bands. A similar analysis has been carried out for the vibron model, an interacting boson model to describe the relative motion in two-body systems, for which part of the results have been obtained analytically [21]. Conclusions and outlook Numerical simulations for the nuclear shell model and the IBM with random interactions suggest that global properties in even-even nuclei, such as J P = 0+ ground states (for both models) and the occurrence of vibrational and rotational bands (for the IBM) may arise from a much broader class of Hamiltonians than the ones usually considered. These unexpected results have sparked a large number of investigations to explain and further explore the properties of random nuclei. Although they have shed light on various aspects of the original problem, i.e. the dominance of 0+ ground states, in our opinion, no definite answer is yet available, and the full implications for nuclear structure physics are still to be clarified. Open questions include, among others, the properties of higher excited states, the transition from ordered, regular features to chaos for an ensemble of Hamiltonians, and the emergence of collective traits from appropriately constrained random shell-model interactions. Other, more general, problems involve the study of the statistical properties of randomly interacting many-body systems and quantum chaos, such as the spectral properties of random matrix ensembles [13, 22, 23], the structure of wave functions [24, 25], the relation with random polynomials [26], the matrix elements of the Hamiltonian [27], and the connection between statistical and dynamical effects in finite many-body systems [19]. The study of random matrix ensembles is an exciting interdisciplinary field whose universal properties have allowed to establish connections between, at first sight, completely unrelated areas of physics and mathematics [28]. It has been known for a long time that random matrix ensembles, such as GOE and GUE, reflect universal properties of complex systems, ranging from level spacings in slow neutron resonances to quantum dots and chaotic billiards [7]. Surprisingly, recent research suggests a link between the statistical distribution of extreme eigenvalues of random matrices and the distribution of prime numbers through the Riemann hypothesis [29], thus relating the properties of random nuclei discussed in this contribution to the most important problem in number theory. 10 We conclude with the observation that the study of random phenomena in nature is too important to be left to chance. Acknowledgements It is a great pleasure to thank Rick Casten, Stu Pittel, David Rowe, Piet van Isacker, and Victor Zamfir for stimulating discussions and their random suggestions and thoughts. This work was supported in part by CONACyT under projects 32416-E and 32397-E, and by DPAGA-UNAM under project IN106400. References [1] R.F. Casten, N.V. Zamfir, and D.S. Brenner, Phys. Rev. Lett. 71, 227 (1993); N.V. Zamfir, R.F. Casten, and D.S. Brenner, Phys. Rev. Lett. 72, 3480 (1994); R.F. 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